High Resolution Infrared Spectroscopy of Complex
Polyatomic Molecules
byP. Bryan Changala
S.B., Massachusetts Institute of Technology, 2013

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2019

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This thesis entitled:High Resolution Infrared Spectroscopy of Complex Polyatomic Molecules
written by P. Bryan Changala
has been approved for the Department of Physics

Prof. Jun Ye

Prof. Carl Lineberger

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

iii
Changala, P. Bryan (Ph.D., Physics)
High Resolution Infrared Spectroscopy of Complex Polyatomic Molecules
Thesis directed by Prof. Jun Ye

Infrared spectroscopy is an essential tool for probing molecular structure and dynamics. Overthe last decade, cavity-enhanced direct frequency comb spectroscopy (CE-DFCS) has emerged asa powerful technique for molecular spectroscopy in the mid-infrared region (3 − 15 µm). CE-DFCS
combines the broad spectral bandwidth and high frequency resolution of frequency combs with
the improved detection sensivity provided by high-finesse optical cavities. At room temperature,
however, all but the simplest molecules suffer from severe spectral congestion, which obscures thedetailed dynamics encoded in spectroscopic fine structure. To overcome this challenge, we demonstrate the integration of CE-DFCS with cryogenic buffer gas cooling, which provides continuous,cold samples of gas-phase molecules at rotational and translational temperatures as low as 10 K. By
significantly reducing internal partition functions and Doppler broadening, we take full advantageof CE-DFCS to reveal the intricate rovibrational structure of complex polyatomic molecules.
The systems investigated in this thesis range from extremely anharmonic molecules to unprecedentedly large carbon cages, illustrating the diverse ways in which a molecule can be complex.
One case study is free internal rotation in nitromethane, a model system for large amplitude nuclear motion. This work has motivated our development of new theoretical tools to predict thehigh resolution spectra of nitromethane and several other floppy molecules. We have also focusedon large molecules outside the domain of traditional high resolution spectroscopy. The culminationof these efforts is the rotationally resolved spectrum of the C60 fullerene. C60 is now both thelargest molecule and the first example of icosahedral symmetry for which rovibrational quantum
state resolution has been achieved. In addition to CE-DFCS, we have constructed a cw-QCL-based
spectrometer to probe and manipulate the quantum states of C60 . This work opens new avenues
for fullerene research and for exploiting large molecules as platforms for quantum science.

Dedication

To my many teachers, with gratitude.

Acknowledgements

There are worse places than JILA to spend six years doing science. I have had a wonderfulexperience working in the Ye lab. Jun has been an inspiring and supportive advisor. I am grateful
to him for not only teaching me what the highest quality research looks like, but for giving me theindependence to explore my own interests while pushing together on scientific goals. Most of mygraduate work in Jun’s group was carried out closely with two outstanding post-docs, Ben Spaun(from 2014 to 2017) and Marissa Weichman (from 2017 to present). Apart from being brilliant
scientists and hard-working colleagues, I have been grateful for their friendship and camaraderie
in the lab. I would also like to thank past members of the mid-IR and XUV frequency comb
groups, including Thinh Bui, Bryce Bjork, Oliver Heckl, Kana Iwakuni, Gil Porat, Stephen Schoun,
Justin Niedermeyer, Christoph Heyl, Craig Benko, and Kevin Cossel, and wish continued successto current members including Jutta Toscano, Qizhong Liang, Chuankun Zhang, and Marissa. I’ve
had the privilege of getting to know many others in the Ye group over the last several years – toonumerous to list here but all excellent scientists and lab mates. I especially enjoyed my many officeconversations with Shimon Kolkowitz.
Part of what makes JILA special is the outstanding administrative and technical staff. Travel,bureacratic issues, purchases, scheduling, etc. always went smoothly due to the hard work of
Amy Allison, Agnieska Lynch, Krista Beck, the entire supply office staff, and many others. The
instrument and electronics shops made crucial contributions to my experiments. I especially wantto acknowledge Hans Green, Todd Asnicar, Terry Brown, and Carl Sauer.
I’ve had the opportunuity to contribute to many outstanding collaborations with scientists

viboth inside and outside JILA. I’ve enjoyed working with Marie-Aline Martin-Drumel, Mike McCarthy, Barney Ellison, Branko Ruscic, David Bross, Jessie Porterfield, David Patterson, John
Doyle, Kevin Lee, Lam Nguyen, Mitchio Okumura, Matt Nava, Allan Oliveira, Julia Lehman, Neil
Reilly, and Carl Lineberger. Bob Field, who gave me my start in spectroscopy at MIT, has continued to act as a valued mentor and advisor. Josh Baraban was an excellent graduate student
mentor to me when I worked in Bob’s group. He moved to Boulder shortly after I did (to post-doc
for Barney Ellison), which gave us the opportunity to continue working together on several theory
projects as well as in the lab, using frequency combs with pyrolysis microreactor jets. Josh remains
a close colleague and friend since starting his own group in Israel. John Stanton has served as an
unofficial advisor of sorts. I am grateful for the many scientific opportunities he has provided me,
including funding my travel to two conferences, and for being a generous promoter of my work.
He and Barney have provided hours of entertainment over many enjoyable dinners at the Toad.
(Barney also merits a special mention for delivering to me one of the only samples of dodecahedranein the world after personally transporting it halfway across the country.)
I’d like to recognize my Comps and Defense committee members, including Carl Lineberger,
David Nesbitt, Eric Cornell, Heather Lewandowski, and Cindy Regal, for their time and effort.
Lastly, I want to thank my family and friends, especially Gargi Datta and Kevin Gilmore;
Amy Nkomo; Mom and Dad; my siblings Alissa, Danielle, and Matt, and siblings-in-law Oliver,
Eoin, and Maggie (all “acquired” during my PhD); and my nephew Sebastian.

Contents

Chapter
1 Introduction

1

2 Cavity-enhanced direct frequency comb spectroscopy of buffer gas cooled molecules

4

2.1

Mid-infrared frequency combs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.1.1

3–5 µm OPO comb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2

6–11 µm DFG comb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2

Absorption spectroscopy with frequency combs . . . . . . . . . . . . . . . . . . . . . 12

2.3

Cavity-enhanced frequency comb spectroscopy
2.3.1

2.4

2.5

2.6

. . . . . . . . . . . . . . . . . . . . . 13

Cavity-comb coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Comb detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.1

FTS detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.2

VIPA spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Cryogenic buffer gas cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5.1

Low-temperature CBGC system . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5.2

High-temperature CBGC system . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5.3

Absorption enhancement cavity . . . . . . . . . . . . . . . . . . . . . . . . . . 30

CE-DFCS/CBGC system performance . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Spectroscopic signatures of unhindered internal rotation in nitromethane
3.1

36

Internal rotation in CH3 NO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

viii

3.2

3.3

3.1.1

Pure torsion level structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.2

Rotation-torsion level structure . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.3

Symmetry, spin statistics, and selection rules . . . . . . . . . . . . . . . . . . 42

High resolution spectra of CH3 NO2 in the 3.3 µm region . . . . . . . . . . . . . . . . 44
3.2.1

Experimental setup and survey spectrum . . . . . . . . . . . . . . . . . . . . 44

3.2.2

The ν3 + ν6 combination band . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.3

CH stretching modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Reduced-dimension calculations of the rotation-torsion structure and CH stretchingvibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4

3.3.1

Method overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3.2

Effect of the torsion barrier height on torsion-rotation energies . . . . . . . . 56

3.3.3

Torsion-mediated coupling of the CH stretch vibrations . . . . . . . . . . . . 59

Torsion-rotation Hamiltonians via curvilinear vibrational Møller–Plesset perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4 Limits to rotationally resolved MIR spectroscopy of large hydrocarbons

67

4.1

Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3

Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5 Rovibrational quantum state resolution of the C60 fullerene

75

5.1

An introduction to C60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2

Generating cold C60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3

Rotationally resolved C60 spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.3.1

Rovibrational wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.3.2

Zeroth order effective Hamiltonians and energies . . . . . . . . . . . . . . . . 85

5.3.3

Spherical top rovibrational selection rules . . . . . . . . . . . . . . . . . . . . 87

ix

5.4

5.3.4

Nuclear spin statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3.5

Rovibrational spectrum simulation . . . . . . . . . . . . . . . . . . . . . . . . 93

Rotational analysis of the 8.5 µm band . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.4.1

R branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.4.2

Q branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.4.3

P branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.5Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6 Quantum cascade laser spectroscopy of C60

105

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2

MIR QCL frequency noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.3

Cavity ring-down spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.4

Optical feedback stabilization QCL spectroscopy . . . . . . . . . . . . . . . . . . . . 110

6.5

Saturation of C60 rovibrational transitions . . . . . . . . . . . . . . . . . . . . . . . . 114

6.6

Conclusions and future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7 High accuracy rovibrational calculations of small molecules

117

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.2

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.2.1

Direct-product grid variational calculations with iterative eigensolvers . . . . 119

7.2.2

Rovibrational perturbation theory based on curvilinear vibrational mean-field
theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.3

7.2.3

Kinetic energy operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.2.4

Coordinate systems and body-fixed frame embedding . . . . . . . . . . . . . . 130

7.2.5

Potential energy surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.3.1

Photoelectron spectroscopy of the methide anion, CH−
3. . . . . . . . . . . . 133

x

7.4

7.3.2

Laboratory and astronomical discovery of disilicon carbide, Si2 C . . . . . . . 135

7.3.3

The ionization energy and equilibrium structure of hydrogen peroxide, H2 O2 140

7.3.4

The molecular structure of gauche-1,3-butadiene, C4 H6 . . . . . . . . . . . . 144

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Bibliography

149

Appendix
A Infrared line lists

172

A.1 Nitromethane, CH3 NO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
A.2 Naphthalene, C10 H8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
A.3 Buckminsterfullerene, C60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
B Spectroscopic constants of the ν3 band of vinyl bromide

182

C Icosahedral symmetry and spherical tensor operators

183

C.1 Icosahedral character table and correlations . . . . . . . . . . . . . . . . . . . . . . . 183
D Spherical top R selection rule186

D.1 Explicit derivation for linear dipole moment operator . . . . . . . . . . . . . . . . . . 186
D.2 A simple physical picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
D.3 Spherical tensor operators and the Wigner-Eckart theorem . . . . . . . . . . . . . . . 191

TablesTable
3.1

Symmetry species of torsional wavefunctions of nitromethane. . . . . . . . . . . . . . 43

3.2

Small amplitude vibrations of nitromethane. . . . . . . . . . . . . . . . . . . . . . . . 44

3.3

Effective Hamiltonian fit for m = 0 levels of ν3 + ν6 . . . . . . . . . . . . . . . . . . . 48

3.4

Rotational parameters of the ν1 and ν10o (m = 0) bands of CH3 NO2 . . . . . . . . . . 54

3.5

Ab initio torsion-rotation parameters of nitromethane. . . . . . . . . . . . . . . . . . 65

4.1

Oven temperatures used to introduce solid samples into CBGC cell. . . . . . . . . . 68

4.2

Rotational fit of the ν29 band of naphthalene. . . . . . . . . . . . . . . . . . . . . . . 70

5.1

Fitted spectroscopic parameters of the C60 R branch . . . . . . . . . . . . . . . . . . 96

7.1

Semi-experimental equilibrium geometry of gauche-butadiene.

7.2

Tunneling parameters of d6 -gauche-butadiene. . . . . . . . . . . . . . . . . . . . . . . 147. . . . . . . . . . . . 146

A.1 Line list for the 2953 cm−1 band (m = 0) of nitromethane. . . . . . . . . . . . . . . . 172
A.2 Line list for the 2953 cm−1 band (|m| = 1) of nitromethane. . . . . . . . . . . . . . . 176
A.3

Line list for the ν1 (m = 0) band of nitromethane. . . . . . . . . . . . . . . . . . . . 176

A.4

Line list for the ν10o (m = 0) band of nitromethane. . . . . . . . . . . . . . . . . . . 177

A.5

Line list for the ν29 band of naphthalene. . . . . . . . . . . . . . . . . . . . . . . . . 179

B.1 Effective Hamiltonian fits for the ν3 band of CH2 CH79 Br and CH2 CH81 Br. . . . . . 182

xii
C.1 Character table for the Ih (M) molecular symmetry group of C60 . . . . . . . . . . . . 183
C.2 Correlation of irreps D(J) of the SO(3) rotation group to those of I. . . . . . . . . . 185

Figures

Figure2.1

Simulated absorption spectrum of vinyl bromide in the CH stretch region. . . . . . .

5

2.2Time and frequency domain structure of a frequency comb. . . . . . . . . . . . . . .

8

2.3

Mid-infrared frequency comb sources. . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.4

Cavity-comb coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5

FTS detection of frequency combs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6

Frequency comb spectroscopy with a VIPA etalon. . . . . . . . . . . . . . . . . . . . 23

2.7

Cryogenic buffer gas cooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.8

A detailed view of the cold cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.9

CE-DFCS absorption spectrum of the ν3 band of vinyl bromide. . . . . . . . . . . . 32

2.10 Rotational and translational temperatures of molecules in the low-temperature CBGC
cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.11 Cooling of C60 in the high-temperature CBGC cell. . . . . . . . . . . . . . . . . . . . 34
3.1

Internal rotation in CH3 NO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2

Rotational energy levels for the m = 0, ±1 torsional states.

3.3

DC electrode geometry in the cold cell. . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4

Absorption spectrum of nitromethane near 3.3 µm . . . . . . . . . . . . . . . . . . . 46

3.5

Reduced term values of the m = 0 states of ν3 + ν6 . . . . . . . . . . . . . . . . . . . 47

3.6

DC Stark effect patterns in nitromethane. . . . . . . . . . . . . . . . . . . . . . . . . 50

. . . . . . . . . . . . . . 41

xiv
3.7

Observed and predicted energies for the m = 0, ±1 levels of the ν3 + ν6 vibrational
state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.8

Assigned transitions of the ν1 band of CH3 NO2 . . . . . . . . . . . . . . . . . . . . . . 54

3.9

Torsion excitation energy and rotational A constant as a function of the effective
barrier height V6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.10 CH bond length coordinates for nitromethane. . . . . . . . . . . . . . . . . . . . . . 60
3.11 Reduced-dimension torsion-vibration energies of the CH stretching modes of nitromethane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1

The ν29 band of cold naphthalene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2

Survey absorption spectrum of cold adamantane, C10 H16 . . . . . . . . . . . . . . . . 71

4.3

The CE-DFCS absorption spectrum of HMT. . . . . . . . . . . . . . . . . . . . . . . 71

4.4

Absorption profile of diamantane, C14 H20 . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5

Vibrational state density for several large carbon-containing molecules. . . . . . . . . 74

5.1

The structure of buckminsterfullerene, C60 . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2

Thermodynamical functions for C60 vibrations. . . . . . . . . . . . . . . . . . . . . . 79

5.3

Absorption from the 8.5 µm band of C60 . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4

The first cold C60 signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.5

Survey spectrum of cold C60 near 8.5 µm (1185 cm−1 ). . . . . . . . . . . . . . . . . . 83

5.6

Spherical top energy levels and IR transitions. . . . . . . . . . . . . . . . . . . . . . . 89

5.7

Detailed views of the 8.5 µm band of C60 . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.8

Fit of the C60 R branch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.9

J assignment via R(J) extrapolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.10 Icosahedral splitting patterns of C60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.1

Frequency noise of the free-running QCL. . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2

QCL current modulation response and PDH frequency stabilization. . . . . . . . . . 108

xv
6.3

QCL cavity ring-down spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.4

QCL optical feedback stabilization and self-locking. . . . . . . . . . . . . . . . . . . . 111

6.5

OFS-locked intensity noise and frequency ramp. . . . . . . . . . . . . . . . . . . . . . 113

6.6

Saturation of cold C60 rovibrational transitions. . . . . . . . . . . . . . . . . . . . . . 115

7.1

Discrete-variable representation (DVR) basis functions.

7.2

Contact transformations of the rovibrational Hamiltonian . . . . . . . . . . . . . . . 126

7.3

Photoelectron spectra and potential curves of CH−
3 /CH3 . . . . . . . . . . . . . . . . 134

7.4

Dispersed fluorescence spectra of Si2 C. . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.5

Potential energy surface, energy levels, and structure of Si2 C. . . . . . . . . . . . . . 138

7.6

Potential energy curve, spectra, and equilibrium structure of H2 O2 . . . . . . . . . . . 142

7.7

Structure and rovibrational energy levels of gauche-butadiene. . . . . . . . . . . . . . 145. . . . . . . . . . . . . . . . 121

D.1 A simple physical picture for the R selection rule. . . . . . . . . . . . . . . . . . . . . 190

Chapter 1

Introduction

Spectroscopy is the primary tool with which we study, probe, and manipulate matter at themolecular scale. The unique spectroscopic signatures of molecules permit detection and monitoringwith a high degree of sensitivity and specificity. The patterns encoded in the spectrum itself reveal
precise details of molecular structure and dynamics. By understanding the nature of this structure,
we can harness molecules as highly controllable complex quantum systems. Advances in our ability
to measure and interpret complex molecular spectra thus open new avenues for a wide range ofproblems in chemical and molecular physics.
The focus of this thesis is high resolution frequency domain spectroscopy of polyatomic
molecules in the mid-infrared (MIR) spectral region (ca. 3 to 10 µm). MIR photons carry sufficientenergy to excite one or more quanta of nuclear vibrational motion. As all polyatomic molecules
possess at least one infrared (IR) active vibrational mode, IR spectroscopy is a universal and essential tool for probing molecular structure and dynamics. “High resolution,” in this context, means
resolving spectroscopic fine structure associated with changes in rotational, and even spin and hyperfine, quantum numbers that occur simultaneously with the primary vibrational transition. Byreaching this level of resolution, it is possible to observe all internal molecular degrees of freedom
in specific, well defined quantum states. The individual transition frequencies between these states
can then be used to determine effective Hamiltonians that connect spectroscopic observations tomolecular scale details, such as geometry, electronic structure, and potential energy surfaces.
Such precise insights have made high resolution infrared spectroscopy an important tool for2studying molecular structure and dynamics. However, its applications have usually been restricted
to small, light molecules. This is because at room temperature systems with even a few heavy
atoms can already occupy millions of rotation-vibration quantum states, leading to severe spectral congestion. Targeting larger, more complex systems requires a high resolution and sensitive
spectrometer coupled with a source of cold, dense molecular gases. This thesis discusses overcoming these experimental challenges by combining two powerful techniques: cavity-enhanced direct
frequency comb spectroscopy (CE-DFCS) and cryogenic buffer gas cooling (CBGC). CE-DFCS exploits the simultaneous broadband and narrow-linewidth properties of infrared frequency combs bycoupling them to high finesse optical cavities to yield high resolution and high sensitivity absorption spectra in a massively parallel fashion. CBGC efficiently quenches translational and internalmotion in molecules to temperatures as low as a few kelvin, drastically reducing spectral congestion
and linewidths, and enhancing absorption signal strengths. This thesis demonstrates harnessingthese tools to obtain high resolution infrared spectra of unprecedentedly complex molecules. The
culmination of this work is the first rotationally resolved spectrum of the C60 fullerene, now by far
the largest molecule for which such measurements have been obtained.
The remaining chapters are organized as follows. Chapter 2 provides an introduction to
CE-DFCS, the frequency comb sources and detection methods used in these studies, and the details of the frequency comb-buffer gas cooling apparatus. One of the first non-trivial molecules
studied with the instrument was nitromethane, CH3 NO2 , which represents a unique example ofcompletely unhindered internal rotation. Chapter 3 analyzes the spectroscopy of this system as acase study in large-amplitude nuclear motion. It also introduces theoretical tools developed to treat
its unusual nuclear motion dynamics. Chapter 4 explores the limits of high resolution spectroscopyof large molecules near 3 µm. We examine the intrinsic challenges to rotationally resolved infrared spectroscopy in the CH stretch region of several large hydrocarbons, including naphthalene,
adamantane, and diamantane. Shifting to the long wavelength infrared region, Chapter 5 presents
the first rotationally resolved frequency comb spectra of C60 near 8.5 µm and discusses the changes
in CBGC conditions required to successfully realize vibrational ground state cooling. The first3
C60 experiments pose as many new spectroscopic puzzles as they solve, and Chapter 6 discusses
on-going C60 experiments to address these questions using continuous wave quantum cascade laser
spectroscopy. Finally, Chapter 7 provides a more comprehensive overview of the theoretical tools
introduced in Chapter 3 and discusses several more of their recent applications for understanding
the spectroscopy of complex polyatomic molecules.

Chapter 2

Cavity-enhanced direct frequency comb spectroscopy of buffer gas cooled
molecules

All spectroscopic experiments consist of three primary components: a light source, a sample
of matter that interacts with the radiation from the light source, and a detector that measuresthis interaction in some way. The many realizations and combinations of such components comprise the large, diverse set of modern spectroscopic techniques [1]. The nature of the physics one
wishes to probe and its underlying spectroscopic signatures dictate the required parameters of agiven spectroscopic experiment: wavelength coverage, frequency resolution and stability, detection
bandwidth and sensitivity, sample density, temperature, pressure, amongst many others.
This chapter details the techniques employed in this thesis for high resolution rovibrational
infrared spectroscopy of gas-phase polyatomic molecules. To appreciate the specific challenges ofthis goal, consider the simulated spectrum in Fig. 2.1 of vinyl bromide (C2 H3 Br), which in many
ways represents a relatively simple “textbook” example of a molecule of its size (6 atoms) andmass (106 or 108 u, depending on the isotopologue). Figure 2.1a shows the three fundamental
CH stretching vibrational bands, which span a 100 cm−1 window near 3050 cm−1 (3.3 µm), at atemperature of 15 K. The peak absorption cross sections are approximately 10−17 cm2 per molecule.
Even at this low temperature, each vibrational band exhibits considerable fine structure, as seen in
Fig. 2.1b for the ν3 band. Taking yet a closer look, Fig. 2.1c shows a small 1-cm−1 -wide portion ofthe ν3 band with dense patterns associated with Ka sub-band structure, nuclear hyperfine effects,
and isotopic mass splittings.

5

Cross section (10-17 cm2)

c
0.4
0.3
0.2
0.1
0
3029.203029.223029.243029.26

3024

3026

3028

3029.283029.30

3029.32

Cross section (10-17 cm2)

b
1.5

ν3

1
0.5
0
3022

3030

3032

Cross section (10-17 cm2)

a
2.5

ν2

2

ν1

ν3

1.5
1
0.5
0

3020

3040

3060

3080

3100

3120

Wavenumber (cm-1)
Figure 2.1: Simulated absorption spectrum of vinyl bromide in the CH stretch region. (a) The threefundamental CH stretch bands, conventionally labeled ν1 , ν2 , and ν3 , span a 100 cm−1 window.
The spectrum is calculated for T = 15 K. (b) A zoom-in of the rotational fine structure of the ν3band. (c) A small 1 cm−1 -wide portion of the ν3 band.

6This example illustrates the general reality of high resolution rovibrational spectroscopy.
Strong fundamental vibrational bands occur over the entire MIR region, with hydride stretchingvibrations (e.g. CH and OH) at the short wavelength end (ca. 2.7 µm to 3.3 µm) and bending, twisting, and heavy atom streching vibrations at the long wavelength end (5 µm to 15 µm)
and extending into the far-IR/terahertz regions (> 15 µm). Efficiently probing a wide variety ofmolecules and multiple vibrational bands requires a light source with broad wavelength tunability
and bandwidth. At the same time, each vibrational band can consist of thousands of individualrovibrational transitions, with line densities easily surpassing 10 to 100 per cm−1 , demanding highspectral resolution. Minimizing spectral congestion and line widths requires preparing cold, lowpressure molecular gases. Furthermore, for linear absorption measurements, which are the focus
here, high detection sensitivity is needed to overcome intrinsically small cross sections.
The general experimental challenges identified for “simple” polyatomic systems like C2 H3 Br
become even more important for the complex polyatomic molecules that are the subject of thiswork. As we will see in later chapters, this complexity occurs as a matter of both kind and degree.
An example of the former is large amplitude motion in nitromethane, CH3 NO2 (Chapter 3). Here,unhindered internal rotation leads to a breakdown of the semi-rigid framework usually employed tomodel rovibrational spectra. Examples of the latter are the large hydrocarbon cages adamantane(C10 H16 ) and diamantane (C14 H20 ), as well as the C60 fullerene (Chapters 4-6). Despite being
relatively rigid molecules, the sheer number of atoms and total mass of these systems leads todensities of internal states that are at the very frontier of what high resolution infrared spectroscopy
can probe.
Returning to the three primary components of spectroscopic experiments, this chapter introduces the techniques of cavity-enhanced direct frequency comb spectroscopy and cryogenic buffer
gas cooling, which provide the necessary light source, detection methods, and sample conditions toaddress the challenges of high resolution rovibrational spectroscopy of large, complex polyatomic
molecules. Infrared frequency combs offer simultaneous high spectral resolution and wide spectral
coverage, while cavity-enhanced detection coupled to broad bandwidth interferometric and disper-

7
sive readout provide high detection sensitivity. These spectrometer capabilities are fully realizedwith the cold, dense samples of gas-phase molecules provided by cryogenic buffer gas cooling. The
remainder of this chapter discusses in detail each of these experimental components.

2.1

Mid-infrared frequency combs
The development of the optical frequency comb has dramatically advanced the fields of pre-cision metrology and spectroscopy [2, 3]. While combs are now a common tool for frequency referencing, they can also be used to interact directly with samples under study, giving rise to a familyof techniques known as direct frequency comb spectroscopy [4]. By coupling frequency combs
to high-finesse optical cavities, cavity-enhanced direct frequency comb spectroscopy (CE-DFCS)
provides simultaneously broad spectral bandwidth, high spectral resolution, and high detection
sensitivity [5, 6]. The extension of CE-DFCS to the mid-infrared spectral region over the last ten
years has enabled these advantages to be exploited for molecular rovibrational spectroscopy [7].
Frequency combs are commonly generated by mode-locked lasers that emit a train of ultrafast
pulses of light with a regular period τ of the order of a few nanoseconds (Fig. 2.2a) [3]. Dispersion
inside the laser oscillator cavity causes a pulse-to-pulse phase slip ∆φce of the carrier field relative
to the pulse intensity envelope. This regular time domain structure leads to the characteristic
comb-like structure in the frequency domain (Fig. 2.2b). Each phase-coherent frequency comb
mode is separated in optical frequency by the repetition rate frep = 1/τ . The spectral bandwidthof the comb is proportional to the inverse of the ultrafast pulse length, typically spanning at least 1
THz. The carrier-envelope phase slip ∆φce shifts the entire comb spectrum by an offset frequency
f0 = frep × ∆φce /2π. The optical frequency of each comb mode is thus νn = n × frep + f0 , where nis the integer comb mode index. Measuring the RF frequencies frep and f0 determines the absolute
frequency position of each optical mode in a calibration-free manner. Active feedback on these two
comb parameters can be used to suppress frequency noise, yielding narrow-line-width comb modes.
MIR frequency combs are typically generated by non-linear wavelength conversion of near-IR(NIR) combs [8–16]. The latter are currently most commonly produced with fiber oscillators doped

8

a
Δφce

E(t)

2Δφce
t

τ

b
I(ν)

0

f0 = frep × Δφce / 2π

frep = 1/ τνn = n × frep + f0

νFigure 2.2: Time and frequency domain structure of an optical frequency comb. (a) Mode-locked
fiber lasers produce ultrafast pulse trains with a repetition period of τ . The pulse-to-pulse carrierenvelope phase slip is ∆φce . (b) In the frequency domain, the comb contains many optical modes,
or “teeth,” at frequencies separated by the repetition rate frep with a uniform frequency offset f0 .

with ytterbium, erbium, or thulium, which provide robust, passive mode-locked operation near 1.0,
1.5, or 2.0 µm, respectively, with repetition rates typically of the order of 100 MHz. Subsequent
fiber amplification can deliver average optical powers of several watts. The correspondingly highpeak intensity of the femtosecond pulses enables efficient non-linear frequency conversion. We use
two MIR comb sources in this work: a 3-5 µm optical parametric oscillator (OPO) [11] (Fig. 2.3a)
and a 6-11 µm difference frequency generation (DFG) comb [15] (Fig. 2.3b).

9amicrowave
ref. signal

idler f0 locking
electronics

frep locking
electronics

fast PZT slow PZT
LP

M1

M2

fan-out
PPLN

PZT

Yb:fiber comb
1 µm, 10 W
frep = 136 MHz

p+i

p

SC-p
highly nonlinear
supercontinuum fiberbEr:fiber oscillator
1.6 μm; frep = 93.4 MHz

Er ampPDH
sidebands
fast
PZT

idler f0

BP filter

Intensity (a.u.)

11× frepi(3-5 µm)

950

cavity mirror
coating1000 1050 1100 1150 1200 1250 1300 1350 1400

Frequency (cm-1)

p

EOM

slow
PZT

pum

LP filter

sign

al

frep feedback

AOM

set f0Er ampTm amp2 μm

supercontinuum
fiber

OP-GaP
DFG
crystal

idler
50 mW
6-11 μm

error signal
from nonlinear
parasitics

Figure 2.3: Mid-infrared frequency comb sources. (a) A high power OPO [11] is synchronously
pumped by a 1 µm Yb:fiber comb. Idler light (i) from 3 µm to 5 µm is generated by tuning thepoling period of a fan-out PPLN crystal. frep is monitored by an oscillator pick-off detector andactively stabilized by feedback onto oscillator cavity PZT elements. A portion of the pump light(p) is broadened in a highly nonlinear fiber. The supercontinuum output (SC-p) is co-aligned withparasitic pump-idler sum frequency light (p + i) and detected on a photodiode. The beat signal at
the idler f0 frequency is used to feed back onto PZT-actuated mirrors (M1, M2) in the OPO cavity.
(b) The DFG comb [15] is generated by a single 1.6 µm Er:fiber oscillator. The oscillator output
is split into two arms, broadened, amplified, and recombined in an OP-GaP crystal to generate
MIR idler light. Adjusting the quasi-phase matching period in the OP-GaP crystal provides idler
tuning from 6 µm to 11 µm. The idler f0 is controlled by the AOM driver frequency. Frequency
sidebands and frep feedback are generated by PZT-actuated mirrors and an EOM in the fiber
oscillator. The time delay between the two DFG pump pulses is actively stabilized by monitoringparasitic nonlinear signals. (Inset) The broad idler spectrum is narrowed with a bandpass (BP)filter centered on C60 absorption band origin near 1185 cm−1 .

10
2.1.1

3–5 µm OPO comb

The high power MIR OPO was originally designed and built in the Ye lab a decade ago by
Adler et al. [11]. The OPO is synchronously pumped by a 1 µm ytterbium (Yb) fiber comb thathas a repetition rate frep = 136.6 MHz and a maximum output power of 10 W. The OPO is basedon a fan-out MgO-doped periodically poled lithium niobate (PPLN) crystal, which sits at the waistof a cavity that is singly resonant at the signal wavelength. By adjusting the quasi-phase matching
period of the crystal, idler output is generated with a center wavelength tunable from 2.8 µm to
4.8 µm, a maximum average power of 1.5 W, and a simultaneous bandwidth up to 0.3 µm. We
typically operate the OPO with idler light from 3.0 µm to 3.3 µm with an average power of 100 mW
to 300 mW.
Stabilization of the optical frequencies of the idler comb requires active locking of the boththe carrier-envelope offset frequency f0 and the repetition rate frep . We lock frep by picking off
a small portion of the 1 µm Yb:fiber comb and measuring its repetition rate directly on a fast
photodiode. The seventh harmonic of the repetition rate 7frep ≈ 956 MHz is mixed with a stable
1 GHz Wenzel quartz oscillator, slowly slaved to a 10 MHz cesium (Cs) clock. The resulting beatnote is used to generate a phase error signal with respect to the RF output of a direct digitalsynthesizer (DDS), which is phase locked to the same RF oscillator. This error signal is then used
to feed back simultaneously on a fast piezo (PZT)-actuated cavity mirror and a slower PZT fiber
stretcher in the fiber laser oscillator.
The f0 frequency of the MIR idler comb is measured by performing an optical beat note
between the 1 µm pump light (p) and the parasitic sum frequency of the pump and idler (p + i)
generated by the OPO crystal. A 150- to 200-mW portion of the pump light is picked off and coupled
into a 15-cm-long piece of highly nonlinear supercontinuum fiber to broaden it to ∼ 780 nm in order
to be spectrally overlapped with the p + i light. The output of the supercontinuum fiber and thep + i light are then spatially and temporally overlapped onto a single photodiode to measure thep − (p + i) optical beat note, which is equal to f0 of the idler comb. This RF signal is used to11generate a phase error signal with another DDS set to the desired f0 value. The error signal isfiltered and fed back onto PZT-actuated mirrors in the OPO cavity.

2.1.2

6–11 µm DFG comb

The DFG frequency comb system [15] was designed and built by our collaborators Kevin Lee
and Martin Fermann. It is based on a 1.5 µm Er-doped fiber oscillator with a 93.4 MHz repetition
rate. frep control is achieved with a fast piezo oscillator end mirror and a slow oscillator fiber
stretcher. An intracavity electro-optic modulator (EOM) provides additional fast-bandwidth frep
control. Oscillator light is split into two arms. The first is amplified in Er fiber to 1.6 W with 110 fs
pulse duration. The second is pre-amplified with Er fiber, recompressed, broadened to 2 µm in ahighly nonlinear fiber, and finally seeded into a Tm fiber amplifier generating 0.6 W with 60 fs pulse
duration. The two arms are spatially and temporally overlapped and focused into an orientation
patterned gallium phosphide (OP-GaP) crystal, which has several quasi-phase matching grating
periods available to tune the spectrum of the DFG from 6 µm to 11 µm, with up to 60 mW of
average power. A delay stage in the 2 µm drive arm is actively servoed by monitoring nonlinear
parasitic output of the OP-GaP crystal. This feedback maintains optimal temporal overlap of thetwo driving pulses in the presence of slow thermal drifts.
The f0 of the DFG frequency comb is equal to the difference of the f0 of the two drivingarms. In this design, both arms originate from the same oscillator and therefore cancel, yieldinga DFG f0 = 0 identically. To shift the f0 to non-zero values, an acousto-optic modulator (AOM)
is placed in one of the arms. By driving the AOM with a stable RF source, the desired f0 can be
simply dialed in and is passively stable. A small portion of the oscillator light is picked off andmeasured on a fast photodiode for frep monitoring. The eleventh harmonic at 11frep ≈ 1027 MHz
is mixed with a stable 1 GHz oscillator, and the beat note is used to generate an frep phase error
with respect to a reference RF signal generated by a DDS (slaved to the same 1 GHz oscillator).
This phase error can be fed back onto the oscillator cavity length piezos to lock frep .

12

2.2

Absorption spectroscopy with frequency combs
An infrared spectrum is obtained by directing comb light through a molecular sample andmeasuring how much each individual comb mode is absorbed. The transmitted intensity I of agiven comb mode is determined by the standard linear absorption (Beer’s Law) expression,
I = I0 exp(−N σ`),(2.1)

where I0 is the incident intensity, N is the number density of the absorbing species, σ is the absorption cross section, and ` is the sample path length. The cross section σ is frequency, temperature,and pressure dependent.
Obtaining the highest signal-to-noise ratio for the absorbed fraction of light A = 1 − I/I0
requires increasing N , σ, or `. For volatile species that are gaseous at room temperature, it isusually easy to reach large number densities N by using high pressure samples. However, collisional
broadening will eventually dominate other contributions to the absorption line width, setting apractical upper limit. The collisionally broadened line width ∆νcol is approximately linear withpressure, ∆νcol = γp, where the broadening coefficient γ is typically 1 − 10 MHz/Torr. At low
pressures, the absorption line width is dominated by Doppler broadening, which has a contributionsr∆νD
8kB T ln 2
T /K=
= 7.16 × 10−7
(2.2)
2
ν0
mc
m/amu
where ∆νD is the full width half maximum (FWHM) of the line, ν0 is the center frequency, kB is
Boltzmann’s constant, T is temperature, m is the molecular mass, and c is the speed of the light.
For example, at 300 K methane has a Doppler line width of about 280 MHz in its strong 3 µm
CH stretch region and a self-broadening coefficient γ ≈ 2 MHz/Torr [17]. At this temperature, theuseful upper density limit is about 1018 cm−3 . Heavier molecules and lower temperatures will resultin smaller Doppler line widths, decreasing the useful density limit. Often, technical considerationsof the absorption cell or a low sample vapor pressure will limit the achievable densities to evensmaller values. The typical molecule density in our experiments is about 1012 cm−3 (see below).
The absorption cross section σ of a rovibrational transition is proportional to the intrinsic

13
strength of the transition (i.e. the squared magnitude of the transition dipole moment), the lower
state population, and the inverse of the line width [18]. The intrinsic line strength is strongest forvibrational transitions for which ∆v = 1, where v is a vibrational quantum number. Therefore,
most of the work here is focused on fundamental vibrational bands (v = 1 ← v = 0), though we will
see some examples of relatively strong combination bands (involving a change in quantum numberof more than one vibrational mode). The lower state population can be concentrated into fewer
quantum states by minimizing the rovibrational partition function Qrot-vib . Both Qrot-vib and theline width, which we assume to be Doppler-limited, increase with temperature. Thus achieving low
temperatures, without unduly compromising number density, is a critical requirement for sensitive
absorption spectroscopy. For the experimental conditions we typically use, we achieve peak crosssections of approximately 10−17 cm2 (see also Fig. 2.1).
The last factor in absorption strength is the path length ` over which the frequency comb
light interacts with the sample. Physical considerations often limit practical table-top cells todimensions less than 1 m. The cell in our experiment has a sample path length of only 6 cm.
Taking the typical values for N and σ discussed above, we can estimate an expected maximum
single pass absorption loss of N σ` = (1012 cm−3 )(10−17 cm2 )(6 cm) = 60 ppm. Such a small
fractional absorption requires stringently low noise baselines to achieve useful signal-to-noise ratios
over practical acquisition times.

2.3

Cavity-enhanced frequency comb spectroscopy
Making significant improvements in N and σ is challenging. The most efficient way to increase

the absorption sensitivity is thus to extend the effective path length of the sample. A simple wayof implementing this is with a multipass cell, such as a White [19] or Herriott [20] configuration,
where light is repeatedly reflected through the absorbing sample. Although the number of passes
can in principle be quite large, technical difficulties often limit it to < 100. Moreover, cumulative
losses from each reflection (e.g. about 1% per reflection on gold mirrors) severely attenuate thetotal power throughput.

14
Large path length enhancement factors and more efficient throughput can instead be achieved
with high-finesse optical cavities [21]. In contrast to multipass cells, optical cavities rely on interference between reflected light to provide long effective path lengths with much less attenuation. Herewe consider the simple case of a symmetric Fabry-Pérot cavity, formed by two identical, parallel
mirrors separated by length ` (Fig. 2.4). Each mirror has a reflectivity R, transmission T , and loss
L (including both absorptive and scattering losses), with R + T + L = 1. Light incident on thecavity is transmitted through the first mirror and reflects back and forth interfering with itself.
If the cavity length ` and the wavelength λ are such that this interference is constructive, thenthe light will resonantly build up and transmit through the cavity. Longitudinal resonances occur
at regularly spaced optical frequencies separated by the free spectral range, FSR = c/2n`, where
n is the index of refraction of the intracavity medium.1 The regularly spaced mode structure of
frequency combs is well suited to coupling into the similarly periodic resonance structure of opticalcavities.
The line width of each cavity resonance is ∆ν = FSR/F, where F is the cavity finesse.
Assuming we have high reflectivity mirrors (1 − R = T + L  1), then the finesse is to a good
approximation

√
π R
π
F≈
≈
.
1−R
T +L

(2.3)

When incident laser light is at a frequency several or more line widths away from a resonance, it isalmost completely reflected back from the cavity. If tuned to a resonance, then it will couple into
the cavity and be transmitted with a power throughput ofτcav =

T
T +L

2
.

(2.4)

The transmitted light experiences many round trips inside the cavity as it gradually leaks out
through the end mirrors. The effective number of passes for light resonant with the cavity is 2F/π.
To illustrate the magnitude of these various cavity parameters, consider a typical MIR1

Our cavities are typically in vacuum or filled with a low density of non-refractive gas such that n = 1 for all
relevant wavelengths. An exception to this is near molecular absorption resonances where dispersive effects can bereadily seen for sufficiently strong transitions.

15
coated dielectric mirror with T = L = 250 ppm, yielding a finesse of F ≈ 6300 and a transmission
coefficient of τcav = 25%. Our cavity design, discussed in more detail below, has a mirror separationof about 54 cm, and therefore an FSR of 280 MHz and a line width of ∆ν = FSR/F = 44 kHz.
Most importantly, the effective path length enhancement factor is 2F/π = 4000. This provides asignificant and crucial boost in absorption sensitivity.

acavity length, ℓtransmission
τcav

incidentlightb

3frep= 2FSR

Cavity resonanceΔν =FSR / F

ν

frep
=
FSR
FSR

optical frequency

νFigure 2.4: Cavity-comb coupling. (a) Frequency comb light is incident on a symmetric FabryPérot cavity of length `. When resonant, a fraction τcav of the light (Eq. 2.4) is transmitted throughthe cavity, making F/π round trips, where F is the cavity finesse. (b) The cavity transmission
resonances (black trace) are spaced by the free spectral range, FSR = c/2`, with a line width (fullwidth-half maximum) of ∆ν = FSR/F. In one case, a frequency comb (red trace) has frep = FSR,
such that each comb mode is resonant with the cavity. In a second case, a frequency comb (bluetrace) with 3frep = 2FSR allows only every third comb mode to be resonant with the cavity,
resulting in a larger effective frep for the transmitted light.

16
2.3.1

Cavity-comb coupling

Efficient cavity-comb coupling requires simultaneously matching the optical frequency of eachcomb mode with that of a cavity resonance [6]. This implies that frep and the cavity FSR must be
related by some integer ratio, m × frep = n × FSR. In the simplest case where m = n = 1, eachcomb mode and cavity resonance are simultaneously coupled in a one-to-one fashion. For otherinteger ratios with m, n, or both greater than 1, only a subset of comb modes will be resonantly
coupled to the cavity (Fig. 2.4b). The obvious disadvantage in this case is that not all the comb
modes can be used for spectroscopy. However, this type of cavity filtering can also be useful.
The transmitted comb has a higher effective frep , and it is therefore easier to individually resolve
transmitted comb modes with a given detection apparatus. This idea is also used for frequency
comb Vernier spectroscopy, where the cavity itself disperses comb modes one-by-one [22]. Detailsof our cavity filtering are discussed further in Section 2.4.
Just as dispersion in the comb laser oscillator cavity induces a non-zero f0 offset frequency,
dispersion from the enhancement cavity mirrors shifts the cavity resonances uniformly by fcav .
Therefore we also require that f0 = fcav . In a typical setup, the fcav value can be measured bytuning the comb f0 until optimal coupling is observed. In the case of the MIR OPO discussed above,
this value is then used as the reference frequency for the f0 phase lock. For the DFG comb, theAO shifter drive frequency is simply set to fcav . The value of fcav depends on the static dispersivematerial properties of the mirror coatings, so that for a given wavelength, it is effectively fixed.
The optical frequencies of the comb modes and cavity resonances are equal to large integer
multiples of the comb frep and cavity FSR, respectively. Therefore, fast relative fluctuations between
frep and the FSR make active feedback necessary. A common tool to achieve tight comb-cavity
locking is the Pound-Drever-Hall (PDH) technique [23]. Here, frequency sidebands are placed on
the comb laser at a modulation frequency fmod larger than the reference cavity line width. A fast
photodiode is used to measure the reflected cavity light intensity signal, which is demodulated atfmod to generate the PDH error signal. This error signal is linearly proportional to the comb-cavity

17frequency detuning over a frequency range approximately equal to the cavity line width. After
passing the error signal through a proportional-integral (PI) filter, it is fed back onto the comb frep
value. The error signal can also be further slowly integrated and fed back onto the cavity length
via piezo actuators to compensate for long term drifts.
For the OPO comb, we generate frequency sidebands by driving a fast piezo actuated mirror
in the fiber oscillator near its 760 kHz mechanical resonance frequency. The error signal is fed backonto the same fast piezo, as well as a tube piezo on the enhancement cavity that servoes its FSR.
For the DFG comb, we use an intracavity EOM in the fiber oscillator for the frequency
sideband generation. In order to induce sufficient phase modulation in the EOM, a passive resonator
is formed with a 100 µH inductor in series with the EOM electrode plates. The intrinsic capacitanceof the EOM crystal (CEOM ≈ 43 pF) together with additional parasitics result in a resonance√frequency of fLC = 1/2π LC = 2.65 MHz, with a Q of about 14. At frequencies below fLC , theresonator passes feedback signals with unity gain. The PDH error signal is fed back onto botha fast piezo actuated oscillator mirror and a slow fiber stretcher. Additional feedback is placedon the oscillator EOM to achieve higher locking bandwidth. The PDH lock only ensures that thedifference between frep and the cavity FSR (or suitable integer multiples) is zero. Once the PDH
lock is actively slaving the comb to the cavity, the comb frep is measured using the 11th harmonic as
discussed above. Its phase error relative to a stable RF reference is fed back onto the enhancement
cavity length to stabilize the absolute value of frep .
The PDH technique continuously locks the comb mode frequencies to their respective cavity
resonances. However, the narrow line width and steep slope of the cavity transmission resonances
can convert residual frequency noise into amplitude noise of the transmitted light. This FM-to-AM
noise conversion can limit the absorption sensitivity. An alternative cavity locking scheme to PDH
locking, called swept cavity locking [6], is relatively immune to FM-to-AM noise conversion andcan provide significant sensitivity improvements. We implement a swept cavity lock by modulating
the enhancement cavity length at 10 kHz via the cavity tube piezo. As the cavity length is swept,
it passes through resonance with the comb light, permitting a transient buildup of power and18transmission burst for each comb mode simultaneously. These transmission bursts occur at twice thecavity length modulation frequency, once for the forward sweep and once for the backward sweep.
The relative time spacing between the transmission bursts when the cavity is being scanned forwardversus backward determines the mean frequency offset of the cavity relative to the comb light. A
small fraction of cavity transmission is picked off, measured on a photodiode, and demodulated at
the cavity sweep frequency with a lock-in amplifier to generate an error signal that is fed back onto
the cavity length piezo. The comb frep only needs to remain within the range of the cavity length
sweep. Therefore the slow feedback bandwidth of the cavity piezo (∼ 1 kHz) is sufficient to maintainthe lock. In the swept cavity method, the amplitude of the transmission burst is determined by
the relatively constant optical slew rate of the swept cavity resonance and insensitive to the cavitycomb optical detuning. Therefore it can provide cavity transmission with relatively low intensity
noise at the cost of reduced duty cycle.

2.4

Comb detection
Two detection methods are used to acquire broadband comb spectra in these experiments.

The first is the familiar technique of Fourier transfrom spectroscoscopy (FTS) and the second
is based on spatial dispersion with a virtually image phased array (VIPA) etalon. This section
discusses the details of each detector and their relative advantages and performance.

2.4.1

FTS detection

We perform FT spectroscopy using a home-built fast scanning interferometer [24] (Fig. 2.5a).
Each arm of the interferometer is double-passed on a translating corner-cube reflector with a physical travel range of 0.7 m. This geometry provides ` = 4 × 0.7 = 2.8 m of optical path length
difference and an instrument resolution of ∆νFT = c/` = 110 MHz (Fig. 2.5b,c). A complete cart
scan (and return) takes 5 − 10 s. If the input to the FTS were a continuous white light source, then∆νFT would determine the ultimate spectral resolution. In the case of a frequency comb, however,the FTS instrument line width is sufficiently narrow to resolve individual comb modes so long as

19
the effective frep < ∆νFT . Once this limit is reached, the effective spectral resolution is determined
only by the line width of the frequency comb (∼ 1 − 10 kHz), thereby providing high resolution
measurements spaced by frep [25]. To fill in the “spectral gap” between the narrow comb modes,
multiple measurements with different frep values are acquired and interleaved. For FTS experiments, we use enhancement cavity filtering that produces an effective frep in the 250 − 300 MHz
range, well into the comb-mode-resolved limit.
An important technical consideration is the instrument line shape of the FTS. We use a
rectangular interferogram window, resulting in a sinc line shape function in the frequency domain.
The outer lobes of the sinc function centered at a given comb mode in general overlap with thecenters of neighboring comb modes, resulting in unwanted cross-talk [25]. We use an automatedrewindowing process to relocate nodes of the instrument line shape function to coincide with thepositions of neighboring comb modes, thereby eliminating cross-talk.
Given our typical FTS cart scanning velocity (5 − 15 cm/s) and optical wavelengths (3 −10 µm), the interferogram signal carrier frequencies are in the 20 − 200 kHz range. The noisebaseline of the FTS in this region is dominated by the intensity noise of the incident comb light.
We perform differential detection to suppress this noise. Each output port of the FTS beamsplitteris measured on a photodiode. The intensity noise is an in-phase common-mode signal on eachphotodiode, while the interferogram signal from one output port is 180◦ out-of-phase relative tothe other port. By subtracting the signal of the two photodiodes, the interferogram signal is doubled
and the common-mode intensity noise is canceled.2 Effective common-mode noise rejection requiresthat the relative gain of the photodiode pair be tuned optimally. This is impractical to do manually
because the average relative power incident on the photodiodes varies over the course of an FTS
cart scan. (As the cart position changes, there are unavoidable drifts in the beam direction andsize.) Therefore, we use a JILA-made autobalancing subtraction circuit that adjusts the gain ofeach photodiode to match their DC levels throughout a cart scan. We typically achieve 20 dB to
2

Uncorrelated noise between√ the two detectors, such as shot and dark noise, are of course not canceled. They
do benefit, however, from a 1/ 2 reduction in their relative amplitude because of the 2× enhancement factor of theinterferogram signal.

a

CW

MIR
COMB

QBS

20

b
PDA
PDB

ABS

PDref

CC

-0.02
35

3

30

2

25

Intensity (arb.)

Intensity (arb.)

c

DAQ

20
15
10
5
0

-0.01

0

0.01

0.02

1
0
-1
-2

1168.18 1168.22 1168.26
Wavenumber (cm )-1-3-100 -80 -60 -40 -20

0

20

40

60

80 100

Optical path length difference (cm)Figure 2.5: FTS detection of frequency combs. (a) MIR comb light and a 1 µm reference cw
Nd:YAG NPRO laser are co-propagated in the FTS interferometer. A quadrant beam splitter(QBS) splits both inputs into two arms that are incident on a scanning corner-cube reflector (CC).
The cw interferogram is measured on a photodiode (PDref ) for optical path length calibration.
The two MIR QBS outputs are each measured on photodiodes (PDA , PDB ). The difference signal
from an autobalanced subtraction (ABS) circuit is acquired with a fast digitizer (DAQ). (b) An
example comb interferogram shows a center burst at zero optical path length difference due tointerference between one ultrafast pulse and itself. Two additional bursts on either side arise frominterference between sequential pulses in time. The zoomed-in window shows the carrier wave of
the MIR light in the center burst region. (c) A narrow portion of the Fourier-transformed spectrum
from the comb interferogram in (b) shows individually resolved comb modes. The spacing between
the peaks (0.009 cm−1 ) is equal to the inverse of the optical path length difference between theinterferogram center bursts (110 cm).

30 dB of common-mode noise rejection, which in some cases is sufficient to reach the shot-noise
limit [26], though we are usually limited by residual non-shot-noise intensity fluctuations. For the
3-5 µm OPO system, we achieve a detection sensitivity of 4.4 × 10−8 cm−1 Hz−1/2 for a single comb

21
mode. Given that the typical bandwidth of the FTS spectrum contains 3300 resolved comb lines,
this corresponds to 7.6 × 10−10 cm−1 Hz−1/2 per spectral element (PSE) [27]. For the DFG system
operating near 8.5 µm, we achieve a sensitity of 2.2 × 10−7 cm−1 Hz−1/2 for a single comb mode.
The spectral bandwidth contains about 5000 resolved comb modes, corresponding to 3.1 × 10−9
cm−1 Hz−1/2 PSE [28].
A conventional absorption measurement requires detecting the light intensity with (I) andwithout (I0 ) the absorber to determine the absorption A = 1 − I/I0 . We typically forego acquiring
two separate interferograms for such a measurement as the cavity-transmitted intensity of the comb
light gradually drifts over the relatively long FTS acquisition time of 5 − 10 s per interferogram.
The absorption spectra for cold species are usually rather sparse with only a minor fraction of
the comb modes being absorbed for a given frep value. Therefore, we can take the comb intensity
envelope as a proxy for the reference intensity I0 . One approach could be to fit this envelope with aslowly varying polynomial or spline function. We take the even simpler route of high-pass filtering
the cavity-transmitted spectrum. This processing is performed for the acquired comb spectrum foreach frep value before interleaving them in the final composite spectrum.

2.4.2

VIPA spectrometer

Two important disadvantages of FTS detection are the relatively long acquisition times (upto ten seconds per interferogram) and the requirement of continuous incident light. The latter
necessitates PDH comb-cavity locking, which as discussed above suffers from FM-to-AM noise
conversion. An alternative to interferometric detection is to use a dispersive element to spatiallyseparate comb modes and subsequently image the broadband spectrum on an infrared camera [6].
This general approach has the advantage of permitting fast, time-resolved, transient detection [29–32] (typically limited by the minimum camera integration time and maximum frame rate) andbeing compatible with the intermittent transmission of the swept-cavity locking scheme.
The primary challenge is constructing a dispersive spectrometer with sufficient resolution to
spatially separate neighboring comb modes. The required resolving power for a diffraction grating

22
is R = mN = λ/∆λ ≈ ν/frep ∼ 105 − 106 , where m is the diffraction order and N is the number ofilluminated grooves. Standard MIR ruled gratings have groove densities on the order of 102 /mm
and operate at low diffraction order (often m = 1). A grating of this type would need to be about
10 m long to provide sufficient resolving power, which is obviously impractical for a table-top
laboratory instrument. Recent advances in high-diffraction-order, high-refractive-index immersion
gratings, however, have made this approach viable [33].
An optical element called a virtually imaged phased array (VIPA) etalon provides the highangular dispersion and resolution necessary for spatial separation of individual comb modes [34–36](Fig. 2.6a). The setup consists of a thin glass parallel plate etalon. The back (output) surface hasa high reflection coating (∼ 95% or more), while the front (input) surface has a ∼ 100% coating.
Input light is cylindrically focused and coupled into the etalon at a gap in the coating on the front
face with a small incidence angle θ. The light bounces back and forth in the etalon, leaking throughthe output face at each bounce. The interference of the reflected beams forms collimated output
light, where the output propagation angle is determined by the optical wavelength. (The angulardispersion is proportional to n cot θ, where n is the index of refraction of the etalon material.) The
angular resolution is determined by the total loss per round trip in the etalon, and is therefore
limited by the bulk material absorption and scattering. With our MIR etalons, VIPA line widths
of 0.5 − 1.0 GHz are achievable, corresponding to a resolving power of ∼ 105 [29, 30, 37]. This
represents a significant improvement over standard diffraction gratings. However, as our comb
frep values are ∼ 100 MHz, some modest cavity filtering is still necessary for comb mode resolved
spectroscopy [27, 38]
At a given output angle, light containing optical frequencies separated by the etalon FSR isoverlapped. Our etalons have FSRs of about 50 GHz, and therefore the overlapping components
are easily separated by placing a low-resolution reflective grating after the etalon at an orthogonal
orientation. This creates a two-dimensional pattern of comb modes, which are then imaged onto
a sensitive infrared camera (Fig. 2.6b). As with FTS spectroscopy, once the comb mode-resolved
limit is reached, the effective resolution is no longer limited by the spectrometer (in this case, theImaging Lens
Infrared Camera

d

10

100
10-1
100

0.25

b

Cell diffusion timecVertical diffraction
Cylindrical VIPA etalon
Reflection grating
lens
Horizontal
diffraction

Absorption noise (%)

a

23
1

101

Equiv. FTS noise102
Δt (ms)

103

104

VIPA
FTS

Absorption

VIPA dispersion

0.20
0.15
0.10
0.05
0

Grating dispersion

2955.9 2955.95 2956 2956.05
Wavenumber (cm-1)

Figure 2.6: Frequency comb spectroscopy with a VIPA etalon. (a) Incident comb light is cylindrically focused into the VIPA etalon, which disperses light in the vertical direction. A low resolution reflection grating disperses overlapping etalon FSRs in the horizontal direction. The twodimensional pattern of comb light is then imaged onto an infrared camera. (Figure reused bypermission from Springer Nature: Applied Physics B: Lasers and Optics,“Cavity-enhanced direct
frequency comb spectroscopy”, M. J. Thorpe and J. Ye. [5]) (b) An example VIPA image showing
resolved comb modes as individual bright spots. The vertical axis corresponds to VIPA dispersion,and the horizontal axis to grating dispersion. Each column fringe is a distinct VIPA order, separated from an adjacent fringe by the VIPA free spectral range (50 GHz). Along a given fringe,
each comb mode spot is separated from its vertical neighbors by 1 GHz via cavity filtering. The
two dark spots (indicated by arrows) correspond to the same absorbed comb mode appearing in
the image in sequential VIPA orders. (c) Fast frame-to-frame subtraction of VIPA images improves
the absorption noise baseline by removing incident comb light intensity fluctuations. At a ∆t = 10
ms subtraction time, we observe a 30× improvement in sensitivity relative to FTS detection forthe same total acquisition time. At shorter times, the noise baseline plateaus, indicating that thecamera dark noise limit has been reached. The cell diffusion time of about 10 ms corresponds to thefastest rate at which the molecular absorber density in the cold cell can be efficiently modulated.
(d) A comparision of FTS and VIPA spectra for a small region of the MIR absorption spectrum of
cold CH3 NO2 .

24
VIPA etalon line width) and instead set by the frequency comb itself.
The VIPA spectrometer detection sensitivity is usually limited by the incident light intensity
noise. Simultaneous differential detection to suppress comb mode intensity noise, as is performed
with the FTS setup above, is in general difficult. While swept-cavity locking improves the baseline
intensity noise by eliminating FM-to-AM noise conversion from the enhancement cavity [6], intrinsic(i.e. pre-cavity) intensity noise of the frequency comb light is not suppressed. To combat residual
intensity noise, we take advantage of the fast frame rate (up to 500 Hz) of the detector camera,
which permits a lock-in type measurement to be performed. Sequential camera frames are acquired,
one with molecular absorption signal present and another without, and subtracted. This eliminates
intensity noise contributions slower than the subraction rate, as shown in Fig. 2.6c. We modulate
the molecular absorption signal by a fast mechanical shutter between the molecule source andthe sample cell (see Section 2.5 below). The modulation rate is ultimately limited by the 10 ms
diffusion time in the cell. At this diffusion-limited modulation rate, we are still able to reachthe camera dark noise limit. For the 3 − 5 µm OPO system, the combination of fast camera
acquisition times and frame-to-frame intensity noise subtraction yield a 30-fold improvement in thecavity transmission noise floor relative to FTS detection normalized to equal acquisition time. It
is important to realize, however, that the swept-cavity locking method reduces the effective path
length of the light by a factor of 1/2 relative PDH locking due to differences in continous versus
impulsive light-cavity coupling [6]. Therefore, the net absorption sensitivity of the VIPA setup is
only about 15× higher than the PDH/FTS detection system (Fig. 2.6d), which is nonetheless asignificant improvement [27].

2.5

Cryogenic buffer gas cooling
The MIR frequency comb techniques discussed in the previous sections provide powerfultools for probing molecular structure and dynamics. Taking full advantage of the high resolution
capabilities of combs for molecular spectroscopy requires preparing molecules in a physical statesuch that the complex structure of their rovibrational spectra is fully revealed. For this reason, room

25
temperature experiments have been largely limited to relatively small systems because moleculeswith even, say, half a dozen heavy atoms already occupy millions of rovibrational quantum states,
yielding intractably congested and unassignable spectra. The ability to produce cold, dense samplesof neutral molecules is therefore crucial to extending high resolution spectroscopy to more complex
molecules.
We address this issue by combining CE-DFCS with cryogenic buffer gas cooling (CBGC) [39–42]. With this technique, a cryogenic cell with a wall temperature Twall ranging from 10 K o 150 K is
filled with an inert buffer gas, usually a noble gas such as helium, neon, or argon, which thermalizes
to Twall . Warm (room temperature or hotter) molecules are injected into the cell where theycollide with the cold buffer gas atoms, approximately thermalizing their rotational and translationaltemperatures to Twall . At these low temperatures, molecular spectra are drastically simplified
relative to room temperature conditions, exhibiting reduced congestion, narrow Doppler line widths,
and larger absorption cross sections. CBGC provides a continuous source of cold molecules withslow lab frame velocities and long interaction times with only modest pumping requirements. These
represent distinct advantages over supersonic expansion cooling, which exhibits fast lab frame jet
velocities and limited interaction times and poses a compromise between low duty cycle pulsed
operation or managing non-trivially large gas loads. As will be discussed in later chapters, efficiently
cooling complex polyatomic molecules with CBGC has permitted the measurement of rotationallyresolved infrared spectra of unprecedentedly large molecules including adamantane (C10 H16 ) in the
3 µm CH stretch region [27, 38] and buckminsterfullerene (C60 ) near 8.5 µm [28].

2.5.1

Low-temperature CBGC system

We use one of two CBGC systems depending on the required cooling capacity and buffer gas
load for the experiment. The low-temperature (10 − 30 K) device [27, 38] is based on a closedcycle liquid-helium two-stage pulse tube cryostat (Cryomech PT410/CP2800), as diagrammed inFig. 2.7a. The first stage has 40 W of cooling capacity at 40 K, and the second stage has 1 W ofcooling capacity at 4 K. The pulse tube head is placed in a room temperature vacuum dewar with26
a base pressure of 1 × 10−6 Torr (when cold). A copper radiation shield mounted to the first stage
reduces the blackbody radiative heat load on the second stage cooler. A hollow aluminum cold
cell, shown in Fig. 2.8, with an interior volume of 6 × 6 × 6 cm3 is mounted to the second stage.
Sensor diodes mounted to the cold cell, second stage, and radiation shield provide temperaturemonitoring. With no gas load, the cold cell reaches a baseline temperature of 3 − 4 K. A small
continuous flow (ca. 10 sccm) of helium buffer gas is pre-cooled to the 4 K stage before entering
the cold cell through an annular slit inlet, resulting in a helium density of 1014 cm−3 to 1015 cm−3
inside the cell.
Helium leaking out of the cell is efficiently cryopumped by charcoal sorbs mounted to the 4 Kstage. The details of the sorb design have a significant impact on their pumping performance [43,44],and it is worthwhile discussing these particulars here. Our “secret formula” comes from John
Doyle’s group [45, 46] via Dave Patterson, who collaborated with us on the construction of the
CBGC system. We adhere activated coconut charcoal (Spectrum C1221, 8-30 mesh) to 1/8”-thicksuper-conductive OFHC (101) copper using Stycast 2850FT epoxy with catalyst 23LV (7.5% byweight).3 Typically, three 3”×6” plates (both sides coated with charcoal) are installed, giving a
total pumping area of about 700 cm2 . With a 10 sccm helium load, the background dewar pressure(measured outside the radiation shield) is about 2 × 10−5 Torr. Using this as an approximation forthe helium pressure near the sorbs, we roughly estimate a pumping speed of 6000 L/s. The sorbs
can run for 3 to 4 hours continuously, from which we also estimate a capacity of about 1500 L·Torr.
Both of these estimates are consistent with the expected performance of coconut charcoal [43, 44].
Once saturated, the pumping speed drops and the background pressure gradually increases
until it reaches about 1 × 10−4 Torr. At pressures higher than this, thermal contact to the room
temperature dewar wall warms the sorbs, releasing helium and causing a runaway “crash”. Usually,
the helium is turned off before a crash is allowed to occur. To regenerate the sorbs, the pump out
valve is opened and the refrigerator is turned off. Once the sorbs are above ca. 40 K almost all of3

For unknown reasons, the Ye group historically has used catalyst 23LV, while the Doyle group reports using
catalyst 24LV [45, 46]. The two catalysts appear to have very similar physical properties [47].

27
the helium has been released, and they can be re-cooled for further use.
In the absence of contamination, we have successfully re-used the same sorbs continuously forover a year. Temporary “clogging” can occur from a vent or leak of atmosphere into the vacuum
dewar. The sorbs can be restored by pumping on them at room temperature for several hours.
(We suspect it helps that the air in Boulder, CO has a relatively low concentration of water vapor.)
The chemical species we introduce into the cold cell are the most potentially problematic source ofcontamination. When the system is cold, these molecules are efficiently cryopumped by the metal
surfaces of the cold cell and radiation shield, and it is likely only a very small fraction reaches thesorbs. However, when the system is eventually warmed up, these molecules are released and cancome into contact with the sorbs. We have found that species that are gas-phase or have a highvapor pressure (at least tens of torr) at room temperature do not cause any observable degradation
to the sorb. Conversely, highly non-volatile solids, such as C60 , remain solid after warming to room
temperature and cause no problems (except for requiring frequent clean-up of the cold cell where thesolid collects). The only permanent degradation we have observed is from certain sublimating solidand non-volatile liquid samples, such as hexamethylenetetramine and indole. After introducing
these species into the cold cell and then warming up, the pumping speed of the sorbs was severely
reduced. Attempts to revive poisoned sorbs by a 150◦ C bakeout overnight recovered only a small
fraction of the peak pumping speed and helium capacity and were ultimately unsuccessful. Finally,
it is important to note that only oil-free turbo and scroll pumps are used for the vacuum dewar,
eliminating another potential source of sorb contamination.
Warm gas-phase molecules enter the cold cell (Fig 2.8) through a 1 cm aperture in the cellwall and quasi-thermalize to Twall with a density of about 1012 cm−3 [27, 38]. For most small
molecules, 10s to 100s of collisions are necessary to cool the internal and translational degrees of
freedom. Buffer gas atom-molecule collisions occur with a mean collision period of 10 − 100 µs,
thereby requiring ca. 1 ms for thermalization. The molecules diffuse in the cell for 10 − 20 ms
before freezing out on the cell walls. Measured rotational and translational temperatures in thislow-temperature cell are typically 10 − 20 K (see Section 2.6). For volatile molecules with sufficientabuffer gas
fill line

pulse tube headcharcoalsorbs
highprecisionscrews
MIR
light

1st
stage
2nd
stage

300 K
vacuum dewar

28bpump out

liquid nitrogen
dewar
vent
fill

35 K radiation
shield (Cu)
10 K buffer gas
cold cell (Al)high reflectivity
cavity mirror

tube piezo

bellows

cavity spacer
molecule source
aperture

Figure 2.7: Cryogenic buffer gas cooling. (a) The low temperature CBGC apparatus is based ona two-stage liquid helium pulse tube refrigerator. The 35 K radiation shield is mounted to the firststage heat exchanger, while the 10 K cold cell is mounted to the second stage. Charcoal sorbs are also
mounted to the second stage to provide high helium pumping speeds. A cavity spacer surrounding
the dewar supports two high reflectivity mirrors that form the MIR enhancement cavity. Precision
screws and a cavity tube piezo provide fine adjustment of the cavity alignment and length. The
cavity optics are vibrationally isolated from the dewar by thin edge-welded bellows. (b) The hightemperature system replaces the helium cold finger with an open liquid nitrogen dewar, and theblackbody radiation shield is removed. The cavity optics and hardware (not shown) are identicalto those in (a).
.

vapor pressure at room temperature, a flow of 1 − 10 sccm is introduced into the cell via a 1/4”-instainless steel tube that terminates 1 − 2 cm in front of the outer surface of the cell wall. For
non-volatile molecules, a small copper oven (2 cm × 2 cm × 4 cm) is used to heat samples to
produce sufficiently high vapor pressures. The oven, with an interior volume of 3 cm3 , is located invacuo just outside the 35 K radiation shield, with a small tube outlet passing through the shield
to guide molecules into the cold cell. The inner diameter (2 mm) and length (3 cm) of the outlet
determine the output conductance of the oven and therefore what vapor pressure is necessary for29

inlet platecold cell

oven

6 cm
cavity mirror
comb light

Figure 2.8: A detailed view of the cold cell. The cell is mounted to a either the liquid helium
or liquid nitrogen cold finger. Pre-cooled buffer gas enters the cell through the annular slit inlet
plate. Warm molecules from an oven source or tube inlet enter the cell and quasi-thermalize to thewall temperature via collisions with buffer gas atoms. Two high reflectivity mirrors (position not
to scale) form a path length enhancement cavity that passes through two side apertures in the cold
cell.

sufficient molecule output flow. The output conductance is high enough to require only modest
oven temperatures (100 ◦ C to 200 ◦ C), but small enough to allow sufficient oven pressure buildup toprovide stable vaporization/sublimation and prevent violent boiling. A small glass window permits
visual monitoring of the interior of the oven while in use. The oven is resistively heated by two
embedded cartridge heaters (50 W, 1/8” OD × 1.25”) and temperature monitored by a type-K
thermocouple.
A “molecule shutter” consisting of a thin aluminum paddle is placed between the moleculetube or oven outlet and the cold cell aperture in order to quickly turn on or off the molecule flow
into the cold cell. The shutter is rotated into and out of the molecule flow with a small DC electricmotor mounted to the 35 K shield. Short current pulses from a home-made driver circuit switch

30
the shutter with an open/close time of 1 ms and a delay of 10 ms.

2.5.2

High-temperature CBGC system

In the high-temperature (80 − 150 K) setup (Fig. 2.7b), the helium cold finger is replacedwith a liquid nitrogen (LN2) dewar without a blackbody radiation shield [28]. LN2 sets a minimum
77 K temperature limit for the cold cell, but is able to provide significantly higher cooling capacity,which permits using higher buffer gas flows and pressures relative to the low-temperature system(up to 100 sccm and 1 Torr background in the dewar). Charcoal cryopumping is replaced with asmall turbo-molecular pump (300 L/s) that is typically choked to an effective pumping speed of lessthan 10 L/s. The high cooling capacity can handle commensurately high radiative heating, thusincreasing the maximum operating temperature of the oven vaporization source. We constructed asecond compact oven source capable of heating solid samples up to 1000 K. It consists of cylindricalcopper body with a detachable heating collar clamped around it. The heating collar has two silicon
nitride ignitors that each deliver a maximum heating power of ca. 1 kW, though they are typicallyoperated at only 60 − 70% of their maximum power to achieve 1000 K oven temperatures. Thisis consistent with the estimated cooling power of the LN2 dewar based on the LN2 boil-off rate of
5 − 10 L/hr, from which we estimate a cooling power about 1 kW.

2.5.3

Absorption enhancement cavity

The absorption enhancement cavity consists of two parallel mirrors with high reflectivity (HR)coatings surrounding the cold cell [27]. Four 1”-thick stainless steel rods spanning either side of
the cryostat dewar fix the gross cavity length (see Fig. 2.7a). The cavity mirrors are mechanicallyisolated from the dewar by a system of edge-welded bellows that form the vacuum connection
between the cavity mirrors and the dewar. The position and alignment of each mirror are adjusted
macroscopically by a set of fine precision screws. One of the cavity mirrors is fitted with a tube piezo
for fine length adjustment and feedback control. The HR mirrors have broadband coatings withpeak finesses of F ≈ 3000 − 6000. We typically achieve on-resonance cavity transmission efficiency

31
of a few percent up to 25%, depending on the mirror losses, over a simultaneous transmission
bandwidth of 50 − 100 cm−1 , limited by the HR mirror dispersion. The cavity spacers limit thepractical range of mirror separations to be ` = 50 − 70 cm, yielding cavity FSRs in the range of
215 to 300 MHz. Cavity filtering is thus unavoidable given the frep values of our two frequency
comb sources (136 MHz for the OPO system and 93.4 MHz for the DFG system). A small gas
inlet is placed near each mirror to provide an optional purge flow that protects the mirrors fromdegradation. This was found necessary with the high-temperature CBGC setup to prevent deposits
of C60 grains from forming on the mirror surface. The output of the enhancement cavity is routed tothe detector systems for transmission read-out. Cavity reflection and transmission photodiodes areplaced before and after the cavity to collect PDH and swept-cavity lock error signals, respectively.

2.6

CE-DFCS/CBGC system performanceThis final section summarizes the technical performance of the combined CE-DFCS and

CBGC apparatus. Returning to the example introduced at the beginning of the chapter, Fig. 2.9ashows the measured FTS absorption spectrum the ν3 band of vinyl bromide acquired with the
low-temperature CBGC cell. This spectrum can be compared to the simulation in Fig. 2.1b. (Thenegative-going signal in the measured absorption spectrum is an artifact of the filtering processapplied to FTS spectra as described in Section 2.4.1.) The dense rotational structure is well resolved,
as can be seen by the observation of hyperfine splittings (Fig. 2.9b) and Ka sub-band structure(Fig. 2.9c). The vibrational band contains well over a 1000 transitions from each isotopologue above
the absorption noise baseline, essentially all of which have been successfully assigned. (Bromine
occurs naturally as 79 Br and 81 Br in approximately 50:50 relative abundance. The results of a full
effective Hamiltonian fit are available in Appendix B.)
The extracted rotational and translational temperature profiles from the vinyl bromide spectrum are shown in Fig. 2.10a,b. A small subset of rotational transitions was used to determine
a rotational temperature of Trot = 17(3) K, while the Doppler-broadened line width indicates atranslational temperature of Ttrans = 18(4) K. Similar measurements for nitromethane (CH3 NO2 )

32
0.20

a

c

0.15

b

0.10

Absorption

0.15

0.05

b

0
3027.66 3027.67 3027.68 3027.69

0.10

0.05

0

0.40

c

0.30
0.20
0.10
0

3023 3024 3025 3026 3027 3028 3029 3030 3031 3032
Wavenumber (cm-1)

3030.00 3030.05 3030.10

Figure 2.9: CE-DFCS absorption spectrum of the ν3 band of vinyl bromide. (a) A survey ofthe entire ν3 band, which spans about 10 cm−1 and contains over 1000 transitions above the noisebaseline. Detailed views of the R(0) and R(9) transitions (marked with arrows) are shown in (b) and(c). (b) The measured R(0) transitions are shown in black (offset for clarity) and the simulations
for C2 H3 79 Br and C2 H3 81 Br are shown in blue and red, respectively. The R(0) transition for eachisotopologue is split by nuclear quadrupole effects into three hyperfine components corresponding
to transitions from the F 00 = 3/2 rovibrational ground state to F 0 = 1/2, 5/2, and 3/2 upper
states, in order of increasing transition frequency. (c) The R(9) region shows clusters of transitions
belonging to different Ka ≤ 4 sub-bands.

yield Trot = 11(1) K and Ttrans = 16(1) K (Fig. 2.10c,d). These results are representative of almost
all molecules we have cooled with the low-temperature CBGC cell. The translational and rotationaltemperatures do not always completely equilibrate, but they are reliably in the 10 K to 20 K range.
Based on the integrated band strengths, we estimate the molecule densities to be in the range of1 − 5 × 1012 cm−3 .
The only molecule we could definitively not cool with the low-temperature CBGC system
was buckminsterfullerene, C60 . The modest cooling power of the low-temperature system turns
out to be no match for the high atom count (60 nuclei), molecular mass (720 u), and oven source
temperature (950 K) of C60 , forcing us to rely on the much higher cooling capacity of the high-Ln(rel. Boltzmann factor)cb0Intensity (norm.)

Trot = 17(3) K-0.5

Δν = 27(3) MHz
Ttrans = 18(4) K

1

-1
-1.5

0.8
0.6
0.4
0.2
0

-2

5

0

10
15
20
State energy (cm-1)

25

-50

d

1.1

1

1

Trot = 11(1) K

0.9

Intensity (norm.)Ln(rel. Boltzmann factor)

a

33

0.8
0.7
0.6
0.5
0.4

0.8

0
50
Frequency offset (MHz)Δν = 33(1) MHz
Ttrans = 16(1) K

0.6
0.4
0.2
0

0

5
10
State energy (cm-1)

15

-50
0
50
Frequency offset (MHz)Figure 2.10: Rotational and translational temperatures of molecules in the low-temperature CBGC
cell. (a) The natural logarithm of the inferred Boltzmann factors for the lower states of the00
C2 H3 79 Br ν3 Ka =0 R-branch with J 00 = 3 − 13. The Boltzmann factors are normalized relative
to the J 00 = 3 level. The fitted slope versus lower state energy indicates a rotational temperatureof 17(3) K. (b) The R(9) transition of the same branch shows a Doppler line width of 27(3) MHz,corresponding to a translational temperature of 18(4) K. (All uncertainties are 2σ.) (c,d) Same as(a,b) for the ν3 + ν6 band of nitromethane, CH3 NO2 .

temperature LN2 cold finger. A full discussion of the challenges associated with C60 and the efforts
to successfully cool it are deferred until Chapter 5. For now, Fig. 2.11 simply presents the final
performance we achieved with the high-temperature CBGC system. The relative rotational state
populations in Fig. 2.11a exhibit a Boltzmann distribution with an effective rotational temperatureof 150(2) K. This is only slightly warmer than the measured wall temperature of 135 K, which
is limited by the radiative heat load from the 950 K oven. (Subsequent tests with heat shields

abTrot = 150(2) K

Absorption (norm.)Ln(rel. Boltzmann factor)

0
-1
-2
-3
-4
0

34

1

Δν = 16.0(6) MHz

0.8
0.6
0.4
0.2
0

100
200
300
State energy (cm-1)

400

-100

-50
0
50
Frequency offset (MHz)100

Figure 2.11: Cooling of C60 in the high-temperature CBGC cell. (a) The relative populations of
the rotational sub-levels exhibit a Boltzmann distribution with Trot = 150(2) K (with a measured
cell wall temperature of 135 K). (b) The R(180) transition centered at 1186.2620 cm−1 has a line
width of 16.0(6) MHz. This is slightly larger than the expected Doppler line width of 12 MHz –possible evidence of collisional broadening.

between the oven and cold cell suggest that the rotational temperatures as low as 100 K might beachievable.)
The absorption peak for the R(180) transition has a measured line width of 16.0(6) MHz,
which is representative of the 15 − 20 MHz range we observe over the entire vibrational band. This
range is somewhat higher than the expected Doppler line width of 12 MHz (assuming that Ttransis equal to the measured Trot value). We find it unlikely that the translational temperature could
be so much higher than both the rotational and cell wall temperatures. Given the relatively highpressures in the cold cell (> 1 Torr), it is entirely plausible this excess line width is instead due tocollisional broadening. Unfortunately, the observed signal-to-noise is insufficient to deconvolve thehomogeneous versus inhomogeneous contributions to the line shape.
We conclude this section with a note on the frequency calibration and accuracy of the CEDFCS spectra. For the 3 µm OPO system, both the FTS and VIPA frequency axes are calibrated bymeasuring CH4 transitions whose precise frequencies are readily available [17]. The FTS calibration
amounts simply to a uniform scaling of the FFT frequency axis, or equivalently, calibrating the

35wavelength of the reference cw laser that is copropagating with the MIR light in the interferometer.
The VIPA calibration is more complex [27]. The reference CH4 transitions are used to parameterize
the 2D mapping of the comb modes in the camera image. Both the VIPA etalon FSR and effective
vertical dispersion are determined in this fashion, and the calibration must be repeated each timethe etalon or grating is re-aligned. The CH4 calibration provides absolute optical frequency accuracyof about 30 MHz. Relative line center frequencies are accurate to within 10 MHz, limited by thefrep scan step size and detection signal-to-noise.
Instead of relying on reference molecules, the frequency comb can also be calibrated byassigning the integer comb mode index and using the fundamental relationship between the comb
optical frequencies and the frep and f0 frequencies in the RF domain,νn = n × frep + f0 .

(2.5)

This is the procedure we use for the 8.5 µm DFG comb. In this case, f0 is simply equal to the AOM
shifting frequency, while the frep value is fixed by the RF phase lock described in Section 2.1.2. A
preliminary calibration is performed that takes advantage of absorption lines from trace intracavity
H2 O [48]. These measurements provide sufficient accuracy to constrain the integer mode index n,
at which point the final absolute optical frequency is determined by Eq. 2.5. The values of frep andf0 are ultimately locked to an atomic time standard with a relative frequency offset error below
10−12 . The absolute frequencies determined in this fashion are verified by checking N2 O and D2 O
line positions [49, 50] obtained in separate measurements. Of course, the actual uncertainty of ourmeasured absorption line centers is dominated not by the absolute frequency calibration, but by
the measured line width and signal-to-noise. For the DFG comb experiments, this (1σ) statisticaluncertainty is 2.5 × 10−5 cm−1 or 0.75 MHz.

Chapter 3

Spectroscopic signatures of unhindered internal rotation in nitromethane

The traditional view of molecular structure is that of a quasi-static arrangement of atoms
undergoing small amplitude fluctuations from a fixed reference geometry. This geometry is determined by the equilibrium configuration of the nuclei on the Born-Oppenheimer potential energy
surface (PES). The equilibrium geometry is typically at the bottom of a deep, isolated well on the
PES, which establishes a physical basis for the quasi-static picture. High resolution spectroscopy
provides precise characterization of molecular structure by connecting spectroscopic observables todetailed properties of the PES in the vicinity of the equilibrium geometry.
Non-rigid molecules exhibiting large amplitude nuclear motion break this structural paradigm.
They experience a relatively flat or shallow PES along one or more internal degrees of freedom.
Even at 0 K, zero-point motion causes significant delocalization of the nuclear wavefunction. Suchfloppy, anharmonic systems lead to a breakdown of traditional spectroscopic models, posing practical difficulties for the analysis and interpretation of spectra. More fundamentally, they challenge
the very notion of static molecular structure. Without the basis of a well defined equilibriumgeometry, one is forced to take a step back and view a molecule as an intrinsically dynamic system.
This chapter explores large amplitude motion (LAM) of the unhindered methyl rotor innitromethane, CH3 NO2 . Section 3.1 introduces the basic rovibrational models for such an internal
rotor. We then report the first high resolution spectra of the CH stretch region near 3 µm in
Section 3.2, including DC Stark measurements that reveal internal-rotation-induced “intramolecularalignment.” The CH stretch vibrations are particularly complex as they couple small amplitude

37
vibrational angular momentum with both total and internal rotation. Section 3.3 presents reduced
dimension rovibrational calculations to better understand this intricate network of coupling. These
calculations provide a useful, but still qualitative, physical picture of the internal rotation dynamics.
Motivated by the lack of accurate and efficient quantitative methods for predicting the rovibrational
structure of non-rigid molecules like CH3 NO2 , we have developed an ab initio method based on
curvilinear vibrational Møller-Plesset perturbation theory [51]. This technique and its successfultreatment of CH3 NO2 are discussed in 3.4. This approach has proved to be a powerful tool for abinitio high resolution spectroscopy of non-rigid molecules in general, and we discuss several of its
other recent applications later in Chapter 7.

3.1

Internal rotation in CH3 NO2
Nitromethane contains a −CH3 methyl group centrally bonded to an −NO2 nitro group,

making it the simplest organic nitro compound. With seven atoms, nitromethane has fifteen internal
degrees of freedom, including fourteen small amplitude vibrations and one internal rotation angle,ρ (see Fig. 3.1a). The MIR frequency comb spectrum probes a subset of the small amplitude
vibrations. Before considering these high energy excitations, we first review the structure of thelow energy torsional and rotational degrees of freedom.

3.1.1

Pure torsion level structure

The simplest effective Hamiltonian describing the internal rotation, or torsion, motion is
Ht = F Jρ2 + V (ρ).

(3.1)

The kinetic energy term F Jρ2 contains the torsional angular momentum operator Jρ = −i∂ρ andthe inverse moment of inertia constant F = ~2 /2Iρ . Iρ is the reduced moment of inertia of thetwo coupled CH3 and NO2 rotors about the internal rotation axis, which is coincident with theprincipal a axis. Given ICH3 /INO2 < 0.1, we expect Iρ ≈ ICH3 , and therefore F to be similar to the
B constant of CH4 . Indeed, microwave spectroscopy has measured F = 5.56 cm−1 [52, 53], while

a

38

b
m=±3

Energy (cm-1)

50

25

m=±2

a

V = 2.1 cm-1

0

b

0°

90°

m=±1
m=0

180° 270° 360°

Figure 3.1: Internal rotation in CH3 NO2 . (a) The CH3 methyl group rotates about the CNbond by the internal rotation angle ρ. The a and b principal axes, which lie in the plane of the
CNO2 frame, are shown. (b) The internal rotation potential energy curve has six-fold symmetry
and a small barrier height of about 2 cm−1 . The energy levels are described approximately by theparticle-on-a-ring expression Em = F m2 , where m = 0, ±1, ±2, . . . is the internal rotation quantum
number. The rotor F constant is about 5 cm−1 . The degeneracy of the |m| = 3n (n > 0) levels isbroken by the six-fold internal rotation barrier.

the B constant of CH4 is 5.24 cm−1 [54].
In the completely unhindered limit, the torsional potential V (ρ) = 0. Ht then becomes
equivalent to a particle-on-a-ring model containing only the kinetic energy term. Requiring 2πperiodicity in ρ, the eigenfunctions of the Jρ2 operator are the simple exponentials,
r
|mi =

1 imρ
e ,
2π

(3.2)

where the m is the torsion quantum number with m = 0, ±1, ±2, . . .. The eigenenergies are
Em = F m2 ,(3.3)

such that states with |m| > 0 are doubly degenerate, reflecting the equivalence of clockwise andcounter-clockwise torsion (Fig. 3.1b). In reality, the torsion potential is non-zero. Symmetry

39
considerations [18] require that it have six-fold symmetry in ρ,V (ρ) =V6
V12
cos(6ρ) +cos(12ρ) + . . . ,
2
2

(3.4)

where lowest order coefficient V6 is dominant, and the remaining terms can be ignored. The cos(6ρ)
operator serves to couple torsional levels with ∆m = ±6, breaking the degeneracy of |m| = 3npairs. The m = ±3 states, in particular, split into non-degenerate (|m = +3i + |m = −3i) and(|m = +3i−|m = −3i) linear combinations separated in energy by 2×hm = +3|V |m = −3i = V6 /2.
Measurement of this energy difference yields an effective barrier height of V6 = 2.10 cm−1 [53].
Except for the m = ±3 case, the ∆m = ±6 energy level differences are large relative to the small
V6 barrier height. Thus, the energy structure still closely resembles a free unhindered rotor, and mremains an approximately good quantum number. Fig. 3.1b shows the V6 potential to scale withthe particle-on-a-ring energy levels.

3.1.2

Rotation-torsion level structure

Rotation of the entire molecular frame leads to a combined rotation-torsion effective Hamiltonian,H = Hr + Ht + Hrt ,(3.5)

where Hr is the asymmetric top rigid rotor Hamiltonian,
Hr = AJa2 + BJb2 + CJc2 ,(3.6)

Ht is as above, and the rotation-torsion coupling term is
Hrt = −2A0 Jρ Ja .

(3.7)

The A, B, and C constants are equal to the inverse of the effective moments of inertia about
the principal a, b, and c axes (see Fig. 3.1a) with one modification: the effective A constant only
includes the a-axis moment of inertia of the CNO2 frame and ignores that of the methyl hydrogenatoms. This arises because the methyl top and the heavy atom frame rotate freely with respect

40
to each other [52, 55]. The rotation-torsion term Hrt accounts for Coriolis coupling between thea-axis component of the total angular momentum, Ja , and the torsion angular momentum, Jρ .
The coupling coefficient A0 is equal to A in the rigid top limit (i.e. when all vibrations other than
the torsion are frozen). In the actual non-rigid case, perturbative contributions from the small
amplitude vibrations lead to a small difference between A and A0 [53].
A convenient zeroth order basis for describing the combined rotation-torsion motion is a direct
product of symmetric top rotational wavefunctions, |J, ka i, and the torsional wavefunctions, |mi(Eq. 3.2). J is the total angular momentum quantum number, and ka = −J, . . . , +J is its projection
on the body-fixed a axis. The symmetric top rotational factor obeys the usual eigenvalue relations,
J2 |J, ka i = J(J + 1)|J, ka i,(3.8)

Ja |J, ka i = ka |J, ka i,(3.9)

where J2 = Ja2 + Jb2 + Jc2 . Ignoring the small torsional potential V (ρ), we can rewrite the rotationtorsion Hamiltonian (Eq. 3.5) as
H = F Jρ2 − 2A0 Ja Jρ +




B+C
B+C 2 B−C 2
A−
Ja2 +
J +
Jb − Jc2 .
22
2(3.10)

These five terms are written in order of approximately decreasing magnitude (F ∼ 5 cm−1 , A0 , A ∼0.4 cm−1 , B ∼ 0.35 cm−1 , C ∼ 0.2 cm−1 ). The zeroth order rotation-torsion wavefunctions
|J, ka i|mi are eigenfunctions of each term in H except for the last term proportional to Jb2 − Jc2 .
If we neglect this so-called asymmetry term, then the eigenenergies areB+C
E(J, ka , m) = F m − 2A ka m + A −
2
2

0



ka2 +

B+C
J(J + 1).
2

(3.11)

Note that in this limit there is a two-fold degeneracy between levels with (ka , m) ↔ (−ka , −m).
Torsional levels of a given value of |m| can be treated approximately separately from eachother, with each supporting an independent manifold of rotational sub-levels. The ground torsional
state, m = 0, behaves like a normal asymmetric top (albeit with a larger than expected A constantas discussed above) because the terms involving Jρ are zero. Its low-J energy patterns are illustrated

41
in Fig. 3.2a. The asymmetry term mixes states with ∆ka = ±2 and splits the two-fold degeneracy
for states with |ka | > 0.

E - J(J+1)×(B+C)/2 (cm-1)

30
25

b
m=0

20
15

±8

10

±6

5
0
-5

30

ka≈0

±2

0

2

±4
asymmetrysplitting

4
J

6

8

E - J(J+1)×(B+C)/2 (cm-1)a-9|m| = 1

25

-7

20
-5

15
10

Coriolis
splitting-3mka≈-1

5

+1

+5

+9

+7

+3

0
-5

0

2

4
J

6

8

Figure 3.2: Rotational energy levels for the m = 0, ±1 torsional states. (a) In the m = 0 torsional
ground state, nitromethane behaves like a normal asymmetric top. For low J values, the energy
manifold is organized by the magnitude of the ka quantum number. The asymmetry term splits thetwo-fold ±ka degeneracy. (For clarity, the figure shows the total energy minus J(J +1)×(B +C)/2.)(b) In the m = ±1 torsional states, Coriolis coupling organizes the rotational manifold by the
product m × ka . Each energy level here is two-fold degenerate. Spin statistics of the two 16 O
atoms mandate that only levels with ka + m = even exist when all small amplitude vibrations arein their ground state. Therefore, in (a) only |ka | = 0, 2, 4, . . . states are shown, and in (b) only|ka | = 1, 3, 5, . . . states are shown.

Excited torsional states, which have |m| > 0, have non-zero torsional angular momentum.
Their energies are largely organized by the product of m and ka because of the Coriolis coupling
term ∝ Ja Jρ . An example for the |m| = 1 manifold is shown in Fig. 3.2b. Here, the asymmetry
term does not split the two-fold degeneracy. This qualitative physical difference between the m = 0
and |m| > 0 manifolds, i.e. that the former is dominated by asymmetry splittings and the latter by42
Coriolis splittings, has important effects on fundamental observables such as the lab frame dipole
moment. We will examine this effect more closely in Section 3.2.

3.1.3

Symmetry, spin statistics, and selection rules

Characterizing the symmetry properies of the vibration-rotation-torsion states is necessary to
predict nuclear spin degeneracy factors and infrared selection rules. Standard point group analysis
is inadequate for treating the non-rigid molecules such as nitromethane, and we must use the moregeneral molecular symmetry (MS) group, or Longuet-Higgins group, based on nuclear permutationinversion operations [18]. Let the three hydrogen nuclei be labeled as H1 , H2 , and H3 , and the twooxygen nuclei as O4 and O5 . The MS group appropriate for nitromethane is generated by threesymmetry operations:• (123), the cylic permutation of H1 , H2 , and H3• (23)∗ , the permutation of H2 and H3 , followed by inversion of all coordinates (nuclear andelectronic) through a space-fixed origin• (45), the permutation of O4 and O5
The group formed by products of these generators is called G12 . (It contains a total of twelve
operations.) The group character table can be found in Table A-24 of Ref. [18]. G12 is isomorphic
to the D3h point group, and the same labels are used for their irreducible representations (irreps).
There are four non-degenerate irreps: A01 , A001 , A02 , and A002 . The 1 /2 label gives the character under(23)∗ (+1 and −1, respectively), while the 0 /00 label gives the character under (45). The two doubly
degenerate irreps are E 0 and E 00 , which have a similar meaning for the 0 /00 labels.
Using standard group theory procedures [18], one can show that the symmetric top rotational
wavefunctions, |J, ka i, with ka = even span the A01 ⊕ A02 irreps, while those with ka = odd span
A001 ⊕ A002 . The symmetries of the torsional wavefunctions are determined by the transformation43
properties of the torsion angle ρ, which are(123)ρ = ρ − 2π/3,(23)∗ ρ = 2π − ρ,

(3.12)(45)ρ = ρ + π.
The irreps of the |mi wavefunctions, Eq. 3.2, are summarized in Table 3.1. The combined torsionrotation symmetry species is determined by the direct product of the individual torsion and rotation
irreps.
Table 3.1: Symmetry species of torsional wavefunctions of nitromethane in the G12 MS group. (nis a non-negative integer.)|m|
0
6n ± 16n ± 26n + 36n + 6|mi irrep.
A01
E 00
E0
A001 ⊕ A002
A01 ⊕ A02

We can now examine the nuclear spin weights of the normal isotopologue 12 CH3 14 N16 O2 .
Following section 8.4.1 of Ref. [18], the nuclear spin degeneracy ratios are 1 : 1 : 1 : 0 : 0 : 0 forrovibronic symmetries A01 : A02 : E 0 : A001 : A002 : E 00 . The exclusion of 00 species occurs because
these are antisymmetric with respect to the (45) operation (permutation of the two 16 O nuclei,
which are spin-0 bosons). Given that the ground states of the electronic and small amplitude
vibrational degrees of freedom are all totally symmetry (A01 ), the net result is that ka + m must be
even (see Fig. 3.2). In vibrationally excited states that transform as A001 , A002 , or E 00 , we must haveka + m = odd.
Using a suitable choice of internal coordinates (see section 15.4.2 of Ref. [18]), the fourteen
small amplitude vibrations have symmetry species
Γvib = 5A01 ⊕ 5A001 ⊕ 4A002 .

(3.13)

Each of these three irreps is IR active. A01 , A001 , and A002 modes have a-, b-, and c-type rotational

44
selection rules, respectively, as well as a strong ∆m = 0 selection rule. A summary of the small
amplitude vibrational modes is provided in Table 3.2. The infrared spectra of the ν2 , ν6 , and ν7
bands have been previously studied at high resolution [56–60].
Table 3.2: Small amplitude vibrations of nitromethane. The mode labeling follows the conventionof Ref. [18].
Mode
Description
ν1Sym. CH stretch
ν2
CN stretch
ν3Sym. NO stretch
ν4Sym. CNO bend
ν5Sym. NCH bend
ν6Asym. NO stretch
ν7
In-plane NO2 rockν9
Out-of-plane NO2 wag
ν10i,o In-plane and out-of-plane CH stretchesν11i,oIn-plane and out-of-plane HCH bends
ν12i,oIn-plane and out-of-plane NCH bends
a Ref. [61]b Ref. [62]c Ref. [63]Symmetry
A01
A01
A01
A01
A01
A001
A001
A002
00
A1 ⊕ A002
A001 ⊕ A002
A001 ⊕ A002

Frequency (cm−1 )
2974a
918a
1397a
657a
1380a
1584a
475a
605a
3042, 3076b
1100, 1131c
1423c

3.2

High resolution spectra of CH3 NO2 in the 3.3 µm region

3.2.1

Experimental setup and survey spectrum

We acquired CE-DFCS absorption spectra of nitromethane using the MIR OPO comb with
FTS and VIPA detection (see Sections 2.1.1, 2.4.1, and 2.4.2). Cold gas-phase nitromethane samples were prepared with the low-temperature CBGC refrigerator, which provided rotational andtranslational temperatures of 11(1) K and 16(1) K, respectively (Fig. 2.10c,d). Two copper electrodes were installed in the cold cell for Stark effect measurements (Fig. 3.3) with DC electric fields
up to 400 V/cm parallel to the cavity axis.
Figure 3.4 shows a 150-cm−1 wide survey spectrum of nitromethane from 2940 cm−1 to3090 cm−1 (3.2 µm to 3.4 µm). This region is dominated by the strong ν3 + ν6 combination
band near 2950 cm−1 . The symmetric CH stretching mode occurs near 2975 cm−1 , while the45Aluminum cold cell+V

-V
Sapphire
spacerCopperelectrode
MIR
light

+V

E-field-V

Figure 3.3: DC electrode geometry in the cold cell. A copper electrode is placed on each of the twocold cell walls perpendicular to the cavity axis. Sapphire discs electrically insulate the electrodes
from the cell wall. A thin piece of indium is placed on either side of each sapphire disc to improve
the thermal contact. Ceramic screws (not shown) are used to mount the electrodes, which generate
a DC electric field up to 400 V/cm parallel to the cavity axis. The apertures in the electrodeplates (ca. 1 cm diameter) and the finite extent of the plates themselves lead to a ∼ 10% field
non-uniformity along the cavity axis.

two components of the asymmetric CH stretching mode, out-of-plane and in-plane, appear near3055 cm−1 and 3085 cm−1 , respectively. Even at low rotational temperatures, this region contains
thousands of individual rovibrational transitions above the noise baseline. The following sections
discuss each vibrational band in more detail.

3.2.2

The ν3 + ν6 combination band

The ν3 + ν6 combination band involves simultaneous excitation of the symmetric and antisymmetric NO stretching modes. The excited state has A001 vibrational symmetry leading to b-typetransitions with ∆Ka = ±1, ±3, · · · , ∆Kc = ±1, ±3, · · · , and ∆m = 0 rotation-torsion selection
rules. Using ground state combination differences derived from microwave data [52, 55], we haveassigned several hundred transitions with |m| ≤ 1. A full line list is included in Appendix A.

46

Nitromethane spectrum multiple E-fields

0.3

0.1

0.08
0.06

0.08

fractional absorption

ν3 + ν6(sym. + asym. NO stretch)

0.06

0.04

Absorption

0.2

0.04

0.02

0.02

0

0

2974.552974.652974.75

2974.56 2974.58 2974.6 2974.62 2974.64 2974.66 2974.68 2974.7 2974.72 2974.74 2974.76

wavenumberν10 (asym. CH stretches)

0.1

ν1 (sym. CH stretch)

0
2940

2950

2960

2970

2980
3050
Wavenumber (cm-1)

3060

3070

3080

3090

Figure 3.4: Absorption spectrum of nitromethane near 3.3 µm. This region is includes the ν3 + ν6
(sym. NO + asym. NO stretch) combination band as well as the ν1 (sym. CH stretch) and ν10 (inplane and out-of-plane asym. CH stretch) fundamental transitions. The inset shows well resolved
rotational fine structure of a small portion of the ν1 band.

The m = 0 transitions are the simplest to analyze as the associated energy levels are structured like a semi-rigid asymmetric top. Upper state term values (energies) were calculated using
the measured transition frequencies and lower state rotation-torsion energies derived from previous
studies [52,55]. A reduced term value plot is shown in Fig. 3.5, where a number of perturbations canbe observed. For the JKa Kc = 313 , 312 , 413 , 514 , and 616 states, the transition is split into a doublet,with two components of similar IR intensity. This indicates that there is complete mixing of eachzeroth order bright state with a perturbing dark state. The magnitudes of the splittings range from
0.14 cm−1 to 0.18 cm−1 , which correspond to the value of the interaction matrix element. Given
the high density of background dark states in this region of the spectrum, it is difficult to identify
the character of the perturbing dark state.
Excluding the perturbed doublets, the remaining upper state term values were fitted to the

47

12

3.5

Ka = 7e,f3(E – 2950 cm-1) - J(J+1)×(B+C)/2

10
8

2.5
Ka = 5f

2

3

4

5

Ka = 5eKa = 3f

6
4

Ka = 3e2

Ka = 1f

0
Ka = 1e

−2

01 2

3

4

5

J

6

7

8

Figure 3.5: Reduced term values of the m = 0 states of ν3 + ν6 . The quantity J(J + 1) × B+C
2
is subtracted from the absolute energy for clearer plotting. States of a given value of Ka = |ka |are connected with solid lines, which represent the eigenvalues of the fitted effective Hamiltonian,Eq. 3.14. Due to 16 O spin statistics, only Ka = odd levels exist in the A001 vibrationally excitedstate. The e/f labels indicate the parity of each state. (e(f ) for levels that are +(−) parity for
J = even.) Black bold markers indicate perturbed states. (Inset) Symmetric doublets appear due
to strong mixing of a zeroth order bright state with a perturbing dark state.

standard Watson A-reduced Hamiltonian (Ir representation) [64]: B−C 2

B+C 2
2J − Ja2 +J+ + J−− ∆J J4 − ∆JK J2 Ja2
2
4

1
2
2
− ∆K Ja4 −δJ J2 + δK Ja2 , J+
+ J−
,
+
2

Ĥ = ν0 + AJa2 +(3.14)

48
where J± = Jb ± iJc and [· · · , · · · ]+ is an anti-commutator. This m = 0 Hamiltonian contains the
non-torsion-dependent terms of the full torsion-rotation Hamiltonian, Eq. 3.10, along with the usual
quartic centrifugal distortion terms [64]. The results of the least-squares fit for this Hamiltonian
are shown in Table 3.3, with the corresponding energy eigenvalues shown as solid curves in Fig. 3.5.
Due to the relatively low J values (J ≤ 8) accessed in the cold spectrum, only the principal
rotational constants (A, B, C) were fitted, with the quartic centrifugal distortion constants held
fixed to their ground state values. Table 3.3 lists the rotational constants for the ground state [55]and ν6 fundamental [58] for comparison with the ν3 + ν6 values measured in this work. The fit
error for the ν3 + ν6 band is about 5× worse than for the ν6 fundamental, suggestive of residual
perturbations. There is a significant negative shift of the A, B, and C constants of ν3 + ν6 relative
to both the ground state and the ν6 fundamental. This effect can also be quantified by the inertial
defect ∆i = Ic − Ia − Ib , where Iα is the effective moment of inertia about the α = a, b, c principal
axis. ∆i changes from +0.203 uÅ2 in the ground state to −0.330 uÅ2 in the ν3 + ν6 level. Sucha large negative shift is consistent with an increase in the effective torsional barrier [65], as the
methyl group becomes “locked-in” with respect to the C–NO2 plane.
Table 3.3: Effective Hamiltonian fit for m = 0 levels of ν3 + ν6 . All parameters are given in cm−1 .
Those in brackets [· · · ] are held fixed to their ground state values. RMSE is the root-mean-square
fit residual. The inertial defect, ∆i = Ic − Ia − Ib , is given in uÅ2 .
Parameter Ground statea
ν00A
0.445037
B
0.351722
C
0.195994∆J × 1060.2048
6∆JK × 10 0.5921∆K × 106−0.2515
δJ × 1060.08229
δK × 1060.52536
RMSE—∆i+0.203a Ref. [55]b This workc Ref. [58]

ν3 + ν6 b
2952.6854(45)0.43990(18)0.34772(30)0.19495(14)[0.2048][0.5921][−0.2515][0.08229][0.52536]
0.0110−0.330ν6 c1583.81163(20)0.4449620(33)0.3516825(26)0.1960255(9)0.2431(23)0.6822(103)−1.5701(93)0.0717(11)0.4573(34)0.0023+0.177

49
We identified several |m| = 1 transitions in addition to those with m = 0. These assignments
were facilitated by observation of their characteristic DC Stark effects. Fig. 3.6 shows an exampleof the different patterns for m = 0 vs. m = ±1 states. In the m = 0 case, the states involved
are non-degenerate and have a vanishing dipole expectation value, hµ~ˆi = 0. Their DC Stark shifts
rely on dipole coupling to other non-degenerate states, leading to a quadratic effect. The m = ±1
states are doubly degenerate, and each pair can be chosen to have non-zero hµ~ˆi of opposite sign,
leading to a linear effect that splits each transition symmetrically into two components.
This effect can be understood in detail by considering the relative sizes of the asymmetry term∝ (Jb2 −Jc2 ) and the Coriolis coupling term ∝ Ja Jρ in the rotation-torsion Hamiltonian, Eq. 3.10, fordifferent rotation-torsion states. Consider, for example, the six rotation-torsion basis states |m, ki
for m = 0, ±1 and k = ±1, which we abbreviate as |0, +i, |+, +i, |−, +i, etc. Neglecting both theasymmetry term and the Coriolis coupling term, these states separate into two groups according
to the value of |m|: a doubly degenerate set for m = 0 and a quadruply degenerate set for m = ±1
(Fig. 3.6c). The Coriolis term, −2A0 Ja Jρ , splits the m = ±1 quartet into two degenerate pairs
according to the relative sign of m and ka : |+, −i with |−, +i and |+, +i with |−, −i. It has noeffect on the m = 0 pair because the eigenvalue of Jρ is zero. On the other hand, the asymmetry22
term, B−C
2 (Jb − Jc ), has a ∆ka = ±2 selection rule and mixes states with ka = +1 ↔ −1 leading to

non-degenerate linear combinations √12 (|m, +i + |m, −i) and √12 (|m, +i − |m, −i) for each valueof m.
In the total Hamiltonian, competition between the Coriolis and asymmetry terms leads tothe different character of the m = 0 and m = ±1 eigenstates. For m = 0, the Coriolis term is“turned off,” so the final eigenstates have the asymmetry doublet structure with equal mixturesof ka = +1 and ka = −1 character. For m = 1, both Coriolis and asymmetry effects are present,
but the former is dominant. (2A0 is an order of magnitude larger than (B − C)/2.) The m = ±1
eigenstates therefore have largely pure ka character of either +1 or −1. The Stark effect operator~ has matrix elements (for ∆J = 0) of the form hm0 , ka0 |µ~µ~ˆ · E~ˆ · E|m,ka i ∝ µa ka δka ka0 δmm0 [66],where µa is the permanent body-frame a-axis dipole moment. The diagonal matrix element of

a

50

b
Measured
Absorption (a.u.)

0.1
Absorption (a.u.)

0 V/cm250 V/cm

0

−0.1

0

Simulated
−500

c

−250
0
250
Offset Freq. (MHz)

Energy500› › “|+,–› |–,+›”|+,– |–,+› › › ›|+,+ |–,– |–,+ |+,–

0.1

~2A’mka−250
0250
Offset Freq. (MHz)›› |+,+ |–,– |–,+ |+,–› › › ›~(B – C)/2|±,+› – |±,–›|±,+ + |±,–› › “|+,+› |–,–›”|0,+› + |0,–› |0,+› + |0,–›|0,+› |0,–›|0,+› |0,–›|0,+› |0,–›|0,+› – |0,–› |0,+› – |0,–›|+,+ |–,–

no asymmetry,no Coriolis

Coriolis
only

asymmetry+ Coriolis

asymmetryonly

no asymmetry,no Coriolis

Figure 3.6: DC Stark effect patterns in nitromethane. (a) The JKa = 55 ← 44 (m = 0) parity
doublet transitions in zero field (black) and 250 V/cm (red). The simulated spectrum is based on
the experimental effective Hamiltonians and a permanent body-frame dipole moment of µa = 3.46
Debye [66]. (b) The Jτ = 21 ← 31 (m = 1) transition splits symmetrically and linearly under anapplied field. (c) Competition between Coriolis coupling and asymmetry splitting leads to different
DC Stark effects for m = 0 and m = ±1 torsion-rotation states, |m, ka i. See text for details.

51
the asymmetry doublet states for m = 0 have contributions that cancel exactly, leading to zero
first-order effect. The diagonal matrix elements of the Coriolis-split states for m = ±1 are non-zero,leading to a linear DC Stark shift.
The essential physics is that Coriolis coupling between the internal torsion motion and thetotal-body rotation leads to an internal alignment of angular momentum in the body-fixed frame,
analogous to rotational motion of a symmetric top. In the absence of Coriolis coupling, the asymmetry term quenches this alignment, as in any ordinary semi-rigid asymmetric top. This is an
example of the unique effects associated with large amplitude motion in polyatomic molecules.
The number of observed |m| = 1 transitions is too small to perform a complete rotational
fit due to the cold experimental temperatures. The subset of observed levels, however, alreadyindicates there is significant disruption of the rotation-torsion structure in the excited ν3 + ν6
state compared to the ground state. For example, Fig. 3.7 compares the observed m = 0 and|m| = 1 upper state energies with calculated predictions based on the ground state rotation-torsion
parameters, including the kinetic energy parameter F , the Coriolis coupling constant A0 , and thesix-fold torsion barrier height V6 . The m = 0 levels are the same as in Fig. 3.5 and show good
agreement with the empirical effective Hamiltonian. This is not the case for the observed |m| = 1
levels, which all belong to the mka ≈ +2 branch. These energies exhibit a downward shift relative
to the predictions, with an average difference of about 0.7 cm−1 for the higher J values. This isevidence of one of several possibilities in the excited vibrational state: (i) it has a higher barrier tointernal rotation, which lowers all |m| = 1 energies; (ii) it has a significantly larger Coriolis constantA0 , which pushes ±mka branches further apart; or (iii) there is some interloping dark state repellingthe observed states downward. Reduced dimension torsion-rotation calculations, discussed below
in Sec. 3.3, providing compelling quantitative evidence for case (i).

52

(E – 2950 cm-1) - J(J+1)×(B+C)/2

15|m|=1, obs|m|=0, obs|m|=1, pred|m|=0, pred
Ka = 7e,f

10

Ka = 5e,f
Ka = 3f

mka = +25

Ka = 3eKa = 1f
0

Ka = 1e

01 2

3

4

5

J

6

7

Figure 3.7: Observed and predicted energies for the m = 0, ±1 levels of the ν3 + ν6 vibrational
state. The measured m = 0 levels (blue #) are shown for Ka (= |ka |) ≤ 7, as in Fig. 3.5. The
measured |m| = 1 levels (red #) are shown only for the nominal mka = +2 branch. All predicted
m = 0 (black ×) and |m| = 1 (black +) energies are shown up to J = 7. (Note that the ν3 + ν6
state has A001 vibrational symmetry such that only m + ka = odd levels have non-zero nuclear spinweight.) The observed |m| = 1 levels show a substantial discrepancy with the predictions calculatedusing ground state rotation-torsion parameters.

53
3.2.3

CH stretching modes

All three CH stretching modes of CH3 NO2 are IR active. The totally symmetric A01 mode ν1 isthe strongest and has the lowest frequency, with a band origin near 2975 cm−1 . The two asymmetricmodes are the out-of-plane component ν10o near 3055 cm−1 and the in-plane component ν10i near3085 cm−1 . The identification of out-of-plane vs. in-plane can be made by the spectroscopic
signatures of the two resulting vibrational symmetries [62]. The out-of-plane mode has A002 symmetry
and therefore c-type rotational selection rules, resulting in dense Q-branch transitions near the band
origin (clearly visible in the survey spectrum, Fig. 3.4). The in-plane mode has A001 symmetry withb-type rotational selection rules and lacks such a strong Q-branch feature.
In total, we have identified nearly a thousand individual rovibrational transitions in thethree bands. Less than 10% of these, however, have been successfully rotationally assigned via
ground state combination differences. Complex zeroth order structure and seemingly widespread
perturbations are reponsible for this low fraction of assignments, line lists for which are provided
in Tables A.3–A.4 (see Appendices).
For the ν1 band, we identified P and R branches accessing the upper state Ka0 = 0e manifold
and P, Q, and R branches accessing the Ka0 = 2f manifold (Fig. 3.8). These transitions provide
sufficient data to fit the band origin and the rotational constants A and (B +C) for the ν1 state (form = 0). The results of this fit are summarized in Table 3.4. There is a substantial (6%) decreasein the effective A parameter. As with ν3 + ν6 , this may be evidence of a corresponding increase in
the effective torsional barrier height. However, given that only two Ka0 manifolds are being fitted
to, it is unclear how significant this value is. Moreover, the relatively high fit residuals (RMSE= 0.066 cm−1 ) suggest there are important interactions unaccounted for, likely involving |m| > 0
states. Although the DC Stark spectra have revealed many |m| > 0 transitions, their assignment
has thus far remained elusive.
The m = 0 transitions of the out-of-plane CH stretch band ν10o proved easier to assign
than those for ν1 . Upper state energies were obtained for Ka0 ≤ 5 (both parity components) and

54
2f

0.07
0.06

P(J)

2f

Q(J)

2f

P(J)

R(J)

R(J)

0e

0e

Absorption

0.05
0.04
0.03
0.02
0.01
0
-0.01
2969 2970 2971 2972 2973 2974 2975 2976 2977
Wavenumber (cm-1)
Figure 3.8: Assigned transitions of the ν1 (m = 0) band of CH3 NO2 . The P/Q/R branches arelabeled with tie lines, where the superscript denotes the upper state Ka quantum number and theusual e/f parity label. (All transitions follow an a-type ∆Ka = 0 selection rule.)

Table 3.4: Rotational parameters of the ν1 and ν10o (m = 0) bands of CH3 NO2 . All values are
given in cm−1 . The difference B − C was constrained to the corresponding ground state value forthe ν1 fit. Quartic centrifugal distortion terms were included for all states, but held fixed to their
ground state values (see Table 3.3).
Parameter Ground statea
ν00A
0.445037
B
0.351722
C
0.195994
RMSE—
a Ref. [55]b This work.

ν1 b2973.35(3)0.4187(71)0.3521(7)0.1963(7)0.066

ν10o b3053.08(2)0.4602(19)0.3520(17)0.1944(13)0.055

J 0 ≤ 6, permitting a fit of all three effective rotational constants (Table 3.4). The fit residuals
for this state are large (RMSE = 0.055 cm−1 ) and similar to those of ν1 . As above, this suggeststhat certain interactions are unaccounted for in the effective Hamiltonian. For the ν10 modes,

55
in particular, torsionally mediated interactions are expected to be important because the torsion
motion transforms the in-plane and out-of-plane coordinates into each other, resulting in strong
Coriolis-like coupling [62, 65].
The in-plane CH stretch band ν10i exhibits the most complicated spectrum of all those considered in this chapter. Using ground state combination differences, we have assigned transitions toseveral m = 0 upper states. The resulting upper state energies, however, bear little resemblance tothe expected rigid asymmetric top zeroth order structure and no attempt at fitting them has beensuccessful. We speculate that the rotation-torsion structure here is even more severely perturbed
by torsion-vibration interactions than for the lower-energy out-of-plane band. This is because thein-plane component is near-resonant with |m| > 0 states (and excited rotational states) of the outof-plane component, leading to a severe reorganization of the zeroth order structure. We examine
these dynamics further with model calculations in the next section.

3.3

Reduced-dimension calculations of the rotation-torsion structure andCH stretching vibrations
Under the Born-Oppenheimer approximation, rotation-vibration energies correspond to theeigenvalues of the nuclear motion Hamiltonian Hnuc . Calculating the eigenvalues (and eigenfunctions) of Hnuc is a highly non-trivial task as the computational expense in general grows exponentially with the number of degrees of freedom. One way to address this problem is to consider asimpler, reduced-dimension Hamiltonian Hred obtained by “freezing” a subset of degrees of freedom, i.e. constraining certain nuclear coordinates to fixed values. The spectrum of Hred can becomputed efficiently and accurately at the cost of approximating the dynamics of the full molecularsystem. For cases in which the active and constrained degrees of freedom are approximately separable, reduced dimension approaches are a useful means for understanding zeroth order nuclear
motion dynamics. In this section, we take advantage of reduced-dimension models to interpret theobserved spectroscopic signatures of internal rotation and CH stretching vibrational dynamics in
CH3 NO2 .

56
3.3.1

Method overview

The first step of a reduced-dimension calculation is to construct the appropriate Hamiltonian,
Hred = T + V (q),(3.15)

where T is the reduced-dimension kinetic energy operator (KEO) and V (q) is the nuclear potential
energy surface (PES). We defer a detailed discussion of calculating the KEO for Chapter 7. For
now, we simply assert that we can obtain a numerically exact KEO for arbitrary reduced-dimension
models with general curvilinear coordinates including all vibration, rotation, and Coriolis contributions. The PES V (q) is a scalar function of the coordinates included in the reduced-dimension
calculation. It can be obtained by fitting a surface to a set of energies obtained by a some electronic
structure method or by constructing a simple model potential.
Once Hred is specified, its eigenvalues and eigenfunctions are computed variationally by diagonalizing its matrix representation in a suitably large basis set. We accomplish this with large-scale
iterative eigensolver methods [67] in combination with direct-product grid discrete variable representation basis sets [68]. Again, the details of these methods can be found in Chapter 7. For the
problem sizes considered here, these techniques provide numerically exact energies and wavefunctions for the Hred models of interest.

3.3.2

Effect of the torsion barrier height on torsion-rotation energies

The rotational analysis of the ν3 + ν6 combination band (Section 3.2.2) revealed two related
facts. First, the effective A constant of the excited vibrational state is 1.2% smaller than that of
the ground state (a relatively large shift). Second, the (|m| = 1) − (m = 0) energy difference in the
excited vibrational state is about 0.7 cm−1 smaller than in the ground state. Both of these effects
can be explained by an increase in the effective torsional barrier height in the excited vibrational
state. Are the effective barrier heights inferred from these shifts mutually consistent? Reduceddimension calculations can help us answer this question.
We consider a (1+3)-dimensional model that includes the torsion angle ρ (see Fig. 3.1) and57
the three body-frame rotations. The remaining fourteen small amplitude internal coordinates areheld fixed to their (ρ-averaged) equilibrium values. The PES V (ρ) is only dependent on the torsion
angle ρ and is represented with a single six-fold term (see discussion above):
V (ρ) =

V6
cos(6ρ),2(3.16)

where V6 is the internal rotation barrier height. For a given value of V6 , we calculate the reduceddimension torsion-rotation energies and extract the lowest torsion excitation energy E(1 − 0) andthe effective rotational A constant. Figure 3.9 shows the barrier height dependence of these two
observables from a series of such calculations. The plots are normalized to their free internalrotation limit (V6 = 0). Each curve shows a similar sigmoidal transition. The torsion excitationenergy E(1 − 0) starts at the free value determined by the torsion F constant (5.6 cm−1 ) andbecomes exponentially small as the rotor moves into the high-barrier tunneling limit. The effective
rotational A constant starts as the inverse moment of inertia of the NO2 frame and changes to thatof the entire CH3 NO2 molecule as the torsion becomes locked in the high-barrier limit. The ratioof these two moments of inertias is equal to the limiting value of A/Afree ≈ 0.92. The turnoverof both curves occurs when the barrier height becomes equal to the energy difference between them = 0 ground state and the m = ±6 excited states, which are the first that the six-fold potential
can couple to the m = 0 state. Thus, we expect the turnover at V6 ∼ F × 62 ≈ 200 cm−1 .
The two barrier-dependence curves allow us to correlate changes in the torsional excitationenergy with changes in the rotational A constant. In the ground vibrational state, the effective V6parameter equals 2.10 cm−1 [53], such that the torsion energy and A are approximately equal totheir free-rotor-limit values. In the excited ν3 + ν6 vibrational state, A decreases to a relative valueofAν3 +ν6Aν3 +ν6
0.43990 cm−1≈=
≈ 0.988Afree
A00.44504 cm−1(see Table 3.3). This value is marked as a horizontal dashed line in the bottom plot of Fig. 3.9.
The corresponding effective barrier height is V6 ≈ 117 cm−1 , marked as vertical dashed line in

58

free
rotor

semi-rigid
vibration

E(1-0)/E(1-0)free

1
0.8

0.84

0.6
0.4
0.2

117 cm-1

0
1

A/Afree

0.98

0.988

0.96
0.94
0.92
0.9

117 cm-1

101
102
103
Effective barrier height, V6 (cm-1)
Figure 3.9: Torsion excitation energy and rotational A constant as a function of the effective
barrier height V6 . The dependence of each observable is shown normalized to its value in the freerotor limit (V6 = 0). The observed value of A/Afree = 0.988 for the ν3 + ν6 vibrational state ismarked with a horizontal dashed line (bottom plot). The corresponding effective barrier height is
V6 = 117 cm−1 . This implies a decrease of the torsion excitation energy to 0.84 times its free rotor
value (top plot).

the same plot. Given this V6 value, we expect the torsional excitation energy to decrease to 0.84
times its free rotor (ground state) value, i.e. to shift down by (1 − 0.84) × F0 ≈ 0.89 cm−1 , whereF0 ≈ 5.56 cm−1 is the ground state internal rotor constant [53]. This value is in reasonably good
agreement with the experimental |m| = 1 energy shift of 0.7 cm−1 in the ν3 + ν6 vibrational state.
One can conclude that the observed torsion-rotation parameters for ν3 + ν6 are roughly consistent59
with an increase in the effective barrier height V6 to slightly more than 100 cm−1 , placing it within
the intermediate region between the low- and high-barrier limits.

3.3.3

Torsion-mediated coupling of the CH stretch vibrations

We can extend the (1+3)-D torsion-rotation calculation by adding the three CH bond length
coordinates, r1 , r2 , and r3 , to model the CH stretching vibrations (see Fig. 3.10) and to understandthe torsion-mediated interactions between them. The PES is now a function of ρ and the threebond lengths. To ensure the PES has proper permutational symmetry, it is convenient to introduce
symmetrized bond length coordinates,1
q1 = √ (r1 + r2 + r3 ) ,31
q10a = √ (2r1 − r2 − r3 ) ,61
q10b = √ (r2 − r3 ) .
2(3.17)The q1 coordinate is totally symmetric (A01 ) and corresponds to the ν1 mode. q10a and q10b transform
as a degenerate E 0 pair. They describe asymmetric CH stretching motions that have a fixedorientation relative to the local CH3 frame. We must perform a ρ-dependent transformation to
generate “in-plane” and “out-of-plane” coordinates [18],q10i = q10a cos ρ − q10b sin ρ,
q10o = q10a sin ρ − q10b cos ρ.

(3.18)The in-plane coordinate q10i transforms as A001 , and the out-of-plane coordinate q10o transforms as
A002 . These correspond to the ν10i and ν10o vibrational modes, respectively. Using these coordinates,
we define a model 4-D torsion-vibration PES as
V =

V6
1
1
1
2
2
cos(6ρ) + k1 q12 + k10i q10i
+ k10o q10o
.
2
2
2
2

(3.19)The V6 parameter is chosen to equal the ground state value of 2.1 cm−1 . The three force constants
k1 , k10i , k10o are tuned to match the reduced-dimension vibrational frequencies to the experimental
anharmonic frequencies.

60

H1

ρ

O4

r1

a

C

r2

b

c

r3

H2

H3
O5

Figure 3.10: CH bond length coordinates for nitromethane. The three stretching coordinates r1 ,r2 , and r3 are defined as the respective C−Hi internuclear distances. ρ = 0 is defined as the C−H1
bond being parallel to the O−N−O plane. (The N atom is eclipsed by the C atom in the drawing.)

The calculated vibration-torsion manifolds for the three CH stretching modes are shownin Fig. 3.11a. The m = 0 levels are the pure vibrational energies of the three modes: symmetricstretch (black), out-of-plane stretch (blue), and in-plane stretch (red) in increasing energy order. For
each vibrational state there is a progression of excited torsional sub-levels whose energies increase
quadratically with m. This quadratic dependence is nearly perfect for the symmetric stretch, while
the two asymmetric stretches show deviations. These are more clearly seen in Fig. 3.11b, where thenominal F m2 torsion energy contribution has been subtracted (with F ∼ 5.6 cm−1 as expected).
The repulsive pattern between the in-plane and out-of-plane manifolds can be modeled with asimple Hamiltonian in the vibrational angular momentum basis. Define the |+i and |−i vibrational
states as1|±i = √ (|ν10i i ± i|ν10o i) ,2

(3.20)and the vibrational angular momentum operator Λ̂ such that
Λ̂|±i = ±|±i.

(3.21)

a

61

b
3800
Energy – Fm2 (cm-1)

Energy (cm-1)

3700
3600
3500
3400
3300
3200
3100

3150
3100
3050
3000
2950

3000
0 1 2 3 4 5 6 7 8 9 10m

0 1 2 3 4 5 6 7 8 9 10m

Figure 3.11: Reduced-dimension torsion-vibration energies of the CH stretching modes of nitromethane. (a) The absolute torsion-vibration energies are plotted for the symmetric stretch(black), out-of-plane asymmetric stretch (blue), and in-plane asymmetric stretch (red) vibrationalmodes as a function of the torsion quantum number m. The dashed curves are the fitted model
Hamiltonian eigenenergies, Eq. 3.24. (b) The same as (a) with the pure torsion energy contribution
F m2 subtracted away.

This vibrational angular momentum is a result of small orbital motions of the H atoms about
their local equilibrium positions. It is distinct from the angular momentum associated with large
amplitude motion of the entire methyl rotor unit. Now consider the following Hamiltonian,
H =ω+


δω
|+ih−| + |+ih−|2+ F Jˆρ2 + ζρ Λ̂Jˆρ .

(3.22)

ω is the mean vibrational frequency of the two asymmetric modes. δω describes the static interactionof the methyl top with the NO2 group, which breaks the degeneracy of the in-plane vs. out-of-planefrequencies. The F Jˆρ2 term is the usual kinetic energy contribution of the methyl rotor. The lastterm ζρ Λ̂Jˆρ accounts for coupling between the small-amplitude vibrational angular momentum andthe large amplitude torsional angular momentum. This is similar to standard Coriolis coupling,
except that in this case the two angular momenta are both internal motions and the body-frame

62
angular momentum does not appear.
The model H is block diagonal in the torsion quantum number. For each value of m, there
is a 2 × 2 block of the form2
 ω + F m + ζρ m

Hm = 

δω/2
ω + F m2 − ζρ m

δω/2


,

(3.23)

with energy eigenvalues±

2

E (m) = ω + F m ±

s

δω
2

2

+ (ζρ m)2 .

(3.24)For example, for m = 0 the diagonal elements become equal. The resulting eigenvalues are thepure vibrational energies ω ± δω and the eigenvectors revert to the expected in-plane |ν10i i andout-of-plane |ν10o i vibrational states. As m increases such that the ζρ Λ̂Jˆρ interaction is dominant
over the static δω splitting, the vibrational states have approximately pure |±i angular momentum
character. This is analogous to the rotation-torsion Coriolis interaction that results in pure ka
character states (Fig. 3.6), but on a larger energy scale.
Fitting Eq. 3.24 to the reduced-dimension torsion-vibration energies yields
ω = 3067 cm−1 ,|δω| = 26.5 cm−1 ,F = 5.6 cm−1 ,|ζρ | = 12.2 cm−1 .
The corresponding E ± (m) curves are plotted as red and blue dashed lines in Fig. 3.11. This
simple model shows excellent agreement with the torsion-vibration energies. The parameter of mostinterest is the torsion-vibration angular momentum coupling constant ζρ . Its relatively large valueof 12.2 cm−1 indicates that this interaction must be taken into account to understand the rotationtorsion fine structure of the asymmetric CH stretch bands. The reduced-dimension estimate of its
magnitude should prove useful for making progress with this analysis.

63

3.4

Torsion-rotation Hamiltonians via curvilinear vibrational Møller–Plesset
perturbation theory
The calculations presented in the previous section are useful tools for understanding thezeroth order structure, dynamics, and interactions at play in nitromethane. The reduced-dimension
approximation, however, is simply too severe to make predictions of spectroscopic accuracy. Thisis especially true of rotational parameters (e.g. A, B, C constants and centrifugal distortion
parameters) measured by microwave and millimeter-wave spectroscopy to high degrees of precision.
In this context, accurate full-dimensional ab initio calculations are necessary both to facilitate
interpretation of experimental results and to provide usefully predictive spectroscopic constants [69].
Direct variational rovibrational calculations of the type employed above became intractably
expensive for molecules the size of nitromethane, which has only seven atoms, due to the exponential
scaling of the computational complexity. Therefore, we must turn to non-variational perturbative
methods that decrease the cost of the calculation without unduly compromising its accuracy. The
standard approach to rovibrational perturbation theory (VPT) is based on a harmonic zeroth orderapproximation to the rectilinear Watson Hamiltonian [70, 71]. VPT fails in two ways for floppy
molecules that undergo large amplitude motion. First, such systems are highly anharmonic. (Thetorsional motion in nitromethane has zeroth order dynamics that are nothing like a harmonic oscillator.) Second, rectilinear (Cartesian) coordinates poorly describe the motion of floppy molecules.
(Torsion in nitromethane is best described by the curvilinear rotation angle of the methyl group,
not by fixed rectilinear displacement vectors required by VPT.) The inadequacy of a second-order
VPT (VPT2) treatment of nitromethane is demonstrated by the wildly inaccurate predictions ofsome of its ground state torsion-rotation parameters, summarized in Table 3.5.
Motivated by the failure of VPT to treat floppy molecules exhibiting large amplitude motion, we have developed an alternative approach based on second order vibrational Møller-Plesset
perturbation theory (VMP2). The key concept is to use a more flexible zeroth order picture thatproperly accounts for anharmonic, curvilinear dynamics. This zeroth order wavefunction |Ψ0 i is a

64
Hartree product of anharmonic 1-D wavefunctions for each vibrational coordinate,|Ψ0 i = φ1 (q1 )φ2 (q2 )φ3 (q3 ) · · · .

(3.25)The qi are general curvilinear coordinates, chosen to maximize the separability of the total vibrational Hamiltonian. The anharmonic 1-D functions φi are optimized variationally by minimizing
the expectation value hΨ0 |H|Ψ0 i. This leads to a set of mean-field equations, which are solved
iteratively until self-consistency is reached, hence the term vibrational self-consistent-field theory(VSCF) [72–74]. The zeroth-order VSCF wavefunction can be refined by applying second-order
Rayleigh-Schrödinger perturbation theory to account for vibrational correlation effects beyond themean-field description. In analogy with electronic structure theory [75], this method is termed
vibrational Møller–Plesset perturbation theory (VMP2) [76]. We have extended curvilinear VMP2
to include rotational and rovibrational terms in the nuclear motion Hamiltonian, thereby generating effective rotational Hamiltonians in addition to anharmonic vibrational frequencies [51] (seeChapter 7).
The two main inputs into a VMP2 calculation are the PES and the choice of curvilinear coordinate system. We employ the full-dimensional CCSD(T)-F12b/HaDZ PES recently constructedby Wang, Carter and Bowman [77]. A minimum energy isomerization path coordinate system
suitable for large amplitude torsion is used as the internal coordinate system. (More details canbe found in Ref. [51]). The primary results of the application to the ground state rotation-torsion
structure of nitromethane are shown in Table 3.5. The curvilinear VMP2 results show uniformly
excellent agreement with the experimental rotation-torsion parameters, including the principal
rotational constants (A, B, and C), the internal rotor constants (A0 and F ), and the quartic centrifugal distortion parameters (∆J , ∆JK , ∆K , δJ , δK ). (These constants are defined in the effective
Hamiltonians in Eqs. 3.10 and 3.14.) This is in striking contrast to VPT2, which is particularly
problematic for quantities involving the a-axis (A, ∆JK , ∆K , and δK ) because it is coincident with
the internal rotation axis. Moreover, VPT2 cannot provide predictions at all for the internal rotor
constants F and A0 . These issues can be traced back to the use of rectilinear coordinates and a

65
harmonic zeroth-order description, which are physically inappropriate for large amplitude motion.
Table 3.5: Ab initio torsion-rotation parameters of nitromethane. Predictions based on curvilinear
VMP2 and standard rectilinear second-order VPT (VPT2) are compared to experimental values.
VPT2 entries marked * are especially unphysical. All values are given in GHz.
Parameter
VMP2
A
13.33
B
10.51
C
5.86
F
166.9
A0
13.25
∆J
×106
5.50
6
∆JK ×1017.84∆K×106 -10.66
δJ
×106
2.24
6
δK
×10
15.77
a Ref. [55]b Ref. [52, 53]

VPT2
12.19*
10.46
5.85
—
—
5.67
952.73*-949.39*2.30-268.95*

Expt.
13.34a10.54a5.88a166.7b
13.28a6.14a17.75a-7.54a2.47a15.75aThe success of curvilinear VMP2 for providing accurate ground state constants for nitromethane
is encouraging. Similar calculations can be performed for excited vibrational states. Effects suchas torsion-vibration angular momentum coupling, as discussed in the previous section, are fully
captured with this approach. These predictions will aid in the continued analysis of the complex
rotation-torsion structure of the CH stretching vibrations.

3.5

Conclusions
The combination of comb spectroscopy and cold molecules results in a remarkable quantity ofdetailed spectroscopic information. Analysis of this data distills a “phonebook” of line positions into
specific structural and dynamical insights, as demonstrated in this chapter for nitromethane. Although many conclusions are to a greater or lesser degree particular to this example, they also reveal
patterns and general physical concepts that can be applied to related molecular systems. Similarly,
analytical and computational tools developed to understand the spectroscopy of nitromethane, suchas the curvilinear VMP2 methods introduced in this chapter, can be applied to a variety of otherproblems. (Examples of such applications are presented in Chapter 7.) As large amplitude motion

66
remains an incompletely understood phenomenon, high resolution infrared spectroscopy continues
to contribute to our understanding of its consequences on molecular structure and dynamics.

Chapter 4

Limits to rotationally resolved MIR spectroscopy of large hydrocarbons

High resolution gas-phase spectroscopy is an essential tool for molecular detection and identification. This is especially true for applications in extraterrestrial and astronomical chemistry,where spectroscopic detection is the primary, and often only, means of identification [78]. Most
astronomical observations of molecules are made via pure rotational transitions in the microwave
and millimeter-wave regime [79], but this requires the presence of a permanent dipole moment.
Many molecules of potential astrochemical importance, such as large polyaromatic hydrocarbons(PAHs) [80] or fullerenes (see Chapter 5), lack a permanent dipole moment by symmetry. One
must instead turn to optical techniques, i.e. vibrational or electronic spectroscopy.
Rotationally resolved infrared studies of large molecules are relatively few in number compared to those of small- to medium-sized molecules of less than about a dozen atoms. One contributing factor is the difficulty of producing gas-phase samples of large, heavy molecules that are
rotationally and vibrationally cold. Even if this can be achieved in the laboratory, the spectra
are often intrinsically congested and complex due to the high density of rovibrational states at theinternal energies excited by infrared photons. These systems are thus an ideal challenge for the
combined capabilities of cryogenic buffer gas cooling and cavity-enhanced direct frequency comb
absorption spectroscopy.
This brief chapter contains several studies of large hydrocarbon molecules of increasing size
and mass. We present rotationally resolved infrared spectra in the CH stretching region (ca. 3 µm)
of naphthalene (C10 H8 ), adamantane (C10 H16 ), and hexamethylenetetramine (HMT, C6 N4 H12 ,68
not strictly a hydrocarbon). We also report our inability to observe rotational structure in the
spectrum of cold diamantane (C14 H20 ), and discuss the general implications of these results forrotationally resolved spectroscopy of large molecules in the context of intramolecular vibrationalenergy redistribution (IVR) [81].

4.1

Experimental details
All spectra discussed in this chapter were recorded using the 3 µm OPO comb with FTS and

VIPA-based detection. Cold (10 to 20 K) samples of each molecular species were prepared withthe low-temperature CBGC cell with He buffer gas. Details of these apparatus are provided in
Sections 2.1.1, 2.4, and 2.5.1 and will not be repeated here.
The target species are crystalline solids at room temperature, requiring an oven source toproduce a sufficient flux of gas-phase molecules. The oven temperature range used in each caseis listed in Table 4.1. For each species, the temperature is below its respective melting point.
Sublimation is sufficiently rapid to produce the necessary flux.
Table 4.1: Oven temperatures used to introduce solid samples into CBGC cell.
SpeciesNaphthalene
Adamantane
HMT
Diamantane

4.2

Oven Temperature (◦ C)
60 − 70
45 − 65120 − 125
110 − 120

Results
Naphthalene, C10 H8 , contains two fused benzene rings making it the simplest PAH. As such,it has received significant spectroscopic attention in both the infrared [82–84] and ultraviolet [85,86]
regions. The ν29 CH stretch band, which has previously been studied with skimmed-molecularbeam optothermal experiments [84], is shown in Fig. 4.1. This CE-DFCS measurement is thefirst rotationally resolved spectroscopy in this wavelength region for naphthalene via direct absorption. We assigned over 150 rovibrational transitions (Table A.5) and used these to determine the

69asymmetric-top rotational parameters of the ν29 state (Table 4.2). The upper state energies andfitted curves are shown in the inset of Fig. 4.1. Fitting just the three principal rotational constants
A, B, and C results in small residuals (RMSE = 4.3 × 10−4 cm−1 ≈ 14 MHz) on the order of theline position uncertainty.1 The band intensity profiles are consistent with a rotational temperatureof 10(3) K.

0.12

ν29

0.10
0.08
0.06

Ered (cm-1)

Fractional Absorption

3065.0Naphthalene (C10H8)

3064.63064.2Data
Fit

3063.8

0

4

J

8

12

0.04
0.02
0

3058

3060

3062

3064

3066

3068

Wavenumber (cm-1)
Figure 4.1: The ν29 band of cold naphthalene. Extensive rotational fine structure is resolved in thebuffer-gas-cooled CE-DFCS spectrum. (Inset) Reduced term energies of ν29 rotational sub-levels
plotted versus J. The fitted effective rotational Hamiltonian (Table 4.2) shows excellent agreementwith the measured rovibrational energies.

Adamantane, C10 H16 , is the simplest member of the diamonoid family of molecules, sonamed for their tetrahedral carbon-cage structure that forms the building block of bulk diamond.
Figure 4.2 shows a 100-cm−1 -wide portion of its spectrum centered around 3.4 µm. This region
contains five strong vibrational bands: the three IR active CH stretch modes (ν20 , ν21 , and ν22 ) as
well as two unassigned bands near 2853.1 cm−1 and 2904.6 cm−1 . Each of these bands contains
1

Although we achieve similar residuals as the fit based on skimmed-beam data by Hewett et al. [84], our rotational
constants disagree by several σ. After comparing combination differences of both sets of data with accurate groundstate energies [82], we speculate that the line positions reported by Hewett et al. suffer from a calibration error.

70
Table 4.2: Rotational fit of the ν29 band of naphthalene. The quartic centrifugal distortion
constants for the upper state were held fixed to the ground state values of Ref. [82]. All values are
given in cm−1 .
Parameter
Ground statea
ν29 b
ν0
0
3064.5942(5)0.104051836(124) 0.104198(30)AB
0.04112733(37)0.0411173(38)C
0.029483552(140) 0.02942455(9)∆J
×109 0.528(49)[0.528]∆JK
×109 1.206(145)[1.206]9∆K×10
5.648(112)[5.648][0.1752]
δJ
×109 0.1752d
δK
×109 1.951d[1.951]
4RMSE ×10
3.1
4.3
a. Ref. [82]. δJ and δK are calculated values.
b. This workc. Ref. [84]

ν29 c3064.58(2)0.104013(17)0.0411023(45)0.0294062(20)
—
—
—
—
—
6.3

extensive rotational fine structure. The Doppler-broadened rovibrational transitions have a line
width of 23(2) MHz, indicating a translational temperature of 17(1) K (Fig. 4.2b). The narrow line
width provides sufficient resolution to reveal intricate tetrahedral centrifugal distortion splittings ofthis heavy spherical top (Fig. 4.2c). HMT, C6 N4 H16 , is a heterocycle cage formed from adamantane
by replacing the four carbon atoms at the vertices of a tetrahedron with nitrogen atoms. Figure 4.3
shows a small portion of its cold absorption spectrum centered near the ν17 CH stretch band. We
observe similarly well resolved and narrow rovibrational transitions as for adamantane.
The next largest diamonoid after adamantane is diamantane, C14 H20 . Its absorption spectrum is shown in Fig. 4.4. Although acquired under nearly identical conditions to the previous
examples, this spectrum shows a quasi-continuous absorption profile with 5-cm−1 -wide vibrational
bands centered around the CH stretch fundamental frequencies. Upon close examination, a limited number of narrow absorption features can be detected that appear at approximately regularintervals of 0.04 − 0.05 cm−1 (Fig. 4.4 inset). This spacing is about equal to 2B, suggesting thatthese features are vestiges of some underlying rotational structure. These transitions have line
widths of 20 − 30 MHz. Assuming this to be a Doppler-broadened feature, that corresponds toFractional Absorption

Adamantane (C10H16)

0.5

ν22

0.4
0.3

-ln(Transmission)

71

a

0.3 b
0.2
0.1

0.2

0

FWHM =
23(2) MHz

ν21

0.3 c

ν20

0.2

Ttrans =17(1) K

0.1
0

-40
0
40
Offset Freq. (MHz)2936.59 2936.63

0.1
0

2852

2856

2860

2906
2910
2914
-1
Wavenumber (cm )

2936

2940

Figure 4.2: Survey absorption spectrum of cold adamantane, C10 H16 . (a) The region spanning2850−2950 cm−1 contains three very bright IR active CH stretch bands (ν20 , ν21 , ν22 ) as well as two
unassigned bands near 2853.1 cm−1 and 2904.6 cm−1 . (b) The narrow Doppler profiles demonstratetranslational cooling to a temperature of Ttrans = 17(1) K. (c) Rovibrational transitions of thisspherical top appear in dense clusters due to tetrahedral fine structure splittings.

Fractional Absorption

0.20

ν17

HMT (C6N4H12)

0.16
0.12
0.08
0.04
0

2948

2952
2956
Wavenumber (cm-1)

2960

Figure 4.3: The CE-DFCS absorption spectrum of HMT in the vicinity of the ν17 CH stretch
band. This spectrum has a similar appearance to that of adamantane (Fig. 4.2) due to the related
structure and identical symmetry of the two molecules.

translational temperatures of 20 K. The width of the rotational contours is also consistent with a

72rotational temperature of 10 − 30 K. This indicates that diamantane is approximately thermalizedwith the He buffer gas bath, and that the continuous absorption profile is an intrinsic feature of its
spectrum.

Fractional Absorption

0.4

Diamantane (C14H20)
0.05 cm-1

0.3
2884.420.2

2884.48 2884.54

0.1
0
2880

2890

2900
2910
2920
Wavenumber (cm-1)

2930

2940

Figure 4.4: Absorption profile of diamantane, C14 H20 . In lieu of the rotational fine structure
observed for other examples, diamantane exhibits a quasi-continuous absorption profile in the CH
stretch region. The strongest vibrational bands are each about 5 cm−1 wide. (Inset) Small portions of the spectrum contain narrow, regularly spaced features suggestive of underlying rotational
structure. These features are only 20 − 30 MHz wide.

4.3

Discussion and conclusion
The observations above represent the largest molecules for which rotationally resolved spectrain 3 µm region have been reported [87, 88], demonstrating the effectiveness of the combined cold
molecule-frequency comb spectrometer. From a practical perspective, the system provides very highdata throughput, collecting millions of individual frequency elements in a short amount of time.
(Altogether, the spectra above required a total of only three hours of data collection.) Gas-phaseinfrared measurements of comparable bandwidth and resolution are otherwise possible only withultra-long FTIR instruments, which typically require the use of synchrotron facilities [82, 83, 87, 89,

7390] and are not suited to table-top experiments.
While our results show just how far high resolution infrared spectroscopy of large molecules
can be taken, they also suggest intrinsic limits to such experiments, as evidenced by the CH
stretch spectrum of diamantane. We interpret its quasi-continuous absorption profile as evidence
of intramolecular vibrational energy redistribution (IVR) between the zeroth-order CH stretchbright states and the manifold of background dark states [81]. The density of vibrational states atthe ∼3000 cm−1 internal energies probed by the infrared photon is about 10× larger for diamantane
than for molecules for which well resolved rotational fine structure is observed, such as adamantane
and naphthalene (see Fig. 4.5). Various studies on IVR rates in hydrocarbons [91–96] suggest
an approximate background state density threshold of ρ ∼ 102 /cm−1 before the onset of severe
spectral fractionation [81]. Diamantane, with ρ ≈ 103 /cm−1 at 3000 cm−1 , is well above thislimit. Based on this rule of thumb, one might expect that adamantane and naphthalene should
also suffer from IVR effects. However, it is important to take into account the ameliorating roleof symmetry [97–99], which reduces the extent of IVR coupling, especially in highly symmetricmolecules like adamantane.
IVR presents a general obstacle for high resolution spectroscopy of large molecules. Continued
progress mandates a shift away from the 3 µm CH stretch region. At the lower internal energies
probed by longer wavelength IR photons, the density of background vibrational states decreases
exponentially. This motivates the development of LWIR frequency comb sources, such as the
6 − 11 µm DFG system described in Section 2.1.2. These tools will enable the exploration of evenlarger and more complex molecular systems with rotationally resolved spectroscopy.

74

Vibrational state density (per cm-1)

108
106
104

C60

C14H20

C10H8

102

C10H16100
10-21000 1500 2000 2500 3000 3500
Energy (cm-1)

Figure 4.5: Vibrational state density for several large carbon-containing molecules, including
adamantane (C10 H16 , blue), naphthalene (C10 H8 , red), diamantane (C14 H20 , black), and buckminsterfullerene (C60 , pink). The curves were calculated using a direct count algorithm [100] withvibrational frequencies from Refs. [101–106]. The approximate IVR threshold of 102 states per
cm−1 [81] is indicated with a dashed horizontal line.

Chapter 5

Rovibrational quantum state resolution of the C60 fullerene

The previous chapter provided specific examples of the general limitations placed on highresolution infrared spectroscopy by large, heavy polyatomic molecules. Some problems are of apurely technical and, some might say, shallow variety: having sufficiently narrow lasers and molecular line widths can eventually reveal even the most congested of spectra. Congestion, however, ismerely a symptom of a deeper and more essential obstacle. The inseparability of the ever multiplying degrees of freedom in large molecules (i.e. IVR) suggests that frequency domain spectroscopy
and an eigenstate-by-eigenstate perspective will have diminishing usefulness in understanding the
dynamics and structure of such complex many-body systems.
This chapter provides a counter-example to this trend in the form of buckminsterfullerene,
C60 (Fig. 5.1). As we will see, the remarkable symmetry and rigidity of C60 make a description of
its rovibrational spectrum and structure at the level of individual quantum states possible. To thebest of the author’s knowledge, C60 represents not only the largest molecule for which a rotationallyresolved spectrum has been obtained, but also the most symmetric, C60 being an example of therelatively rare icosahedral (Ih ) point group. Section 5.1 provides an overview of the structure andthermodynamics of C60 , illustrating why its size and mass make rotationally resolved spectroscopy
so formidable. As demonstrated in prior chapters, these challenges are overcome with a combinationof CBGC and CE-DFCS. However, unlike previous experiments, the particularly difficult task ofcooling C60 requires the high-temperature, high cooling power CBGC setup introduced in Chapter 2.
Section 5.2 describes in more detail the production of cold, dense samples of gas-phase C60 with

76
this device and the DFG frequency comb used to measure its infrared spectrum. An effective
rovibrational Hamiltonian model is introduced in Section 5.3. We use this to extract detailed
molecular scale insights into C60 . Finally, the chapter concludes with a discussion of some remaining
spectroscopic puzzles presented by these first frequency comb experiments.

Figure 5.1: The structure of buckminsterfullerene, C60 . Sixty carbon atoms lie at the vertices ofa truncated icosahedron (a soccer ball shape), making C60 an example of the icosahedral Ih point
group. Each carbon atom site is chemically indistinguishable to all others.

5.1

An introduction to C60
The existence of a stable, hollow cage of sixty carbon atoms was first postulated and studied

theoretically in the 1970s and early 1980s by several groups independently [107–110]. However,it was not until 1985 that the seminal mass spectrometry experiments of Kroto et al. provided
clear evidence for C60 [111]. The apparent stability of the molecule quickly led to the conjecturethat it formed a closed spherical cage with 20 hexagonal faces and 12 isolated pentagonal faces,

77
i.e. the truncated icosahedral structure shown in Fig. 5.1, which inspired the name “buckminsterfullerene,” and soon thereafter “buckyball,” in honor of the geodesic structures of the architect
Buckminster Fuller [112]. Subsequent infrared and 13 C-NMR studies of isolated C60 in the early
1990s confirmed its highly symmetric Ih structure [113–118]. Since then, a wide variety of spectroscopic and analytical techniques, including x-ray and electron diffraction [119, 120], optical Raman
and neutron scattering [121–127], matrix and helium droplet isolation spectroscopy [128–133], andphotoelectron spectroscopy [134–136], have greatly contributed to our understanding of this uniquemolecule. Spectroscopy in particular has played an essential role in the astronomical identificationof C60 and its derivatives [132, 137–141], and high resolution laboratory data may help resolve
current uncertainty regarding its astrophysical state [132]. However, despite intense interest, norotationally resolved spectrum of gas-phase C60 has been reported to date.
The high degree of symmetry exhibited by C60 makes it relatively difficult to probe withelectromagnetic radiation. Like all spherical tops, C60 possesses no permanent dipole moment,1
and therefore has no pure rotational microwave spectrum. Although centrifugal distortion effects
in some spherical tops, like tetrahedral CH4 , permit very weak pure rotational transitions [143],even these are strictly electric dipole forbidden in icosahedral tops such as C60 due to the presence of a center of symmetry. We must therefore turn to the optical regime, i.e. electronic orvibrational spectroscopy. The first strong electronic absorption features only appear relatively highin the singlet excitation manifold [144]. Vibronic coupling within the singlet manifold and fastintersystem crossing to triplet states [145,146] will potentially obscure spectroscopic fine structure,
but nonetheless would be very interesting to explore. Lower lying states have either orbital- or
spin-forbidden (or both) transitions from the ground electronic state and have very weak oscillator
strengths [144].
This leaves IR spectroscopy as the most viable approach to make the first observations ofrovibrational quantum state-resolved structure in C60 . With N = 60 atoms, each molecule has
3N −6 = 174 vibrational modes. Under the Ih point group (the character table for which is provided
1

In fact, the lowest rank multipole moment allowed for C60 is the 64-pole (a hexacontatetrapole) [142].

78
in Appendix C), only those modes that transform as the T1u irreducible representation are IR active.
There are four such sets of T1u vibrations, each comprised of three triply degenerate modes. Thesefour T1u bands are readily observed in low resolution condensed- or gas-phase experiments andoccur at 7.0, 8.5, 17.4, and 19.1 µm [131]. We choose to focus on the 8.5 µm band, which has amodest integrated band intensity of 9.4 km mol−1 [131] and is conveniently located in the centerof the operating range of our DFG frequency comb.
The remaining obstacle to obtaining rotationally resolved spectra of C60 is its formidable thermodynamics. The vibrational and rotational degrees of freedom pose complementary challenges.
Although the rigidity of the C60 carbon frame leads to relatively high vibrational frequencies fora molecule of its size (spanning 260 to 1600 cm−1 [106]), the sheer number of vibrational modes
results in a steep rise in the density of states with increasing internal energy. Conversely, while
there are only 3 rotational degrees of freedom, the large moment of inertia, ca. 6 × 103 uÅ2 , results
in a very small rotational energy level spacing. (The fundamental J = 0 − 1 spacing is approxiately
h×170 MHz.) Both of these contribute to a large internal partition function, particularly the vibrational contribution Zvib , which is shown in Fig. 5.2. At the temperature of our C60 oven, whichapproaches 1000 K, there are Zvib ≈ 1029 thermally occupied vibrational states! Even at 300 K,
Zvib is greater than 103 . It is necessary that most of the population is cooled to the vibrational
ground state, i.e. Zvib ∼ 1, to avoid spectral congestion from hot band transitions. This requires
internal temperatures of 150 K or lower. Concomitant with the rapidly increasing partition functionis the large internal energy stored in the vibrational modes. At 1000 K, each C60 molecule contains
60000 cm−1 (7.5 eV) of vibrational energy. To achieve vibrational ground state cooling, all of thisenergy must be efficiently removed.

5.2

Generating cold C60
We turn once again to CBGC to generate cold samples of gas-phase C60 . The reader isreferred back to Fig. 2.8, which shows the geometry of the C60 oven source and cryogenic cold cell.
A 1 − 10 mg min−1 flow of C60 molecules thermalized to the source temperature of 950 K exits

79

10

10

105

104

Partition Function

104

103

103
102

10

2

101

101
100

0

200

400
600
800
Temperature (K)Average Energy (cm-1)

5

6

100
1000

Figure 5.2: Thermodynamical functions for C60 vibrations. The vibrational partition function (bluedashed line, left vertical axis) increases super-exponentially with temperature as more and morevibrational modes “unfreeze.” The corresponding average vibrational energy (red solid line, rightvertical axis) reaches several tens of thousands of cm−1 by 1000 K, reflecting the large heat capacity
per molecule.

the oven into cold cell. The cell intersects the axis of the optical enhancement cavity, which has afinesse of 6000 and a FSR of 280 MHz. DFG comb light tuned to cover the 8.5 µm absorption bandof C60 is continously coupled into the optical cavity and monitored with the FTS spectrometer.
Our first attempts to cool vibrationally hot C60 molecules used the low-temperature helium
CBGC setup (see Section 2.5.1), which successfully cooled all attempted species in the past. Although deposits of vaporized material from the oven clearly indicated that material was entering
the cell, there was no detectable absorption signal from C60 , or any other absorber, above a 1 ppm
single pass loss baseline. While a variety of explanations could account for the absence of signal,
including thermal degradation of C60 into amorphous carbon, the primary suspect was insufficiently
high buffer gas density, which leads to inadequate cooling for both translational and internal states.
With poor translational cooling, C60 molecules ballistically travel from the oven to the cold cell
walls and freeze out, instead of slowly diffusing throughout the cell, thereby inhibiting a build-up

80
of C60 density. With poor internal state cooling, too many excited vibrational states are occupied,leading to broadening, congestion, and dilution of the absorption cross-section. Indeed, in recently
attempted supersonic jet absorption spectroscopy experiments, inefficient vibrational cooling was
also speculated to be the reason for the lack of detectable C60 absorption [147]. The large massdifference between C60 and the buffer gas helium atoms likely further reduces the collisional coolingefficiency in our experiment.
A serendipitous accident in lab provided support for the “collisionally starved” explanation
and a path forward for C60 cooling. One afternoon, an undetected short of a thermocouple leadon the C60 oven resulted in a temperature reading that was about one-half of the actual oventemperature. This led us to signficiantly overheat the oven, and the resulting radiative heat load
warmed the cryo-sorbs responsible for pumping helium buffer gas. They quickly experienced a hardcrash, releasing a significant amount of desorbed helium. In a few seconds, the dewar backgroundpressure increased from less than 10−4 Torr to about 10−1 Torr, and a strong, broad absorption
signal immediately appeared in the FTS comb spectrum, as shown in Fig. 5.3. This signal wasstrongly dependent on the oven temperature and located at the expected band origin position [131],
indicating C60 – in some form – was almost certainly the carrier. Using different buffer gas species,
we were able to reproduce the signal, with the band center being unaffected by the buffer gas species,
while the shape and width did exhibit some slight dependence. The optimal dewar pressure was
between 80 and 500 mTorr, depending on buffer gas. In order to run the low-temperature CBGC
cell at these high background pressures, the helium refrigerator had to be turned off and we reliedon mechanical vacuum pumps. The traces in Fig. 5.3b were collected after the cold cell had warmed
to a temperature of 100 K.
The physical state of C60 giving rise to the broad absorption feature is still uncertain. One
hypothesis is that it is from a C60 -buffer gas van der Waals complex or even a (C60 )n “supercluster” [148, 149]. Although we cannot entirely rule out these possibilities, we believe the simplest
and most likely case is that it is simply from “warm” C60 , i.e. partially cooled from the initial oventemperature, but not completely thermalized to the buffer gas bath (ca. 100 K). This reasoning

a

50

1

Oven off
Oven onAbsorption (norm.)

Comb intensity (arb.)

81

b

40
30
20

0.8
0.6
0.4

10

0.2

0

0

1140 1150 1160 1170 1180 1190 1200 1210

Wavenumber (cm-1)

He
Ar
N2

1178 1180 1182 1184 1186 1188 1190
Wavenumber (cm-1)

Figure 5.3: Absorption from the 8.5 µm band of C60 . (a) The raw comb spectral intensity envelope
shows a clear absorption dip from C60 centered near 1185 cm−1 . The periodic fringes are dueto etalons, while the handful of sharp absorption peaks are from ambient water vapor. (b) The
normalized absorption profiles in “warm” conditions for several buffer gas species including helium,
argon, and molecular nitrogen. The buffer gas temperature is 100 K.

suggests that increasing not only the number of buffer gas-C60 collisions but also the average energy
transfer per collision will help drive the hot C60 molecules to thermalization with the cold buffer
gas.
After extensively exploring a variety of buffer gas parameters, including the input flow rate,
the background dewar pressure (controlled independently of input flow rate by adjusting the pumping speed), the position of the oven relative to the annular slit inlet plate (see Fig. 2.8), and thebuffer gas species, we were eventually able to observe narrow absorption resonances in place of thebroad absorption envelope. Figure 5.4 illustrates these signatures for several different buffer gas
species. The strongest signal was obtained with argon buffer gas, suggesting it is the most efficientcollisional cooling partner for C60 . (Note that the spectra in Fig. 5.4 were acquired with a single
frep position, and only a fraction of the C60 absorption resonances overlap with the 280 MHz-spaced
cavity resonances.)
As indicated above, the small cooling capacity of the helium refrigerator is insufficient tooperate at such high buffer gas pressures for any extended period of time. We addressed this82

Absorption intensity (arb.)

0.10
0.08
0.06
0.04

He (80 mTorr, 20 K)
Ne (700 mTorr, 50 K)
N2 (500 mTorr, 90 K)
Ar (60 mTorr, 100 K)

0.02
01181 1182 1183 1184 1185 1186 1187 1188
Wavenumber (cm-1)

Figure 5.4: The first cold C60 signals. The broad, continuous absorption observed with helium
buffer gas is compared to optimized signals obtained with neon, nitrogen, and argon buffer gas.
The traces are offset for clarity. With the last three buffer gas species, narrow, well resolved
absorption resonances were observed. The strong peak near 1185 cm−1 is the Q branch of the C60
vibrational band. The broad line near 1187 cm−1 is from ambient water. The remaining “weeds”
are transitions from the C60 R and P branches. These spectra were all obtained at a single comb
frep value, such that the absorption spectrum is only sampled at the 280 MHz spacing provided by
the enhancement cavity. (The negative over-shoot is an artifact of the filtering process applied to
the FTS comb spectra.)issue by constructing the high-temperature CBGC system described in Section 2.5.2. The primarydifference is the replacement of the helium refrigerator with an LN2 cold finger, which provides
sufficient cooling capacity to run indefinitely at the required buffer gas pressures, enabling repeatedfrep scans and averaging. Of course, this also restricts the minimum achievable temperature to the77 K boiling point of LN2.
A survey spectrum of the full C60 band acquired with the high-temperature CBGC cell isshown in Fig. 5.5. The cold spectrum contains data acquired over 20 hours of total averaging

83
time. The 280 MHz comb mode spacing set by the cavity FSR was split into 89 subdivisions
such that the final optical frequency step size in the spectrum was about 3 MHz. The optimum
cooling conditions were found to be an argon buffer gas flow rate of 50 − 100 sccm and a 250 mTorr
background dewar pressure. The oven was placed 0.5 − 1.0 cm from the cold cell inlet aperture.
The extracted rotational temperature is 150(2) K, only slightly warmer than the measured cell wall
temperature of 135 K (see Fig. 2.11). A comparison of the warm and cold spectra in Fig. 5.5 showsa clear red shift of the warm spectrum relative to the band origin of the cold spectrum (located at
the strong Q branch). This is consistent with the warm spectrum being assigned to vibrationally
hot C60 , as hot bands are usually red shifted due to vibrational anharmonicity [131]. The strongest
absorption in the cold spectrum (the peak of the Q branch) corresponds to a single-pass absorptionof 25 ppm.

Warm
Cold
Sim.

Absorption (norm.)

2.0

1.5

1.0

0.5

P

Q

R

0
1182

1183

1184
1185
Wavenumber (cm-1)

1186

1187

1188

Figure 5.5: Survey spectrum of cold C60 near 8.5 µm (1185 cm−1 ). Under optimized conditions(see text), the broad, continuous absorption from warm C60 (red, top trace) collapses to a narrow,well-resolved vibrational band from cold C60 in its vibrational ground state (blue, middle trace). A
zeroth order simulation (black, bottom trace) shows the extended rotational fine structure of theP, Q, and R branches. (The simulation temperature is 150 K.)

84

5.3

Rotationally resolved C60 spectroscopy

5.3.1

Rovibrational wavefunctions

The rotational fine structure observed in the cold spectrum encodes a wealth of details about
the quantum structure of C60 . To lowest order, the rotations of C60 are those of a spherical top withtotal angular momentum operator J. These can be described with the symmetric top wavefunctions|J, k, mi, which obey the standard eigenvalue relations
J2 |J, k, mi = J(J + 1)|J, k, mi,(5.1)

Jz |J, k, mi = k|J, k, mi,(5.2)

JZ |J, k, mi = m|J, k, mi,(5.3)

where factors of ~ have been suppressed. J = 0, 1, 2, . . . is the total angular momentum quantum
number. Jz is the body-fixed z-axis projection operator of J, and k = −J, . . . , +J is the body-fixedprojection quantum number. Similarly, JZ is the lab-frame Z-axis projection operator of J, andm = −J, . . . , +J is the corresponding lab-frame projection quantum number.
The triply degenerate T1u vibrational mode that gives rise to the 1185 cm−1 absorption
band can be described with the vibrational normal coordinates qx , qy , and qz . To lowest order,these three vibrational modes are equivalent to a 3D isotropic harmonic oscillator with vibrational
angular momentum `. The components of ` are defined as`x = qy pz − qz py ,(5.4)

with cyclic permutations for `y and `z . px,y,z are the conjugate vibrational momenta of the coordinates qx,y,z . The vibrational quantum states describing this triplet of T1u modes are |n, `, k` i,where n is the total number of vibrational quanta, ` = n, n − 2, n − 4, . . . is the vibrational angularmomentum quantum number, and k` = −`, . . . , +` is the body-fixed projection quantum number

85
for `. The vibrational angular momentum operator obeys the expected eigenvalue relations`2 |n, `, k` i = `(` + 1)|n, `, k` i,(5.5)`z |n, `, k` i = k` |n, `, k` i.

(5.6)

Combined rovibrational wavefunctions can be written in an uncoupled representation bytaking the direct product |J, k, mi|n, `, k` i. These functions are simultaneously eigenfunctions of
J2 , Jz , JZ , `2 , and `z . As might be expected when encountering multiple angular momenta, it isuseful to define a composite vector, the so-called “pure rotational” angular momentum,
R ≡ J − `.

(5.7)

Eigenfunctions of R can be constructed from the uncoupled basis |J, k, mi|n, `, k` i using standard
angular momentum coupling relations [150]. The coupled functions are |R, kR , J, `, n, mi, where R isthe angular momentum quantum number for R and kR = −R, . . . , +R is its body-fixed projection.
The allowed values of R satisfy the usual triangle inequality, |J − `| ≤ R ≤ J + `. The lower stateof the observed infrared band is the vibrational ground state with n = ` = 0. Thus, in the groundstate, R = J. The upper state of the infrared band is the first excited state of the T1u mode, withn = ` = 1. Here, R takes the values J and |J ± 1|.

5.3.2

Zeroth order effective Hamiltonians and energies

The rotational energy levels of each vibrational state are determined by an effective rotational
Hamiltonian. For the ground vibrational state, the simplest zeroth order Hamiltonian is that ofrigid spherical top,Ĥgr = B 00 J2 ,(5.8)

where B 00 is the ground state rotational constant. The ground state wavefunctions |R = J, kR , J, 0, 0, miare eigenfunctions of this Hamiltonian,Ĥgr |R = J, kR , J, 0, 0, mi = Egr (J)|R = J, kR , J, 0, 0, mi,

(5.9)

86
where
Egr (J) = B 00 J(J + 1).

(5.10)

At first glance, this may appear similar to the rotational level structure of a diatomic or linear
molecule, which also has the simple J(J + 1) dependence. However, it is important to note thata spherical top contains an extra rotational degree of freedom and associated degeneracy relative
to a linear rotor. In addition to the usual gJ = (2J + 1) degeneracy of the lab-frame projection
quantum number m, there is an additional gR = (2R + 1) degeneracy factor of the body-fixedprojection quantum number kR . For the ground state with J = R, the total degeneracy factor isgJ gR = (2J + 1)2 .
In the vibrationally excited state, the presence of non-zero vibrational angular momentum
leads to a slightly more complicated effective Hamiltonian,Ĥex = ν0 + B 0 J2 − 2B 0 ζ(J · `),(5.11)

where ν0 is the vibrational band origin and B 0 is the excited state rotational constant. B 0 differs
very slightly from B 00 due to changes in the molecule’s effective moments of inertia upon vibrational
excitation. The last term in Eq. 5.11 accounts for Coriolis coupling between the total angularmomentum J and the vibrational angular momentum `. The strength of this coupling is determinedby the Coriolis constant ζ, which encodes the geometric details of the vibrational mode and cantake values −1 ≤ ζ ≤ 1 [150]. By making use of the definition Eq. 5.7, we can rewrite the coupling
term as−2J · ` = (J − `)2 − J2 − `2
= R2 − J2 − `2 .

(5.12)(Note that J and ` commute with each other.) The coupled wavefunctions for the upper state|R = {J, |J ± 1|}, kR , J, 1, 1, mi are now clearly eigenfunctions of Ĥex with eigenvalues
Eex (R, J) = ν0 + B 0 J(J + 1) + B 0 ζ [R(R + 1) − J(J + 1) − `(` + 1)] .

(5.13)

87
The three possible values of R = J, J ± 1 sort the excited state energies naturally into three
manifolds with distinct J-dependencies(+)

R=J +1(0)

R=J

Eex (J) = EJ + 2B 0 ζJ,
Eex (J) = EJ − 2B 0 ζ,

(5.14)(−)

Eex (J) = EJ − 2B 0 ζ(J + 1), R = J − 1,
where EJ = ν0 + B 0 J(J + 1) is the pure vibrational and rigid rotor contribution to the total
(+)

(0)

(−)

energy. Physically, these three manifolds, which we label T1u , T1u , and T1u , correspond to states
where J and ` are mutually antiparallel, perpendicular, and parallel, respectively. As in the groundvibrational state, each excited state level has a total degeneracy of gJ gR = (2J + 1)(2R + 1).

5.3.3

Spherical top rovibrational selection rules

We now examine the rovibrational selection rules for C60 to predict the expected transitions
between the energy levels derived in the previous section. As with all atoms and molecules thereis the usual ∆J = 0, ±1 total angular momentum selection rule for electric dipole transitions (theonly type we consider here). The high symmetry of spherical tops leads to another approximate,
but typically quite strong, selection rule for the R and kR quantum numbers. If we assume thatthe body-fixed dipole moment can be expanded only linearly in the T1u vibrational coordinates, i.e.
µ~ bf ∝ ~q,(5.15)

where ~q = (qx , qy , qz )T , then it can be shown (see Appendix D) that the following commutation
equalities hold between any body-frame component of ` or J and any lab-frame component of thedipole moment operator µ~ L:[`α , µi ] = [Jα , µi ],(5.16)

where α = x, y, z is a body-frame axis, and i = X, Y, Z is a lab-frame axis. We therefore have
[Rα , µi ] = 0,(5.17)

88
and in particular[R2 , µi ] = 0,(5.18)[Rz , µi ] = 0.

(5.19)

These final two commutation relations mean that in T1u fundamental vibrational bands in sphericaltops, the eigenvalues of R2 and Rz cannot change, yielding the spherical top IR selection rules∆R = ∆kR = 0.

(5.20)

Combined, the R- and J-selection rules result in a highly constrained set of transitions between
the ground state and excited T1u state. For a given lower state |R = J, kR , J, n = ` = 0i (omitting
the m quantum number) there are at most three allowed transitions, one P-type (∆J = −1), one
Q-type (∆J = 0), and one R-type (∆J = +1),
|R = J, kR , J, n = ` = 0i → |R = J, kR , J − 1, n = ` = 1i,|R = J, kR , J, n = ` = 0i → |R = J, kR , J,n = ` = 1i,|R = J, kR , J, n = ` = 0i → |R = J, kR , J + 1, n = ` = 1i.
For the edge case of the lower state |R = J = 0i, there is only a single (R-type) transition to the|R = 0, J = 1i upper state.
These transitions are summarized in the level diagram shown in Fig. 5.6. Note that a given
rotational level of the excited vibrational state is accessed via an IR transition from only a single rotational level of the ground vibrational state. This restriction is due to the spherical top R-selection
rule. In most molecules, which are less symmetrical, the situation is different: a given upper stateis accessed by transitions from multiple lower states. This means that we are unable to determine
ground state combination differences via the zeroth order allowed transitions in spherical tops. Thisin turn reduces the number of parameters in the effective Hamiltonians that are determinable fromthe measured spectrum.

89

T1u(+)

T1u(0)T1u(−)

E(+) = νvib + BJ(J+1) + 2BζJ

E(0) = νvib + BJ(J+1) − 2Bζ

E(−) = νvib + BJ(J+1) − 2Bζ(J+1)

R=J+1

R=J
J

R

13

13

12

11

11

10

10

9

9

8

8

7

7

6

6

5

5
4
3
2
1

4
3
2
1
0

J

R

13

14

12

12

12

13

11

11

11

12

10

10

10
9
8
7
6
5
4
3
2
1
0

11
10
9
8
7
6
5
4
3
2
1

9

9

8
7
6
5
4
3
2
1

8
7
6
5
4
3
2
1

J

ΔJ = -1
ΔR = 0

R

12

13

P

R=J−1

J

Q

ΔJ = 0
ΔR = 0R

ΔJ = +1
ΔR = 0

R

13

13

12

12

11

11

10

10

9

9

8
7
6
5
4
3
2
1
0

8
7
6
5
4
3
2
1
0

Ag vibrational ground state
Figure 5.6: Spherical top energy levels and IR transitions. The lower half of the diagram shows
rotational levels of the ground vibrational state. The upper half shows those for the excited vibra(+)(0)(−)
tional state, sorted into three Coriolis manifolds (T1u , T1u , and T1u ) according to the relativevalues of the R and J quantum numbers. Example P, Q, and R transitions are shown. Solid blacklines indicate rotational levels that have non-zero nuclear spin degeneracy factors, while gray lines
indicate rotational levels that have nuclear spin degeneracy factors of zero (for 12 C60 ).

90
5.3.4

Nuclear spin statistics

An unpurified sample of C60 contains many isotopologues due to the large number (i.e. sixty)
of possible carbon substitution sites. Taking only the 12 C and 13 C nuclear isotopes to have nonnegligible abundance, the 12 C60−n 13 Cn isotopologue fraction isPn = (1 − p)60−n pn




60
,
n

(5.21)

where p = 1.1% is the natural abundance of 13 C [151]. Therefore, only about 52% of natural C60
is pure 12 C60 , 34% is singly substituted 12 C59 13 C, and 11% is doubly substituted 12 C58 13 C2 . The
remaining 3% is more highly substituted. Note that there is only a single isotopomer of 12 C59 13 C
(which has Cs symmetry) because each carbon site of the truncated icosahedron is structurally
equivalent. There are several 12 C58 13 C2 isotopomers owing to the different relative positions of two
substitution sites.
The nuclear spin statistics of each of these isotopologues have important effects on theirrovibrational structure and spectra by modifying the gR degeneracy factor. We consider first the
major 12 C60 isotopologue, which contains sixty identical spin-0 12 C nuclei. A permutation P of any
subset of these bosonic nuclei must leave the total molecular wavefunction |Ψtot i unchanged,
P |Ψtot i = +|Ψtot i.

(5.22)

Pure rotation point group elements, such as E (identity) and Cn (a rotation of 2π/n about someaxis), correspond to permutations of 12 C nuclei and must follow Eq. 5.22. Point group elementsi (body-frame inversion), Sn (improper rotations), and σ (reflections) correspond to permutationinversion operations P ∗ , i.e. they involve a pure permutation P combined with inversion of allcoordinates through some space-fixed origin. These operations must follow
P ∗ |Ψtot i = ±|Ψtot i(5.23)

depending on whether |Ψtot i is a state of total + parity or total − parity, respectively. Inspectionof the Ih character table (Table C.1) shows that the only irreducible representations of Ih that meet

91
these requirements are Ag (for + parity states) and Au (for − parity states). We therefore requirethat |Ψtot i belong to one of these two irreducible representations.
The total molecular wavefunction can be factored into electronic, rotation-vibration, andnuclear spin factors,|Ψtot i = |ψel i|ψrovib i|ψnuc i.

(5.24)

We consider here only the electronic ground state, which is closed-shell and has Ag symmetry.
Each 12 C nucleus is spin-0, and therefore the total nuclear spin wavefunction is trivially Ag as well.
This means that the symmetry of the total molecular wavefunction is determined by that of therovibrational factor, i.e. the |R, kR , J, `, n, mi wavefunctions.
To determine the rovibrational symmetry of the |R, kR , J, `, n, mi wavefunctions, let us review
some fundamental results from the interface of angular momentum theory and group theory. Recall
that the set of all rotations about lab-fixed axes forms a group isomorphic to SO(3) [18, 152]. Welabel this particular version of the rotation group SO(3)L (‘L’ for ‘lab-fixed’). For a given value of
total angular momentum J, the 2J + 1 wavefunctions |R, kR , J, `, n, mi with m = −J, . . . , J span
the D(J) irreducible representation of SO(3)L . All the symmetry properties with respect to SO(3)Lare determined by the values of J and m.
In a similar fashion, we define a second group by the set of all rotations about molecule-fixedaxes and call it SO(3)M (‘M’ for ‘molecule-fixed’). For a fixed orientation of the molecular framein space, there is a one-to-one equivalency between the elements of SO(3)L and SO(3)M . However,this mapping changes if the orientation of the molecule in space changes, and the two groups
have distinct meanings.2 Just as J and m determine the lab-frame symmetry properties, R and kR
determine the molecule-frame symmetry properties. In particular, for a given value of R, the 2R+1
wavefunctions |R, kR , J, `, n, mi with kR = −R, . . . , R span the D(R) irreducible representation ofSO(3)M .
2

For example, if the x, y, z molecule-fixed axes are aligned with the X, Y, Z lab-frame axes, respectively, then arotation of π/2 about z, say, which is an element of SO(3)M , is the same as a rotation of π/2 about Z, an elementof SO(3)L . If the molecule is reoriented in space such that the z-axis is parallel to lab frame X-axis, then that sameelement of SO(3)M is equal to a different element of SO(3)L (i.e. π/2 about X).

92
For a perfectly spherical molecule, SO(3)M would indeed be the true rotational symmetrygroup. C60 is of course not a perfect sphere, and its rotation group is the icosahedral group I, which
is a sub-group of SO(3)M . (The full molecular symmetry group Ih is just a direct product of I with
the inversion group {E, i}.) We can determine the icosahedral symmetries of the |R, kR , J, `, n, miwavefunctions by using the correlations of the irreducible representations of SO(3)M and I [153].
This procedure is carried out in Appendix C and the results summarized in Table C.2. The
additional g/u label for the full Ih group is determined by the character under the i operation,
which comes only from the electronic and vibrational factors (in this case, the parity of `). All
rotational levels of the groud vibrational state are g, and all rotational levels of the excited T1uvibrational state are u.
We can now take all these results together. Bose-Einstein statistics imply that only rovibrational states with Ag/u symmetry are allowed. Table C.2 tells us which |R, kR , J, `, n, mi wavefunctions correlate to Ag/u . For R = 0, 6, 10, 12, 15, 16, 18, 20 − 22, and 24 − 28, we can construct onlya single linear combination of kR states that transform as Ag/u . All other values of R < 30 haveno linear combination of kR states that transforms as Ag/u and thus do not exist. For R > 30, we
can always find at least one linear combination of kR states that transform as Ag/u , but this is far
fewer than the original gR = 2R + 1 degeneracy. In the high-R limit, only 1 in 60 states is allowed.
These extreme nuclear spin statistics are accounted for in Fig. 5.6, where levels that do not exist
in 12 C60 are faded out. For low values of R, “holes” appear in the spectrum where missing lines
would be expected in the absence of spin statistics. For higher values of R, the degeneracy factors
result in characteristic integer-ratio intensity patterns.
A single 13 C substitution destroys the indistinguishability of the remaining fifty-nine 12 C
nuclei, as they can now be “labeled” by their position relative to the unique 13 C nucleus. Therefore,
there are no spin-statistical effects expected in the 12 C59 13 C spectrum. Of the several 12 C58 13 C2
isotopomers, most are asymmetric and do not exhibit any statistical effects. Those in which thetwo 13 C nuclei are in symmetrical positions will have some non-trivial spin statistics (similar to
those of ortho/para H2 O), though not as extreme as for the case of 12 C60 . We do not consider

93
those here.

5.3.5

Rovibrational spectrum simulation

Taking the results of the last several sections, namely the zeroth order rovibrational energies,
selection rules, and spin statistics, leaves the simulated infrared spectrum plotted in black in Fig. 5.5.
The spectrum consists of a strong central Q branch, composed of many overlapped transitions,surrounded by extended P and R branches containing transitions evenly spaced by approximately(B 00 + B 0 )(1 − ζ) ≈ 0.0078 cm−1 . Visual inspection of the survey spectrum shows that the simulated
Q and R branches qualitatively agree with the appearance of the observed spectrum, while the P
branch does not. The following sections will examine each region of the spectrum in more detail.

5.4

Rotational analysis of the 8.5 µm band

5.4.1

R branch

The observed R branch contains a highly uniformly spaced progression of well resolved rovibrational transitions. We observe about three hundred individual features above the absorption
noise baseline, a subset of which are illustrated in Fig. 5.7a. As discussed above, the R(J) transitions correspond to those from the ground vibrational state to the “R = J − 1” manifold of theexcited vibrational state (Fig. 5.6), with ∆J = +1, ∆R = 0 selection rules. Using the zeroth orderenergies, we expect these transitions to occur at frequencies(−)
ν[R(J)] = Eex(J + 1) − Egr (J)= ν0 + B 0 (J + 1)(J + 2) − 2B 0 ζ(J + 2) − B 00 J(J + 1)
= ν0 + (2B̄ + ∆B)(1 − 2ζ) +J[2B̄(1 − ζ) + ∆B(2 − ζ)] +J 2 ∆B,(5.25)

where B̄ = (B 0 + B 00 )/2 is the mean of the ground and excited state rotational constants and∆B = B 0 − B 00  B̄ is their difference. The primary J-dependence is through the linear term,

94a

J = 160

170

180

190

Absorption (arb.)

2001186.101186.151186.20 1186.25 1186.30
Wavenumber (cm-1)

1186.40Q(J)1184.845

1184.855

1184.78 1184.81 1184.84

Wavenumber (cm-1)

Absorption (arb.)c
Absorption (arb.)b1186.35

1182.35 1182.45 1182.55 1183.65

1183.75 1183.85

Wavenumber (cm-1)

Figure 5.7: Detailed views of the 8.5 µm band of C60 . (a) The measured (blue, top) and simulated(black, bottom) R(J) transitions are shown from J = 160 to 200. The characteristic intensity
alternation patterns are a signature of 12 C60 nuclear spin statistics. (b) The band origin region
shows several features, including the 12 C60 Q branch extending from 1184.845 to 1184.855 cm−1 .
The inset shows a fit to Q branch contour using a quartic centrifugal distortion model (see text).
The remaining features are speculatively assigned to 12 C59 13 C. (c) The P branch region contains
weaker and more congested transitions that are in qualitatively disagreement with the zeroth ordersimulation. This is possible evidence of icosahedral fine structure or dark state perturbers.

whose coefficient is dominated by the 2B̄(1 − ζ) contribution. The spacing between adjacent
R(J) transitions should be approximately equal to this quantity. There is also a small quadratic
dependence on J proportional to ∆B.
Approximately three hundred R(J) transitions with J ≈ 60 − 360 were assigned and fitted

95(see Fig. 2.11b) to determine their center frequencies. These frequency positions were then used to
fit the parameters in Eq. 5.25. The frequencies and fit residuals are shown in Fig. 5.8, and the fitted
spectroscopic constants are summarized in Table 5.1. The complete line list and raw spectrum isaLine Position (cm-1)

available in a permanent online data repository [154].

1187
1186
1185

-1

-4

Residual (cm x 10 )

b

1188

100

150

200

250

300

350

100

150

200 250
J

300

350

J

4
2
0
-2
-4

Figure 5.8: Fit of the C60 R branch. (a) The R(J) line positions are shown for J ≈ 60 − 360.
They exhibit a nearly linear dependence on J. (b) The residuals from fitting to Eq. 5.25 have aroot-mean-square value of 7.4 × 10−5 cm−1 and show at least two possible avoided crossings nearJ = 220 and J = 270. The individual error bars are the 1σ line center uncertainties, which have amean value of 2.5 × 10−5 cm−1 .

We confirmed the correct J assignment in two ways. First, the fitted spectroscopic parameters
were used to extrapolate the R(J) line positions to the fictitious “R(−1)” transition. This should
occur at ν[R(−1)] = ν0 − (2B̄ + ∆B)ζ, which is coincident with the zeroth order Q(1) position

96
Table 5.1: Fitted spectroscopic parameters of the C60 R branch with Eq. 5.25. The fit residuals
have a total root-mean-square error of 7.4×10−5 cm−1 . The 1σ uncertainty in the fitted parameters
is shown in parentheses in units of the last digit.
Parameter

Value (cm−1 )ν0 + (2B̄ + ∆B)(1 − 2ζ)
2B̄(1 − ζ) + ∆B(2 − ζ)∆B

1184.86196(3)0.0078300(3)−2.876(6) × 10−7

up to small differences of O(∆B). This agreement is shown in Fig. 5.9. A second confirmation isprovided by the nuclear spin statistics intensity patterns, which have a very strict and characteristic
dependence on J. As can be seen in Fig. 5.7a, the observed intensity patterns are in agreementwith those predicted by the symmetry analysis described in Section 5.3.4. Based on both frequency
and intensity information, we can have confidence in the correct J assignment and therefore theAbsorption (arb.)

validity of the spectroscopic constants in Table 5.1.

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Q(J)

-2

-1

0

1

2 R(J)

1184.85 1184.86 1184.87 1184.88
Wavenumber (cm-1)
Figure 5.9: J assignment via R(J) extrapolation. The extrapolated position of the fictitious R(−1)
transition coincides with the onset of the Q branch near 1184.855 cm−1 , confirming the correct R
branch J assignment.

The rotational fine structure of the R branch spectrum provides detailed structual information

97
about isolated C60 molecules. Our initial analysis does not allow an indepenent determination of
B 00 and ζ due to the restrictive ∆R = 0 spherical top selection rule. However, if we assume a range
2

1 ~of ζ = −0.30 to −0.45 based on theoretical calculations [155], we can estimate B 00 = hc
2I ≈ 0.0027

to 0.0030 cm−1 , where I is the effective moment of inertia of the ground vibrational state. Fora spherical shell of mass m and radius r, the moment of inertia is I = 23 mr2 , yielding a C60
radius in the range of r = 3.4 to 3.6 Å. This is consistent with a previous gas-phase electron
diffraction measurement of 3.557(5) Å, which includes thermal averaging effects that lengthen themeasured radius relative to that of the vibrational ground state [119]. Further analysis of therotational fine structure of 12 C60 (and eventually 12 C59 13 C) will be necessary to constrain B 00 andζ independently and completely determine the gas-phase structural parameters. High resolution
spectra in combination with the large range of observed J values provide highly precise spectroscopic
constants, which will ultimately provide an equally precise structure determination. Our measured
value of ∆B implies that the effective C60 radius increases by only 0.005% upon excitation of theobserved vibrational mode, which is primarily of a surface-tangent C−C bond stretching character.
Furthermore, the narrow IR transition linewidths (mostly Doppler-limited to about 20 MHz) to theexcited vibrational state provide a lower bound for its IVR lifetime of at least 8 ns, despite being
embedded in a dense manifold of dark vibrational states (see Fig. 4.5).

5.4.2

Q branch

The Q branch region of the 8.5 µm band is shown in Fig. 5.7b. This 0.1 cm−1 -wide window
contains several features with complex rotational contours. We assign the blue-most feature, shownin the inset Fig. 5.7b, to 12 C60 . The Q(J) transitions access the central Coriolis manifold of theexcited vibrational state, which consists of rotational states satisfying R = J (Fig. 5.6). According
to the zeroth order energy expressions (Eqs. 5.10,5.14), these Q(J) transitions should occur at

98
frequencies(0)ν[Q(J)] = Eex(J) − Egr (J)= ν0 + B 0 J(J + 1) − 2B 0 ζ − B 00 J(J + 1)
= ν0 − (2B̄ + ∆B)ζ + ∆B J(J + 1).

(5.26)

Due to the small quadratic J-dependence proportional to ∆B, these transitions lie close to one
another, resulting in an unresolved rotational branch contour. According to the zeroth order model,
the Q(J) frequencies should be monotonically red-degraded (as ∆B < 0). However, the observed
contour shows a clear band head associated with an initial red degradation towards lower frequencies
and then a reversal back towards higher frequencies, suggesting the presence of centrifugal distortion(CD) effects, which arise from the flexibility of the carbon skeleton.
The simplest addition to the zeroth order effective Hamiltonians is a quartic scalar CD
operator, −DJ4 , which contributes −DJ 2 (J + 1)2 to the total rovibrational energy. The quartic
CD constant can take different values for the ground and excited vibrational states, D00 and D0
respectively, and their difference contributes a new quartic term to the Q(J) transition frequencies
ν[Q(J)] = ν0 − (2B̄ + ∆B)ζ + ∆B J(J + 1) − ∆D J 2 (J + 1)2 ,(5.27)

where ∆D = D0 − D00 . A rotational temperature Trot and a line width must also be included to fit
the intensity profile of the Q branch contour. Using such a model, a contour can be successfully
fitted to the measured profile (dashed line shown in the inset of Fig. 5.7b). Here, the values of theband origin, ∆B, and the line width are fixed their values determined in the R branch fit, while∆D and Trot are optimized by a non-linear least squares fit to −2.27(1) × 10−12 cm−1 and 103(1) K,
respectively. However, it is unclear how meaningful the fitted value of Trot is. For example, therotational temperature from a fit to the R branch intensities yields Trot = 150(2) K (Fig. 2.11).
This latter value is expected to be more reliable as it is derived from the relative intensities ofmany individually resolved transitions, rather than the total contour of the many unresolved Q(J)transitions. If Trot is held fixed to the R branch-derived value, the fitted value of ∆D from the Q

99branch contour changes slightly to −2.10(3) × 10−12 cm−1 . For a fixed value of ∆B, the value of∆D is largely determined by the width of the Q branch contour, explaining why ∆D is relatively
insensitive to Trot in the fit. Therefore, a ∆D value in the range of −2.0 to −2.5 × 10−12 cm−1
seems like a reasonable conservative estimate.
Canonical rovibrational perturbation theory indicates that the quartic CD constant should
be of the order of magnitude of B 3 /ω 2 , where ω is a vibrational frequency [156]. Taking B ∼0.003 cm−1 and ω ∼ 500 cm−1 yields D ∼ 10−13 cm−1 . To confirm this order of magnitude
prediction, we computed the density functional theory (DFG) harmonic force constants of C60
using the B3LYP functional with a 6-31g basis set, as implemented in the NWChem package [157].
This calculation predicts D = 5.8 × 10−13 cm−1 . The R branch transition frequencies also provide
information on the magnitude of D. Here, the presence of quartic CD manifests in a J 3 dependenceof the line positions. An attempt to include a cubic term in the R branch fit shows that there is
no statistically significant J 3 dependence over the J = 60 − 360 range to within our frequency
measurement uncertainty, suggesting an upper bound of |D| < 1 × 10−11 cm−1 , consistent withthe theoretical estimates. If we assume the DFG prediction to be correct for the ground state
constant D00 , then the Q branch contour implies that the upper state value D0 is several times
larger in magnitude than D00 and of opposite sign. This situation is certainly plausible, but would
imply that local perturbations or higher order CD effects cause a significant change to the effective
Q branch D0 value. Unfortunately, it is difficult to explore these details with only the relatively
crude contour information of the unresolved Q branch. Nonetheless, it is clear from the observed
structure that effects beyond the simple zeroth order rigid spherical top are important.
As can be seen in Fig. 5.7b, there are three features red-shifted from the 12 C60 Q branchcontour. While there are multiple explanations for them, including vibrational hot bands, we
believe the most likely possibility is that they belong to the 12 C59 13 C isotopologue, which occurs
at a 2:3 abundance relative to the pure 12 C isotopologue. A single 13 C substitution breaks thetriple degeneracy of the T1u vibrational mode, resulting in three closely spaced vibrational origins
of comparable intensity [158]. The lack of clear 12 C59 13 C signatures in the R branch region, on

100
the other hand, can be explained by the spectroscopic splittings that result from the inequivalent
moments of inertia. This has a relatively smaller effect on the already unresolved Q branch contours.
We do not attempt a detailed analysis of the putative 12 C59 13 C features here due to the significantly
more complex rotational fine structure expected for this asymmetric isotopologue [158].

5.4.3

P branch

In contrast to the simple structure of the R branch, the P branch exhibits a less regular,
more congested pattern of transitions. Both the number and position of absorption features in theobserved spectrum disagree with the zeroth order simulation, which is based on the spectroscopic
parameters determined from the R branch fit. In the survey spectrum of this band (Fig. 5.5), there
is an obvious asymmetry in the absorption intensity envelope of the P branch relative to the R
branch. Beginning at the band origin near 1184.9 cm−1 and moving down in frequency, some low-J
features do begin to appear, though they are relatively weak and irregular as shown in the righthalf of Fig. 5.7c. As one continues through the P branch, the absorption intensity vanishes before
re-emerging at the high-J tail (left half of Fig. 5.7c). In this last portion, the absorption lines haveapproximately the same spacing as in the R-branch, but are offset from the positions expectedfrom the R branch fit. The apparent decrease in the integrated absorption cross section may infact just be an artifact of the high-pass filtering performed on the cold C60 FTS spectrum (seeSection 2.4.1), which removes any broad, quasi-continuous component. Therefore, one explanation
for the appearance of the P branch is that whatever mechanism is responsible for the splitting of
the low-J transitions continues to intensify in the middle of the branch. Here, the congestion isso severe that the spectrum is continuous and unresolvable, and thus subtracted from the highpass filtered signal. As J increases into the tail of the branch, the splitting mechanism causes thetransitions to re-coalesce with their neighbors (a “rotational echo” of sorts), explaining why they
have a similar spacing as, but a systematic offset from, the zeroth order simulation.

101

5.5

Discussion
One of the clearest results of our initial spectroscopic analysis is the confirmation of thecharacteristic 12 C60 rotational intensity patterns, representing the most extreme example of nuclear
spin statistical effects in a molecular system. This observation illustrates the consequences ofsymmetry in C60 in a fundamentally different way than previous spectroscopic studies. In the latter
case, the icosahedral symmetry of C60 has been established based on observed selection rules andenergetics [113–118, 159]. However, these effects are classical in nature and boil down to certain
matrix element integrals being equivalent to each other or to zero at the level of experimental
sensitivity. In contrast, nuclear spin statistics effects are fundamentally quantum mechanical innature, deriving from the simultaneous indistinguishability of all sixty 12 C nuclei and the perfecticosahedral symmetry of their geometrical arrangement.
The most glaring spectroscopic puzzle uncovered so far is the qualitatively different behaviorof the R and P branches of the 8.5 µm band. The R branch exhibits no deviation from the simplest
possible zeroth order structure, except for what appear to be very weak, local avoided crossings.
This in and of itself is a remarkable fact given the very high values of angular momentum observedin the spectrum, i.e. J ∼ 360. This “textbook” regularity of the R branch structure makes it
difficult to find a consistent explanation for the much more complex structure observed in the P
branch. As shown in Fig. 5.6, the spherical top IR selection rules imply that the P, Q, and R
branch transitions all access mutually exclusive sets of upper states. It is therefore possible that(+)

some effect perturbs the P branch (connected to the T1u manifold) while sparing the R branch(−)(connected to the T1u manifold). Such a perturbation could fall into one of two cases: (i) a local
perturbation caused by resonant interactions with one or a few accidentally degenerate dark state(s)or (ii) non-local perturbations with the entire rovibrational manifold that manifest as high order
Coriolis and centrifugal distortion terms in each vibrational state’s effective Hamiltonian. In case(i), it is difficult to imagine that any state that interacts strongly enough to so thoroughly perturb
(+)

(−)

the T1u manifold does not also do so to the T1u manifold. The total spin-rovibronic symmetry of

102each excited state sub-level is Au , and therefore there is no rigorous symmetry selection rule that
protects one manifold, but not the other.
In case (ii), the icosahedral symmetry severely restricts the form that high order terms cantake in effective rovibrational Hamiltonians [18, 160]. In the ground vibrational state, where ` = 0,
the rotational Hamiltonian only contains powers of the total angular momentum J and its bodyfixed components. The simplest high order contributions are of the form (J2 )n , representing thequartic (n = 2), sextic (n = 3), etc. spherical centrifugal distortion operators.3 These operators
have rank-0 tensor character under SO(3)M , and the Wigner-Eckart theorem prevents them frombreaking the degeneracy of the 2J + 1 body-fixed projections of J [152, 161].
Splittings of rovibrational transitions must come from non-spherical operators (rank k > 0).
We still require that these transform as the totally symmetric representation Ag . The correlation
properties between SO(3)M and Ih (Table C.2) tell us that the first allowed non-spherical operator
has rank-6 character. It takes the form
√
(6)

Ω

=

√ 

11 (6)
7(6)(6)Ω5 − Ω−5 ,
Ω0 +
5
5

(5.28)(k)where Ωq is the q th component of a spherical tensor operator of rank k [162,163]. The eigenvalues ofthis operator, and therefore the expected first-order energy splittings caused by it, are independentof the physical details of the molecule. They are determined only by the algebraic tensor structureof Ω(6) . Figure 5.10 illustrates the patterns of such icosahedral splittings for C60 rotational states.
The Ω(6) term is the simplest non-spherical tensor operator one can consider. In the vibrationally excited state, the presence of non-zero vibrational angular momentum ` introduces many
additional operators resulting in a complicated menagerie of icosahedral splitting patterns. Numerical exploration of the allowed possibilities, however, suggests that none affects the spectrum
in such a way that the R branch is left unperturbed over the entire observed J = 60 − 360 range
while simultaneously accounting for the P branch behavior. Observation of low-J lines, which are
below the comb spectrum noise baseline for J < 60, will undoubtedly assist in sorting out these3

Only even powers of J can appear to preserve time-reversal symmetry [18].

b
Hg
J=8
gR = 17

T2g
Gg

Eigenvalue of Ω(6)/J3(J+1)3

a

103

1.5
1
0.5
0
-0.5

Hg

-1

0

20

40

J

60

80

100

Figure 5.10: Icosahedral splitting patterns of C60 via the Ω(6) operator. (a) Under sphericalsymmetry, the J = 8 rotational level would contain gR = 2J + 1 = 17 degenerate components,
corresponding to the different projections of J in the body-fixed frame. These split into multiplets ofreduced degeneracy according to the symmetry correlations between SO(3)M and Ih (see Table C.2).
(b) The complete eigenvalue structure of Ω(6) as a function of angular momentum quantum number
J. The characteristic clustering patterns are related to the semi-classical rotational energy surfacesof a deformable icosahedral top [162, 163].

complications.

5.6

Conclusions
The high resolution spectroscopy reported in this chapter provides a first glimpse at theunique rovibrational structure and symmetry of C60 . Beyond the detailed spectroscopic aspects
particular to C60 , rovibrational quantum state resolved measurements establish the possiblity ofusing fullerenes as platforms to explore quantum science. For example,• Small fullerenes straddle the border between molecular-scale structure and bulk materials [164]. What can the vibrational and electronic structure of C60 tell us about phononsand electronic transport in extended systems?
• C60 in its rest frame and ground rovibrational state represents a perfect atomic lattice at

104absolute zero temperature. The pure 13 C60 isotopologue is therefore a pristine instance ofa spherical spin-1/2 lattice [165, 166]. Can rovibrational quantum state control of single
13 C

60 molecules be used to study and manipulate the many-body dynamics of a large spin

network?
• Fullerenes present the unique capability of encapsulating atoms and small molecules within
their hollow core [167]. Can such endo-fullerenes be used to functionalize C60 in useful ways,
such as for cooling and trapping? Can such systems be used as models for understanding
the interactions, dynamics, and energy flow between small molecules and an extended bathsystem?
Precision spectroscopy lays the ground work for these and other exciting new avenues in molecularscience.

Chapter 6

Quantum cascade laser spectroscopy of C60

6.1

Introduction
Frequency comb spectroscopy enables the rapid acquisition of high resolution, broadband

measurements with a high detection sensitivity. The large ratio between the comb bandwidth andrepetition rate (ca. 103 –105 :1), however, results in an average power per comb mode that is typicallyquite small, even when accounting for intracavity build up. This often restricts low-power MIR
combs to act as passive spectroscopic probes. Single-frequency continous-wave (cw) lasers, on theother hand, provide complementary capabilities to frequency combs. By concentrating intensity intoa single optical frequency, cw lasers can efficiently pump MIR transitions to transfer population
between rovibrational quantum states. Moreover, one can take advantage of a large variety ofcavity-enhanced cw-laser spectroscopy techniques for ultrasensitive detection [21] . These benefits,of course, come at the cost of reduced spectral bandwidth.
This chapter summarizes our on-going exploration of C60 spectroscopy with the developmentof a MIR quantum cascade laser (QCL)-based spectrometer operating near 8.5 µm. Section 6.2
describes the technical aspects of our QCL device and our initial attempts at frequency stabilization.
Cavity ring-down experiments are then discussed in Section 6.3. In Section 6.4, we move ourattention to true cw spectroscopy via optical feedback stabilization, resulting in the discovery ofsaturation of C60 rovibrational transitions, described in Section 6.5. Finally, Section 6.6 presents
some concluding remarks on future experimental directions.

106

6.2

MIR QCL frequency noise
QCLs are semi-conductor laser devices with a one-dimensional quantum well structure that

permits the lasing wavelength to be engineered over a wide range of the MIR region [168, 169].
We use a distributed feedback (DFB)-QCL [170] (Alpes Lasers) that provides stable single-mode
operation near 8.5 µm with an average output power of up to 30 mW. The laser output is temperature/current tunable from 1183 to 1189 cm−1 , which covers most of the C60 vibrational band
(Fig. 5.5). Unlike traditional diode lasers, frequency tuning of DFB-QCLs is performed by thermal tuning of the refractive index of the active region, which ultimately limits the wavelength
modulation response bandwidth to less than 100 kHz (see below).
Figure 6.1a shows the free-running frequency noise density of the QCL measured using theside of an N2 O absorption peak as a frequency discriminator. The derived optical line width isabout 1 MHz, which is significantly broader than the 50 kHz line width of the optical enhancement
cavity used for the absorption measurements. The effect of these frequency fluctuations as the laser
wavelength is scanned over a cavity resonance can be seen in both the cavity transmission and PDH
reflection signals shown in Fig. 6.1b.
The free-running laser frequency must be stabilized to avoid detrimental relative laser-cavity
noise. We initially attempted to feed back on the laser frequency via current modulation using a
PDH error signal from the spectroscopy cavity. This strategy proved unsuccessful due to the limited
frequency feedback bandwidth of the QCL. As shown in Figs. 6.2a and b, the QCL frequency
modulation response drops by 50% (relative to DC) by 100 kHz, where it has already acquired30◦ of phase delay. This results in simply not enough feedback bandwidth to narrow a 1 MHz
free-running line width to a 50 kHz cavity. We also tried an intermediate approach using a shortpre-stabilization cavity with a line width of 1 MHz, comparable to that of the laser. The QCL could
be successfully locked to the pre-stabilization cavity, demonstrating a significant reduction in its
low-frequency wavelength jitter (Fig. 6.2c). Nevertheless, this feedback loop was still bandwidthlimited to about 50 kHz due to the slow thermal frequency response of the laser. The remaining

a

107

104

7 MHz

Intensity noise baseline
Measured frequency noise
Estimate from current noise

103

102

PDH
Trans.

0.6

Amplitude (arb.)

Frequency noise (Hz/Hz1/2)

b
0.5
0.4
0.3
0.2
0.1
0
103

104

105

Frequency (Hz)-1.5 -1 -0.5

0

0.5

1

1.5

Time (ms)Figure 6.1: Frequency noise of the free-running QCL. (a) The frequency noise spectral density(red) is measured using an N2 O absorption line as a frequency discriminator. The blue horizontal
line indicates the expected frequency noise assuming that it is determined only by the current drivernoise (which, as can be seen, it is not). (b) Cavity transmission and reflected PDH signals during alaser frequency scan. The PDH sidebands at ±3.5 MHz provide a horizontal frequency calibration
scale.

laser frequency noise above 50 kHz results in essentially no change to the short-time line width andtherefore no improvement in the laser-spectroscopy cavity coupling performance.

6.3

Cavity ring-down spectroscopy
Cavity ring-down spectroscopy (CRDS) [171] offers a way to work around QCL frequency

fluctuations. During the ring-down measurement, the laser does not interact with the cavity, andthe signal is therefore immune to relative laser-cavity frequency noise, as well as laser intensity noise.
We configured the QCL setup for CRDS operation as shown by the diagram in Fig. 6.3a. Laserlight is repeatedly injected into the cavity by sweeping either the cavity length or QCL current.
The average cavity-laser offset is stabilized by a swept-cavity lock feedback loop (see Section 2.3.1).
As laser light comes into resonance, it builds up in the cavity and is detected by a transmission
photodiode. A pulse detector circuit monitors when the cavity build-up reaches a given thresholdvalue. At this point, the laser is switched off resonance by a short current square pulse, and theCurrent
Intensity
Frequency

10-1
10

10

0
-60
-120
-180
-240
-300
-360

4

10

6

Current
Intensity
Frequency

10

2

10
Frequency (Hz)
4

10

6

103
Frequency noise (Hz/Hz1/2)

Relative response

100

2Phase delay (deg.)

b

108

c

a

10

2

101

fPI = 1 kHz

500 kHz

100
10-1
103

baseline

104
105
Frequency (Hz)Figure 6.2: QCL current modulation response and PDH frequency stabilization. (a) This plotshows the amplitude response of the QCL current driver output, the QCL optical output intensity,and the QCL optical frequency as a function of the current driver modulation input signal frequency.
All traces have been normalized to their respective DC response. The intensity response is flat out
to 1 MHz, while the frequency response begins to roll off steeply past 10 kHz. (b) The phase
response curves associated with (a). (c) The in-loop QCL frequency noise for the PDH lock to
the pre-stabilization cavity (∆νcav = 1 MHz). Several curves are shown for PI corner frequencies
ranging from 1 to 500 kHz. The feedback loop bandwidth is approximately 50 kHz, which is limitedby the frequency response function measured in (a).

free ring-down curve is acquired by a fast digitizer (Keysight LXI-L4532A) and transferred to aPC.1 The ring-down signal is fitted to a single exponential to extract the cavity lifetime τ , which isconverted to the single-pass cavity loss. A slow current scan is applied to the QCL to acquire the
CRDS spectrum over a small frequency window.
The CRDS spectrum of the Q-branch region of 12 C60 is shown in Fig. 6.3b. The maximum
additional loss above the empty cavity baseline is about 25 ppm, consistent with the previous directabsorption CE-DFCS measurements. The CRDS Allan deviation, shown in Fig. 6.3c, exhibits an
optimal averaging time of 10–20 s before the onset of slow baseline drifts. The corresponding
single-pass absorption sensitivity is 2.5 × 10−8 Hz−1/2 . By comparison, the DFG comb system
1
Ideally, instead of a current pulse, one would use an AOM acting as a fast shutter to block the laser from thecavity. Our commercial 8.5 µm AOM, however, exhibits a shut-off time of > 1.5 µs, which is not short enoughcompared to the 3 µs ring-down time of the cavity.

109

a

Spectroscopy cavity

ISO

mode
matching

Short PZT

cavity error

sweep

QCL

cavitytransmission

Long PZTPI filter

sweep

Lock-In

Current Mod

Pulse
detector

current square pulse

trigger

integrator
reset
slow current scanb

DAQ

Digit.

PCc
C60
Q-branch

12

515
510
505
500
495

laser frequency

490
485

0

200 400 600 800
QCL current offset (arb.)

Allan deviation (ppm)Single pass loss (ppm)

520

1000

10-1

10-2

100

101
102
Averaging lab time (s)Figure 6.3: QCL cavity ring-down spectroscopy (CRDS). (a) Diagram of the CRDS setup. QCLlight is routed through an optical isolator (ISO) and mode matching optics before being injected
into the enhancement cavity. The average laser-cavity frequency offset is stabilized with a sweptcavity feedback lock. Ring-down events are triggered by a pulse detector monitoring the cavity
transmission. The detector circuit sends a square pulse to the QCL current to quickly move thelaser off resonsance with the cavity. The ring-down signals are acquired by a fast digitizer andtransferred to a PC for processing and fitting. A DAQ device controls the slow current scans andresets the swept-cavity lock integrator when the scan needs to hop to the next cavity mode. (b) The
CRDS spectrum of the 12 C60 Q branch. The strongest feature is the Q-branch head, corresponding
to an additional 25 ppm of intracavity loss. The plot spans an optical frequency window of about
1 GHz. (c) Allan deviation curve for the CRDS measurements. The optimal averaging time isabout 10–20 s, after which the measurement becomes sensitive to slow drifts in the cavity lifetime.

110
achieves 3.1 × 10−9 cm−1 Hz−1/2 PSE (Section 2.4.1), which with a 6-cm cold cell translates to
1.9 × 10−8 Hz−1/2 PSE, about 20% better than the CRDS measurement. One drawback of CRDS
is the low duty cycle. The cavity light has a lifetime of τ ≈ 3 − 4 µs and ring-downs can only be
acquired (in our setup) at a rate of about 1 kHz. Furthermore, the peak transmitted power during
the ring-down is only about 1% of the saturation limit of the photodiode, resulting in an integrated
power duty cycle of less than 0.01%. This limits the maximum achievable sensitivity. In principle,
one would hope to reach the shot noise limit, which at the saturation power of our photodiodes(∼200 µW) and cavity finesse (F = 6000) translates to a single-pass absorption sensitivity of
4 × 10−12 Hz−1/2 , a full three orders of magnitude better than the CRDS performance. The point,of course, is that one should not waste any photons, and this can only be achieved with a cw
measurement.

6.4

Optical feedback stabilization QCL spectroscopy
As demonstrated above, the limited frequency modulation bandwidth of the QCL renders

electronic feedback insufficient for a tight cw laser-cavity lock. An alternative solution is to use
optical feedback, which can provide significantly faster feedback bandwidth. Although optical feedback is usually detrimental to the performance and stability of semiconductor lasers, under certain
circumstances it can used advantageously to perform self-locking of the laser to an external cavity [172–175]. A simplified diagram of the optical feedback stabilization (OFS) setup is illustrated
in Fig. 6.4a. The key requirements for successful self-locking to the cavity are to (i) minimize opticalfeedback from light that is non-resonant with the cavity and (ii) control the optical phase of the
feedback. The first requirement is met by replacing our usual two-mirror cavity with a three-mirror
V-shaped cavity. QCL light is coupled into the cavity at the apex mirror. Because the incident
beam is not perperpendicular to the mirror, the non-resonant cavity reflections are deflected away,
while intracavity circulating light is able to retrace the optical path back to the laser. The opticalphase is optimized by controlling the physical distance between the QCL and the cavity at thesub-λ scale. This is accomplished with a combination of a fast, short-throw, bullet-mount piezo

111

a

QCL

current feed-forward

frequency
ramp

crossedpolarizers

fast PZT

non-resonant
reflection

cavity length
cavity transmission

V-cavity

delay length feed-forward

0.6

Δνfb(MHz)

0.4
0.2
0

c
2
0
-2
-10

β=10-4-0.2-0.4
lock range-0.6-10 -5 0 5 10δνfree(MHz)

0

10

6
Cavity transmission (V)bdelay stage

β≈1×10-2

543
β≈3×10-3

2
1
0
0

50
100
150
200
Free-running frequency νfree (MHz)Figure 6.4: QCL optical feedback stabilization (OFS) and self-locking. (a) A simplified diagramof the QCL-OFS setup. The V-cavity geometry permits resonant cavity light to return to the laser,
while deflecting non-resonant reflections. The inset shows the custom V-shaped vacuum mountthat holds the two closer-spaced cavity mirrors. The feedback power ratio is controlled with a pairof crossed polarizers, which act as a variable attenuator. A fast PZT-mounted mirror and a slow
delay stage together provide control of the optical feedback phase. When the OFS lock is active, afrequency scan is performed by ramping the cavity length, with simultaneous feed-forward signals
sent to the QCL current and delay-length piezos. (b) The relationship between the free-runninglaser-cavity detuning δνfree and the output frequency detuning ∆νfb in the presence of opticalfeedback with a power ratio of β = 10−4 . The OFS locking range extends between the two turning
points of the curve. The dashed line corresponds to the ∆νfb = δνfree turning curve without opticalfeedback. The inset shows a zoomed-out view. (c) The measured cavity transmission during a
QCL current scan for several values of β in the range 3 × 10−3 to 1 × 10−2 . The horizontal axis iscalibrated to the free-running frequency νfree , with each curve offset such that the capture-point is
at νfree = 0.

112
mirror [176] and a slow, long-throw, piezo-actuated delay stage.
One of the key parameters controlling OFS is the ratio β = Pfb /Ptot between the power ofthe feedback light Pfb recoupled into the QCL laser mode and the total laser output power Ptot .
Figure 6.4b shows the relationship between the free-running optical frequency of the laser νfree andthe actual output frequency νfb in the presence of optical feedback, simulated for β = 10−4 andassuming that the QCL laser cavity has an optical length of n` = 5 mm and a finesse of 5 [173]. (Wedo not know the actual parameters of the QCL laser cavity. These are simply reasonable guesses.)
Over the OFS locking range, which is ∝ β 1/2 , the laser frequency is pulled to center of the cavity
resonance and free-running frequency fluctuations are suppressed. The self-locking effect is easily
observed by scanning the QCL current (i.e. the free-running frequency) and monitoring the cavity
transmission signal. This is shown in Fig. 6.4c for several values of β (tuned by adjusting a pairof crossed polarizers between the laser and external cavity). As νfree enters the capture range of
the OFS, it quickly locks to the cavity resonance with a sharp turn-on in the cavity transmission.
As νfree continues to scan through the lock range, it moves from one side of the cavity resonanceto the other, passing through a transmission maximum when the actual detuning ∆νfb relative tothe cavity resonance is zero. Once νfree reaches the end of the lock range, it quickly falls out of
resonance and the cavity transmission drops to zero. At our maximum achievable feedback power
ratio, which we estimate to be β = 10−2 , the locking range is about 190 MHz, i.e. 95% of thecavity FSR = 200 MHz. This performance is similar to other recent demonstrations of OFS with
QCLs [177–180].
We actively stabilize the optical feedback phase by applying a f = 20 − 35 kHz dither toνfree via QCL current modulation. The cavity transmission signal is then demodulated to isolate
the 1f component, which is filtered and fed back onto the fast delay PZT, maintaining optimal
OFS conditions. The residual intensity noise (RIN) spectrum of the cavity transmission with anactive OFS lock is shown in Fig. 6.5a, along with the RIN spectrum of the incident cavity light(“Reference”) with and without optical feedback. At frequencies below 1 kHz, residual lasercavity frequency fluctuations add intensity noise to the cavity transmission, but at frequencies

113
between 1 kHz and the cavity half-width of 35 kHz, the incident and transmitted cavity light havecomparable amplitude noise. It is also evident that as the optical feedback corrects for free-runningfrequency fluctuations, it imprints intensity noise onto the laser output. DC cavity transmission has
a baseline RIN of about 2 × 10−4 Hz−1/2 , corresponding to a single-pass absorption sensitivity of
1×10−7 Hz−1/2 (the V-cavity has a finesse of about 3000). By adding another stage of modulation to
perform wavelength-modulation spectroscopy, the detection band could be moved to the 1–10 kHz
window, where the detection sensitivity would be as low as 5 × 10−10 Hz−1/2 , about a factor of 40×
better than the CE-DFCS or QCL-CRDS sensitivities.

a

b
1
10

Signal (norm.)

RIN (Hz-1/2)

-3

10-4
10

-5

10-6
10

-7

100

Cavity trans.
Reference
Reference (no OF)

101

102 103 104
Frequency (Hz)Cavitytrans.

0.8

70 MHz0.6
0.4

Cavity length
0.66

0.2
0
105

0.62

Delay length

0

0.58
0.116

0.120

0.05 0.1 0.15 0.2 0.25
Time (s)Figure 6.5: OFS-locked intensity noise and frequency ramp. (a) RIN spectra are plotted for thecavity transmission (blue), reference incident cavity light (red), and incident cavity light without
optical feedback (black). The cavity roll-off (half-width) frequency is 35 kHz. The narrow spikes
are from a 20 kHz current dither used for the delay-length stabilization servo. (b) An OFSlock frequency ramp containing an N2 O absorption line. The cavity transmission (red) shows the
70 MHz-wide Doppler profile with a narrow 700 kHz Lamp dip (inset). The cavity length ramp
(blue) and delay-length feed-forward signal (black) are also plotted.

A slow ramp is applied to the optical cavity length to perform a frequency scan. The OFSlock forces the laser frequency to follow the moving cavity resonance, while the 1f dither lock
maintains the optimal feedback phase. Feed-forward signals are applied to both the QCL current
and delay-length piezos to keep their respective locks within their operation range. The maximum

114
cavity ramp range is about 220 MHz (limited by the throw of the slow delay-length piezo). Thisis larger than the 200 MHz FSR of the cavity, so by hopping the laser between sequential cavity
resonances, a wider spectral window can be continuously stitched together. Figure 6.5b shows asingle OFS-locked cavity ramp containing a strong N2 O absorption peak. The cavity transmission
signal, the cavity length ramp, and the delay length feed-forward signal are plotted. The transitionis partially saturated revealing a narrow Lamb dip feature, which has a FWHM of 700 kHz, limitedby transit time and power broadening.

6.5

Saturation of C60 rovibrational transitions
Upon the demonstration of successful frequency ramps with N2 O, we immediately searched for

C60 transitions. Spectrally broad absorption, observed as attenuation of the DC cavity transmission,
was detected and attributed to the warm C60 “blob” (see Fig. 5.5). However, there were no
detectable signatures of narrow transitions from cold C60 , not even from the strong Q-branchhead. After the possibility of frequency calibration errors was quickly ruled out, we were left tosuspect that the C60 molecules were not being properly cooled, which could explain why only broad
background absorption was detected. We spent several weeks investigating this issue, focusing on
the C60 oven and buffer gas annular slit inlet. These components were modified and re-built multiple
times. The buffer gas flow rates and pumping speeds were also carefully recalibrated. None of theseactions resulted in any change in the observed signal.
After eliminating all other possibilities, we considered the prospect that cold C60 transitions
might be saturated, given that our typical OFS operating conditions result in an intracavity circulating power of up to 700 mW. Indeed, by attenuating the QCL light incident on the cavity,
narrow absorption resonances were recovered. Figures 6.6a and b show an R(J) transition near1186.2 cm−1 . Increasing the intracavity circulating power from Pcav = 40 mW to 160 mW resultsin the complete disappearance of the absorption signal. The C60 Q branch is shown in Figs. 6.6c
and d. At Pcav = 15 mW, the expected Q-branch contour begins to take shape, but this structure
already begins to wash out at Pcav = 30 mW. The effective single-pass loss for the Q-branch headCavity transmission (norm.)

c

1.010

Pcav = 40 mW

1.005
1.000
0.995
0.995

R(J)

0.985

0

4
1
2
3
Cavity displacement (μm)

Cavity transmission (norm.)

b

115
1.010

Pcav = 160 mW

1.005
1.000
0.995
0.995
0.985

0

4
1
2
3
Cavity displacement (μm)

d

1.03

Pcav = 15 mW

1.02
1.01

Q head

1.00
0.99
0.98
0

4
1
2
3
Cavity displacement (μm)

Cavity transmission (norm.)

Cavity transmission (norm.)a1.02 Pcav = 30 mW
1.00
0.98

0

4
1
2
3
Cavity displacement (μm)

Figure 6.6: Saturation of cold C60 rovibrational transitions. (a) An OFS-locked cavity scan with
Pcav = 40 mW shows absorption from an R(J ≈ 170) transition near 1186.2 cm−1 . The two traces
are from the upward (red) and downward (blue) portions of the cavity length ramp. (b) Same as(a) with Pcav = 160 mW. (c) A scan near 1184.85 cm−1 with Pcav = 15 mW shows the partiallyresolved Q-branch head. (d) Same as (c) with Pcav = 30 mW, showing a washed out Q-branchcontour.

at Pcav = 15 mW is only about half the expected magnitude, suggesting that it is still partiallysaturated.
Although the observation of C60 saturation came as a surprise, in hindsight it might have been
anticipated. The integrated intensity of the 8.5 µm band is about 9.4 km mol−1 [131], corresponding
to a vibrational transition dipole of 0.05 D. The CE-DFCS data suggests that the IR transitions
are collisionally broadened to about 20 MHz implying a saturation intensity of ∼102 W/cm2 .
The maximum Pcav = 700 mW with a cavity waist of w0 = 1.5 mm2 yields a peak intensity

116
of I = 2Pcav /πw02 ≈ 20 W/cm2 . This back-of-the-envelope calculation suggests that saturation
effects should not have been entirely unexpected with cw OFS-locking. On the other hand, thepeak intensities in the CRDS measurements are 25–50× lower, and in the CE-DFCS measurements
about 104 × lower (per comb mode), explaining why no saturation effects are observed in thoseexperiments.

6.6

Conclusions and future directions
The discovery of C60 saturation ultimately opens up new experimental possibilities. For

example, efficient vibrational population transfer is a pre-requisite for hole-burning and doubleresonance techniques. In the near-term, however, it represents a significant experimental barrier
for intracavity absorption measurements. Engineering low intracavity powers with an OFS lock ispractically difficult. The OFS process fundamentally relies on laser power being coupled into theexternal cavity and then back into the laser cavity. Attenuating light before it enters the external
cavity or decreasing the build-up factor of the cavity compromises OFS performance. One way
to overcome this challenge is to take advantage of sideband spectroscopy. The ideal technique
is probably NICE-OHMS [181], which requires frequency sidebands at a modulation frequency
equal to the cavity FSR (200 MHz). Unfortunately, the limited frequency response bandwidth of
QCLs makes this impractical to accomplish with direct current modulation, and external phasemodulators such as EOMs are not widely available at 8.5 µm. A compromise approach is to use
AM sidebands. While this strategy sacrifices various advantages of FM spectroscopy, it is relatively
straightforward to current modulate a QCL at 200 MHz. Only a small fraction of the total power(probably < 1%) needs to be placed in the AM sidebands. The high intensity carrier interacts
with one cavity mode generating a stable OFS lock, while AM sidebands on either side couple toadjacent cavity modes to perform a linear absorption measurement below the saturation intensity.
This approach is currently being implemented in the Ye lab and promises to help us take the next
step in C60 QCL spectroscopy.

Chapter 7

High accuracy rovibrational calculations of small molecules

7.1

Introduction
Gas-phase, frequency domain spectroscopy probes the molecular eigenstates associated withelectronic, vibrational, and rotational motion. The interpretation, assignment, and analysis of such
spectra, ranging from the microwave to optical regions, is aided enormously by accurate ab initio
calculations. Reliable predictions of anharmonic vibrational frequencies and rotational constants
are only the starting point: treatment of higher order effects such as centrifugal distortion, Coriolis
interactions, tunneling splittings, and various types of perturbations is often necessary to successfully describe a high resolution spectrum [182]. Certain quantities of interest are intractable or
impossible to extract directly from experimental spectra alone and require theoretical input. For
example, calculated nuclear zero-point motion effects are crucial to equilibrium structure determination [183, 184] and high-level thermochemistry protocols [185–187].
Ab initio approaches are often based on solving the time-independent Schrödinger equation(SE) for the total molecular Hamiltonian,
Hmol (re , Rn ) = Helec (re ; Rn ) + [Tnuc (Rn ) + Unuc (Rn )] .
Hmol contains an electronic part Helec (re ; Rn ), which depends on both the electronic re and nuclear Rn coordinates, and a nuclear part that includes the nuclear kinetic energy Tnuc (Rn ) andinternuclear Coulomb repulsion Unuc (Rn ). The Born-Oppenheimer approximation [188] splits themolecular problem into two halves by adiabatically separating the electronic and nuclear degrees

118
of freedom. In this framework, one first solves an electronic SE,
Helec (re ; Rn )|ψe (re ; Rn )i = Ee (Rn )|ψe (re ; Rn )i,(7.1)

which depends parametrically on Rn . The solution to this problem constitutes the extensive fieldof electronic structure theory and quantum chemistry.
We focus here on the second step, namely the nuclear motion SE,
Hnuc |ψnuc (Rn )i = [Tnuc + V (Rn )] |ψnuc (Rn )i = Enuc |ψnuc (Rn )i,(7.2)

which describes rovibrational dynamics occuring on an effective potential energy surface (PES)
V (Rn ) = Ee (Rn ) + Unuc (Rn ). In polyatomic molecules, this is a high-dimensional, many-bodyproblem. Its solution is especially complex for non-rigid, floppy systems that exhibit extreme anharmonicity and large-amplitude nuclear motion. Many molecules of fundamental physical andchemical interest, including reactive transients, radicals and ions, and exotic astrochemical species,
fall into this category.1 We have already encountered one example earlier in Chapter 3 on theinfrared spectrocopy of nitromethane. Automated “blackbox” tools, such as those based on standard vibrational perturbation theory [70, 191], fail – often catastrophically so – for these types ofsystems [51]. The development of accurate and efficient alternative methods to tackle them remainsa challenge.
This chapter discusses the computational methods we use to understand and interpret therovibronic spectra of small- to medium-sized polyatomic molecules. Section 7.2 describes thesetools, which include implementations of relatively well established techniques along with newly
developed approaches and extensions. In Section 7.3, we explore several applications to floppy,
anharmonic molecules including the methide anion [192], disilicon carbide [193–195], hydrogen
peroxide [196, 197], and gauche-1,3-butadiene [198]. Each of these examples is framed by the goalof interpreting and quantitatively understanding measured spectra. Indeed, most of these projects
are the result of close collaboration with experimental spectroscopists. Ultimately, we hope todemonstrate the important benefits of close contact between experiment and theory [182].
1

An excellent survey of this topic can be found in two recent special issues of the Journal of Molecular Spectroscopy
organized in honor of the late Jon Hougen [189, 190], who made numerous fundamental contributions to this field.

119

7.2

Methods
In this section, we give an overview of the construction of nuclear motion Hamiltonians andthe determination of their energy eigenvalues and eigenfunctions. The Hamiltonian has two parts:
the kinetic energy operator (KEO) and the potential energy surface (PES). The KEO is determined
only by the choice of coordinates used to describe the nuclear degrees of freedom, while the PES isbuilt from fitting high-dimensional functions to data sets of quantum chemical electronic energies.
The strategies employed to determine the spectrum of the Hamiltonian include both variational andperturbative techniques. The former involves iterative eigensolvers combined with sparse directproduct grid representations of the Hamiltonian. For systems that are too large to treat with thisapproach, we have developed a version of rovibrational perturbation theory based on curvilinear
vibrational mean-field theory. The software used to perform each of these various parts of a nuclear
motion calculation has been organized into a freely available package called Nitrogen [199].

7.2.1

Direct-product grid variational calculations with iterative eigensolvers

The most general way to compute the spectrum of a rovibrational Hamiltonian is via variational methods [200, 201]. In this approach, wavefunctions are represented as linear combinationsof a set of basis functions. The matrix representation of the Hamiltonian operator in this basis set
is diagonalized to compute its eigenvalues (energies) and eigenvectors (wavefunctions). By systematically increasing the size of the basis set, numerically exact energies and wavefunctions can beconverged. The calculation is limited only by the size of the Hamiltonian matrix one can build anddiagonalize.
The most straightforward basis set for multi-dimensional wavefunctions is a direct product of
one-dimensional basis functions for each degree of freedom [201, 202]. These take the general form
(1)

(2)

(n)

Φi1 i2 ···in (~q) = φi1 (q1 )φi2 (q2 ) · · · φin (qn ),(7.3)

where each vibrational coordinate qk , k = 1 . . . n, has its own independent set of 1-D basis func(k)tions φik (qk ). (We consider rotational motion later.) The desired many-body wavefunctions are120
represented as an n-index sum over the direct product basis,
Ψ(~q) =

XX
i1

···

i2

X

ci1 i2 ···in Φi1 i2 ···in (~q),

(7.4)

in

where the coefficients ci1 i2 ···in are equal to the components of the eigenvectors of the Hamiltonian(k)matrix. As the number of basis functions φik for each coordinate approaches completeness in 1-D,
so does the many-dimensional direct-product basis.
The challenge with this approach is that the number of direct-product basis functions grows
exponentially with the number of coordinates. For example, each 1-D basis set typically needs aminimum of 10 functions for accurate results, yielding a total of 10n direct-product basis functions.
A five-atom molecule, such as CH4 , has n = 3N − 6 = 9 internal coordinates, so its minimum
basis set contains 109 functions. Most importantly, the Hamiltonian operator in this basis is a
109 × 109 matrix, which would require about 1010 gigabytes of memory to store on a computer.
Even if this were possible, direct diagonalization methods of m × m matrices scale as O(m3 )and are impractical for such large matrices. Furthermore, only the lowest energy eigenvalues are
typically desired, which represent a very small fraction of the complete eigenvalue spectrum of the
Hamiltonian matrix. Calculating the entire set of eigenvalues is therefore a vast waste of resources.
The problem of calculating and storing the Hamiltonian matrix is avoided by using iterative
diagonalization methods [203–205], in particular the Lanczos algorithm [206], which is applicableto Hermitian matrices. Iterative methods focus the computational effort on calculating the lowest
energy eigenvalues and not the entire unnecessary spectrum. They also require only that one
calculate the Hamiltonian matrix-vector product, w = Hv, where v is an arbitrary vector of basis
function coefficients. The key point is that matrix-vector products (MVPs) can be calculated
without the explicit construction of the matrix H itself. This drastically reduces the memory
requirements, making the direct-product basis approach viable. We forgo a detailed discussion of
iterative diagonalization algorithms here. We most often employ the so-called thick-restart Lanczos
method, details of which can be found in Ref. [67].
In general, the efficiency of iterative techniques is determined by the cost of MVPs with

121

q

Figure 7.1: Discrete-variable representation (DVR) basis functions. The two DVR functions (redsolid line and blue dashed line) are associated with a one-dimensional grid of twenty-five points(black dots). Each DVR function is highly localized about its respective grid point and has a nodeat all other grid points.

the Hamiltonian. Discrete-variable representation (DVR) basis sets [68, 207–209] are an essential
tool for reducing the computational expense of this step. DVRs are coordinate-representationbasis functions that are highly localized about discrete values (i.e. grid points) of the coordinate
variable [68] (see Fig. 7.1). One definition of DVRs is the eigenfunctions of the coordinate operator
in a finite basis set of orthogonal polynomials, and therefore they may be thought of as finite basis
approximations of Dirac delta functions. Indeed, a fundamental property of DVRs is that matrix
representations of scalar functions are approximately diagonal,Z

dq φ∗i (q)f (q)φj (q) ≈ δij f (qi ),(7.5)

where φi (φj ) is the DVR basis function centered on grid point qi (qj ). This diagonal approximationis related to Gaussian quadrature of integrals, and the associated error rapidly decreases as thenumber of DVR basis functions (i.e. the number of grid points) increases [68].
Using DVR basis sets with iterative diagonalization is advantageous for two reasons. First,
the diagonal property results in a highly sparse Hamiltonian matrix with a small fraction of non-zero
matrix elements. Second, these matrix elements can be calculated without performing expensive integrals. The only information needed is the values of the PES and KEO coefficients at the DVR grid

122
points. These properties make MVPs and iterative diagonalization simple and efficient. Additional
techniques such as pre-conditioned spectral transformations [205, 210, 211] and symmetry-adapted
Lanczos [212] can further improve the computational efficiency.
The above discussion focuses on the pure vibrational problem (J = 0). Rotations are easily
included by attaching a set of symmetric top rotational basis functions to each DVR grid point.
Other than modifying the Hamiltonian MVP routine to account for rotational and rovibrationalterms in the KEO, the iterative diagonalization algorithm is essentially unchanged. Both the size ofthe combined rovibrational direct-product basis and the cost of MVPs are proportional to 2J + 1,
the number of symmetric top functions for a given value of J.

7.2.2

Rovibrational perturbation theory based on curvilinear vibrational meanfield theory

Although iterative techniques and DVRs make direct-product-basis variational calculations
orders of magnitude more efficient than direct matrix construction and diagonalization, their coststill scales exponentially with the number of coordinates, limiting them to small molecules of
up to four or five atoms. (We note that recent advances [213–215] have substantially pushed
this dimensionality limit for some cases.) For medium-sized molecules, it it advantageous to use
approximate methods that are even more efficient while retaining enough accuracy to be useful forspectroscopy.
One of the most popular approximate methods is second-order vibrational perturbation theory (VPT2) [69, 70, 191]. The zeroth-order description in VPT2 is a set of uncoupled harmonic
oscillator vibrations and rigid-top rotations. Perturbative corrections from anharmonicity androtation-vibration coupling are used to generate anharmonic vibrational frequencies and effective
rotational parameters. Although VPT2 is based on the formally exact Watson Hamiltonian [71], it
relies on rectilinear normal coordinates and single-reference Eckart embedding [216]. This renders
it best suited for molecules that undergo small harmonic displacements from a single, well defined
equilibrium configuration. VPT2 thus fails for floppy or highly anharmonic molecules [51, 191].

123
This section introduces a more general rovibrational perturbative method that goes beyond
the Watson Hamiltonian/VPT2 framework. Non-rigid systems demand the use of curvilinear coordinates, general representations of the PES and KEO beyond power series/Taylor expansions,and more flexible quasi-Eckart frame embeddings not tied to a single reference geometry (whichin cases like nitromethane (Chapter 3) may not even be well defined). Prior work on perturbative [217, 218] and hybrid variational-perturbative [219, 220] approaches have addressed some ofthese issues. We base our approach on vibrational self-consistent field theory (VSCF) [72–74, 221]and second-order vibrational Møller–Plesset perturbation theory (VMP2) [76,222]. VMP2 correctsa zeroth-order VSCF mean-field wavefunction for vibrational correlation effects using RayleighSchrödinger perturbation theory, in analogy to electronic structure MP2 [75]. While VSCF-based
methods were originally developed for the rectilinear Watson Hamiltonian, they have since been
applied to a variety of curvilinear coordinate systems [223–230]. The contribution of the present
work is to extend the curvilinear VSCF/VMP2 approach by accounting for the rotational androtation-vibration coupling terms of the nuclear motion Hamiltonian [51]. Rotational and rovibrational effective Hamiltonians are calculated by applying a second-order contact, or Van Vleck,transformation [231] to the zeroth order VSCF Hamiltonian.
The zeroth-order VSCF wavefunction ansatz is a Hartree product for the n vibrational coordinates,Ψ0 (~q) = ψ1 (q1 )ψ2 (q2 ) · · · ψn (qn ),(7.6)

or in ket notation,|Ψ0 i = |1i|2i · · · |ni =

Y

|ki.

(7.7)

k

Variational optimization of |Ψ0 i leads to a 1-D Schrödinger equation for each degree of freedom,ĥk |ki = εk |ki,

(7.8)

124
where





Y
Y
ĥk =  hl| Hv  |li
l6=k

(7.9)

l6=k

is the one-body Hamiltonian for vibrational coordinate k, computed by averaging the full vibrational
Hamiltonian Hv over the other degrees of freedom l 6= k. As in other mean-field theories, this setof equations is solved iteratively until self-consistency is reached. We use underlying DVR basis
sets for each of these 1-D problems.
Second-order vibrational Møller–Plesset perturbation theory (VMP2) is used to correct the
VSCF wavefunction |Ψ0 i. The VMP2 energy is computed using standard Rayleigh-Schrödinger
perturbative corrections,
(0)

(2)

(7.10)

E0 = hΨ0 |Hv |Ψ0 i,(7.11)

E0VMP2 = E0 + E0 ,
(0)
(2)

E0 =

X

|hΨ0 |Hv |vi|2
(0)

(0)

,

(7.12)|vi6=|Ψ0 i E0 − Ev

where the second-order sum is over virtual excitation wavefunctions |vi formed from higher-energy
eigenfunctions of the one-body problems, Eq. 7.8. (These form a direct-product basis set that completely spans the one-body functional space.) The zeroth order wavefunction |Ψ0 i is not necessarily
the ground state. Excited vibrational states can be targeted equally as well.
This approach provides approximate wavefunctions and energies for the pure vibrational
Hamiltonian Hv = Tv +V , where Tv is the vibrational KEO and V is the PES. To calculate rovibrational states with J > 0, one must consider the full nuclear motion Hamiltonian H = Hv + Tr + Trv ,which contains additional KEO terms for rotational motion (Tr ) and rotation-vibration coupling(Trv ). These terms are typically several orders of magnitude smaller than the vibrational part of
the Hamiltonian and therefore may be folded into the perturbative treatment. This procedure isformally carried out with a contact or Van Vleck transformation [231], with the final product being
a rotational or rovibrational effective Hamiltonian for the state(s) of interest [232, 233].
We provide only a brief overview here of contact transformations. (Further details can be

125found, for example, in Appendix C of Ref. [234].) We first define the zeroth-order Hamiltonian
using the VSCF wavefunctions as
H (0) =

X

Ev(0) |vihv|,(7.13)

v(0)

where Ev = hv|Hv |vi are the zeroth-order energies. Letting ∆H = Hv − H (0) , we partition H as
H = H (0) + ∆H + Trv + Tr
= H (0) + λH 0 ,(7.14)(7.15)

where the perturbation term H 0 = ∆H + Trv + Tr appears with an order-sorting parameter λ, which
is formally equal to 1.
Consider the Hamiltonian matrix representation with zeroth-order rovibrational wavefunctions |vi|ri. |vi is a VSCF wavefunction (|Ψ0 i or an excited virtual state), while |ri is somerotational basis function (usually a symmetric top wavefunction, but the actual choice does notmatter). A schematic drawing of H in this basis is shown in Fig. 7.2. The Hamiltonian is organized
into vibrational blocks hv|H|vi along the diagonal. Off-diagonal coupling from H 0 between different
vibrational blocks is of O(λ). A contact transformation is a unitary transformation of the original
Hamiltonian to form a modified Hamiltonian H̃ of the form,
H̃ = e−iλS HeiλS ,(7.16)

where S is a Hermitian operator. S is chosen to make H̃ approximately block-diagonal by eliminating off-diagonal coupling up through some given order in λ. Here, we go to second-order such thatthe off-diagonal coupling is reduced to O(λ2 ). The new diagonal block hv|H̃|vi is also modified
from the original hv|H|vi by changes of O(λ2 ). At this point, the remaining small off-diagonal
coupling is neglected, and the isolated block for the vibrational state of interest is numericallydiagonalized to generate rovibrational energy eigenvalues and wavefunctions. The size of each diagonal vibrational block is only (2J + 1) × (2J + 1), so its direct diagonalization is inexpensive.
In practice, the transformation operator S is never explicitly constructed. Simple equations relate

126
the matrix elements of the effective Hamiltonian H̃ in terms of those of the original Hamiltonian
H (see Refs. [51, 233, 234]).

ab

Figure 7.2: Contact transformations of the rovibrational Hamiltonian. (a) A unitary transformation
with exp(iλS) eliminates off-diagonal coupling through λ and transforms the diagonal block (H)vv
into the effective Hamiltonian (H̃)vv . (b) Resonantly interacting vibrational states are grouped together into a multi-state block. The contact transformation generates a larger effective Hamiltonianthat accurately treats the non-perturbative interactions between the resonant states.

This basic procedure works well when individual vibrational states are well-separated in energy and only interact weakly with each other. Large-amplitude motion, however, often results in
low-lying vibrationally excited states that have energy differences on the same scale as rotational
excitations. This leads to non-perturbative, resonant rotation-vibration interactions. An excellent

127
example is the rotation-torsion manifold in nitromethane (Chapter 3). In this case, the resonantly interacting vibrational states can be grouped together into a multi-state block. The contact
transformation eliminates coupling between this enlarged block and the rest of the rovibrational
manifold, resulting in a multi-state effective Hamiltonian (Fig. 7.2b). This effective Hamiltonian
accurately treats the non-perturbative interactions among the resonant vibrational states. We will
see another example of the need for multi-state effective Hamiltonians in the section below abouttunneling gauche-butadiene.

7.2.3

Kinetic energy operators

We now consider the detailed form of the curvilinear Hamiltonians used in the variational andperturbative methods discussed above. Relative to rectilinear coordinates, which form the basisof the Eckart-Watson Hamiltonian [71], curvilinear coordinates provide an improved treatment ofmolecular vibrations and large-amplitude motion. These benefits come at the cost of a significantly
more complex analytical form of the nuclear motion KEO [235–238]. However, the use of DVR basis
functions, which require only numerical evaluation of the KEO instead of close-formed analyticalexpressions, obviates these drawbacks [239–242].
We define a curvilinear coordinate system for an N -atom molecule by specifying the Cartesian
positions ~xi of each nucleus i = 1 . . . N in a body-fixed, center-of-mass frame as a function of ninternal vibrational coordinates qk , k = 1 . . . n. For full-dimensional problems, n = 3N − 6, though
one can also consider reduced-dimension models that use only a subset n < 3N − 6 of the available
degrees of freedom. The n internal coordinates are collectively denoted as ~q. The body-frame
Cartesian position functions ~xi (~q) generate an associated metric tensor g, an (n + 3) × (n + 3)

128
symmetric matrix with elements given by
gkl =

gαl =

N
X
i
N
X

mi ∂k ~xi · ∂l ~xi ,(7.17)

mi (êα × ~xi ) · ∂l ~xi ,(7.18)

mi (êα × ~xi ) · (êβ × ~xi ).

(7.19)

i

gαβ =

N
X
i

Here and throughout, k, l, m, etc. are vibrational indices taking values 1 . . . n, and α, β, γ, etc.
are rotational indices, taking values x, y, and z (i.e. the three body-fixed axes). ∂k is shorthandfor ∂/∂qk , êα is a unit vector parallel to the body-fixed α axis, and mi is the mass of the ith atom.
We also define the determinant g = det(g) and inverse G = g−1 of the metric tensor.
Following Podolsky [243], the rovibrational KEO can then be written as
T = Tv + Tr + Trv ,(7.20)

which includes a vibrational term,
Tv =~2 X −1/4 †g∂k Gkl g 1/2 ∂l g −1/4 ,2(7.21)


−~2 X
iJα iJβ
Gαβ
,
,
4
~
~ +

(7.22)

kla rotational term,
Tr =

αβ

and a rotation-vibration coupling term,
Trv =

iJγ
−~2 X
(−∂k† Gkγ + Gkγ ∂k ),2~

(7.23)

kγ

where the operator Jα is the body-frame projection of the total angular momentum J along the
α-axis. (The Hermitian conjugate derivatives ∂k† act to the left.) Matrix elements of this KEO arecalculated using the so-called Dirac volume element dq1 dq2 · · · dqn [243, 244] instead of the usual
Euclidean volume element that includes the Jacobian determinant of the curvilinear coordinate
system, i.e. the normalization convention is
Z

Z
···

dq1 · · · dqn ψ ∗ (~q)ψ(~q) = 1.

(7.24)

129
The above form of the KEO is convenient for the iterative direct-product DVR grid approach(Section 7.2.1). Matrix elements of complicated expressions such as g −1/4 ∂k† Gkl g 1/2 ∂l g −1/4 aresimple to evaluate by repeated insertion of the truncated identity operator and use of the diagonalDVR approximation [239].
For the VSCF-based methods, it turns out to be useful to push the derivative operators
appearing in Tv to the edges of each term (i.e. ∂k all the way to right and ∂k† to the left). Doing so
rearranges Tv into an equivalent form [229, 230],Tv =
~2 X †~2 X 
∂k Gkl ∂l +Ul ∂l + ∂l† Ul + VT ,22
kl

(7.25)lwhere the single-derivative coefficients are
1X
Ul = −
4



k

∂k g
g


Gkl ,

(7.26)and the kinetic pseudo-potential is
~2 X
VT =
32
kl



∂k g
g



∂l g
g


Gkl .

(7.27)The determinant derivative is conveniently evaluated using the relation (∂k g)/g = tr(G∂k g).
The VSCF calculations require that matrix element integrals of the various functions in
the KEO, such as Gkl , Ul , and VT , as well as the PES, be computed between Hartree productbasis states. These KEO coefficients in general depend on all n internal coordinates, leading tovery high dimensional integrals. For example, molecules as large as butadiene, considered below,
would require 24-dimensional sums, which are simply impractical to compute. A common method
to address this problem is to expand the various high-dimensional functions in a many-body (or“n-mode”) expansion [221, 229]. For some scalar function F (q1 , q2 , . . . , qn ), this exansion is
F (q1 , . . . , qn ) = f0 +Xkfk (qk ) +Xk<lfkl (qk , ql ) +

X
k<l<m

fklm (qk , ql , qm ) + · · · ,(7.28)

130
where
f0 = F (q1ref , . . . , qnref ),fk = F (q1ref , . . . , qk , . . . , qnref ) − f0 ,fkl = F (q1ref , . . . , qk , ql , . . . , qnref ) − fk − fl − f0 ,fklm = F (q1ref , . . . , qk , ql , qm . . . . , qnref ),− fkl − fkm − flm − fk − fl − fm − f0 ,(7.29)and so on. (q1ref , . . . , qnref ) is a reference geometry about which the expansion is computed (usually
the equilibrium configuration). Many-body expansions break up a complicated high-dimensional
function into smaller parts of lower dimensionality. By taking the expansion to include all terms
up to n-body contributions, the exact original function F is recovered, but nothing is gained. It is
often accurate, however, to truncate the expansion at three- or four-body terms, which drasticallyreduces the dimensionality of the matrix element integrals of F . The small errors introduced by
the many-body expansion can be systematically controlled by checking that computed energies areconverged with respect to the maximum number of n-body terms.

7.2.4

Coordinate systems and body-fixed frame embedding

The procedures above allow us to construct the numerically exact KEO for an arbitrary
set of internal coordinates ~q and body-fixed frame embedding (implicitly defined by the Cartesian
position functions ~xi (~q)). The question remains, how do we choose these coordinates optimally?
For iterative direct-product DVR grid calculations, we often use primitive valence coordinates
such as internuclear distances, bond angles, and dihedral angles [237]. Simple orthogonal coordinate
systems like Jacobi and Radau coordinates are also convenient [245, 246]. For reduced-dimension
calculations it is of course necessary to choose coordinates that accurately describe the dynamics
of the sub-system of interest. The computational efficiency and accuracy of full-dimensional directproduct grid calculations, however, is not strongly dependent on the choice of coordinates as longas they compactly span the energetically relevant regions of configuration space.

131
On the other hand, the choice of coordinates is critically important for VSCF-based calculations. In general, one wants to find a coordinate system that minimizes coupling and leads toan approximately separable vibrational Hamiltonian, Hv (~q) ≈ h1 (q1 ) + h2 (q2 ) + · · · . This ensuresthat the VSCF Hartree product is an accurate zeroth-order wavefunction and that the various
many-body expansions will be accurate for low expansion orders. For molecules with a single equilibrium configuration, the ground and lowest excited vibrational states are usually best describedwith the normal coordinates associated with the PES minimum. These are found by a (curvilinear)
GF harmonic analysis [247]. Systems with more than one dynamically accessible minimum, which
describes most of the examples below, require a more elaborate treatment such as a reaction path(RP) coordinate system [224, 248–252]. A RP coordinate system is based on the steepest descent
trajectory that connects neighboring minima through intermediate saddle points on the PES. At
each point along the steepest descent RP, a modified GF calculation is performed to determine the“instantaneous” normal modes orthogonal to the RP at that position. In this way, the RP coordinate system smoothly interpolates between the different normal mode coordinates associated witheach local minimum. It provides an approximately globally separable coordinate system, which is
crucial to making VSCF/VMP2 efficient and accurate.
In addition to choosing a coordinate system that minimizes the coupling between different
vibrations, VSCF/VMP2 requires that the body-fixed frame embedding results in small rotationvibration coupling terms Trv . This embedding is defined by how the molecular body-fixed frame isrotated in space for a given value of the internal coordinates ~q. For most cases, the best choice ofembedding is the Eckart frame [216], which eliminates rotation-vibration coupling near the equilibrium geometry. Implementing Eckart frame embedding for arbitrary curvilinear Hamiltonians isin general a complicated, non-trivial problem [241, 242, 253–256]. The approach we take is basedon the elegant quaternion algebra method recently introduced by Krasnoshchekov et al. [257]. It
allows for the simple numerical evaluation of exact Eckart-frame KEOs. Further algorithmic details
can be found in Ref. [51].

132
7.2.5

Potential energy surfaces

Last but not least, an accurate potential energy surface based on high-level quantum chemical methods is the foundation for any nuclear motion calculation. The techniques discussed above
require the PES to be evaluated over coordinate grids that can contain up to millions of points.
Performing an individual electronic structure calculation at each of these grid points is impractical, and the usual approach is to instead fit an analytical representation of the PES to a smaller
data set of pre-computed electronic energies, of which perhaps only thousands to tens of thousands are necessary. We rely on single-reference coupled-cluster (CC) theory – in particular its
CCSD(T) variant [258] – in combination with the Dunning correlation-consistent basis sets [259]or the atomic natural orbital (ANO) basis sets of Almlöf and Taylor [260], which perform well for
vibrational frequency calculations [261]. All electronic structure calculations for our in-house PES’s
are performed with the CFOUR package [262].
One advantage of DVR basis sets is that the PES only needs to be numerically evaluatedon grid points. The coordinates used to describe the PES are therefore entirely independent of
the “dynamical” coordinates used in the nuclear motion calculation itself, providing substantial
flexibility in the underlying analytical representation of the PES. A high-dimensional polynomial insome set of internal coordinates is typically fit via linear-least-squares to a set of electronic energies.
The method of permutationally invariant polynomials (PIPs) is particularly useful for generating
surfaces that rigorously enforce nuclear permutation symmetry [263]. We make extensive use ofthis approach and have developed a suite of tools for constructing, fitting, and coding PIP surfaces
combined with automated differentiation [199, 264].

7.3

Applications
The first half of this chapter introduced methods for high accuracy ab initio rovibrational

calculations. In the second half, we apply these to several molecular systems. Most of these projects
were carried out in close coordination with experimental collaborators, as acknowledged in each

133
section, as well as two essential theory collaborators, Josh Baraban (Ben-Gurion University) andJohn Stanton (University of Florida).

7.3.1

Photoelectron spectroscopy of the methide anion, CH−
3

The methide anion, CH−
3 , is the simplest carbanion species [265]. As the conjugate base ofCH4 , it plays an important role in constraining thermochemical cycles that determine the gas-phase
acidity of CH4 , itself an important thermochemical reference [266]. CH−
3 is isoelectronic to NH3 andH3 O+ and shares with them a non-planar, trigonal pyramidal structure with tunneling inversion
between the two equivalent enantiomers [267]. Characterizing this fundamental structural motif is,however, challenging due to the difficulty in isolating this reactive species in the gas phase [268–270].
−
This section discusses the high-resolution photoelectron spectra of CH−
3 (and CD3 ) recently

measured in the Lineberger lab as part of an intra-JILA collaboration [192]. The experimentmeasures the electron kinetic energy (eKE) of photo-detached electrons generated via CH−
3 + hν →CH3 + e− , where hν is represents a photon from a pulsed nanosecond laser source. The derived
electron binding energies (eBE = hν − eKE) provide a map of the vibrational energy levels of theanion and neutral species. The experimental spectrum (Fig. 7.3a) exhibits an extended Franck–
Condon progression in the ν2 umbrella mode associated with the large change in geometry between
the pyramidal C3v anion and the planar D3h neutral. The goal of the calculations is to accurately
simulate this spectrum, particularly features associated with the inversion tunneling splitting in
the anion.
We used a reduced-dimension treatment to calculate the vibrational levels of CH−
3 and CH3 .
The umbrella inversion angle φ and the three C–H bond lengths were included, while the HCH
bond angles were constrained to be equal. A 4-D anion potential energy surface was constructed
initially at the CCSD(T)/d-aug-pVTZ level of theory. This surface was further refined with corrections for a variety of higher order effects including triple augmentation, core correlation, triples andnon-iterative quadruples correlation, diagonal Born-Oppenheimer corrections, scalar relativistic
corrections, and zero-point motion in the neglected HCH bend angles (see supplementary infor-

a

CH3– photoelectron spectrum

134b

CH3– /CH3 potential energy curves

Potential energy

Intensity (a.u.)Electron binding energy (cm-1)

Δinv
Electron binding energy (eV)

Inversion angle φ (deg.)

Figure 7.3: Photoelectron spectra and potential curves of CH−
3 /CH3 . (a) Upper panel: the photoelectron spectra are shown for two photon energies, 1.165 eV (red) and 0.383 eV (black). The
inset shows a raw velocity-mapped photoelectron image (see Ref. [192] for details). Lower panel:
The simulated Franck–Condon spectrum. (b) The potential energy curves (not to scale) of CH−
3(black) and CH3 (green) as a function of the umbrella inversion angle φ. The vertical arrows indicate Franck–Condon-allowed transitions from the v = 0+ (red) and v = 0− (blue) anion tunnelingstates, which are split in energy by ∆inv . (Figures adapted with permission from A. Oliveira et al.,J. Am. Chem. Soc. 137:12939 (2015). Copyright 2015 American Chemical Society.)mation of Ref. [192] for details). For the neutral, we used the ab initio force field calculated by
Schwenke [271]. The reduced-dimension vibrational levels of the anion and neutral were calculatedon these surfaces using the iterative diagonalization direct-product DVR grid method.
Figure 7.3a demonstrates the excellent agreement between the experimental photoelectron
spectrum with a simulation using the Franck–Condon overlap factors between the calculated anion
and neutral vibrational wavefunctions. Peaks “A” through “I” correspond to transitions from thev = 0+ /0− ground state tunneling doublet of the anion to successively higher overtones of the ν2umbrella inversion mode of the neutral. The calculated spectrum also confirms the assignmentof peak “a” as a hot band from the v = 1+ anion vibrational level. Figure 7.3b illustrates thecontrasting level structure of the double-well anion potential curve vs. the single-well neutral. The

135
energy splitting ∆inv between the v = 0+ and v = 0− tunneling levels of the anion corresponds
to the frequency of interconversion between the two enantiomer wells. From the positions of theexperimental peaks, this tunneling splitting is determined to be ∆inv = 21(5) cm−1 , consistent
with the calculated value of 25.0 cm−1 . (Similar results are obtained for the deuterated CD−
3species [192].)−1
The inversion tunneling splitting for CH−
3 is intermediate to those of NH3 (0.79 cm , thefamous ammonia maser transition [272, 273]) and H3 O+ (55.34 cm−1 [274]). These only very
roughly correlate with the corresponding effective 1-D inversion barrier heights: 661 cm−1 for CH−3
as inferred from our 4-D surface vs. 650 cm−1 for H3 O+ and 1782 cm−1 for NH3 [275]. Thisillustrates why concepts such as 1-D effective barrier heights should be considered with a grain ofsalt. Tunneling dynamics are sensitive to the complete multi-dimensional potential energy surface.
Vibrational motion calculations that account for such multi-dimensional dynamics, such as thosereported here, are ultimately necessary for spectroscopically accurate ab initio results.

7.3.2

Laboratory and astronomical discovery of disilicon carbide, Si2 C

Small silicon- and carbon-containing molecules are key building blocks for the formationof interstellar dust [276] and thought to be important intermediates in the generation of solid
SiC [277]. Of the four triatomics containing only silicon and carbon (Sim Cn , m + n = 3), theleast well-studied is disilicon carbide, Si2 C, despite evidence of it being unusually stable [278] andan abundant Si-bearing species in space [279]. Quantum chemical studies predict Si2 C to havea bent geometry with an Si–C–Si bond angle of about 115◦ , but a low barrier to linearity of
only ∼ 800 cm−1 [280–282]. The highly anharmonic rovibrational structure of such “quasi-linear”
molecules is of considerable fundamental interest [283]. There has been no prior spectroscopic
characterization of Si2 C, however, with the exception of limited matrix-IR [284, 285] and lowresolution electronic excitation spectra [286].
This gap has recently been addressed by a series of spectroscopic studies carried out in the
McCarthy lab at the Harvard-Smithsonian Center for Astrophysics characterizing the rovibrational

136
structure of the ground electronic state of Si2 C [193,194]. As part of a joint theoretical-experimental
collaboration, we carried out extensive ab initio calculations to aid in the assignment, analysis, andprediction of these spectra. The project originated with observations of the ground 1 A1 electronic
state of Si2 C acquired by laser-induced dispersed fluorescence (DF) from excited electronic statesin the 380–395 nm region. The DF spectra, shown in Fig 7.4, exhibit extended Franck–Condon
progressions in the ν2 bending mode consistent with the large change in geometry between the nearlinear excited state and the bent ground state (∠SiCSi = 115◦ , see Fig. 7.5a). Weaker stretch-bend
combinations are also observed. We constructed an Si2 C PES at the frozen-core CCSD(T)/PVQZ
level of theory and used this to compute the lowest 200 vibrational states, which extend up to
about 4000 cm−1 above the vibrational ground state, as well as rotational sub-levels up to J ≤ 3
for vibrational states below 2500 cm−1 . With guidance from these calculations, essentially all
emission lines in the DF spectra could be assigned with excellent agreement between the observedand calculated vibrational energies (see supplementary material of Ref. [193]).
The barrier to linearity reveals itself spectroscopically by the observed decrease and thenincrease of the effective bending frequency as the barrier energy is approached and then exceeded.
This “Dixon-dip” behavior is readily seen in plots of the bending interval as a function of the vibrational energy, where a minimum occurs in the vicinity of the linear saddle point [287] (Fig. 7.5b).
From the (v1 = 0, v2 = n, v3 = 0) bending progression, we extract an experimental effective barrier height of 783(48) cm−1 [288], in good agreement with theory (802(9) cm−1 ). Excitation in
the symmetric Si–C stretching mode ν1 decreases the effective barrier by about 300 cm−1 , while
excitation in the antisymmetric stretch ν3 increases it by about 360 cm−1 . These changes are
consistent with the changes in the calculated harmonic frequencies at the saddle-point. The spectroscopic signatures of the barrier to linearity are also apparent in the experimental and calculated
eigenvalue lattices for the (Ka , ν2 ) levels, shown in Fig. 7.5c. (Ka is the usual a-axis rotational
quantum number.) As the ν2 bending mode is excited up to and beyond the barrier to linearity, theKa -dependence transitions from quadratic to linear. This is a quintessential example of quantum
monodromy, where a critical point in the Hamiltonian due to the linear saddle point prevents the137* Las er

* Las er

13

1 32 1

Relative wavenumber (cm-1)

Figure 7.4: Dispersed fluorescence (DF) spectra of Si2 C. The main plot shows the DF spectrum
after excitation at 25515 cm−1 of a parallel transition within the Ka sub-level. Essentially allfeatures can be assigned with guidance from variational nuclear motion calculations. Vibrational
assignments are summarized in Table S1 of Ref. [193]. Inset: DF spectrum of Si2 C at 26242 cm−1
excitation. This is a perpendicular transition with emission to Ka = 1 and 3 sub-levels, which
diverge as the barrier to linearity is approached. (Figure reproduced from N. Reilly et al., J. Chem.
Phys. 142:231101 (2015), with the permission of AIP Publishing.)

a

138

b
800
600

ΔE ν2 (cm-1)

Energy (cm-1)

700
500
400
300
200
100
0

100

120

140

160

180

SiCSi angle (degrees)cd

E (ν2) (cm-1)b

10
9

E (ν2) (cm-1)

8

1.693(1) Å
114.87(5)˚

a

7
6
5
4
3
2
1
v2 = 0Figure 7.5: Potential energy surface, energy levels, and structure of Si2 C. (a) The potential energy
surface as a function of the Si–C–Si bending angle, with a minimum near 115◦ and a barrier tolinearity of ca. 800 cm−1 . The 1-D bending wavefunctions for states below the barrier are shown.
(b) The bending energy level interval, representative of the effective bending frequency, is plottedversus vibrational energy for the pure bending progression (v1 , v2 , v3 ) = (0, n, 0), as well as forthe symmetric stretch (1, n, 0) and antisymmetric stretch (0, n, 2) combinations. A characteristic
dip occurs at the energy of the barrier. (c) (v2 , Ka ) eigenvalue lattice for v2 ≤ 10 and Ka ≤ 3.
The transition from quadratric to linear Ka -dependence near v2 = 7 is a signature of quantum
monodromy. (d) The structures of Si2 C (this work) and the three other silicon carbides Sim Cn
with m + n = 3. (Figures b,c reproduced from N. Reilly et al., J. Chem. Phys. 142:231101 (2015),
with the permission of AIP Publishing; Figure d adapted with permission from M. McCarthy etal., J. Phys. Chem. Lett. 6:2107 (2015). Copyright 2015 American Chemical Society.)

139
existence of a uniquely defined set of global quantum numbers [289].
Guided by the accurate rovibrational variational calculations, a new search was conductedby our collaborators for the pure rotational spectrum of Si2 C using Fourier transform microwave(FTMW) spectroscopy [194]. Detection of Si2 C by FTMW has proven challenging because of
its sparse rotational spectrum (with half the levels missing from nuclear spin statistics) and thesensitivity of its rotational constants to small changes in its geometry, especially the obtuse Si–C–Si
bending angle. Our calculations predicted a strong rotational transition (20,2 → 11,1 ) near 38 GHz
for the Si13 CSi isotopologue, which was quickly found in the laboratory. Two additional lines weredetected at exactly the predicted frequencies for the 29 Si and 30 Si isotopic shifts, confirming thecarrier to be Si2 C. Ultimately, between eight and twelve transitions from each of 28 Si2 12 C andthe 13 C, 29 Si, and 30 Si singly substituted isotopologues were measured, permitting a fit of their
ground state rotational constants. The experimental rotational constants were then corrected withcalculated zero-point motion shifts to obtain semi-experimental equilibrium constants, which werethen used to derive the precise semi-experimental equilibrium structure shown in Fig. 7.5d. By
comparing this structure to other triatomic silicon carbides, it can be inferred that when silicon
is the central atom, it prefers a small bending angle (< 90◦ ), while a central carbon atom prefers
only slightly bent or linear geometries. These structural patterns illustrate the predominantly
single bonding nature of silicon via p orbitals vs. the multiple bonding of carbon via sp or sp2
hybridization [194].
The laboratory identification of Si2 C immediately led to its astronomical discovery in the
carbon-rich star IRC+10216 [195]. These observations have shown that, along with SiC2 , Si2 C
is the most abundant species containing a Si–C bond in the dust formation region of this star,
indicating that it plays a key role in the formation of dust grains. Since this initial discovery,
additional astronomical and laboratory observations have been made [290], including rotational
satellite transitions from vibrationally excited Si2 C, which greatly benefited from our variationallycalculated vibration-rotation constants [276]. Rapid progress continues to be made on the spectroscopy of Si2 C, including the first rotationally resolved infrared measurements of the ν3 band

140
recently reported by Witsch et al. [291].

7.3.3

The ionization energy and equilibrium structure of hydrogen peroxide, H2 O2

In the past fifteen years, the advent of Active Thermochemical Tables (ATcT) [292, 293] has
dramatically improved the accuracy of fundamental thermodynamic quantities of small molecules.
The ATcT approach relies on thermochemical networks, which explicitly account for the high degree of interdependency between the thermodynamics of even very different molecules. All available
information, including both experimental and high-level theoretical data (with calibrated uncertainties), is taken into account by the network to provide self-consistent, accurate values of bondenergies, enthalpies of formation and reaction, etc. with meaningful uncertainties. In some cases,
computational results are used to reanalyze published data for inclusion in ATcT. In this section, we
discuss such an example for the ionization energy (IE) of hydrogen peroxide, H2 O2 . This work was
carried out in collaboration with the “ATcT Task Force” led by Branko Ruscic (Argonne National
Laboratory).
The IE of H2 O2 has potentially important impact on the ions of compounds involved inhydrogen combustion chemistry, as well as several other key chemical species [197, 294]. In 2016,
Schio et al. studied H2 O2 via threshold photoelectron spectroscopy (TPES), reporting an IE of
10.685 ± 0.005 eV [294]. This value disagrees by many σ with the (then current) ATcT value of
10.637 ± 0.006 eV [295], the largest contributors to which a photoionization mass spectrometry(PIMS) measurement, 10.631 ± 0.007 eV [296], and an early photoelectron spectroscopy measurement, 10.62 ± 0.02 eV [297]. The TPES value is based on the assignment of peak “a” in Fig. 7.6b to
a vibrational hot band involving the torsional mode, placing the vibrational origin (the 0–0 transition) between peaks “a” and “b”. This assignment is largely based on harmonic Franck–Condon
simulations carried out by the authors of Ref. [294]. The ground state of H2 O2 , however, exhibits
large-amplitude nuclear motion (Fig. 7.6a) and a simple harmonic simulation is inadequate.
We have reanalyzed the TPES spectrum using simulations based on variational anharmonic
vibrational wavefunctions and energies of the neutral molecule and cation. For the neutral, we

141
use the potential energy surfaces previously reported by Koput et al. [298, 299], while for thecation we constructed a new surface at the FC-EOMIP-CCSD/ANO0 level of theory [260, 300].
We converged full-dimensional vibrational wavefunctions with the iterative diagonalization directproduct DVR grid method. Franck–Condon factors were calculated with the overlap integrals of thecomputed wavefunctions. The resulting photoelectron spectrum simulation is shown in Fig. 7.6b.
The simulated spectrum has been shifted so that the origin band coincides with peak “a”. In the
low-energy region of the spectrum, the first unresolved feature actually contains the 401 hot band(ν4 is the torsional mode) and the origin band. Features up to 11.1 eV can also be assigned (seeTable 2 of Ref. [197]). The excellent qualitative agreement with experimental spectrum confirms
the correct origin assignment. This alternative interpretation of the TPES spectrum leads to arevised IE of 10.649 ± 0.005 eV.
In the older PIMS measurement by Litorja and Ruscic [296], the reported IE of 10.631 ±0.007 was based on the half-height of the first step in the onset of the photoionization efficiency(PIE) curve, shown in Fig. 7.6c. The anharmonic simulations suggest, however, that the strong
401 hot band is probably biasing this determination to too small a value. In light of this, we havereanalyzed the PIMS PIE curve in comparison to the integrated Franck–Condon simulation at
300 K (convoluted with the appropriate instrument line shape). The simulations indicate that the
PIMS value should be slightly increased to 10.645 ± 0.010 eV, which is now consistent with therevised TPES value of 10.649 ± 0.005 eV.
A careful analysis of the experimental spectra also requires one to consider shifts due toassymetric rotational envelopes. A completely quantitative analysis of this effect requires knowledgeof the the photoelectron partial wave amplitudes, which is outside the scope of this study. However,an analysis of the rotational contours under various scenarios suggests that the peak of the origin
band contour will indeed be blue-shifted by about 7 meV relative to the true IE, though witha large uncertainty of ±6 meV. This shift is mostly due to the large difference in the value of(B + C)/2 in the cation (1.01 cm−1 ) and neutral (0.86 cm−1 ). Taking this into account leaders tofurther refinement of the PIMS and TPES IE values: 10.638 ± 0.012 eV and 10.642 ± 0.008 eV,

142Photoionization Efficiencyc

800 2+
700
600 1–
500 1+
400
300
0–
200
+
100 v=0060 120 180 240 300
Dihedral angle (degrees)

Photoelectron IntensitybEnergy (cm-1)arevisedoriginassignment

10.5

d

10.6

10.60
10.64
Photon Energy (eV)

a’

b’’

b’

10.7 10.8 10.9 11.0
Photon Energy (eV)

11.1

H

1.4524(3) 0.9617(2)

H

10.56

b

a’’

a

113.6(3)original IE estimate(half-height)

c

O

O

99.76(6)10.68

Figure 7.6: Potential energy, spectra, and equilibrium structure of H2 O2 . (a) The large-amplitude
torsion angle is delocalized over a double-well potential energy curve. The ground torsional state
has two tunneling components, v = 0+ and 0− split by 11 cm−1 . (b) The TPES spectrum from
Ref. [294] (black) together with the simulated stick spectrum (this work). Blue features originate
from the 0+ ground state, green features from 0− , and red features from higher torsional states(1+ , 1− , 2+ , . . .). The stick spectrum was simulated at a vibrational temperature of 200 K. (c) PIE
curve from Ref. [296] (black), along with integrals of the simulated spectrum (at 300 K). The threesimulated curves are shown assuming an IE of 10.635 eV (blue), 10.645 eV (red), and 10.655 eV(green), without including effects from rotational contour shifts (see text). The original IE estimate
at the half-height of the PIE step is indicated. (d) The semi-experimental equilibrium geometry of
H2 O2 . Bond lengths are in Å, and angles are in degrees. (Figures b, c adapted with permission fromP. B. Changala et al., J. Phys. Chem. A, 121:8799 (2017). Copyright 2017 American Chemical
Society; Figure d reused with permission from J. Baraban et al., J. Mol. Spectrosc., 343:92 (2018).
Copyright 2018 Elsevier).

143
respectively. Together with a new ab initio IE calculated as part of this work (see Ref. [197] fordetails), a revised ATcT analysis results in a final composite IE value of 10.641 ± 0.006 eV. Thisvalue ends up being very similar to the previous best ATcT estimate (10.637 ± 0.006 eV [295]), but
this appears to be due to a fortuitous cancellation of the corrections associated with the vibrational
hot band analysis and rotational contour shifts.
During the course of this work, it was realized that the equilibrium structure of H2 O2 had
surprisingly never been completely determined [301–303], despite sufficient spectroscopic data being
available [304–308]. The main challenge is the large difference between the equilibrium re structure
and the effective ground state r0 structure, making it critical to account for effects of large-amplitude
zero-point torsional motion [303, 309]. We calculated the rovibrational zero-point corrections using
se )the same methods as above for the H2 O2 IE study. Semi-experimental rotational constants (Beqwere determined by subtracting the ab initio zero-point motion correction from the measured
ground state constants for HOOH, HOOD, and DOOD [304–308]. Figure 7.6d shows the semise values [196]. Without the inclusion of zeroexperimental equilibrium geometry fitted to the Beq

point motion corrections, the fitted HOOH dihedral angle is 6◦ larger, illustrating the important
effects of large-amplitude torsional motion. It is interesting to note that the equilibrium OH bondlength in H2 O2 (0.9617(2) Å) is much closer to that of H2 O (0.9578(1) Å [310]) than the OH radical(0.969628(9) Å [311]), consistent with the fact that the hybridization in H2 O2 is more similar tothat in H2 O than to OH.

144
7.3.4

The molecular structure of gauche-1,3-butadiene, C4 H6The Diels–Alder cycloaddition reaction between a conjugated diene and a dienophile is one of
the most important ring-forming reactions in chemistry [312]. The classic example is the additionof 1,3-butadiene, H2 C=CH–CH=CH2 , and ethylene, H2 C=CH2 , to form cyclohexene,+

Although the trans isomer of butadiene is the thermodynamically most stable, it must isomerize to acis conformation before proceeding through the aromatically stabilized, pericyclic planar transition
state. The planarity of cis-butadiene itself, however, has not been established. In fact, a numberof quantum chemical studies [313–317] and gas-phase Raman experiments [318–320] have led tothe conclusion that “cis”-butadiene actually exhibits a non-planar gauche geometry [321], yet adefinitive structural characterization of this molecule has remained elusive.
In another joint theoretical-experimental collaboration with the McCarthy and Pattersongroups, a combination of sensitive microwave techniques and high accuracy rovibrational calculations has been used to conclusively show that cis-butadiene possesses a non-planar gauche
equilibrium geometry and undergoes facile tunneling interconversion between its two enantiomericforms [198]. Prior attempts to measure the microwave spectrum of gauche-butadiene have been
unsuccessful due to multiple experimental challenges. First, the gauche conformer lies 2.93 kcal
mol−1 above the more stable trans conformer [322]. At room temperature this translates into a
relative population of only about 1%. Second, it possesses a very small dipole moment (ca. 0.09 D)
leading to weak rotational transitions, the frequencies of which are sensitive to the C=C–C=C
torsional angle τ (Fig. 7.7a). These experimental challenges were overcome with a combinationof cavity-enhanced FTMW combined with a supersonic expansion and chirped-pulse FTMW ina cryogenic buffer gas cell (similar to the buffer gas cell apparatus used for the frequency comb
spectroscopy described in this thesis). Further experimental details can be found in Ref. [198].
Spectral searches were initially guided by high-level ab initio calculations. With 10 atoms and

145

c

H3

H2
H1

τ

b

Term Energy - 4J(J+1) (GHz)

a

25

0–

20

K=0f

15

K=1f

10

0+

5
0

0

2

1

J

3

5

4

6

V(τ) (kcal mol-1)

0+

2 12

0–

2 02
1 11

4
16.63

1 +v = 0 -+

16.65

3 03

2
H

CH2

H

CH2

0
-180 -135 -90 -45

16.64

2 02

45 90 135 180

1 10

1 11

τ

0

2 11
2 12

1 01

16.66 GHz

0 00
Δ inv

~55 GHz1 01
J K aK c

0 00

0+

0–

C=C-C=C angle, τ (degrees)

Figure 7.7: Structure and rovibrational energy levels of gauche-butadiene. (a) The C=C–C=C
torsional dihedral angle of the butadiene carbon backbone is labeled τ . (b) The torsion potential
energy curve has a global minimum at the trans geometry (τ = 180◦ ) and two symmetry-equivalentlocal minima at the gauche geometry (τ ≈ ±35◦ ). The expanded view shows the rotational sublevels of the ground state tunneling doublet and the allowed pure rotational transitions. The dashedarrows indicate the transitions shown in the measured inset spectrum. The 0+ and 0− manifolds
are separated in energy by the tunneling frequency ∆inv . (c) Reduced rotational term energy plotof the ground state tunneling doublet for d6 butadiene. The K = 1f manifold of the 0+ tunnelingcomponent crosses the K = 0f manifold of the 0− tunneling component at J = 4 resulting inmagnified Coriolis perturbations. E + /O+ /E − /O− rovibrational symmetries are represented byfilled circles/open circles/open squares/filled squares, respectively. (Figures reused with permission
from J. Baraban et al. Angew. Chem. Int. Ed., 57:1821 (2018).)

146
24 vibrational modes, butadiene is too large for a brute-force direct-product DVR grid calculation.
We therefore turned to the rotational curvilinear VMP2 method to calculate accurate rovibrationalparameters. We constructed a full-dimensional FC-CCSD(T)/ANO1 PIP PES and the associatedcurvilinear reaction path coordinate system. The gauche form of butadiene has a double-wellpotential energy curve along the τ torsion angle as shown in Fig. 7.7b. The ground vibrational
state splits into v = 0+ and 0− tunneling components. Both tunneling components support anindependent manifold of rotational sub-levels, and transitions within these appear as closely spaced
doublets in the microwave spectrum. Rotational constant predictions calculated via rotational
VMP2 guided targeted spectroscopic searches, which soon led to the observation of transitions
near the predicted frequencies. The observed transitions were used to fit rotational constantsfor several isotopologues of gauche-butadiene. These measured rotational constants were finallycorrected with theoretical zero-point motion shifts to determine the semi-experimental equilibrium
structure summarized in Table 7.1. The fitted dihedral angle of τ = 33.8(13)◦ unambiguously
establishes the non-planarity of gauche-butadiene.
Table 7.1: Semi-experimental equilibrium geometry of gauche-butadiene. Bond lengths are in Å,and angles are in degrees. Numbers in brackets [ ] are fixed to ab initio values (see Ref. [198] fordetails).
Parameterr(C–C)
r(C=C)∠C–C=C∠C=C–C=C (τ )r(C–H1 )∠H1 –C=C∠H1 –C=C–C
r(C–H2 )∠H2 –C=C∠H2 –C=C–H3
r(C–H3 )∠H3 –C–C∠H3 –C–C=C

Value
1.48(2)
1.33(1)
124.1(5)33.8(13)1.08(1)123(2)
2(1)[1.07983]
118(1)[2.410]1.083(5)116.2(6)−146.4(7)Microwave transitions between the v = 0+ and 0− tunneling components are symmetry

147
forbidden, so that the 0+ /0− tunneling splitting ∆inv cannot be directly probed. Multi-state
rotational VMP2 calculations, however, reveal that the two tunneling components interact via a
C × (Ja Jc + Jc Ja )-type Coriolis term, with an interaction constant C = 20 − 30 MHz depending on
the isotopologue. This interaction results in the largest spectroscopic perturbations near crossingpoints of the 0+ /0− rotational manifolds (Fig. 7.7c). By a careful fit of the microwave transition
frequencies to an interacting two-state model, both the tunneling frequency ∆inv and the Coriolis
interaction coefficient C can be determined. Table 7.2 shows the results of this analysis for the
fully deuterated d6 isotopologue. The excellent agreement between the observed and calculated
tunneling parameters is indicative of the robust zeroth-order treatment that curvilinear VSCF
provides. A similar tunneling-Coriolis interaction mechanism has also been used to analyze therecently identified microwave spectrum of gauche-isoprene, a subtituted butadiene derivative [323].
Table 7.2: Tunneling parameters of d6 -gauche-butadiene. ∆inv is the tunneling inversion splitting(the energy difference between the v = 0+ and 0− vibrational states). C is the coefficient of the(Ja Jc + Jc Ja ) Coriolis interaction operator.
Parameter∆inv / cm−1
C / MHz

Expt.
0.55
23.4

Calc.
0.58
23.3

At first glance, the non-planar geometry of gauche-butadiene might be justified by invoking
repulsive steric interactions between the terminal hydrogen atoms. Wiberg et al. have recently
suggested, however, that upon closer examination this effect is probably too weak to account forthe observed structure [324]. Instead, one must consider terminal π-interactions. At the planar
cis configuration, 1,4-π interactions are Hückel antiaromatic, destabilizing the molecule, while attwisted geometries the π system has a Möbius topology, stabilizing the π-electron system. A
detailed study of a large number of heterodienes shows that this mechanism is consistent with theobserved effects [324]. This is an excellent example of new insights into molecular interactions
inspired by the precise structural information made available by the combination of high resolution
spectroscopy and high level rovibrational theory [182].

148

7.4

Conclusions
This chapter has highlighted several applications of high-accuracy rovibrational calculations

to the spectroscopy of complex polyatomic molecules. Each of these examples featured floppy
molecules that exhibit highly anharmonic, large-amplitude nuclear motion. Direct-product DVR
grid techniques provide a flexible, numerically exact method to treat these systems, but they
are restricted to relatively small molecules. Curvilinear rotational-VMP2 provides quantitative
predictions via a less expensive perturbative approach. For large systems, like butadiene, there arefew, if any, comparable alternative methods.
The high quality zeroth-order picture of curvilinear VSCF suggests it may be useful for applications beyond pure spectroscopy. We are currently exploring extensions to it to study quantum
reaction dynamics and resonance states via complex scaling [325–329], as well as thermodynamical
properties such as partition functions with thermal variants [330]. It may be possible to combine thebenefits of curvilinear VSCF with other techniques. For example, the VSCF Hamiltonian could be
used for pre-conditioned spectral transformations with iterative diagonalization methods [205, 211]or to generate an importance sampling trial function for diffusion Monte Carlo techniques [331,332].
Ultimately, a good zeroth-order picture that accurately captures the physics of strong anharmonicity forms an excellent starting point for a variety of nuclear motion calculations.

Bibliography[1] Quack, M. and Merkt, F., eds. Handbook of High-resolution Spectroscopy. John Wiley &Sons (2011).
[2] Udem, T., Holzwarth, R., and Hänsch, T. W.
416(6877):233–237 (2002).

Optical frequency metrology.

Nature,[3] Cundiff, S. T. and Ye, J. Femtosecond optical frequency combs. Rev. Mod. Phys., 75(1):325–
342 (2003).
[4] Stowe, M. C., Thorpe, M. J., Pe’er, A., Ye, J., Stalnaker, J. E., Gerginov, V., and Diddams,
S. A. Direct frequency comb spectroscopy. Adv. At. Mol. Opt. Phy., 55:1–60 (2008).
[5] Thorpe, M. and Ye, J. Cavity-enhanced direct frequency comb spectroscopy. Appl. Phys. B,
91(3-4):397–414 (2008).
[6] Adler, F., Thorpe, M. J., Cossel, K. C., and Ye, J. Cavity-enhanced direct frequency comb
spectroscopy: technology and applications. Ann. Rev. Anal. Chem., 3:175–205 (2010).
[7] Weichman, M. L., Changala, P. B., Ye, J., Chen, Z., Yan, M., and Picqué, N. Broadband
molecular spectroscopy with optical frequency combs. J. Mol. Spectrosc., 355:66–78 (2019).
[8] Foreman, S. M., Jones, D. J., and Ye, J. Flexible and rapidly configurable femtosecond pulse
generation in the mid-IR. Opt. Lett., 28(5):370–371 (2003).
[9] Erny, C., Moutzouris, K., Biegert, J., Kühlke, D., Adler, F., Leitenstorfer, A., and Keller, U.
Mid-infrared difference-frequency generation of ultrashort pulses tunable between 3.2 and 4.8µm from a compact fiber source. Opt. Lett., 32(9):1138–1140 (2007).
[10] Gambetta, A., Ramponi, R., and Marangoni, M. Mid-infrared optical combs from a compact
amplified Er-doped fiber oscillator. Opt. Lett., 33(22):2671–2673 (2008).
[11] Adler, F., Cossel, K. C., Thorpe, M. J., Hartl, I., Fermann, M. E., and Ye, J. Phase-stabilized,
1.5 W frequency comb at 2.8–4.8 µm. Opt. Lett., 34(9):1330–1332 (2009).
[12] Ruehl, A., Gambetta, A., Hartl, I., Fermann, M. E., Eikema, K. S. E., and Marangoni, M.
Widely-tunable mid-infrared frequency comb source based on difference frequency generation.
Opt. Lett., 37(12):2232–2234 (2012).
[13] Gambetta, A., Coluccelli, N., Cassinerio, M., Gatti, D., Laporta, P., Galzerano, G., andMarangoni, M. Milliwatt-level frequency combs in the 8-14 µm range via difference frequency
generation from an Er:fiber oscillator. Opt. Lett., 38(7):1155–1157 (2013).

150[14] Heckl, O. H., Bjork, B. J., Winkler, G., Changala, P. B., Spaun, B., Porat, G., Bui, T. Q., Lee,K. F., Jiang, J., Fermann, M. E., Schunemann, P. G., and Ye, J. Three-photon absorption
in optical parametric oscillators based on OP-GaAs. Opt. Lett., 41(22):5405–5408 (2016).
[15] Lee, K. F., Hensley, C. J., Schunemann, P. G., and Fermann, M. E. Midinfrared frequency
comb by difference frequency of erbium and thulium fiber lasers in orientation-patterned
gallium phosphide. Opt. Express, 25(15):17411–17416 (2017).
[16] Iwakuni, K., Porat, G., Bui, T. Q., Bjork, B. J., Schoun, S. B., Heckl, O. H., Fermann, M. E.,
and Ye, J. Phase-stabilized 100 mW frequency comb near 10 µm. Appl. Phys. B, 124(7):128(2018).
[17] Gordon, I., Rothman, L., Hill, C., Kochanov, R., Tan, Y., Bernath, P., Birk, M., Boudon, V.,
Campargue, A., Chance, K., Drouin, B., Flaud, J.-M., Gamache, R., Hodges, J., Jacquemart,D., Perevalov, V., Perrin, A., Shine, K., Smith, M.-A., Tennyson, J., Toon, G., Tran, H.,
Tyuterev, V., Barbe, A., Császár, A., Devi, V., Furtenbacher, T., Harrison, J., Hartmann,
J.-M., Jolly, A., Johnson, T., Karman, T., Kleiner, I., Kyuberis, A., Loos, J., Lyulin, O.,
Massie, S., Mikhailenko, S., Moazzen-Ahmadi, N., Müller, H., Naumenko, O., Nikitin, A.,
Polyansky, O., Rey, M., Rotger, M., Sharpe, S., Sung, K., Starikova, E., Tashkun, S., Auwera,
J. V., Wagner, G., Wilzewski, J., Wcislo, P., Yu, S., and Zak, E. The HITRAN2016 molecular
spectroscopic database. J. Quant. Spectrosc. Radiat. Transf., 203:3–69 (2017).
[18] Bunker, P. R. and Jensen, P. Molecular Symmetry and Spectroscopy. NRC Research Press,
Ottawa, Canada, 2nd edition (1998).
[19] White, J. U. Long Optical Paths of Large Aperture. J. Opt. Soc. Am., 32(5):285–288 (1942).
[20] Herriott, D. R. and Schulte, H. J. Folded Optical Delay Lines. Appl. Opt., 4(8):883–889(1965).
[21] Gagliardi, G. and Loock, H.-P., eds. Cavity-Enhanced Spectroscopy and Sensing. Springer(2014).
[22] Gohle, C., Stein, B., Schliesser, A., Udem, T., and Hänsch, T. W. Frequency Comb VernierSpectroscopy for Broadband, High-Resolution, High-Sensitivity Absorption and Dispersion
Spectra. Phys. Rev. Lett., 99(26):263902 (2007).
[23] Drever, R. W. P., Hall, J. L., Kowalski, F. V., Hough, J., Ford, G. M., Munley, A. J., andWard, H. Laser phase and frequency stabilization using an optical resonator. Appl. Phys. B,
31(2):97–105 (1983).
[24] Adler, F., Maslowski, P., Foltynowicz, A., Cossel, K. C., Briles, T. C., Hartl, I., and Ye,
J. Mid-infrared Fourier transform spectroscopy with a broadband frequency comb. Opt.
Express, 18(21):21861–72 (2010).
[25] Maslowski, P., Lee, K. F., Johansson, A. C., Khodabakhsh, A., Kowzan, G., Rutkowski,
L., Mills, A. A., Mohr, C., Jiang, J., Fermann, M. E., and Foltynowicz, A. Surpassing thepath-limited resolution of Fourier-transform spectrometry with frequency combs. Phys. Rev.
A, 93(2):021802 (2016).
[26] Foltynowicz, A., Ban, T., Maslowski, P., Adler, F., and Ye, J. Quantum-Noise-Limited
Optical Frequency Comb Spectroscopy. Phys. Rev. Lett., 107(23):233002 (2011).

151[27] Changala, P. B., Spaun, B., Patterson, D., Doyle, J. M., and Ye, J. Sensitivity and resolution
in frequency comb spectroscopy of buffer gas cooled polyatomic molecules. Appl. Phys. B,
122(12):292 (2016).
[28] Changala, P. B., Weichman, M. L., Lee, K. F., Fermann, M. E., and Ye, J. Rovibrational
quantum state resolution of the C60 fullerene. Science, 363:49–54 (2019).
[29] Fleisher, A. J., Bjork, B. J., Bui, T. Q., Cossel, K. C., Okumura, M., and Ye, J. Mid-Infrared
Time-Resolved Frequency Comb Spectroscopy of Transient Free Radicals. J. Phys. Chem.
Lett., 5(13):2241–2246 (2014).
[30] Bjork, B. J., Bui, T. Q., Heckl, O. H., Changala, P. B., Spaun, B., Heu, P., Follman, D.,
Deutsch, C., Cole, G. D., Aspelmeyer, M., Okumura, M., and Ye, J. Direct frequency comb
measurement of OD + CO → DOCO kinetics. Science, 354:444–448 (2016).
[31] Bui, T. Q., Bjork, B. J., Changala, P. B., Heckl, O. H., Spaun, B., and Ye, J. OD + CO →D + CO2 branching kinetics probed with time-resolved frequency comb spectroscopy. Chem.
Phys. Lett., 683:91–95 (2017).
[32] Bui, T. Q., Bjork, B. J., Changala, P. B., Nguyen, T. L., Stanton, J. F., Okumura, M., andYe, J. Direct measurements of DOCO isomers in the kinetics of OD + CO. Science Advances,4(1):eaao4777 (2018).
[33] Iwakuni, K., Bui, T. Q., Niedermeyer, J. F., Sukegawa, T., and Ye, J. Comb-resolved spectroscopy with immersion grating in long-wave infrared. Opt. Express, 27(3):1911 (2019).
[34] Shirasaki, M. Large angular dispersion by a virtually imaged phased array and its application
to a wavelength demultiplexer. Opt. Lett., 21(5):366 (1996).
[35] Xiao, S. and Weiner, A. M. 2-D wavelength demultiplexer with potential for ≥1000 channels
in the C-band. Opt. Express, 12(13):2895 (2004).
[36] Diddams, S. A., Hollberg, L., and Mbele, V. Molecular fingerprinting with the resolved modesof a femtosecond laser frequency comb. Nature, 445(7128):627–630 (2007).
[37] Thorpe, M. J., Adler, F., Cossel, K. C., de Miranda, M. H., and Ye, J. Tomography of a supersonically cooled molecular jet using cavity-enhanced direct frequency comb spectroscopy.
Chem. Phys. Lett., 468(1-3):1–8 (2009).
[38] Spaun, B., Changala, P. B., Patterson, D., Bjork, B. J., Heckl, O. H., Doyle, J. M., and Ye,
J. Continuous probing of cold complex molecules with infrared frequency comb spectroscopy.
Nature, 533:517–520 (2016).
[39] Messer, J. K. and De Lucia, F. C. Measurement of Pressure-Broadening Parameters for the
CO-He System at 4 K. Phys. Rev. Lett., 53(27):2555–2558 (1984).
[40] Patterson, D., Tsikata, E., and Doyle, J. M. Cooling and collisions of large gas phasemolecules. Phys. Chem. Chem. Phys., 12(33):9736–41 (2010).
[41] Patterson, D. and Doyle, J. M. Cooling molecules in a cell for FTMW spectroscopy. Mol.
Phys., 110:1757–1766 (2012).

152[42] Piskorski, J., Patterson, D., Eibenberger, S., and Doyle, J. M. Cooling, spectroscopy andnon-sticking of trans-stilbene and Nile Red. 15:3800–4 (2014).
[43] Tobin, A. G., Sedgley, D. W., Batzer, T. H., and Call, W. R. Evaluation of charcoal sorbents
for helium cryopumping in fusion reactors. J. Vac. Sci. Technol. A, 5(1):101–105 (1987).
[44] Sedgley, D. W., Tobin, A. G., Batzer, T. H., and Call, W. R. Characterization of charcoals
for helium cryopumping in fusion devices. J. Vac. Sci. Technol. A, 5(4):2572–2576 (1987).
[45] Hutzler, N. R., Parsons, M. F., Gurevich, Y. V., Hess, P. W., Petrik, E., Spaun, B., Vutha,
A. C., DeMille, D., Gabrielse, G., and Doyle, J. M. A cryogenic beam of refractory, chemicallyreactive molecules with expansion cooling. Phys. Chem. Chem. Phys., 13(42):18976–85 (2011).
[46] Hutzler, N. R., Lu, H.-I., and Doyle, J. M. The buffer gas beam: an intense, cold, and slow
source for atoms and molecules. Chem. Rev., 112(9):4803–27 (2012).
[47] Stycast 2850FT Technical Data Sheet. http://www.hitek-ltd.co.uk/media/downloads/
5284/Stycast%202850%20FT%20Cat%209%2011%2023LV%20and%2024LV%20(2013).pdf. Accessed: March 20, 2019.
[48] Tennyson, J., Bernath, P. F., Brown, L. R., Campargue, A., Császár, A. G., Daumont,L., Gamache, R. R., Hodges, J. T., Naumenko, O. V., Polyansky, O. L., Rothman, L. S.,
Vandaele, A. C., Zobov, N. F., Al Derzi, A. R., Fábri, C., Fazliev, A. Z., Furtenbacher, T.,
Gordon, I. E., Lodi, L., and Mizus, I. I. IUPAC critical evaluation of the rotational–vibrational
spectra of water vapor, Part III: Energy levels and transition wavenumbers for H2 16 O. J.
Quant. Spectrosc. Radiat. Transf., 117:29–58 (2013).
[49] Olson, W. B., Maki, A. G., and Lafferty, W. J. Tables of N2 O absorption lines for thecalibration of tunable infrared lasers from 522 cm−1 to 657 cm−1 and from 1115 cm−1 to1340 cm−1 . J. Phys. Chem. Ref. Data, 10(4):1065–1084 (1981).
[50] Camy-Peyret, C., Flaud, J. M., Mahmoudi, A., Guelachvili, G., and Johns, J. W. C. Linepositions and intensities in the ν2 band of D2 O improvied pumped D2 O laser frequencies. Int.
J. Infrared Millimeter Waves, 6(3):199–233 (1985).
[51] Changala, P. B. and Baraban, J. H. Ab initio effective rotational and rovibrational Hamiltonians for non-rigid systems via curvilinear second order vibrational Møller–Plesset perturbation
theory. J. Chem. Phys., 145:174106 (2016).
[52] Sørensen, G., Pedersen, T., Dreizler, H., Guarnieri, A., and Cox, A. P. Microwave spectra ofnitromethane and D3-nitromethane. J. Mol. Struct., 97:77–82 (1983).
[53] Sørensen, G. and Pedersen, T. Symmetry and microwave spectrum of nitromethane. Stud.
Phys. Theor. Chem., 23:219–236 (1983).
[54] Barnes, W. L. Measurement and Analysis of the ν3 Band of Methane. J. Chem. Phys.,
56(10):5160 (1972).
[55] Rohart, F. Microwave spectrum of nitromethane internal rotation Hamiltonian in the low
barrier case. J. Mol. Spectrosc., 57(2):301–311 (1975).

153[56] Hazra, A., Ghosh, P., and Kshirsagar, R. Fourier Transform Infrared Spectrum and Rotational Structure of the A-Type 917.5 cm−1 Band of Nitromethane. J. Mol. Spectrosc.,
164(1):20–26 (1994).
[57] Hazra, A. and Ghosh, P. Assignment of the m 6= 0 Transitions in the ν4 Band of Nitromethaneby the Symmetric Top Approximation Method. J. Mol. Spectrosc., 173(1):300–302 (1995).
[58] Pal, C., Hazra, A., Ghosh, P. N., and Kshirsagar, R. High resolution Fourier transform
infrared spectrum and vibration-rotation analysis of the B-type 1584 cm−1 band of nitromethane. J. Mol. Struct., 407(2-3):165–170 (1997).
[59] Dawadi, M. B., Degliumberto, L., Perry, D. S., Mettee, H. D., and Sams, R. L. High-resolution
infrared spectroscopy of the asymmetric NO stretch band of jet-cooled nitromethane andassignment of the lowest four torsional states. J. Mol. Spectrosc., 343:85–91 (2018).
[60] Dawadi, M. B., Twagirayezu, S., Perry, D. S., and Billinghurst, B. E. High-resolution Fouriertransform infrared synchrotron spectroscopy of the NO2 in-plane rock band of nitromethane.
J. Mol. Spectrosc., 315:10–15 (2015).
[61] McKean, D. and Watt, R. Vibrational spectra of nitromethanes and the effects of internalrotation. J. Mol. Spectrosc., 61(2):184–202 (1976).
[62] Cavagnat, D. and Lespade, L. Internal dynamics contributions to the CH stretching overtone
spectra of gaseous nitromethane NO2 CH3 . J. Chem. Phys., 106(19):7946 (1997).
[63] Hill, J. R., Moore, D. S., Schmidt, S. C., and Storm, C. B. Infrared, Raman, and coherent
anti-Stokes Raman spectroscopy of the hydrogen/deuterium isotopomers of nitromethane. J.
Phys. Chem., 95(8):3037–3044 (1991).
[64] Watson, J. K. G. Aspects of quartic and sextic centrifugal effects on rotational energy levels.
In J. Durig, ed., Vibrational Spectra and Structure, Vol. 6, chapter 1. Elsevier, Amsterdam(1977).
[65] Halonen, M., Halonen, L., Callegari, A., and Lehmann, K. K. Molecular Beam Infrared
Spectrum of Nitromethane in the Region of the First C-H Stretching Overtone. J. Phys.
Chem. A, 102(46):9124–9128 (1998).
[66] Tannenbaum, E., Myers, R. J., and Gwinn, W. D. Microwave Spectra, Dipole Moment, andBarrier to Internal Rotation of CH3 NO2 and CD3 NO2 . J. Chem. Phys., 25:42 (1956).
[67] Wu, K. and Simon, H. Thick-Restart Lanczos Method for Large Symmetric Eigenvalue
Problems. SIAM J. Matrix Anal. Appl., 22:602–616 (2000).
[68] Light, J. C. and Carrington Jr., T. Discrete-Variable Representations and their Utilization.
Adv. Chem. Phys., 114:263 (2000).
[69] Puzzarini, C., Stanton, J. F., and Gauss, J. Quantum-chemical calculation of spectroscopic
parameters for rotational spectroscopy. Int. Rev. Phys. Chem., 29(2):273–367 (2010).
[70] Mills, I. M. Vibration-Rotation Structure in Asymmetric- and Symmetric-Top Molecules. In
K. N. Rao and C. W. Mathews, eds., Molecular Spectroscopy: Modern Research, chapter 3.2,
pages 115–140. Academic Press, New York (1972).

154[71] Watson, J. K. G. Simplification of the molecular vibration-rotation Hamiltonian. Mol. Phys.,
15:479–490 (1968).
[72] Bowman, J. M. Self-consistent field energies and wavefunctions for coupled oscillators. J.
Chem. Phys., 68:608–610 (1978).
[73] Bowman, J. M. The self-consistent-field approach to polyatomic vibrations. Acc. Chem. Res.,
19(7):202–208 (1986).
[74] Gerber, R. B. and Ratner, M. A. Self-Consistent-Field Methods for Vibrational Excitations
in Polyatomic Systems. Adv. Chem. Phys., 70:97–132 (1988).
[75] Møller, C. and Plesset, M. S. Note on an Approximation Treatment for Many-Electron
Systems. Phys. Rev., 46(7):618–622 (1934).
[76] Norris, L. S., Ratner, M. A., Roitberg, A. E., and Gerber, R. B. Møller–Plesset perturbation
theory applied to vibrational problems. J. Chem. Phys., 105:11261–11267 (1996).
[77] Wang, X., Carter, S., and Bowman, J. M. Pruning the Hamiltonian Matrix in MULTIMODE:
Test for C2 H4 and Application to CH3 NO2 Using a New Ab Initio Potential Energy Surface.
J. Phys. Chem. A, 119(47):11632–11640 (2015).
[78] Tennyson, J. Astronomical Spectroscopy. World Scientific (Europe), 3rd edition (2019).
[79] Ziurys, L. M. Millimeter and Submillimeter Wave Spectroscopy and Astrophysical Applications. In Handbook of High-resolution Spectroscopy. John Wiley & Sons, Ltd (2011).
[80] Tielens, A. Interstellar Polycyclic Aromatic Hydrocarbon Molecules. Annu. Rev. Astron.
Astrophys., 46(1):289–337 (2008).
[81] Nesbitt, D. J. and Field, R. W. Vibrational Energy Flow in Highly Excited Molecules: Roleof Intramolecular Vibrational Redistribution. J. Phys. Chem., 100(31):12735–12756 (1996).
[82] Pirali, O., Goubet, M., Huet, T. R., Georges, R., Soulard, P., Asselin, P., Courbe, J., Roy, P.,
and Vervloet, M. The far infrared spectrum of naphthalene characterized by high resolution
synchrotron FTIR spectroscopy and anharmonic DFT calculations. Phys. Chem. Chem.
Phys., 15:10141–50 (2013).
[83] Albert, S., Albert, K. K., Lerch, P., and Quack, M. Synchrotron-based highest resolution
Fourier transform infrared spectroscopy of naphthalene (C10 H8 ) and indole (C8 H7 N) and its
application to astrophysical problems. Faraday Discuss., 150 (2011).
[84] Hewett, K. B., Shen, M., Brummel, C. L., and Philips, L. A. High resolution infrared
spectroscopy of pyrazine and naphthalene in a molecular beam. J. Chem. Phys., 100:4077(1994).
[85] Majewski, W. and Meerts, W. Near-UV spectra with fully resolved rotational structure ofnaphthalene and perdeuterated naphthalene. J. of Mol. Spectrosc., 104:271–281 (1984).
[86] Joo, D.-L., Takahashi, R., O’Reilly, J., Katô, H., and Baba, M. High-Resolution Spectroscopyof Jet-Cooled Naphthalene: the 000 and 3310 Bands of the Ã1 B1u ← X̃ 1 Ag Transition. J. Mol.
Spectrosc., 215:155–159 (2002).

155[87] Pirali, O., Boudon, V., Oomens, J., and Vervloet, M. Rotationally resolved infrared spectroscopy of adamantane. J. Chem. Phys., 136(2):024310 (2012).
[88] Brumfield, B. E., Stewart, J. T., and McCall, B. J. Extending the limits of rotationallyresolved absorption spectroscopy: pyrene. J. Phys. Chem. Lett., 3(15):1985–1988 (2012).
[89] Pirali, O., Boudon, V., Carrasco, N., and Dartois, E. Rotationally resolved IR spectroscopyof hexamethylenetetramine (HMT) C6 N4 H12 . Astron. Astrophys., 561:A109 (2014).
[90] Pirali, O. and Boudon, V. Synchrotron-based Fourier transform spectra of the ν23 and ν24
IR bands of hexamethylenetetramine C6 N4 H12 . J. Mol. Spectrosc., 315:37–40 (2015).
[91] Coveleskie, R. A., Dolson, D. A., and Parmenter, C. S. A direct view of intramolecularvibrational redistribution in S1 p-difluorobenzene. J. Chem. Phys., 72:5774–5775 (1980).
[92] Hopkins, J. B., Langridge-Smith, P. R. R., and Smalley, R. E. Ground state vibrationalrandomization in alkylbenzenes. J. Chem. Phys., 78(6):3410–3414 (1983).
[93] Parmenter, C. S. and Stone, B. M. The methyl rotor as an accelerating functional group for
IVR. J. Chem. Phys., 84(8):4710–4711 (1986).
[94] McIlroy, A. and Nesbitt, D. J. Vibrational mode mixing in terminal acetylenes: Highresolution infrared laser study of isolated J states. J. Chem. Phys., 92(4):2229–2243 (1990).
[95] McIlroy, A. and Nesbitt, D. J. Large amplitude skeletal isomerization as a promoter of
intramolecular vibrational relaxation in CH stretch excited hydrocarbons. J. Chem. Phys.,101(5):3421 (1994).
[96] Lehmann, K. K., Pate, B. H., and Scoles, G. On the measurement of the rate of intramolecular vibrational energy redistribution via high resolution molecular beam optothermal spectroscopy. Laser Chem., 11:237 (1991).
[97] Lederman, S. M., Runnels, J. H., and Marcus, R. A. Densities of vibrational states of given
symmetry species and their use in statistical estimates involving Coriolis and anharmonic
effects. J. Phys. Chem., 87:4364–4367 (1983).
[98] Quack, M. On the densities and numbers of rovibronic states of a given symmetry species:
Rigid and nonrigid molecules, transition states, and scattering channels. J. Chem. Phys.,
82:3277 (1985).
[99] Pechukas, P. Comment on “Densities of vibrational states of given symmetry species”. J.
Phys. Chem., 88(4):828–828 (1984).
[100] Baer, T. and Hase, W. L. Unimolecular Reaction Dynamics. Oxford University Press, NewYork (1996).
[101] Sellers, H., Pulay, P., and Boggs, J. E. Theoretical prediction of vibrational spectra. 2. Force
field, spectroscopically refined geometry, and reassignment of the vibrational spectrum ofnaphthalene. J. Am. Chem. Soc., 107(23):6487–6494 (1985).
[102] Mitra, S. S. and Bernstein, H. J. Vibrational spectra of naphthalene-d0 , -α-d4 , and -d8
molecules. Can. J. Chem., 37(3):553–562 (1959).

156[103] Jensen, J. O. Vibrational frequencies and structural determination of adamantane. Spectrochim. Acta A, 60(8-9):1895–905 (2004).
[104] Ramachandran, G. and Manogaran, S. Vibrational spectra of adamantanes X10 H16 anddiamantanes X14 H20 (X=C, Si, Ge, Sn): A theoretical study. THEOCHEM, 766(2-3):125–
135 (2006).
[105] Jenkins, T. and Lewis, J. A Raman study of adamantane (C10 H16 ), diamantane (C14 H20 )and triamantane (C18 H24 ) between 10 K and room temperatures. Spectrochim. Acta A,
36(3):259–264 (1980).
[106] Schettino, V., Pagliai, M., Ciabini, L., and Cardini, G. The Vibrational Spectrum of Fullerene
C60 . J. Phys. Chem. A, 105(50):11192–11196 (2001).
[107] Osawa, E. Superaromaticity. Kagaku, 25:854–863 (1970).
[108] Davidson, R. A. Spectral analysis of graphs by cyclic automorphism subgroups. Theo. Chim.
Acta, 58(3):193–231 (1981).
[109] Bochvar, D. A. and Gal’pern, E. G. . Proc. Acad. Sci. USSR, 209:239–241 (1973).
[110] Stankevich, I. V., Nikerov, M. V., and Bochvar, D. A. The Structural Chemistry of Crystalline
Carbon: Geometry, Stability, and Electronic Spectrum. Russ. Chem. Rev., 53(7):640–655(1984).
[111] Kroto, H. W., Heath, J. R., O’Brien, S. C., Curl, R. F., and Smalley, R. E. C60 : Buckminsterfullerene. Nature, 318(6042):162–163 (1985).
[112] Kroto, H. Symmetry, space, stars and C60 . Rev. Mod. Phys., 69(3):703–722 (1997).
[113] Ajie, H., Alvarez, M. M., Anz, S. J., Beck, R. D., Diederich, F., Fostiropoulos, K., Huffman, D. R., Kraetschmer, W., Rubin, Y., Schriver, K. E., Sensharma, D., and Whetten,
R. L. Characterization of the soluble all-carbon molecules C60 and C70 . J. Phys. Chem.,
94(24):8630–8633 (1990).
[114] Taylor, R., Hare, J. P., Abdul-Sada, A. K., and Kroto, H. W. Isolation, separation andcharacterisation of the fullerenes C60 and C70 : the third form of carbon. J. Chem. Soc.
Chem. Commun., 0(20):1423 (1990).
[115] Krätschmer, W., Lamb, L. D., Fostiropoulos, K., and Huffman, D. R. Solid C60 : a new formof carbon. Nature, 347(6291):354–358 (1990).
[116] Krätschmer, W., Fostiropoulos, K., and Huffman, D. R. The infrared and ultraviolet absorption spectra of laboratory-produced carbon dust: evidence for the presence of the C60
molecule. Chem. Phys. Lett., 170(2-3):167–170 (1990).
[117] Yannoni, C. S., Johnson, R. D., Meijer, G., Bethune, D. S., and Salem, J. R. Carbon-13
NMR study of the C60 cluster in the solid state: molecular motion and carbon chemical shiftanisotropy. J. Phys. Chem., 95(1):9–10 (1991).
[118] Yannoni, C. S., Bernier, P. P., Bethune, D. S., Meijer, G., and Salem, J. R. NMR determination of the bond lengths in C60 . J. Am. Chem. Soc., 113(8):3190–3192 (1991).

157[119] Hedberg, K., Hedberg, L., Bethune, D. S., Brown, C. A., Dorn, H. C., Johnson, R. D., andDe Vries, M. Bond lengths in free molecules of buckminsterfullerene, C60 , from gas-phaseelectron diffraction. Science, 254(5030):410–412 (1991).
[120] Liu, S., Lu, Y.-J., Kappes, M. M., and Ibers, J. A. The Structure of the C60 Molecule: X-Ray
Crystal Structure Determination of a Twin at 110 K. Science, 254(5030):408–410 (1991).
[121] Parker, S. F., Bennington, S. M., Taylor, J. W., Herman, H., Silverwood, I., Albers, P.,
and Refson, K. Complete assignment of the vibrational modes of C60 by inelastic neutron
scattering spectroscopy and periodic-DFT. Phys. Chem. Chem. Phys., 13(17):7789 (2011).
[122] Pintschovius, L. Neutron studies of vibrations in fullerenes. Rep. Prog. Phys., 59(4):473–510(1996).
[123] The vibrational Raman spectra of purified solid films of C60 and C70 . Chem. Phys. Lett.,
174(3-4):219–222 (1990).
[124] Bethune, D. S., Meijer, G., Tang, W. C., Rosen, H. J., Golden, W. G., Seki, H., Brown,
C. A., and de Vries, M. S. Vibrational Raman and infrared spectra of chromatographicallyseparated C60 and C70 fullerene clusters. Chem. Phys. Lett., 179(1-2):181–186 (1991).
[125] Prassides, K., John, T., Dennis, S., Hare, J. P., Tomkinson, J., Kroto, H. W., Taylor, R., andWalton, D. R. Inelastic neutron scattering spectrum of the fullerene C60 . Chem. Phys. Lett.,
187(5):455–458 (1991).
[126] Cappelletti, R. L., Copley, J. R. D., Kamitakahara, W. A., Li, F., Lannin, J. S., and Ramage,D. Neutron measurements of intramolecular vibrational modes in C60 . Phys. Rev. Lett.,
66(25):3261–3264 (1991).
[127] Affholter, K. A., Henderson, S. J., Wignall, G. D., Bunick, G. J., Haufler, R. E., and Compton,
R. N. Structural characterization of C60 and C70 fullerenes by small-angle neutron scattering.
J. Chem. Phys., 99(11):9224–9229 (1993).
[128] Sassara, A., Zerza, G., and Chergui, M. Phosphorescence of C60 in rare gas matrices. Chem.
Phys. Lett., 261:213–220 (1996).
[129] Sassara, A., Zerza, G., Chergui, M., and Leach, S. Absorption Wavelengths and Bandwidths for Interstellar Searches of C60 in the 2400−4100 ÅRegion. Astrophys. J. Supp. Ser.,135(2):263–273 (2001).
[130] Sogoshi, N., Kato, Y., Wakabayashi, T., Momose, T., Tam, S., DeRose, M. E., and Fajardo,
M. E. High-Resolution Infrared Absorption Spectroscopy of C60 Molecules and Clusters inParahydrogen Solids. J. Phys. Chem. A, 104(16):3733–3742 (2000).
[131] Iglesias-Groth, S., Cataldo, F., and Manchado, A. Infrared spectroscopy and integrated molar
absorptivity of C60 and C70 fullerenes at extreme temperatures. Mon. Not. R. Astron. Soc.,
413(1):213–222 (2011).
[132] Brieva, A. C., Gredel, R., Jäger, C., Huisken, F., and Henning, T. C60 as a probe forastrophysical environments. Astrophys. J., 826(2):122 (2016).

158[133] Absorption spectroscopy of C60 molecules isolated in helium droplets. Chem. Phys. Lett.,
276:393–398 (1997).
[134] Wang, X.-B., Ding, C.-F., and Wang, L.-S. High resolution photoelectron spectroscopy ofC−
60 . J. Chem. Phys., 110(17):8217 (1999).
[135] Wang, X.-B., Woo, H.-K., and Wang, L.-S. Vibrational cooling in a cold ion trap:
Vibrationally resolved photoelectron spectroscopy of cold C−
60 anions. J. Chem. Phys.,
123(5):051106 (2005).
[136] Huang, D.-L., Dau, P. D., Liu, H.-T., and Wang, L.-S. High-resolution photoelectron imaging
of cold C−
60 anions and accurate determination of the electron affinity of C60 . J. Chem. Phys.,140(22):224315 (2014).
[137] Cami, J., Bernard-Salas, J., Peeters, E., and Malek, S. E. Detection of C60 and C70 in ayoung planetary nebula. Science, 329(5996):1180–2 (2010).
[138] Sellgren, K., Werner, M. W., Ingalls, J. G., Smith, J. D. T., Carleton, T. M., and Joblin, C.
C60 in reflection nebulae. Astrophys. J., 722(1):L54–L57 (2010).
[139] Campbell, E. K., Holz, M., Gerlich, D., and Maier, J. P. Laboratory confirmation of C+
60 as
the carrier of two diffuse interstellar bands. Nature, 523(7560):322–323 (2015).
[140] Campbell, E. K. and Maier, J. P. Perspective: C+
60 and laboratory spectroscopy related todiffuse interstellar bands. J. Chem. Phys., 146(16):160901 (2017).
[141] Maier, J. P. and Campbell, E. K. Fullerenes in Space. Angew. Chem. Int. Ed., 56(18):4920–
4929 (2017).
[142] Gelessus, A., Thiel, W., and Weber, W. Multipoles and Symmetry. J. Chem. Ed., 72(6):505(1995).
[143] Dorney, A. and Watson, J. Forbidden rotational spectra of polyatomic molecules. J. Mol.
Spectrosc., 42(1):135–148 (1972).
[144] Orlandi, G. and Negri, F. Electronic states and transitions in C60 and C70 fullerenes. Photochem. Photobiol. Sci., 1(5):289–308 (2002).
[145] Haufler, R. E., Chai, Y., Chibante, L. P. F., Fraelich, M. R., Weisman, R. B., Curl, R. F.,
and Smalley, R. E. Cold molecular beam electronic spectrum of C60 and C70 . J. Chem. Phys.,
95(3):2197–2199 (1991).
[146] Haufler, R., Wang, L.-S., Chibante, L., Jin, C., Conceicao, J., Chai, Y., and Smalley, R.
Fullerene triplet state production and decay: R2PI probes of C60 and C70 in a supersonicbeam. Chem. Phys. Lett., 179:449–454 (1991).
[147] Stewart, J. T., Brumfield, B. E., Gibson, B. M., and McCall, B. J. Inefficient Vibrational
Cooling of C60 in a Supersonic Expansion. ISRN Phys. Chem., 2013:1–10 (2013).
[148] Hansen, K., Müller, R., Hohmann, H., and Campbell, E. Stability of clusters of fullerenes.
Z. Phys. D, 40(1-4):361–364 (1997).

159[149] Hansen, K., Hohmann, H., Müller, R., and Campbell, E. E. B. Icosahedra of icosahedra: The
stability of (C60 )13 . J. Chem. Phys., 105(14):6088 (1998).
[150] di Lauro, C. Spectra of Spherical Top Molecules. In Rotational Structure in Molecular
Infrared Spectra, pages 225–245. Elsevier (2013).
[151] Rumble, J. R., ed. CRC Handbook of Chemistry and Physics. CRC Press/Taylor & Francis,
Boca Raton, FL, 99th edition (2018).
[152] Zare, R. N. Angular Momentum. John Wiley & Sons (1988).
[153] Bunker, P. R. and Jensen, P. Spherical top molecules and the molecular symmetry group.
Mol. Phys., 97(1-2):255–264 (1999).
[154] Changala, P. B., Weichman, M. L., Lee, K. F., Fermann, M. E., and Ye, J. External data for “Rovibrational quantum state resolution of the C60 fullerene,” Version 1,
https://doi.org/10.7910/DVN/RVVAN6. Harvard Dataverse (2018).
[155] Weeks, D. E. and Harter, W. G. Rotation-vibration scalar coupling zeta coefficients andspectroscopic band shapes of buckminsterfullerene. Chem. Phys. Lett., 176:209–216 (1991).
[156] Hecht, K. T. Vibration-rotation energies of tetrahedral XY4 molecules. J. Mol. Spectrosc.,5(1-6):390–404 (1961).
[157] Valiev, M., Bylaska, E. J., Govind, N., Kowalski, K., Straatsma, T. P., van Dam, H. J. J.,
Wang, D., Nieplocha, J., Apra, E., Windus, T. L., and de Jong, W. A. NWChem: a comprehensive and scalable open-source solution for large scale molecular simulations. Comput.
Phys. Commun., 181:1477 (2010).
[158] Weeks, D. E. Isotopic and anharmonic perturbations to the dipole active vibrational modesof buckminsterfullerene. J. Chem. Phys., 96(10):7380–7393 (1992).
[159] Johnson, R. D., Meijer, G., and Bethune, D. S. C60 has icosahedral symmetry. J. Amer.
Chem. Soc., 112(24):8983–8984 (1990).
[160] Boudon, V., Champion, J.-P., Gabard, T., Loëte, M., Michelot, F., Pierre, G., Rotger, M.,
Wenger, C., and Rey, M. Symmetry-adapted tensorial formalism to model rovibrationaland rovibronic spectra of molecules pertaining to various point groups. J. Mol. Spectrosc.,228(2):620–634 (2004).
[161] Wigner, E. P. Group Theory and its Application to the Quantum Mechanics of Atomic Spectra.
Academic Press, New York (1959).
[162] Harter, W. G. and Weeks, D. E. Rovibrational spectral fine structure of icosahedral molecules.
Chem. Phys. Lett., 132(4-5):387–392 (1986).
[163] Harter, W. G. and Weeks, D. E. Rotation–vibration spectra of icosahedral molecules. I.
Icosahedral symmetry analysis and fine structure. J. Chem. Phys., 90(9):4727–4743 (1989).
[164] Georgakilas, V., Perman, J. A., Tucek, J., and Zboril, R. Broad Family of Carbon Nanoallotropes: Classification, Chemistry, and Applications of Fullerenes, Carbon Dots, Nanotubes,
Graphene, Nanodiamonds, and Combined Superstructures. Chem. Rev., 115(11):4744–4822(2015).

160[165] Harter, W. G. and Reimer, T. C. Nuclear spin weights and gas phase spectral structure of
12 C
13 C
60 and60 buckminsterfullerene. Chem. Phys. Lett., 194(3):230–238 (1992).
[166] Chen, C. C. and Lieber, C. M. Synthesis of pure 13 C60 and determination of the isotope effectfor fullerene superconductors. J. Amer. Chem. Soc., 114(8):3141–3142 (1992).
[167] Popov, A. A., Yang, S., and Dunsch, L. Endohedral Fullerenes. Chem. Rev., 113(8):5989–6113(2013).
[168] Faist, J., Capasso, F., Sivco, D. L., Sirtori, C., Hutchinson, A. L., and Cho, A. Y. Quantum
Cascade Laser. Science, 264:553–556 (1994).
[169] Curl, R. F., Capasso, F., Gmachl, C., Kosterev, A. A., McManus, B., Lewicki, R., Pusharsky,
M., Wysocki, G., and Tittel, F. K. Quantum cascade lasers in chemical physics. Chem. Phys.
Lett., 487:1–18 (2010).
[170] Faist, J., Gmachl, C., Capasso, F., Sirtori, C., Sivco, D. L., Baillargeon, J. N., and Cho, A. Y.
Distributed feedback quantum cascade lasers. Appl. Phys. Lett., 70(20):2670–2672 (1997).
[171] Berden, G. and Engeln, R., eds. Cavity ring-down spectroscopy: techniques and applications.
John Wiley & Sons (2009).
[172] Dahmani, B., Hollberg, L., and Drullinger, R. Frequency stabilization of semiconductor lasers
by resonant optical feedback. Opt. Lett., 12(11):876 (1987).
[173] Laurent, P., Clairon, A., and Breant, C. Frequency noise analysis of optically self-locked
diode lasers. IEEE J. Quantum Electron., 25(6):1131–1142 (1989).
[174] Ohshima, S. and Schnatz, H. Optimization of injection current and feedback phase of an
optically self-locked laser diode. J. Appl. Phys., 71(7):3114–3117 (1992).
[175] Burkart, J., Romanini, D., and Kassi, S. Optical feedback stabilized laser tuned by singlesideband modulation. Opt. Lett., 38:2062 (2013).
[176] Briles, T. C., Yost, D. C., Cingöz, A., Ye, J., and Schibli, T. R. Simple piezoelectric-actuatedmirror with 180 kHz servo bandwidth. Optics Express, 18(10):9739 (2010).
[177] Maisons, G., Carbajo, P. G., Carras, M., and Romanini, D. Optical-feedback cavity-enhanced
absorption spectroscopy with a quantum cascade laser. Opt. Lett., 35:3607 (2010).
[178] Hamilton, D. J. and Orr-Ewing, A. J. A quantum cascade laser-based optical feedback cavityenhanced absorption spectrometer for the simultaneous measurement of CH4 and N2 O in air.
Appl. Phys. B, 102:879–890 (2011).
[179] Gorrotxategi-Carbajo, P., Fasci, E., Ventrillard, I., Carras, M., Maisons, G., and Romanini,
D. Optical-feedback cavity-enhanced absorption spectroscopy with a quantum-cascade laser
yields the lowest formaldehyde detection limit. Appl. Phys. B, 110:309–314 (2013).
[180] Fasci, E., Coluccelli, N., Cassinerio, M., Gambetta, A., Hilico, L., Gianfrani, L., Laporta, P.,
Castrillo, A., and Galzerano, G. Narrow-linewidth quantum cascade laser at 8.6 µm. Opt.
Lett., 39:4946 (2014).

161[181] Ye, J., Ma, L.-S., and Hall, J. L. Ultrasensitive detections in atomic and molecular physics:
demonstration in molecular overtone spectroscopy. J. Opt. Soc. Am. B, 15(1):6 (1998).
[182] Martin-Drumel, M., Baraban, J. H., Changala, P. B., Stanton, J. F., and McCarthy, M. C.
The Hunt for Elusive Molecules: Insights from Joint Theoretical and Experimental Investigations. Chem.: Eur. J., 25(30):7243–7258 (2019).
[183] Bak, K. L., Gauss, J., Jørgensen, P., Olsen, J., Helgaker, T., and Stanton, J. F. The accurate
determination of molecular equilibrium structures. J. Chem. Phys., 114(15):6548–6556 (2001).
[184] Pawlowski, F., Jørgensen, P., Olsen, J., Hegelund, F., Helgaker, T., Gauss, J., Bak, K. L.,
and Stanton, J. F. Molecular equilibrium structures from experimental rotational constants
and calculated vibration–rotation interaction constants. J. Chem. Phys., 116(15):6482–6496(2002).
[185] Pfeiffer, F., Rauhut, G., Feller, D., and Peterson, K. A. Anharmonic zero point vibrationalenergies: Tipping the scales in accurate thermochemistry calculations? J. Chem. Phys.,
138(4):044311 (2013).
[186] Karton, A., Rabinovich, E., Martin, J. M. L., and Ruscic, B. W4 theory for computational thermochemistry: In pursuit of confident sub-kJ/mol predictions. J. Chem. Phys.,
125(14):144108 (2006).
[187] Tajti, A., Szalay, P. G., Császár, A. G., Kállay, M., Gauss, J., Valeev, E. F., Flowers, B. A.,
Vázquez, J., and Stanton, J. F. HEAT: High accuracy extrapolated ab initio thermochemistry.
J. Chem. Phys., 121(23):11599–11613 (2004).
[188] Born, M. and Oppenheimer, R. Zur Quantentheorie der Molekeln. Ann. Phys., 389:457–484(1927).
[189] Spectroscopy of large amplitude vibrational motions, on the occasion of Jon Hougen’s 80thbirthday - Part I. In L.-H. Xu and I. Kleiner, eds., J. Mol. Spectrosc. vol 342, pages 1–138(2017).
[190] Spectroscopy of large amplitude vibrational motions, on the occasion of Jon Hougen’s 80thbirthday - Part II. In L.-H. Xu and I. Kleiner, eds., J. Mol. Spectrosc. vol 343, pages 1–126(2018).
[191] Barone, V. Anharmonic vibrational properties by a fully automated second-order perturbative
approach. J. Chem. Phys., 122:014108 (2005).
[192] Oliveira, A. M., Lu, Y.-J., Lehman, J. H., Changala, P. B., Baraban, J. H., Stanton, J. F., andLineberger, W. C. Photoelectron Spectroscopy of the Methide Anion: Electron Affinities of−
CH3 and CD3 and Inversion Splittings of CH−
3 and CD3 . J. Am. Chem. Soc., 137(40):12939–
45 (2015).
[193] Reilly, N. J., Changala, P. B., Baraban, J. H., Kokkin, D. L., Stanton, J. F., and McCarthy,
M. C. The ground electronic state of Si2 C: Rovibrational level structure, quantum monodromy, and astrophysical implications. J. Chem. Phys., 142:231101 (2015).

162[194] McCarthy, M. C., Baraban, J. H., Changala, P. B., Stanton, J. F., Martin-Drumel, M.-A.,
Thorwirth, S., Gottlieb, C. A., and Reilly, N. J. Discovery of a Missing Link: Detection and
Structure of the Elusive Disilicon Carbide Cluster. J. Phys. Chem. Lett., 6:2107–2111 (2015).
[195] Cernicharo, J., McCarthy, M. C., Gottlieb, C. A., Agúndez, M., Prieto, L. V., Baraban, J. H.,
Changala, P. B., Guélin, M., Kahane, C., Martin-Drumel, M. A., Patel, N. A., Reilly, N. J.,
Stanton, J. F., Quintana-Lacaci, G., Thorwirth, S., and Young, K. H. Discovery of SiCSi in
IRC+10216: A Missing Link Between Gas and Dust Carriers of Si–C Bonds. Astrophys. J.
Lett., 806(1):L3 (2015).
[196] Baraban, J. H., Changala, P. B., and Stanton, J. F. The equilibrium structure of hydrogen
peroxide. J. Mol. Spectrosc., 343:92–95 (2018).
[197] Changala, P. B., Nguyen, T. L., Baraban, J. H., Ellison, G. B., Stanton, J. F., Bross, D. H.,
and Ruscic, B. Active Thermochemical Tables: The Adiabatic Ionization Energy of Hydrogen
Peroxide. J. Phys. Chem. A, 121:8799–8806 (2017).
[198] Baraban, J. H., Martin-Drumel, M.-A., Changala, P. B., Eibenberger, S., Nava, M., Patterson, D., Stanton, J. F., Ellison, G. B., and McCarthy, M. C. The Molecular Structureof gauche-1,3-Butadiene: Experimental Establishment of Non-planarity. Angew. Chem. Int.
Ed., 57(7):1821–1825 (2018).
[199] NITROGEN, Numerical and Iterative Techniques for Rovibronic Energies with General Internal Coordinates, a program by P. B. Changala, http://www.colorado.edu/nitrogen.
[200] Bowman, J. M., Carrington, T., and Meyer, H.-D. Variational quantum approaches forcomputing vibrational energies of polyatomic molecules. Mol. Phys., 106(16-18):2145–2182(2008).
[201] Carrington, T. Perspective: Computing (ro-)vibrational spectra of molecules with more thanfour atoms. J. Chem. Phys., 146(12):120902 (2017).
[202] Bramley, M. J. and Carrington, T. A general discrete variable method to calculate vibrationalenergy levels of three- and four-atom molecules. J. Chem. Phys., 99(11):8519–8541 (1993).
[203] Lehoucq, R., Gray, S., Zhang, D.-H., and Light, J. Vibrational eigenstates of four-atom
molecules: a parallel strategy employing the implicitly restarted Lanczos method. Comp.
Phys. Comm., 109(1):15–25 (1998).
[204] Yu, H.-G. and Nyman, G. A spectral transform Krylov subspace iteration approach toquantum scattering. Chem. Phys. Lett., 298(1-3):27–35 (1998).
[205] Huang, S.-W. and Carrington, T. A new iterative method for calculating energy levels andwave functions. J. Chem. Phys., 112:8765 (2000).
[206] Lanczos, C. An Iteration Method for the Solution of the Eigenvalue Problem of Linear
Differential and Integral Operators. J. Res. Natl. Bur. Stand., 45(255) (1950).
[207] Lill, J., Parker, G., and Light, J. Discrete variable representations and sudden models inquantum scattering theory. Chem. Phys. Lett., 89(6):483–489 (1982).

163[208] Light, J. C., Hamilton, I. P., and Lill, J. V. Generalized discrete variable approximation inquantum mechanics. J. Chem. Phys., 82(3):1400 (1985).
[209] Colbert, D. T. and Miller, W. H. A novel discrete variable representation for quantum
mechanical reactive scattering via the S-matrix Kohn method. J. Chem. Phys., 96:1982(1992).
[210] Mátyus, E., Šimunek, J., and Császár, A. G. On the variational computation of a large numberof vibrational energy levels and wave functions for medium-sized molecules. J. Chem. Phys.,131(7):074106 (2009).
[211] Poirier, B. and Carrington, T. Accelerating the calculation of energy levels and wave functions
using an efficient preconditioner with the inexact spectral transform method. J. Chem. Phys.,
114(21):9254 (2001).
[212] Wang, X.-G. and Carrington, T. A symmetry-adapted method for calculating energy levels
with different symmetries from a single set of iterations. J. Chem. Phys., 114(4):1473 (2001).
[213] Leclerc, A. and Carrington, T. Calculating vibrational spectra with sum of product basis functions without storing full-dimensional vectors or matrices. J. Chem. Phys., 140(17):174111(2014).
[214] Thomas, P. S., Carrington, T., Agarwal, J., and Schaefer, H. F. Using an iterative eigensolver
and intertwined rank reduction to compute vibrational spectra of molecules with more thana dozen atoms: Uracil and naphthalene. J. Chem. Phys., 149(6):064108 (2018).
[215] Carrington, T. Using Iterative Eigensolvers to Compute Vibrational Spectra. pages 217–243.
John Wiley & Sons, Ltd (2018).
[216] Eckart, C. Some Studies Concerning Rotating Axes and Polyatomic Molecules. Phys. Rev.,
47:552–558 (1935).
[217] McCoy, A. B. and Sibert III, E. L. Perturbative calculations of vibrational (J=0) energy
levels of linear molecules in normal coordinate representations. J. Chem. Phys., 95(5):3476(1991).
[218] McCoy, A. B. and Sibert III, E. L. Perturbative Studies of the Vibrations of Polyatomic
Molecules Using Curvilinear Coordinates. In Advances in Molecular Vibrations and Collision
Dynamics, Vol. 1A, pages 255–282. JAI Press (1991).
[219] Pavlyuchko, A. I., Yurchenko, S. N., and Tennyson, J. A hybrid variational-perturbation
calculation of the ro-vibrational spectrum of nitric acid. J. Chem. Phys., 142(9):094309(2015).
[220] Pavlyuchko, A. I., Yurchenko, S. N., and Tennyson, J. Hybrid variational-perturbation
method for calculating ro-vibrational energy levels of polyatomic molecules. Mol. Phys.,
113:1559–1575 (2015).
[221] Carter, S., Culik, S. J., and Bowman, J. M. Vibrational self-consistent field method for
many-mode systems: A new approach and application to the vibrations of CO adsorbed on
Cu(100). J. Chem. Phys., 107(24):10458–10469 (1997).

164[222] Christiansen, O. Møller–Plesset perturbation theory for vibrational wave functions. J. Chem.
Phys., 119:5773–5781 (2003).
[223] Carter, S. and Handy, N. C. The vibrations of H2 O2 , studied by “multimode,” with a large
amplitude motion. J. Chem. Phys., 113:987–993 (2000).
[224] Bowman, J. M., Huang, X., Handy, N. C., and Carter, S. Vibrational Levels of Methanol Calculated by the Reaction Path Version of MULTIMODE, Using an ab initio, Full-Dimensional
Potential. J. Phys. Chem. A, 111:7317–7321 (2007).
[225] Horn, T. R., Gerber, R. B., and Ratner, M. A. Vibrational states of very floppy clusters:
Approximate separability and the choice of good curvilinear coordinates for XeHe2 , I2 He. J.
Chem. Phys., 91(3):1813–1823 (1989).
[226] Zuñiga, J., Bastida, A., Requena, A., and Hidalgo, A. Self-consistent-field calculation ofvibrational bound states for triatomic molecules using transformed Jacobi coordinates. J.
Phys. Chem., 95:2292–2297 (1991).
[227] Bounouar, M. and Scheurer, C. The impact of approximate VSCF schemes and curvilinear coordinates on the anharmonic vibrational frequencies of formamide and thioformamide.
Chem. Phys., 347(1):194–207 (2008).
[228] Scribano, Y., Lauvergnat, D. M., and Benoit, D. M. Fast vibrational configuration interaction using generalized curvilinear coordinates and self-consistent basis. J. Chem. Phys.,
133(9):094103 (2010).
[229] Strobusch, D. and Scheurer, C. Hierarchical expansion of the kinetic energy operator in curvilinear coordinates for the vibrational self-consistent field method. J. Chem. Phys., 135:124102(2011).
[230] Strobusch, D. and Scheurer, C. The hierarchical expansion of the kinetic energy operator
in curvilinear coordinates extended to the vibrational configuration interaction method. J.
Chem. Phys., 135:144101 (2011).
[231] Van Vleck, J. H. On σ-Type Doubling and Electron Spin in the Spectra of Diatomic Molecules.
Phys. Rev., 33:467–506 (1929).
[232] Lukka, T. J. and Kauppi, E. Seminumerical contact transformations: From internal coordinate rovibrational Hamiltonian to effective rotational Hamiltonians. Framework of themethod. J. Chem. Phys., 103(15):6586 (1995).
[233] Field, R. W., Baraban, J. H., Lipoff, S. H., and Beck, A. R. Effective Hamiltonians forElectronic Fine Structure and Polyatomic Vibrations. In M. Quack and F. Merkt, eds.,
Handbook of High-Resolution Spectroscopy, page 1461. John Wiley & Sons, Chichester, UK(2011).
[234] Gordy, W. and Cook, R. L. Microwave Molecular Spectra. John Wiley & Sons, New York,
3rd edition (1984).
[235] Exact quantum molecular Hamiltonians. Mol. Phys., 72:1–31 (1991).

165[236] Lukka, T. J. A simple method for the derivation of exact quantum-mechanical vibrationrotation Hamiltonians in terms of internal coordinates. J. Chem. Phys., 102:3945–3955 (1995).
[237] Lauvergnat, D. and Nauts, A. Exact numerical computation of a kinetic energy operator incurvilinear coordinates. J. Chem. Phys., 116(19):8560 (2002).
[238] Watson, J. K. G. The molecular vibration-rotation kinetic-energy operator for general internal
coordinates. J. Mol. Spectrosc., 228:645–658 (2004).
[239] Mátyus, E., Czakó, G., and Császár, A. G. Toward black-box-type full- and reduceddimensional variational (ro)vibrational computations. J. Chem. Phys., 130(13):134112 (2009).
[240] Fábri, C., Mátyus, E., and Császár, A. G. Rotating full- and reduced-dimensional quantum
chemical models of molecules. J. Chem. Phys., 134(7):074105 (2011).
[241] Fábri, C., Mátyus, E., and Császár, A. G. Numerically constructed internal-coordinate Hamiltonian with Eckart embedding and its application for the inversion tunneling of ammonia.
Spectrochim. Acta A, 119:84–9 (2014).
[242] Szalay, V. Aspects of the Eckart frame ro-vibrational kinetic energy operator. J. Chem.
Phys., 143(6):064104 (2015).
[243] Podolsky, B. Quantum-Mechanically Correct Form of Hamiltonian Function for Conservative
Systems. Phys. Rev., 32(5):812–816 (1928).
[244] Dirac, P. A. M. The Physical Interpretation of the Quantum Dynamics. Proc. Roy. Soc. A,
113:621–641 (1927).
[245] Mladenovic, M. Rovibrational Hamiltonians for general polyatomic molecules in sphericalpolar parametrization. I. Orthogonal representations. J. Chem. Phys., 112:1070 (2000).
[246] Gatti, F. and Iung, C. Exact and constrained kinetic energy operators for polyatomic
molecules: The polyspherical approach. Phys. Rep., 484(1):1–69 (2009).
[247] Wilson Jr., E. B., Decius, J. C., and Cross, P. C. Molecular Vibrations. Dover, New York(1980).
[248] Kraka, E. Reaction path Hamiltonian and the unified reaction valley approach. Comp. Mol.
Sci., 1(4):531–556 (2011).
[249] Nauts, A. and Chapuisat, X. The description of molecular large-amplitude motions in termsof curvilinear coordinates associated with the reaction path. Chem. Phys., 76(3):349–366(1983).
[250] Miller, W. H., Handy, N. C., and Adams, J. E. Reaction path Hamiltonian for polyatomic
molecules. J. Chem. Phys., 72:99 (1980).
[251] Page, M. and McIver, J. W. On evaluating the reaction path Hamiltonian. J. Chem. Phys.,
88:922 (1988).
[252] Jackels, C. F., Gu, Z., and Truhlar, D. G. Reaction-path potential and vibrational frequencies
in terms of curvilinear internal coordinates. J. Chem. Phys., 102(8):3188 (1995).

166[253] Szalay, V. Eckart ro-vibrational Hamiltonians via the gateway Hamilton operator: Theory
and practice. J. Chem. Phys., 146(12):124107 (2017).
[254] Lauvergnat, D., Luis, J. M., Kirtman, B., Reis, H., and Nauts, A. Numerical and exact kinetic
energy operator using Eckart conditions with one or several reference geometries: Application
to HONO. J. Chem. Phys., 144(8):084116 (2016).
[255] Dymarsky, A. Y. and Kudin, K. N. Computation of the pseudorotation matrix to satisfy the
Eckart axis conditions. J. Chem. Phys., 122(12):124103 (2005).
[256] Yachmenev, A. and Yurchenko, S. N. Automatic differentiation method for numerical construction of the rotational-vibrational Hamiltonian as a power series in the curvilinear internal
coordinates using the Eckart frame. J. Chem. Phys., 143(1):014105 (2015).
[257] Krasnoshchekov, S. V., Isayeva, E. V., and Stepanov, N. F. Determination of the Eckart
molecule-fixed frame by use of the apparatus of quaternion algebra. J. Chem. Phys.,
140(15):154104 (2014).
[258] Raghavachari, K., Trucks, G. W., Pople, J. A., and Head-Gordon, M. A fifth-order perturbation comparison of electron correlation theories. Chem. Phys. Lett., 157(6):479–483(1989).
[259] Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms
boron through neon and hydrogen. J. Chem. Phys., 90(2):1007 (1989).
[260] Almlöf, J. and Taylor, P. R. General contraction of Gaussian basis sets. I. Atomic naturalorbitals for first- and second-row atoms. J. Chem. Phys., 86(7):4070 (1987).
[261] McCaslin, L. and Stanton, J. Calculation of fundamental frequencies for small polyatomic
molecules: a comparison between correlation consistent and atomic natural orbital basis sets.
Mol. Phys., 111(9-11):1492–1496 (2013).
[262] CFOUR, Coupled-Cluster techniques for Computational Chemistry, a quantum-chemical program package by J.F. Stanton, J. Gauss, M.E. Harding, P.G. Szalay with contributions from
A.A. Auer, R.J. Bartlett, U. Benedikt, C. Berger, D.E. Bernholdt, Y.J. Bomble, L. Cheng,
O. Christiansen, M. Heckert, O. Heun, C. Huber, T.-C. Jagau, D. Jonsson, J. Jusélius, K.
Klein, W.J. Lauderdale, F. Lipparini, D.A. Matthews, T. Metzroth, L.A. Mück, D.P. O’Neill,
D.R. Price, E. Prochnow, C. Puzzarini, K. Ruud, F. Schiffmann, W. Schwalbach, C. Simmons, S. Stopkowicz, A. Tajti, J. Vázquez, F. Wang, J.D. Watts and the integral packages
MOLECULE (J. Almlöf and P.R. Taylor), PROPS (P.R. Taylor), ABACUS (T. Helgaker,
H.J. Aa. Jensen, P. Jørgensen, and J. Olsen), and ECP routines by A. V. Mitin and C. van
Wüllen. For the current version, see http://www.cfour.de.
[263] Braams, B. J. and Bowman, J. M. Permutationally invariant potential energy surfaces inhigh dimensionality. Int. Rev. Phys. Chem., 28(4):577–606 (2009).
[264] Gong, J. Z., Matthews, D. A., Changala, P. B., and Stanton, J. F. Fourth-order vibrational perturbation theory with the Watson Hamiltonian: Report of working equations andpreliminary results. J. Chem. Phys., 149(11):114102 (2018).
[265] Squires, R. R. Gas-phase carbanion chemistry. Acc. Chem. Res., 25(10):461–467 (1992).

167[266] DePuy, C. H., Gronert, S., Barlow, S. E., Bierbaum, V. M., and Damrauer, R. The gas-phase
acidities of the alkanes. J. Am. Chem. Soc., 111(6):1968–1973 (1989).
[267] Špirko, V. and Bunker, P. The inversion potential and rotation-inversion energy levels of
H3 O+ and CH−
3 . J. Mol. Spectrosc., 95:226–235 (1982).
[268] Ellison, G. B., Engelking, P. C., and Lineberger, W. C. An experimental determination of thegeometry and electron affinity of methyl radical. J. Am. Chem. Soc., 100:2556–2558 (1978).
[269] Graul, S. T. and Squires, R. R. Gas-phase reactions of the methyl anion. J. Am. Chem. Soc.,
111:892–899 (1989).
[270] Mitchell, S. E., Conklin, P. M., and Farley, J. W. First observation of autodetachment
lifetimes of methide, CH−
3 . J. Chem. Phys., 118:11017 (2003).
[271] Schwenke, D. W. A theoretical study of the ro-vibrational spectrum of the X state of CH3 .
Spectrochim. Acta A, 55(3):731–738 (1999).
[272] Cleeton, C. E. and Williams, N. H. Electromagnetic Waves of 1.1 cm Wave-Length and theAbsorption Spectrum of Ammonia. Phys. Rev., 45:234–237 (1934).
[273] Gordon, J. P., Zeiger, H. J., and Townes, C. H. The Maser—New Type of Microwave Amplifier, Frequency Standard, and Spectrometer. Phys. Rev., 99:1264–1274 (1955).
[274] Yu, S., Drouin, B. J., Pearson, J. C., and Pickett, H. M. Terahertz Spectroscopy and Global
Analysi of H3 O+ . Astrophys. J. Suppl. S., 180(1):119–124 (2009).
[275] Rajamäki, T., Miani, A., and Halonen, L. Six-dimensional ab initio potential energy surfaces
for H3 O+ and NH3 : Approaching the sub-wavenumber accuracy for the inversion splittings.
J. Chem. Phys., 118(24):10929–10938 (2003).
[276] McCarthy, M. C., Gottlieb, C. A., and Cernicharo, J. Building blocks of dust: A coordinatedlaboratory and astronomical study of the archtype AGB carbon star IRC+10216. Journal of
Molecular Spectroscopy, 356:7–20 (2019).
[277] Zetterling, C.-M., ed. Process Technology for Silicon Carbide Devices. IET, The Institutionof Engineering and Technology, Michael Faraday House, Six Hills Way, Stevenage SG1 2AY,
UK (2002).
[278] Photolysis experiments on SiC mixed clusters: From silicon carbide clusters to silicon-doped
fullerenes. J. Chem. Phys., 110:6927–6938 (1999).
[279] Takano, S., Saito, S., and Tsuji, T. Silicon Chemistry in the Outer Envelope of IRC-10216:
The Detailed Distribution of SiC2 . Publ. Astron. Soc. Japan, 44:469–480 (1992).
[280] Spielfiedel, A., Carter, S., Feautrier, N., Chambaud, G., and Rosmus, P. Theoretical Studyof the Electronic States of Si2 C. J. Phys. Chem., 100:10055–10061 (1996).
[281] Barone, V., Jensen, P., and Minichino, C. Vibro-rotational analysis of Si2 C from an ab initio
potential energy surface. A comparison between perturbative and variational methods. J.
Mol. Spectrosc., 154:252–264 (1992).

168[282] Koput, J. Ab Initio Potential Energy Surface and Vibration-Rotation Energy Levels of Disilicon Carbide, CSi2 . J. Mol. Spectrosc., 342:83–91 (2017).
[283] Bunker, P. R. Quasilinear and Quasiplanar Molecules. Ann. Rev. Phys. Chem., 34:59–75(1983).
[284] Kafafi, Z. H., Hauge, R. H., Fredin, L., and Margrave, J. L. Infrared matrix isolation spectrumof the disilicon carbide (Si2 C) molecule. J. Phys. Chem., 87:797–800 (1983).
[285] Presilla-Márquez, J. D. and Graham, W. R. M. Fourier transform vibrational spectroscopyof Si2 C in solid Ar. J. Chem. Phys., 95:5612–5617 (1991).
[286] Steglich, M. and Maier, J. P. Electronic transitions of jet-cooled SiC2 , Si2 C2 (n = 1−3), Si3 Cn(n = 1, 2), AND SiC6 H4 between 250 and 710 nm. The Astrophysical Journal, 801(2):119(2015).
[287] Dixon, R. N. Higher vibrational levels of a bent triatomic molecule. Trans. Faraday Soc.,
60:1363 (1964).
[288] Baraban, J. H., Changala, P. B., Mellau, G. C., Stanton, J. F., Merer, A. J., and Field, R. W.
Spectroscopic characterization of isomerization transition states. Science, 350(6266):1338–
1342 (2015).
[289] Child, M. Quantum monodromy and molecular spectroscopy. Contemp. Phys., 55:212–221(2014).
[290] Cernicharo, J., Guélin, M., Agúndez, M., Pardo, J. R., Massalkhi, S., Fonfrı́a, J. P., Velilla
Prieto, L., Quintana-Lacaci, G., Marcelino, N., Marka, C., Navarro, S., and Kramer, C. IRC+10216 as a spectroscopic laboratory: improved rotational constants for SiC2 , its isotopologues, and Si2 C. Astron. Astrophys., 618:A4 (2018).
[291] Witsch, D., Lutter, V., Breier, A. A., Yamada, K. M. T., Fuchs, G. W., Gauss, J., and Giesen,
T. F. Infrared Spectroscopy of Disilicon-Carbide, Si2 C: The ν3 Fundamental Band. J. Phys.
Chem. A, 123(19):4168–4177 (2019).
[292] Ruscic, B., Pinzon, R. E., Morton, M. L., von Laszevski, G., Bittner, S. J., Nijsure, S. G.,
Amin, K. A., Minkoff, M., and Wagner, A. F. Introduction to Active Thermochemical Tables:
Several “Key” Enthalpies of Formation Revisited. J. Phys. Chem. A, 108:9979–9997 (2004).
[293] Ruscic, B., Pinzon, R. E., von Laszewski, G., Kodeboyina, D., Burcat, A., Leahy, D., Montoy,
D., and Wagner, A. F. Active Thermochemical Tables: thermochemistry for the 21st century.
J. Phys. Conf. Ser., 16:561–570 (2005).
[294] Schio, L., Alagia, M., Dias, A. A., Falcinelli, S., Zhaunerchyk, V., Lee, E. P. F., Mok, D.
K. W., Dyke, J. M., and Stranges, S. A Study of H2 O2 with Threshold Photoelectron
Spectroscopy (TPES) and Electronic Structure Calculations: Redetermination of the First
Adiabatic Ionization Energy (AIE). J. Phys. Chem. A (2016).
[295] Ruscic, B. and Bross, D. H. Active Thermochemical Tables (ATcT) values based on version
1.122 of the Thermochemical Network. 2016; online at http://ATcT.anl.gov.

169[296] Litorja, M. and Ruscic, B. A photoionization study of the hydroperoxyl radical, HO2 , andhydrogen peroxide, H2 O2 ,. J. Electron Spectros. Relat. Phenomena, 97(1-2):131–146 (1998).
[297] Ashmore, F. S. and Burgess, A. R. Study of some medium size alcohols and hydroperoxides
by photoelectron spectroscopy. J. Chem. Soc., Faraday Trans., 73:1247 (1977).
[298] Koput, J., Carter, S., and Handy, N. C. Ab initio prediction of the vibrational-rotationalenergy levels of hydrogen peroxide and its isotopomers. J. Chem. Phys., 115(18):8345–8350(2001).
[299] Malyszek, P. and Koput, J. Accurate ab initio potential energy surface and vibration-rotationenergy levels of hydrogen peroxide. J. Comp. Chem., 34(5):337–345 (2013).
[300] Stanton, J. F. and Gauss, J. Analytic energy derivatives for ionized states described by theequation-of-motion coupled cluster method. J. Chem. Phys., 101(10):8938–8944 (1994).
[301] Khachkuruzov, G. A. and Przhevalskii, I. N. Molecular constants of hydrogen peroxide. 4:
Structural parameters. Opt. Spektrosk., 36(172-174) (1974).
[302] Cremer, D. and Christen, D. On the r0 - and re -structures of H2 O2 . J. Mol. Spectrosc.,74(3):480–482 (1979).
[303] Koput, J. On the structure and the torsional potential function of hydrogen peroxide. J.
Mol. Spectrosc., 115:438–441 (1986).
[304] Helminger, P., Bowman, W. C., and De Lucia, F. C. A study of the rotational-torsional
spectrum of hydrogen peroxide between 80 and 700 GHz. J. Mol. Spectrosc., 85(1):120–130(1981).
[305] Bowman, W. C., De Lucia, F. C., and Helminger, P. The higher K−1 states of hydrogen
peroxide. J. Mol. Spectrosc., 87:571–574 (1981).
[306] Hillman, J. J. On the submillimeter spectrum of hydrogen peroxide. J. Mol. Spectrosc.,
95:236–238 (1982).
[307] Flaud, J. M., Johns, J. W., Lu, Z., Winnewisser, G., and Klein, H. The torsion-rotation
spectrum of D2 O2 . Can. J. Phys., 79:367–374 (2001).
[308] Herberth, D., Baum, O., Pirali, O., Roy, P., Thorwirth, S., Yamada, K. M., Schlemmer, S.,
and Giesen, T. F. Far infrared Fourier-transform spectroscopy of mono-deuterated hydrogen
peroxide HOOD. J. Quant. Spec. Rad. Transf., 113:1127–1133 (2012).
[309] Koput, J. An ab initio study on the equilibrium structure and torsional potential energyfunction of hydrogen peroxide. Chem. Phys. Lett., 236:516–520 (1995).
[310] Császár, A. G., Czakó, G., Furtenbacher, T., Tennyson, J., Szalay, V., Shirin, S. V., Zobov,
N. F., and Polyansky, O. L. On equilibrium structures of the water molecule. J. Chem. Phys.,
122(21):214305 (2005).
[311] Amano, T. Difference frequency laser spectroscopy of OH and OD: Simultaneous fit of theinfrared and microwave lines. J. Mol. Spectrosc., 103:436–454 (1984).

170[312] Diels, O. and Alder, K. Synthesen in der hydroaromatischen Reihe. Justus Liebigs Ann.
Chem., 460:98–122 (1928).
[313] Rice, J. E., Liu, B., Lee, T. J., and McMichael Rohlfing, C. The structure of cis-butadiene.
Chem. Phys. Lett., 161(3):277–284 (1989).
[314] Alberts, I. L. and Schaeffer, H. F. The second stable conformer of 1,3-butadiene. Geometry
optimizations with configuration interaction and coupled cluster methods. Chem. Phys. Lett.,
161(4-5):375–382 (1989).
[315] Guo, H. and Karplus, M. Ab initio studies of polyenes. I. 1,3-butadiene. J. Chem. Phys.,
94:3679–3699 (1991).
[316] Panchenko, Y. N. and De Maré, G. R. Vibrational analysis of buta-1,3-diene and its deutero
and 13C derivatives and some of their rotational isomers. J. Struct. Chem., 49:235–244 (2008).
[317] Feller, D. and Craig, N. C. High level ab initio energies and structures for the rotamers of
1,3-butadiene. J. Phys. Chem. A, 113:1601–7 (2009).
[318] Engeln, R., Consalvo, D., and Reuss, J. Evidence for a gauche minor conformer of 1,3butadiene. Chem. Phys., 160:427–433 (1992).
[319] Boopalachandran, P., Craig, N., Groner, P., and Laane, J. Gas-phase Raman spectra andthe potential energy function for the internal rotation of 1,3-butadiene and its isotopologues.
J. Phys. Chem. A, 115:8920–7 (2011).
[320] Boopalachandran, P., Craig, N. C., and Laane, J. Gas-phase Raman spectra of s-trans- ands-gauche-1,3-butadiene and their deuterated isotopologues. J. Phys. Chem. A, 116:271–81(2012).
[321] Krasnoshchekov, S. V., Craig, N. C., Boopalachandran, P., Laane, J., and Stepanov, N. F.
Anharmonic Vibrational Analysis of the Infrared and Raman Gas-Phase Spectra of s-transand s-gauche-1,3-Butadiene. J. Phys. Chem. A, 119:10706–10723 (2015).
[322] Saltiel, J., Sears, D. F., and Turek, A. M. UV Spectrum of the High Energy Conformer of
1,3-Butadiene in the Gas Phase. J. Phys. Chem. A, 105:7569–7578 (2001).
[323] Porterfield, J. P., Westerfield, J. H., Satterthwaite, L., Patterson, D., Changala, P. B., Baraban, J. H., and McCarthy, M. C. Rotational Characterization of the Elusive gauche-Isoprene.
J. Phys. Chem. Lett., 10:1981–1985 (2019).
[324] Wiberg, K. B., Rablen, P. R., and Baraban, J. H. Butadiene and Heterodienes Revisited. J.
Org. Chem., 83:8473–8482 (2018).
[325] Reinhardt, W. P. Complex Coordinates in the Theory of Atomic and Molecular Structure
and Dynamics. Ann. Rev. Phys. Chem., 33:223–255 (1982).
[326] Christoffel, K. M. and Bowman, J. M. Complex coordinate, self-consistent field calculationsof vibrational resonance energies. J. Chem. Phys., 76:5370–5374 (1982).
[327] Seideman, T. and Miller, W. H. Transition state theory, Siegert eigenstates, and quantum
mechanical reaction rates. J. Chem. Phys., 95(3):1768–1780 (1991).

171[328] Rom, N., Ryaboy, V., and Moiseyev, N. Cumulative reaction probability by the complex
coordinate scattering theory. J. Chem. Phys., 98:6327–6331 (1993).
[329] Ryaboy, V. and Moiseyev, N. Cumulative reaction probability from Siegert eigenvalues:
Model studies. J. Chem. Phys., 98:9618–9623 (1993).
[330] Roy, T. K. and Prasad, M. D. A thermal self-consistent field theory for the calculation ofmolecular vibrational partition functions. J. Chem. Phys., 131:114102 (2009).
[331] McCoy, A. B. Diffusion Monte Carlo approaches for investigating the structure and vibrational spectra of fluxional systems. Int. Rev. Phys. Chem., 25:77–107 (2006).
[332] Petit, A. S. and McCoy, A. B. Diffusion Monte Carlo in internal coordinates. J. Phys. Chem.
A, 117:7009–18 (2013).
[333] PGOPHER, A Program for Simulating Rotational, Vibrational and Electronic Spectra, C.
M. Western, University of Bristol, http://pgopher.chm.bris.ac.uk.
[334] de Kerckhove Varent, D. Contribution a l’etude de la molecule de bromure de vinyle enspectroscopie hertzienne. II. Bromure de vinyle monodeutere et substitue en 13 C, second
ordre du couplage quadripolaire, structure de la molecule. Ann. Soc. Sci. Brux., T84:277–292(1970).
[335] Hayashi, M., Ikeda, C., and Inagusa, T. Microwave spectrum, structure, and nuclear
quadrupole coupling constant tensor of vinyl bromide and iodide. J. Mol. Spectrosc.,139(2):299–312 (1990).

Appendix A

Infrared line lists

A.1

Nitromethane, CH3 NO2Tables A.1 and A.2 list the rovibrational transition frequencies for the m = 0 and |m| = 1

components, respectively, of the ν3 + ν6 band (Section 3.2.2). Tables A.3 and A.4 list the m = 0transitions for the ν1 and ν10o bands (Section 3.2.3).
Table A.1: Line list for the 2953 cm−1 band (m = 0) of nitromethane, assigned as ν3 + ν6 .
Transitions are specified with the lower state (00 ) and upper state (0 ) asymmetric top quantum
numbers J, Ka , and Kc . E 00 (E 0 ) is the lower (upper) state energy in cm−1 . An offset of 2950 cm−1
has been subtracted from the IR line positions and upper state energies. “Component 1” and“component 2” refer to members of a mixed bright state/dark state doublet.
E0
m0 J 0 Ka0 Kc0 Line Pos.
0 1 1 10.89693.31820 1 1 1
1.76903.31860 1 1 1
3.31843.31840 1 1 01.1362
3.46400 1 1 02.9161
3.46380 2 1 2−0.0584
4.26370 2 1 2
1.32874.26380 2 1 2
1.93584.26370 2 1 2
3.71594.26360 2 1 10.72204.69290 2 1 12.27194.69320 2 1 1
3.14374.69340 3 1 30.84035.54360 3 1 3
1.57305.54400 3 1 33.99445.5440(continued on next page)a Ground state energies from Ref. [52].
m00 J 00 Ka00 Kc00
0 2 2 00 2 0 20 0 0 00 2 2 10 1 0 10 3 2 10 3 0 30 2 2 10 1 0 10 3 2 20 2 2 00 2 0 20 4 0 40 3 2 20 2 0 2E 00a
2.421311
1.549651
0.000000
2.327828
0.547713
4.322080
2.9351102.327828
0.547713
3.970934
2.421311
1.549651
4.703276
3.970934
1.549651

Comment

component 1

173

Table A.1: (continued)
m00 J 00 Ka00 Kc00
0 4 0 40 3 2 20 2 0 20 4 2 30 3 2 10 3 0 30 2 2 1
0 4 2 30 3 2 10 3 0 30 2 2 1
0 4 2 20 4 4 00 3 2 20 2 2 00 4 4 10 3 2 10 2 2 10 5 0 50 4 2 30 3 0 30 5 2 40 4 2 20 4 0 40 3 2 20 5 2 40 4 2 20 4 0 40 3 2 20 5 4 1
0 5 2 30 4 4 1
0 4 2 30 3 2 10 5 4 20 4 4 00 4 2 20 3 2 20 6 0 60 5 2 40 4 0 40 6 2 50 5 2 3E 00a
4.703276
3.970934
1.549651
6.082186
4.322080
2.9351102.327828
6.082186
4.322080
2.9351102.327828
6.839377
8.311898
3.970934
2.421311
8.295303
4.322080
2.327828
6.860890
6.082186
2.9351108.619788
6.839377
4.703276
3.970934
8.619788
6.839377
4.703276
3.970934
11.352928
9.856035
8.295303
6.082186
4.322080
11.234530
8.311898
6.839377
3.970934
9.409937
8.619788
4.703276
11.560383
9.856035

m0 J 0 Ka0 Kc0 Line Pos.
E0
0 3 1 3
0.97695.68020 3 1 31.70965.68050 3 1 34.13105.68060 3 1 20.33266.41480 3 1 22.09306.41510 3 1 23.47996.41500 3 1 24.08696.41470 3 1 20.4721
6.55430 3 1 22.23276.55470 3 1 2
3.6193
6.55440 3 1 24.22656.55440 3 3 10.66787.50720 3 3 1−0.8046
7.50730 3 3 1
3.5361
7.50700 3 3 15.08607.50730 3 3 0−0.7425
7.55280 3 3 03.23097.55300 3 3 05.22487.55260 4 1 40.4886
7.34950 4 1 4
1.2671
7.34930 4 1 44.41457.34960 4 1 3−0.0580
8.56180 4 1 31.72228.56160 4 1 33.85848.56170 4 1 34.59088.56180 4 1 30.11858.73830 4 1 3
1.89908.73830 4 1 34.03518.73840 4 1 34.76798.73880 4 3 2
−1.55639.79660 4 3 2−0.0594
9.79660 4 3 2
1.50139.79660 4 3 2
3.71489.79690 4 3 25.47469.79670 4 3 1−1.2089 10.0256
0 4 3 11.7136 10.0255
0 4 3 13.1866 10.0260
0 4 3 1
6.0545 10.0255
0 5 1 50.08839.49830 5 1 50.87839.49810 5 1 54.79539.49850 5 1 4−0.3965 11.1639
0 5 1 41.3077 11.1637(continued on next page)a Ground state energies from Ref. [52].

Comment
component 2

component 1

component 2

weak

component 1, blended

component 2

component 1

174

Table A.1: (continued)
m00 J 00 Ka00 Kc00
0 5 0 50 4 2 30 6 2 50 5 2 30 5 0 50 4 2 30 6 4 20 6 2 40 5 4 20 5 2 40 4 2 20 6 4 30 5 4 1
0 5 2 30 4 4 1
0 4 2 30 6 6 00 5 4 20 4 4 00 6 6 10 5 4 10 4 4 10 7 0 70 6 2 50 5 0 50 7 0 70 6 2 50 5 0 50 7 2 60 6 2 40 6 0 60 5 2 40 7 2 50 6 4 30 6 2 50 5 2 30 7 4 40 6 4 20 6 2 40 5 4 20 5 2 40 7 4 40 6 4 2E 00a
6.860890
6.082186
11.560383
9.856035
6.860890
6.082186
15.137207
13.256243
11.234530
8.619788
6.839377
14.743171
11.352928
9.856035
8.295303
6.082186
17.788197
11.234530
8.311898
17.786379
11.352928
8.295303
12.350829
11.560383
6.860890
12.350829
11.560383
6.860890
14.895396
13.256243
9.409937
8.619788
17.012438
14.743171
11.560383
9.856035
18.767500
15.137207
13.256243
11.234530
8.619788
18.767500
15.137207

m0 J 0 Ka0 Kc0 Line Pos.
E0
0 5 1 44.3030 11.1639
0 5 1 45.0816 11.1638
0 5 1 4−0.2190 11.3414
0 5 1 41.4853 11.3413
0 5 1 44.4808 11.3416
0 5 1 45.2591 11.3413
0 5 3 3
−2.5377 12.5996
0 5 3 3−0.6579 12.5984
0 5 3 31.3649 12.5994
0 5 3 33.9795 12.5993
0 5 3 35.7600 12.5994
0 5 3 2
−1.5973 13.1459
0 5 3 2
1.7934 13.1463
0 5 3 23.2907 13.1467
0 5 3 24.8515 13.1468
0 5 3 2
7.0647 13.1468
0 5 5 1−2.6539 15.1343
0 5 5 13.9001 15.1346
0 5 5 1
6.8224 15.1343
0 5 5 0−2.6450 15.1414
0 5 5 03.7881 15.1411
0 5 5 06.8458 15.1411
0 6 1 6−0.3695 11.9813
0 6 1 60.4209 11.9812
0 6 1 65.1207 11.9816
0 6 1 6−0.2185 12.1324
0 6 1 6
0.5715 12.1319
0 6 1 6
5.2709 12.1318
0 6 1 5−0.7442 14.1512
0 6 1 50.8948 14.1511
0 6 1 54.7417 14.1516
0 6 1 55.5319 14.1517
0 6 3 4−1.1530 15.8594
0 6 3 41.1160 15.8591
0 6 3 44.2986 15.8590
0 6 3 46.0030 15.8591
0 6 3 30 6 3 31.7430 16.8802
0 6 3 33.6240 16.8802
0 6 3 35.6459 16.8805
0 6 3 30 6 3 30 6 3 31.6553 16.7925(continued on next page)a Ground state energies from Ref. [52].

Comment

component 2

blendedblendedcomponent 1

component 2, overlap

weak

175

Table A.1: (continued)
m00 J 00 Ka00 Kc00
E 00a m0 J 0 Ka0 Kc0
0 6 2 4 13.256243
0 6 3 30 5 4 2 11.234530
0 6 3 30 5 2 48.619788
0 6 3 30 7 6 1 21.948226
0 6 5 20 6 6 1 17.786379
0 6 5 20 6 4 3 14.743171
0 6 5 20 5 4 1 11.352928
0 6 5 20 7 6 2 21.927772
0 6 5 10 6 6 0 17.788197
0 6 5 1
0 6 4 2 15.137207
0 6 5 10 5 4 2 11.234530
0 6 5 10 8 0 8 15.683641
0 7 1 70 7 2 6 14.895396
0 7 1 70 6 0 69.409937
0 7 1 70 8 2 7 18.622625
0 7 1 60 7 2 5 17.012438
0 7 1 60 7 0 7 12.350829
0 7 1 60 6 2 5 11.560383
0 7 1 60 8 2 6 21.143107
0 7 3 50 7 4 4 18.767500
0 7 3 50 7 2 6 14.895396
0 7 3 50 6 2 4 13.256243
0 7 3 50 7 4 3 19.614558
0 7 3 40 7 2 5 17.012438
0 7 3 40 6 4 3 14.743171
0 7 3 40 7 4 3 19.614558
0 7 3 40 7 2 5 17.012438
0 7 3 40 6 4 3 14.743171
0 7 3 40 8 8 0 30.831142
0 7 7 10 6 6 0 17.788197
0 7 7 10 8 8 1 30.830978
0 7 7 00 7 6 1 21.948226
0 7 7 00 6 6 1 17.786379
0 7 7 00 9 0 9 19.408388
0 8 1 80 7 0 7 12.350829
0 8 1 8a Ground state energies from Ref. [52].

Line Pos.
3.5361E0
16.7924−3.28090.88073.9241
7.3146−3.2034
0.9366
3.58727.4901−0.73910.04925.5346−1.1604
0.44985.11175.9019−1.5966
0.77824.6502
6.28961.56844.17076.4402
1.36253.96586.2355−4.56238.4802−4.5623
4.32118.4830−1.14785.909118.6673
18.6671
18.6673
18.6675
18.724418.7248
18.724418.7247
14.9445
14.9445
14.9445
17.462217.4623
17.4625
17.4623
19.5465
19.5457
19.5456
19.5458
21.1830
21.1832
21.1834
20.9771
20.9782
20.9787
26.2688
26.2684
26.2687
26.269326.269418.2606
18.2599

Comment
overlap

blendedtentative

blendedtentative

tentative

very weak, blended
very weak, blended

tentative
tentative

176
Table A.2: Line list for the 2953 cm−1 band (|m| = 1) of nitromethane, assigned as ν3 + ν6 .
Transitions are specified with the lower state (00 ) and upper state (0 ) J quantum number and τ , anenergy ordering index for a given J and m (counting only non-zero spin-weighted levels). E 00 (E 0 )is the lower (upper) state energy in cm−1 . An offset of 2950 cm−1 has been subtracted from theIR line positions and upper state energies.
|m00 | J 00 τ 00
E 00a |m0 | J 0 τ 0 Line Pos.
1 3 1
7.545719
1 2 10.79041 2 1
6.459007
1 2 1
1 1 1
5.390459
1 2 1
2.94571 4 1
9.383413
1 3 10.38001 3 28.197563
1 3 1
1 2 1
6.459007
1 3 1
3.30431 5 1 11.564673
1 4 1
0.0002
1 4 2 10.534994
1 4 1
1.02981 3 2
8.197563
1 4 1
3.36731 3 1
7.545719
1 4 1
4.01931 6 1 14.126832
1 5 1−0.7197
1 5 2 13.134167
1 5 1
0.27291 4 1
9.383413
1 5 1
4.02401 7 1 17.076453
1 6 1−1.0788
1 5 1 11.564673
1 6 1
4.43301 8 1 20.415573
1 7 1−1.42811 7 2 19.500138
1 7 1−0.5125
1 6 1 14.126832
1 7 1
4.8611a Ground state energies from Ref. [52].

E08.3361

Comment
very weak

8.3361
9.76349.7633
11.5648
11.5648
11.5648
11.5650
13.4071
13.4071
13.4074
15.9976
15.9976
18.9874
18.9876
18.9879

Table A.3: Line list for the ν1 (m = 0) band of nitromethane. Transitions are specified with thelower state (00 ) and upper state (0 ) asymmetric top quantum numbers J, Ka , and Kc . E 00 (E 0 ) is
the lower (upper) state energy in cm−1 . An offset of 2970 cm−1 has been subtracted from the IR
line positions and upper state energies.
J 00 Ka00 Kc00
2 0 2
0 0 0
3 0 3
1 0 1
4 0 4
2 0 2
5 0 5
3 0 3
6 0 6
4 0 4

E 00a J 0 Ka0 Kc0 Line Pos.
1.549651
1 0 1
2.44330.000000
1 0 1
3.99282.935110
2 0 2
1.99230.547713
2 0 2
4.37964.703276
3 0 3
1.5774
1.549651
3 0 3
4.73116.860890
4 0 41.18022.935110
4 0 45.1061
9.409937
5 0 50.76474.703276
5 0 55.4713(continued on next page)a Ground state energies from Ref. [52].

E03.99303.99284.92744.92736.2807
6.28078.04108.0412
10.1746
10.1746

177

Table A.3: (continued)
J 00 Ka00 Kc00
E 00a J 0 Ka0 Kc0 Line Pos.
7 0 7 12.350829
6 0 6
0.36155 0 56.860890
6 0 6
5.85178 0 8 15.683641
7 0 7−0.0576
6 0 6
9.409937
7 0 7
6.21609 0 9 19.408388
8 0 8−0.47407 0 7 12.350829
8 0 8
6.5840
10 0 10 23.525068
9 0 9−0.8898
8 0 8 15.683641
9 0 9
6.95173 2 23.970934
2 2 1
1.40482 2 02.421311
2 2 12.95444 2 36.082186
3 2 2
1.05443 2 1
4.322080
3 2 22.81512 2 12.327828
3 2 2
4.80915 2 48.619788
4 2 30.70164 2 2
6.839377
4 2 3
2.48203 2 23.970934
4 2 35.35056 2 5 11.560383
5 2 40.31205 2 39.856035
5 2 4
2.01674 2 3
6.082186
5 2 45.79077 2 6 14.895396
6 2 5−0.0986
6 2 4 13.256243
6 2 5
1.53975 2 48.619788
6 2 5
6.17728 2 7 18.622625
7 2 6−0.5235
7 2 5 17.012438
7 2 6
1.08696 2 5 11.560383
7 2 6
6.53919 2 8 22.741677
8 2 7−0.9491
8 2 6 21.143107
8 2 70.64977 2 6 14.895396
8 2 7
6.8973a Ground state energies from Ref. [52].

E012.7123
12.7126
15.6260
15.6259
18.934418.9348
22.6353
22.6353
5.37575.37577.1366
7.13727.13699.32149.32149.3214
11.8724
11.8727
11.8729
14.7968
14.7959
14.7970
18.0991
18.099318.0995
21.7926
21.7928
21.7927

Table A.4: Line list for the ν10o (m = 0) band of nitromethane. Transitions are specified with thelower state (00 ) and upper state (0 ) asymmetric top quantum numbers J, Ka , and Kc . E 00 (E 0 ) is
the lower (upper) state energy in cm−1 . An offset of 3050 cm−1 has been subtracted from the IR
line positions and upper state energies.
J 00 Ka00 Kc00
2 2 1
1 0 1
2 2 0
0 0 0

E 00a
2.32780.54772.42130.0000J 0 Ka0 Kc0 Line Pos.
E0
1 1 1
1.41183.7396
1 1 1
3.19163.73931 1 0
1.46283.88411 1 0
3.88373.8837(continued on next page)a Ground state energies from Ref. [55].

Comment

178

Table A.4: (continued)
J 00 Ka00 Kc00
3 2 2
2 0 2
3 2 1
2 2 1
1 0 1
4 2 3
3 0 3
4 2 2
3 2 2
2 0 2
4 4 1
3 2 1
2 2 1
4 4 0
2 2 0
5 2 4
4 0 4
5 2 3
4 2 3
3 0 3
5 4 2
4 2 2
3 2 2
5 4 1
4 4 1
3 2 1
6 2 5
5 0 5
6 2 5
5 0 5
6 2 4
5 2 4
4 0 4
6 4 3
5 2 3
4 2 3
6 4 2
5 4 2
4 2 2
6 6 1
4 4 1
6 6 0
4 4 0

E 00a
3.97101.54974.32212.32780.54776.08222.9351
6.83953.97101.54978.29544.32212.32788.31202.42138.61974.70329.85616.08222.9351
11.2347
6.83953.971011.35318.29544.3221
11.56026.8608
11.56026.860813.25628.61974.703214.7434
9.85616.082215.1375
11.2347
6.839517.7865
8.295417.7884
8.3120J 0 Ka0 Kc0 Line Pos.
E0
2 1 2
0.67914.65012 1 23.10014.64982 1 10.8052
5.12732 1 12.7992
5.12702 1 14.5793
5.12703 1 3−0.0225
6.05973 1 3
3.12466.05973 1 20.05066.89013 1 22.91886.88983 1 25.34056.89023 3 1−0.1496
8.14583 3 13.82358.14563 3 1
5.81798.1457
3 3 0−0.1394
8.17263 3 05.75108.17234 1 4−0.7495
7.87024 1 43.16747.87064 1 3−0.7800
9.07614 1 32.99369.07584 1 36.14099.07604 3 2−0.7982 10.4365
4 3 2
3.5968 10.4363
4 3 2
6.4652 10.4362
4 3 1−0.7575 10.5956
4 3 1
4 3 1
6.2729 10.5950
5 1 5−1.58649.97385 1 5
3.11289.97365 1 5−1.7127
9.84755 1 52.98669.84745 1 4−1.5958 11.6604
5 1 4
3.0410 11.6607
5 1 4
6.9572 11.6604
5 3 3−1.4608 13.2826
5 3 3
3.4258 13.2819
5 3 3
7.2000 13.2822
5 3 2−1.3996 13.7379
5 3 2
5 3 2
6.8982 13.7377
5 5 1−1.7785 16.0080
5 5 1
7.7114 16.0068
5 5 0−1.7791 16.0093
5 5 0
7.6973 16.0093(continued on next page)a Ground state energies from Ref. [55].

Comment

weakoverlapped with 615 ← 625very weak
blended, weak
blended, weak

179

Table A.4: (continued)
J 00 Ka00 Kc00
E 00a J 0 Ka0 Kc0 Line Pos.
7 2 6 14.8950
6 1 6
6 0 6
9.40966 1 6
3.07637 2 5 17.0123
6 1 5−2.33966 2 5 11.5602
6 1 5
3.11285 0 56.86086 1 57.81227 4 4 18.7676
6 3 4−2.12026 2 4 13.2562
6 3 4
3.39105 2 48.61976 3 48.0271
7 4 3 19.6150
6 3 3−2.0864
6 4 3 14.7434
6 3 3
5 2 3
9.85616 3 37.67187 2 6 14.8950
7 1 6
3.34726 0 6
9.40967 1 68.8327a Ground state energies from Ref. [55].

A.2

E0

Comment

12.4859
14.6727
14.6730
14.6730
16.6474
16.6472
16.6468
17.5286(no comb. diff. found)see 515 ← 505

17.5279
18.242218.2423

Naphthalene, C10 H8Table A.5: Line list for the ν29 band of naphthalene. Transitions are specified with the lower state(00 ) and upper state (0 ) asymmetric top quantum numbers J, Ka , and Kc . Line positions are givenin cm−1 .
J 00 Ka00 Kc00
1 0 1
1 0 1
1 1 0
1 1 0
1 1 1
1 1 1
2 0 2
2 0 2
2 0 2
3 0 3
3 0 3
3 1 2
3 1 2
3 1 2
3 1 3
3 1 3
3 1 3
3 2 1

J 0 Ka0 Kc0
1 1 0
2 1 2
1 0 1
2 2 1
0 0 0
2 0 2
1 1 1
2 1 1
3 1 3
3 1 2
4 1 4
3 0 3
3 2 1
4 2 3
2 0 2
3 2 2
4 0 4
2 1 2

Line Pos. J 00 Ka00 Kc00 J 0 Ka0 Kc03064.5194
8 0 8
7 1 7
3064.4017
8 0 8
8 1 7
3064.6690
8 0 8
9 1 9
8 1 8
7 0 7
3064.2528
3064.7279
8 1 8
8 2 7
3064.5172
8 1 8
9 0 9
3064.6708
8 2 6
8 1 7
3064.5063
8 2 6
8 3 5
3064.3481
8 2 6
9 3 7
3064.4839
9 0 9
8 1 8
9 0 9
9 1 8
3064.2969
3064.7039
9 0 9 10 1 103064.4141
9 1 8
8 2 7
3064.1400
9 1 8
9 2 7
3064.8398
9 1 8 10 2 9
9 1 9
8 0 8
3064.3515
3064.3646
9 1 9
9 2 8
3065.0374
9 1 9 10 0 10
(continued on next page)Line Pos.
3065.0889
3064.2144
3064.0293
3065.0972
3064.1656
3064.0332
3064.8343
3064.3302
3063.7335
3065.1466
3064.1433
3063.9680
3065.1692
3064.2996
3063.8850
3065.1533
3064.1138
3063.9724

180

Table A.5: (continued)
J 00 Ka00 Kc00
3 2 1
3 2 1
4 0 4
4 0 4
4 0 4
4 1 4
4 1 4
4 1 4
4 2 2
4 2 2
4 2 2
4 3 1
4 3 1
4 3 1
5 0 5
5 0 5
5 0 5
5 1 4
5 1 4
5 1 4
5 1 4
5 1 5
5 1 5
5 1 5
5 2 3
5 2 3
5 2 3
5 3 2
5 3 2
5 3 2
5 3 3
5 3 3
5 3 3
6 1 5
6 1 5
6 1 5
6 1 5
6 2 5
6 2 5
6 2 5
6 2 5
6 2 5
7 0 7
7 0 7

J 0 Ka0 Kc0
3 1 2
4 3 2
3 1 3
4 1 3
5 1 5
3 0 3
4 2 3
5 0 5
4 1 3
4 3 1
5 3 3
3 2 2
4 2 2
5 4 2
4 1 4
5 1 4
6 1 6
4 2 3
5 0 5
5 2 3
6 2 5
4 0 4
5 2 4
6 0 6
5 1 4
5 3 2
6 3 4
4 2 3
5 2 3
6 4 3
4 2 2
5 2 4
6 4 2
5 2 4
6 0 6
6 2 4
7 2 6
5 1 4
6 1 6
6 3 4
7 1 6
7 3 4
6 1 6
7 1 6

Line Pos. J 00 Ka00 Kc00 J 0 Ka0 Kc03064.7743
9 2 7
9 1 8
3063.9725
9 2 7
9 3 6
3064.8222
9 2 7 10 3 83064.4500 10 0 109 1 93064.2470 10 0 10 10 1 93064.8909 10 0 10 11 1 113064.3267 10 1 99 2 83064.2925 10 1 9 10 0 103064.7658 10 1 9 10 2 83064.2657 10 1 9 11 2 103063.9096 10 2 8 10 1 93065.2256 10 2 8 10 3 73064.9241 10 2 8 11 3 99 1 83063.7607 10 2 93064.8940 10 2 9 10 1 103064.4041 10 2 9 10 3 83064.1955 10 2 9 11 1 103064.8170 11 0 11 10 1 103064.7827 11 0 11 11 1 103064.4220 11 0 11 12 1 123064.0499 11 1 10 10 2 93064.9396 11 1 10 11 0 113064.2950 11 1 10 11 2 93064.2236 11 1 10 12 2 113064.7658 11 1 11 10 0 103064.2802 11 1 11 11 2 103063.8542 11 1 11 12 0 123065.3033 11 2 10 10 1 93064.9089 11 2 10 11 3 93063.6914 11 2 10 12 1 113065.2799 12 0 12 11 1 113064.9507 12 0 12 12 1 113063.6880 12 0 12 13 1 133064.9107 12 1 11 11 2 103064.8387 12 1 11 12 2 103064.4118 12 1 11 13 2 123064.0101 12 1 12 11 0 113065.1368 12 1 12 12 2 113064.9290 12 1 12 13 0 133064.2260 13 0 13 12 1 123064.1843 13 0 13 13 1 123063.7038 13 0 13 14 1 14
3065.0268 13 1 12 12 2 113064.2825 13 1 12 13 0 13
(continued on next page)Line Pos.
3064.8850
3064.3341
3063.7039
3065.2060
3064.0759
3063.9096
3065.2435
3065.1025
3064.2364
3063.8360
3064.9458
3064.3234
3063.6748
3065.2970
3065.1220
3064.1169
3063.8679
3065.2650
3064.0113
3063.8503
3065.3122
3065.1663
3064.1655
3063.7834
3065.2660
3063.9994
3063.8503
3065.3446
3064.0738
3063.8019
3065.3226
3063.9465
3063.7897
3065.3763
3064.0906
3063.7271
3065.3226
3063.9401
3063.7897
3065.3810
3063.8849
3063.7296
3065.4386
3065.2883

181

Table A.5: (continued)
J 00 Ka00 Kc00
7 0 7
7 1 6
7 1 6
7 1 6
7 1 6
7 1 7
7 1 7
7 1 7
7 2 5
7 2 5
7 2 5
7 3 4
7 3 4
7 3 4
7 3 4

A.3

J 0 Ka0 Kc0
8 1 8
6 2 5
7 0 7
7 2 5
8 2 7
6 0 6
7 2 6
8 0 8
7 1 6
7 3 4
8 3 6
6 2 5
7 2 5
7 4 3
8 4 5

Line Pos.
3064.0862
3065.0019
3064.9024
3064.3887
3063.9715
3065.0419
3064.2130
3064.0943
3064.7970
3064.3164
3063.7670
3065.4827
3064.8698
3064.1352
3063.5621

J 00 Ka00 Kc00
13 1 12
13 1 13
13 1 13
13 1 13
13 2 11
13 2 11
13 2 11
13 2 12
13 2 12
13 2 12
14 0 14
14 0 14
14 0 14
14 1 14
14 1 14
14 1 14J 0 Ka0 Kc0
14 2 13
12 0 12
13 2 12
14 0 14
12 3 1013 3 1014 3 1212 1 11
13 3 1114 1 13
13 1 13
14 1 1315 1 15
13 0 13
14 2 13
15 0 15

Line Pos.
3063.6706
3065.3810
3063.8809
3063.7296
3065.4484
3064.2002
3063.5742
3065.4482
3063.9725
3063.6747
3065.4385
3063.8224
3063.6694
3065.4385
3063.8209
3063.6694

Buckminsterfullerene, C60
A line list of the assigned R-branch transitions of the 8.5 µm band of C60 , as well as the raw

spectrum itself, is located in a permanent online data repository [154].

Appendix B
Spectroscopic constants of the ν3 band of vinyl bromide

Table B.1 below contains the fitted spectroscopic parameters for the ν3 band of vinyl bromide(see Fig. 2.1). The fit was performed using the standard Watson A-reduced quartic Hamiltonian [64]and nuclear quadrupole hyperfine terms with the PGOPHER program [333].
Table B.1: Effective Hamiltonian fits for the ν3 band of CH2 CH79 Br and CH2 CH81 Br. The
standard Watson A-reduced quartic Hamiltonian (Ir representation) was used to fit the measured
transition frequencies. All values are given in cm−1 , except for χaa , χbb−cc , and |χab |, which are
given in MHz. All ground state (v = 0) constants are taken from Ref. [334], again except for χaa ,χbb−cc , and |χab |, which are taken from Ref. [335]. Values in [ ] brackets are held fixed during thefit. 1σ uncertainties are specified in parentheses; for ν0 values, the uncertainty corresponds to theestimated absolute frequency accuracy of our calibration procedure.
79 Br

81 Br

Parameter

Ground state

ν3

Ground state

ν3

ν0
A
B
C
∆J
∆JK
∆K
δJ
δK
χaa
χbb−cc|χab |

01.810093
0.1388471
0.1288373
0.534−0.10216
0.42230.57
0.3922470.9837.04
246.143027.4152(10)1.804461(7)0.1387183(4)0.1286733(5)0.520(12)−0.1188(29)0.1624(40)[0.57][0.3922]469.55(237)31.59(733)[246.14]01.809641
0.1380359
0.1281362
0.528−0.10146
0.42060.57
0.3841393.5830.86
204.173027.4041(10)1.804053(9)0.1379096(5)0.1279755(5)0.553(13)−0.1018(47)0.2835(51)[0.57][0.3841]394.13(237)
22.68(716)[204.17]

×107
×105
×104
×108
×106

RMS error×104

2.20

2.24

Appendix C

Icosahedral symmetry and spherical tensor operators

This appendix summarizes key results from a group theoretical analysis of icosahedral C60 .

C.1

Icosahedral character table and correlations
Table C.1 reproduces the character table of the icosahedral Ih molecular symmetry group(adapted from Table A-16 of Ref. [18]). It is useful to note that Ih is a direct product group of
I and the inversion group {E, i}. I is the icosahedral rotation group, containing only the purerotation symmetry operations of the icosahedron. Transformation properties of the symmetric top

Table C.1: Character table for the Ih (M) molecular symmetry group of C60 . “φ” is the equivalentrotation angle corresponding to the given group element class. The symmetries of the body-fixedangular momentum operators (Jx,y,z ) and dipole moment operators (x, y, z) are indicated. The labframe dipole moment operators each belong to the symmetry given by Γ∗ , the parity-antisymmetricirreducible representation (Au for Ih ).
Ih : E 12C5 12C52 20C3
2π
4π
2π
φ: 0
5
5
3
Ag : 11
1
1
Au : 11
1
1
T1g : 3η+
η−0
T1u : 3η+
η−0−+
T2g : 3η
η0
T2u : 3η−η+0Gg : 4−1−1
1
Gu : 4−1−1
1
Hg : 500−1
Hu : 50 √ 0−1
Note: η ± = (1 ± 5)/2.

15C2
π
1
1
−1
−1
−1
−1
0
0
1
1

i
0
1
−1
3
−3
3
−3
4
−4
5
−5

3
12S10

12S10

20S6

2π
5

4π
5

2π
3

1
−1
η+
−η +
η−
−η −
−1
1
0
0

1
−1
η−
−η −
η+
−η +
−1
1
0
0

1
−1
0
0
0
0
1
−1
−1
1

15σ :
π:
1:
−1 : Γ∗−1 : (Jx , Jy , Jz )1 : (x, y, z)
−1 :
1:
0:
0:
1:
−1 :

184
rotational wavefunctions |Jkmi can be determined by recognizing that they transform as the theirreducible representation (irrep) D(J) of the rotation group SO(3), and then correlating between
D(J) and the irreps of I, which is a subgroup of SO(3). This is facilitated by the fact that thecharacter of D(J) under a symmetry element is determined solely by the angle of the equivalentrotation (φ in Table C.1) via

1
sin
(J
+
)φ
2
χ(J) (φ) =.
sin(φ/2)(C.1)

Once the characters of D(J) are determined, the reduction in terms of the irreps of I is straightforward. These results are summarized in Table C.2 (and have been derived previously by others; see,for example, Table 3 of Ref. [153]). The correlations between irreps of SO(3) and I determine muchof the fundamental structure of C60 . For example, the fact that the lowest non-vanishing multipole
moment of C60 is the hexacontatetrapole (64-pole) is due to the fact that the first D(J>0) that
correlates to A is for J = 6 (26 = 64). These correlations are also critical in determining nuclear
spin statistics and the spherical tensor properties of rovibrational effective Hamiltonians for C60 .

185

Table C.2: Correlation of irreps D(J) of the SO(3) rotation group to those of I, the icosahedral
rotation group. For J = 30n + mod(J, 30), the representation is equal to that of mod(J, 30) plusn × (1A + 3T1 + 3T2 + 4G + 5H).
J : A
0 : 1
1 :
2 :
3 :
4 :
5 :
6 : 1
7 :
8 :
9 :
10 : 111 :
12 : 113 :
14 :
15 : 1
16 : 117 :18 : 119 :20 : 1
21 : 122 : 123 :
24 : 1
25 : 1
26 : 127 : 128 : 129 :
30 : 2

T1

T2

G

1

1
1

H

1
1
1
1
1
1
1
2
1
1
1
2
2
2
1
2
2
3
2
2
2
3
3
3
2
3
3

1
1
1
1
1
1
1
2
1
2
1
2
2
2
2
2
2
3
2
3
2
3
3
3
3

1
1
1
2
1
1
2
2
2
2
2
2
3
3
2
3
3
3
4
3
3
4
4
4
4

1
1
1
1
2
1
2
2
2
2
3
2
3
3
3
3
4
3
4
4
4
4
5
4
5
5
5

Appendix D
Spherical top R selection rule

In this appendix, we derive the R selection rule introduced in Section 5.3.3 for the case ofa linear dipole moment function. Then we discuss the basis of this rule in terms of both a simple
physical picture and some general arguments based on icosahedral symmetry and the Wigner-Eckart
theorem.

D.1

Explicit derivation for linear dipole moment operator
Let {qx , qy , qz } be triply degenerate vibrational coordinates for an IR active fundamental.

Their conjugate momenta are {px , py , pz }, with [qx , px ] = i, [qx , py ] = 0, etc. The molecule bodyfixed frame components of the vibrational angular momentum are`x = qy pz − qz py(D.1)

and cyclic permutations thereof. The commutation relations between the components of the vibrational angular momentum and the coordinates are

and cyclic permutations thereof.

[`z , qx ] = iqy ,(D.2)[`z , qy ] = −iqx ,(D.3)[`z , qz ] = 0,

(D.4)

187
The body-fixed dipole moment can be expanded to first order in the triply degenerate vibrational coordinates qx 
µ~ bf ∝  qy  .




qz

(D.5)

The lab-fixed dipole moment is related to the body-fixed one byµ~ L = S −1 · µ~ bf ,(D.6)

where the direction cosine matrix S −1 is, in terms of the standard Euler angles, cθcφcχ − sφsχ −cθcφsχ − sφcχ sθcφ S −1 =  cθsφcχ + cφsχ −cθsφsχ + cφcχ sθsφ  .




−sθcχ
sθsχ
cθ

(D.7)

(Cosine and sine are abbreviated as c and s, respectively.)
We now compute the commutators of the components of the vibrational and total angularmomentum with µ~ L . We begin with the z component of the vibrational angular momentum:

188





 qx 
−1 [`z , µ~ L ] = [`z , S ·  qy ]




qz


 qx 
−1= S · [`z ,  qy ]




qz


 iqy 
−1 
= S ·  −iqx 




0




 qy (cθcφcχ − sφsχ) + qx (cθcφsχ + sφcχ) = i  qy (cθsφcχ + cφsχ) + qx (cθsφsχ − cφcχ)  .
−qy (sθcχ) − qx (sθsχ)(D.8)

We now evaluate the commutator with the z component of the total angular momentum

189
Jz = −i∂χ :[Jz , µ~ L ] = [−i∂χ , S −1 · µ~ bf ]= −i[∂χ , S −1 ] · µ
~ bf




 −cθcφsχ − sφcχ −cθcφcχ + sφsχ 0 = −i  −cθsφsχ + cφcχ −cθsφcχ − cφsχ 0  · µ
~

 bf


+sθsχ
sθcχ
0

 



 −cθcφsχ − sφcχ −cθcφcχ + sφsχ 0   qx 

 


 

= −i  −cθsφsχ + cφcχ −cθsφcχ − cφsχ 0  ·  qy 

 


 

+sθsχ
sθcχ
0
qz


 qx (cθcφsχ + sφcχ) + qy (cθcφcχ − sφsχ) = i  qx (cθsφsχ − cφcχ) + qy (cθsφcχ + cφsχ) −qx (sθsχ) − qy (sθcχ)(D.9)

We can now see that[`z , µ~ L ] = [Jz , µ~ L](D.10)[Rz , µ~ L ] = [Jz − `z , µ~ L ] = 0.

(D.11)and therefore

It must be that Rx and Ry also have vanishing commutators with µ~ L (our choice of body-fixedz axis was arbitrary to begin with), so that R2 = Rx2 + Ry2 + Rz2 completely commutes with the labframe dipole moment operator,[R2 , µ~ L ] = 0,

(D.12)and dipole transitions must therefore satisfy ∆R = 0. The z component relation,[Rz , µ~ L ] = 0,also implies ∆kR = 0, where kR is the body-fixed z projection of R.

(D.13)

190

D.2

A simple physical picture
We can approximate the carbon cage of C60 as a spherical elastic shell depicted in Fig. D.1a.

A fixed point on the sphere is marked with a red dot. Vibrational motion deforms the surface andproduces an oscillating dipole moment oriented along the direction of deformation (Fig. D.1b). The
electric field of incident radiation exerts a torque on the oscillating vibrational dipole causing thedeformation to rotate (Fig. D.1c), but leaving the orientation of the shell itself unchanged. The
deformation is able to move along the surface of the sphere freely, like a traveling wave, without
the body of the sphere moving. This can only occur because the deformation has no preferredorientatation.

a

b

δ+

c
δ+

µ

µ
δ-

δ-

d

e

δ+

f

δ+

µ

µ
δ-

δFigure D.1: A simple physical picture for the R selection rule. (a) A spherical shell is deformed,
producing (b) an oscillating vibrational dipole. (c) Rotation of this dipole by an electric fieldrotates the deformation, but not the underlying spherical shell. (d) An aspherical shell can also
be deformed, producing (e) a similar oscillating dipole along its long axis. (f) Rotation of this
dipole requires that the entire body move, as the orientation of the vibration is fixed relative tothe body-fixed frame.

The three-fold degeneracy of the T1u IR active vibrations of C60 leads them to behave like theabove deformation of a spherical shell. The vibrational angular momentum ` is associated with the191rotation of the deformation, while the “pure rotational” angular momentum R is associated withthe rotation of the spherical frame itself. The vibrational dipole is a “slippery handle.” Because it
can rotate freely with respect to the frame, an electric field can only exert a torque on ` and not
R. Spectroscopically, this translates to the ∆R = ∆kR = 0 selection rules.
To help solidify this physical picture, consider the more common case of a less symmetricalmolecule, illustrated as an oblong shell in Fig. D.1d. A vibrational deformation along the long axisof the shell again produces an oscillating vibrational dipole in the same direction (Fig. D.1e). Unlike
the spherical case, however, this non-degenerate vibration has a definite preferred orientation withrespect to the shell. Vibrations along the short axis occur at a different frequency, and so theycannot “resonantly” travel freely between the long and short axes. Electric radiation that torques
the vibrational dipole moment must rotate the deformed shell as a whole (Fig. D.1f). Thus, in nonspherical molecules, vibrational excitations can occur with a simultaneous change in the angularmomentum of the entire molecular frame. (An example of this type of oblong shell is a linear
molecule such as acetylene, HCCH. The deformation along the long-axis would correspond to theantisymmetric CH stretching mode. Such a Σ − Σ transition has ∆J = ±1 selection rules.)D.3

Spherical tensor operators and the Wigner-Eckart theorem
The R selection rule can also be understood in the general formalism of spherical tensor

operators. The triply generate vibrational coordinates {qx , qy , qz } behave as a vector, i.e. thecomponents of a proper rank-1 tensor, with respect to the SO(3)M rotation group generated by the
body-fixed components of R. Therefore, in the linear dipole approximation µ~ bf ∝ (qx , qy , qz )T is
also a rank-1 vector in SO(3)M . When rotating body-fixed quantities into the lab frame, we use
the direction cosine matrix S −1 (Eq. D.7). Each row of S −1 itself transforms as a rank-1 vector inSO(3)M . Therefore, rotating the body-fixed dipole moment operator into the lab frame, Eq. D.6,
amounts to performing three separate tensor couplings between a pair of rank-1 vectors to generate
a rank-0 operator. Each row of S −1 is coupled with µ~ bf to form a single lab frame component of µ
~ L,each of which is a spherical rank-0 operator in SO(3)M . We can then make use of the Wigner-Eckart

192
theorem [152, 161], from which it follows that matrix elements of the form
0hR, kR , J, m|T̂q(k) |R0 , kR
, J 0 , m0 i,(D.14)

0 + q, where k and q are the rank and tensor
are zero unless |R0 − k| ≤ R ≤ R0 + k and kR = kR(k)component of the operator T̂q

in SO(3)M . For the linear dipole approximation, each lab-framecomponent of µ
~ L is a spherical operator with k = q = 0, leading to the ∆R = ∆kR = 0 selection
rule.
Wait a minute, one might protest. Isn’t the lab-frame dipole moment operator µ~ L a vector?
It is and it isn’t. This is an excellent example of the difference between the SO(3)M and theSO(3)L rotation groups. SO(3)M is generated by the body-fixed frame components of R, while
SO(3)L is generated by the lab frame components of J. Each lab frame component of µ
~ L is a rank0 spherical tensor operator in SO(3)M (under the linear dipole approximation), while the threecomponents together transform as a rank-1 vector operator in SO(3)L . Electric dipole transitions
therefore have ∆R = 0 and ∆J = 0, ±1 selection rules. The two rotation groups co-exist, providing
complementary descriptions of the properties of spherical tensor operators.
We can now see that the “slipperiness” of the vibrational transition dipole for spherical topsin our simple physical picture above is associated with its rank-0 tensor properties. However, foricosahedral tops like C60 , the true molecular symmetry group is not quite SO(3)M , but rather the
lower symmetry icosahedral group Ih . Therefore, the lab frame dipole moment operators may havehigher rank contributions that break the ∆R = 0 selection rule. As can be seen in the Ih character
table (Table C.1), the lab frame dipole moment operators each transform as Au . Table C.2 indicatesthat Au operators can have not only rank-0 contributions, but also rank-6, 10, 12, etc. Such higher
rank contributions in µ
~ L must come from corresponding terms in the body-frame dipole moment
operator µ~ bf , which transforms as T1u . The linear approximation restricts µ~ bf to transform as arank-1 vector, but, in general, rank-5, 6, 7, . . . contributions are possible. These high rank terms
are expected to be small, making the ∆R = 0 selection rule quite strong.