Photoionization, Photodissociation, andLong-Range Bond Formation in Molecular
Rydberg Statesby
Edward Lees Hamilton
B.A., Calvin College, 1997

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
2003

This thesis entitled:
Photoionization, Photodissociation, and Long-Range Bond Formation in Molecular
Rydberg States
written by Edward Lees Hamilton
has been approved for the Department of Physics

Chris H. Greene

John Bohn

Date

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

iii
Hamilton, Edward Lees (Ph.D., Physics)
Photoionization, Photodissociation, and Long-Range Bond Formation in Molecular Rydberg States
Thesis directed by Prof. Chris H. Greene

The Rydberg spectra of atoms and small molecules offers an experimentally convenient probe for exploring the exchange of energy between Rydberg electrons and otherforms of electronic, vibrational, and rotational excitation. This thesis investigates a series of special topics in the field of molecular Rydberg spectra, using a diverse set oftheoretical techniques all designed to take advantage of the computational efficiency ofthe sorts of scattering parameterizations commonly associated with the field of quantum
defect theory. In particular, I consider various mechanisms by which Rydberg electrons
participate in the formation (bonding) and destruction (dissociation) of molecular states.
First, I review the methodology of multichannel quantum defect theory in molecular systems, demonstrating its versatility in reducing a complicated set of channelcoupled solutions into a physically observable photoionization spectrum with exceptionally high resolution, even in regions characterized by complex resonant structures withstrong energy dependence. The utility of the Fano frame transformation is discussed,
two approaches to the problem of extracting resonant effects via the delay of asymptotic
boundary conditions are presented, and a case study featuring the molecular hydrogenisotopomer HD is examined in detail.
Second, I turn to the question of Rydberg electrons in the presence of both anionic core and a neutral perturbing particle, extending certain basic features of theabove philosophy to a two-center geometry. This system is predicted to give rise toa potential well that supports bound states, with a potential curve minimum existingat many hundreds or thousands of Bohr radii. The problem is first handled at theiv
level of a zero-range potential approximation, where the solution can be written bymeans of degenerate perturbation theory. This approach is compared to a more robust,
but computationally expensive, description of the interaction in terms of a finite range
model potential, requiring diagonalization of the Hamiltonian with respect to an L 2basis. Some properties of these states are also noted. Next, a more powerful but
difficult formulation using the Coulomb Green’s function, subject to limiting boundary
conditions at the position of the core and perturber, is derived. Finally, a semiclassicalinterpretation, corresponding to the trajectories of a point particle electron moving
classically in a Coulombic field, is examined in detail.
Third, I return to the case of the diatomic Rydberg spectrum, this time extending the solution to accommodate dissociation pathways through the use of a Siegert
pseudostate basis. Previously developed methods of treating the competition between
ionization and dissociation are reviewed and evaluated. The Siegert basis is defined,
together with an efficient procedure for its calcuation, and some of its unconventional
properties are explicitly noted. The Siegert-MQDT method is applied to several reactive scattering or half-scattering processes, including photodissociation, dissociative
ionization, and dissociative recombination.

Dedication

To my grandfather, Edward H. Hamilton, who unlike myself was not merely
acquainted with electrons, but could also teach them to perform useful tricks.

vi

Acknowledgements

A thesis is not so much the accomplishment of the mind of its author as it is
the work of dozens of other minds being filtered through his own. The hope is that, inthe course of the filtering, all that other work can directed toward a sharper focus, likewater through a funnel or light through a lens.
My initial year at Colorado was challenging, and it seems appropriate to express
my appreciation for those who helped me survive it. David Nesbitt, in addition to
being a fine teacher (and based on his impromptu in-class barbershop performance, alsoa fine vocalist!), was a source of early encouragement. Carl Lineberger helped ease mytransition by guaranteeing me a position my first summer out, when I was still figuring
out where I wanted to go and what I wanted to do when I got there. My office mate Jeff
Wright rescued me from certain of the more fiendish problems in Mr. Jackson’s famous
textbook on rather short notice, and displayed a reassuring combination of humor andgood sense when the answers weren’t so apparent to either of us.
My research colleagues have provided constant support in both professional, scientific, and personal capacities. Hugo van der Hart deserves credit for introducing me tob-splines, the beginning of a long and profitable relationship. (With the splines, I mean,although Hugo and I get along pretty well too.) Mark Baertschy was always available
to explain why my eigenvalues were coming out complex, and other aracane curiosities
of linear algebra. I only knew Jeff Stephens for a couple of months, but after using somany of his old codes to help me get off the ground my first year, I feel like I owe him

viiseveral years’ worth of gratitude. But of the many other associates I could thank, Brian
Granger stands out as a talented physicist and a decent person. He was always readywith a whiteboard and marker to walk me through the derivation, or photocopy his oldnotes from those beautifully organized binders. I owe him a special non-research-relatedcommendation for mailing me a package across country. Trust me, it was important.
Last and foremost, Chris Greene has devoted an enormous investment of timeand patience to teaching me just a small corner of his encyclopedic knowledge of AMO
physics, and I can’t begin to express my thanks for having access to his expertise. He
is among the rare academics to have a talent for both instruction and research, without
respecting any boundary between them; he lives in both worlds at once, the equivalent ofwave-particle duality in academic science. If there is any uncertainty relation enforcing
a tradeoff between the quality and scope of his work, then one can only assume thatthe h-bar that sets its bound must be infinitesimally small.
This research was supported with a grant from the National Science Foundation.

viiiContents

Chapter
1 Introduction

1

1.1

Historical development of experimental technique . . . . . . . . . . . . .

4

1.2

Emerging applications for Rydberg state theory . . . . . . . . . . . . . .

6

1.3

Outline of presented topics

8. . . . . . . . . . . . . . . . . . . . . . . . .

2 General review of molecular MQDT

10

2.1

Degrees of freedom in molecular physics . . . . . . . . . . . . . . . . . .

12

2.2

Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.3

Channel elimination: Traditional and energy-smoothed approaches . . .

23

2.4

Application: HD photoionization spectrum . . . . . . . . . . . . . . . .

29

3 Long-range Rydberg state: Approximate and model approaches

40

3.1

The Fermi pseudopotential approximation . . . . . . . . . . . . . . . . .

41

3.2

Degenerate perturbation theory for the hydrogenic case . . . . . . . . .

43

3.3

Omont’s generalized delta-function approximation . . . . . . . . . . . .

47

3.4

Resonance scattering effects . . . . . . . . . . . . . . . . . . . . . . . . .

48

3.5

Finite range nonlocal pseudopotential . . . . . . . . . . . . . . . . . . .

53

4 Long-range Rydberg states: Green’s function method
4.1

The Coulomb Green’s function . . . . . . . . . . . . . . . . . . . . . . .

57
59

ix
4.2

Hydrogenic solution for S- and P-wave scattering . . . . . . . . . . . . .

61

4.3

Solution in the presence of an external field . . . . . . . . . . . . . . . .

73

4.4

Semiclassical interpretation . . . . . . . . . . . . . . . . . . . . . . . . .

76

5 Siegert states

92

5.1

Dissociative channels in molecular MQDT: General considerations . . .

92

5.2

Established methods for handling dissociative channels . . . . . . . . . .

94

5.2.1

Jungen eigenphase method . . . . . . . . . . . . . . . . . . . . .

94

5.2.2

Stephens-Greene box averaging method . . . . . . . . . . . . . .

98

5.2.3

Two-dimensional R-matrix method . . . . . . . . . . . . . . . . .

99

5.3

Siegert pseudostates: Basic concepts . . . . . . . . . . . . . . . . . . . . 109

5.4

Siegert pseudostates: Single channel Green’s function method . . . . . . 120

5.5

Extending MQDT to a Siegert pseudostate basis: Theory . . . . . . . . 125

5.6

Extending MQDT to a Siegert pseudostate basis: Results and discussion 132

6 Conclusion

137

Bibliography

145

x

TablesTable
2.1

H2 rovibrational levels . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

5.1

Relative yields for R-matrix calculation resonances . . . . . . . . . . . . 107

5.2

Resonance parameters calculated with the SPS method . . . . . . . . . 133

xiFiguresFigure2.1

Ab initio H2 potential energy curves . . . . . . . . . . . . . . . . . . . .

15

2.2

H2 quantum defect curves . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.3

Energy independent model H2 potential energy curves . . . . . . . . . .

18

2.4

HD photoionization, experiment vs. theory, 125500-126000 cm-1 . . . .

32

2.5

HD photoionization, experiment vs. theory, 126000-126500 cm-1 . . . .

33

2.6

HD photoionization, experiment vs. theory, 126500-127000 cm-1 . . . .

34

2.7

HD photoionization, experiment vs. theory, 127000-127500 cm-1 . . . .

35

2.8

Very high resolution HD experimental spectrum . . . . . . . . . . . . . .

39

2.9

MQDT-calculated oscillator strength . . . . . . . . . . . . . . . . . . . .

39

3.1

Surface plot of the 3 S scattered state.

. . . . . . . . . . . . . . . . . . .

46

3.2

Potential curves for the 3 Σ state. . . . . . . . . . . . . . . . . . . . . . .

50

3.3

Potential curves for the 3 Π state. . . . . . . . . . . . . . . . . . . . . . .

51

3.4

Triplet phase shifts for 87 Rb. . . . . . . . . . . . . . . . . . . . . . . . .

52

3.5

Surface plot of the 3 P scattered state. . . . . . . . . . . . . . . . . . . .

54

3.6

Comparison of the Fermi, finite range nonlocal, and Green’s function
methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4.1

Sigma potential curves for Rb-Rb near n=30 . . . . . . . . . . . . . . .

74

4.2

Stark map for atomic Rb . . . . . . . . . . . . . . . . . . . . . . . . . .

75

xii
4.3

Stark level spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

4.4

Avoided crossings with the degenerate manifold . . . . . . . . . . . . . .

78

4.5

Stark splittings as a function of field at fixed R . . . . . . . . . . . . . .

79

4.6

Elliptic hydrogenic eigenstates . . . . . . . . . . . . . . . . . . . . . . . .

83

4.7

Potential curves from peturbation theory in elliptic eigenstates . . . . .

84

4.8

Contour wavefunction plots a function of R . . . . . . . . . . . . . . . .

86

4.9

Elliptic-coordinate bound state expansion coefficients . . . . . . . . . . .

87

4.10 Classical trajectories contributing to path interference . . . . . . . . . .

90

4.11 Countour plot comparison of the semiclassical and quantum Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

5.1

R-matrix configuration space partition . . . . . . . . . . . . . . . . . . .

96

5.2

R-matrix model potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.3

R-matrix photoionization cross-section . . . . . . . . . . . . . . . . . . . 108

5.4

R-matrix photodissociation cross-section . . . . . . . . . . . . . . . . . . 108

5.5

R-matrix total photoabsorption cross-section . . . . . . . . . . . . . . . 108

5.6

Spectrum of eigenstates for the Siegert pseudostate equation . . . . . . . 116

5.7

Detailed spectrum of SPS eigenstates near the real axis . . . . . . . . . 117

5.8

Siegert state basis truncation . . . . . . . . . . . . . . . . . . . . . . . . 131

5.9

Dissociative photoionization for H 2 . . . . . . . . . . . . . . . . . . . . . 134

5.10 Dissociative recombination in a model potential calculation . . . . . . . 135

Chapter 1

Introduction

The progress of scientific thought has, at several critical junctures in history,
depended essentially on something that can only be regarded, with the advantage ofhindsight, as an exceptional stroke of good luck. The theory of celestial mechanics, as
first developed by Johannes Kepler, involved several levels of fortunate happenstance:
Kepler was lucky to have selected Mars for his studies, with its notably elliptical orbit
capable of detection even by the rudimentary measurements of his era; he was luckythat the law he was in the process of discovering involved orbits that traced conic
sections, a class of geometric constructs that had been the object of extensive study by
the ancient Greeks for purely aesthetic reasons; and he was lucky that the motion ofplanets in the solar system, due to the mass disparity between them and the sun, could
be described so well at the level of approximation as a set of noninteracting two-bodysystems. Without the benefit of Kepler’s good fortune, Copernicus would not have been
so quickly vindicated, nor would the foundation for Newton’s subsequent developmentof gravitation been so firmly established.
By coincidence, it would be another researcher named Johannes who, three centuries later, traced Kepler’s steps in recognizing an empirical law derived from the same
inverse-square force law as Kepler’s orbital mechanics– although, like Kepler, Johannes
Rydberg had little appreciation of the physical basis for his empirical formula. (Ironically, his doctoral dissertation, written in mathematics before his interests turned to2physics, had been on the subject of conic sections.) As with Kepler’s fortuitous choiceof Mars, Rydberg had the benefit of several spectral lines of hydrogen (what we now
know as the Balmer series) that lay well within the visible region of the spectrum.
These lines had in 1885 been identified by yet another Johann, the Swiss schoolteacher
Johann Jacob Balmer, as being well-characterized by the formula λ ∝ n 2 /(n2 − 22 )[1], although Rydberg was not aware of this at the time he began his study. Rydberg,
working from a larger library of spectroscopic data for hydrogen and the alkali metals,
was able to successfully generalize Balmer’s form of the transition frequency in 1888,
with the now-familiar result [2]

n
1
1
=
−
.
2
N0
(m1 + c1 )(m2 + c2 )2(1.1)

The names of the variables are here intentionally presented using Rydberg’s original
choice of symbols: n is the wavenumber of the emitted light, N 0 is the eponymous constant named in Rydberg’s honor, and m 1 and m2 are positive integers. The appearanceof the additional constants ci was, happily enough, the only modification necessary toextend the hydrogenic formula to the analysis of alkali spectra. And this, of all theprovidential manifestations of natural simplicity considered thus far, is the one withwhich this thesis shall be most properly concerned.
It would be an excusable generalization for one to observe that, in practical terms,
Rydberg states are the only multiparticle states that are quantitatively understood atanything approaching the celebrated level of success achieved by quantum mechanics
for ground state energies and wavefunctions. The tools of quantum chemistry thathave so far been developed for the treatment of many particle systems rely chiefly
on variational approaches that minimize variables (most commonly the total energy)
subject to constraints, and as such generalize poorly to excited states. Only in the lasttwenty years have techniques for excited state ab initio calculation begun to achieve

3
reasonable success, and even then only at great computational expense [3]. The Rydberg
states arising from a Coulombic potential, uniquely and fortuitously, pass over into alimit which reduces the complex electronic correlations into a simply parameterized form
that reflects a nearly-exact integrability of the Schrödinger equation for the electronic
potential. As a consequence, even molecular Rydberg spectra with a dizzying arrayof closely spaced resonances are still naturally tractable with respect to spectroscopic
assignment. The more highly excited a Rydberg state becomes, the more it acquires thecharacter of a perturbed hydrogenic state, and thus the more regular and predictable
the associated structure is expected to become.
With sufficiently accurate spectroscopic methods, deviations from the general expression given above begin to emerge, reflecting a variety of subtle perturbative effects
on the spectrum that would be difficult to detect through other methods. The analysis
of Rydberg spectra effectively extracts quantitative information about the spatial distribution of electrons and nuclei in the core, as well as the partitioning of energy between
possible modes of core state excitation. One important class of perturbative effects is
associated with core anisotropy, arising from either the electrostatic multipole momentsof the core geometry or induced polarization of the core electrons. Other small spectroscopic shifts may be attributed to relativistic modification of the motion and interactionof the Rydberg electron, including the so-called Casimir force. These corrections are
all manifestations of the alteration of the long-range Coulombic potential, and can be
expressed rather intuitively, albeit often non-trivially, as additional terms in the electronic Hamiltonian defining Rydberg motion. While these terms may possess complex
tensorial character, they are essentially adiabatic in nature, and thus remain amenable
to treatment within the familiar framework of adiabatic approximations.
Another important class of spectroscopic signatures for Rydberg-core interaction
involves the sensitivity of the Rydberg electron to short-range many-particle dynamics
in the immediate vicinity of the core structure. Due to the disproportionality of the

4spatial probability distribution of the highly excited electron relative to that of the (atmost weakly excited) core, the volume over which such interactions can contribute isnecessarily small. For Rydberg states with more than a few quanta of angular momentum, the resultant centrifugal barrier shields the core entirely, and all short-range
effects are buried deep inside of the inner turning point of the effective potential. For
any case where the Rydberg wave function extends even slightly into the core volume,
however, the core may exert substantial effects on the solution within that volume, andthereby alter the stationary superposition of hydrogenic solution states in the asymptotic Coulomb region as well. Further discussion of the origin of these effects will bedeferred until Chapter 2; for the moment, it suffices to note that they require a fundamentally non-Born-Oppenheimer description. When the electron is far from the core,
it has relatively little kinetic energy, and its motion cannot be considered “fast” on thetime scale of core dynamics; when the electron is close to the nucleus, it has enormous
kinetic energy due to proximity to the singularity of the Coulomb potential, and entersand exits the core volume on a time scale much faster than any adiabatic rearrangementof energy within the core. In summary, one may instructively observe that the Rydberg
electron lives in one region of space where the Born-Oppenheimer potential is valid but
the single particle approximation fails due to strong correlation with the other electronsand coupling with the core degrees of freedom, and one region of space where the single
particle approximation is valid but the Born-Oppenheimer approximation fails due tothe decoupling of the slow Rydberg electron from the geometry and orientation of thecore state.

1.1

Historical development of experimental technique
The separation between lines arising from the manifolds of a hydrogenic energy

spectrum diminishes rapidly with increasingly primary quantum number n, as n13 . The
difference between the n = 99 and n = 100 manifolds is already less than half a5wavenumber. Selective excitation of a particular state either within or near a particular
manifold (when, for example, the usual selection rules are broken by the introduction ofa small external static electric field) demands even greater control over the energy andlinewidth of the incident light. The extraction of detailed structure within the Rydberg
spectrum has thus been dependent upon advances in the efficiency with which light
can be produced at both high intensity and narrow bandwidth. Prior to the advent ofmodern laser optics, this required the use of dispersive instruments such as prisms anddiffraction gratings to isolate monochromatic components of a broadband source. The
task was further complicated by the necessity of working under vacuum conditions dueto the strong continuum absorption of common background gas components in the ultraviolet, where Rydberg transitions of atoms and small molecules are most commonly
observed. Molecular oxygen becomes opaque below 1850 Å, and molecular nitrogen
below 990 Å. (For a survey of some early difficulties of the development of spectroscopy
in the VUV, see [4].)In light of these experimental difficulties, it is understandable that high-resolution
Rydberg spectroscopy did not attain sufficient resolution to detect small (i.e., on theorder of a wavenumber) structure and shifts until the late 1960s and early 1970s. The
earliest high resolution discrete absorption spectrum of molecular hydrogen was that
conducted by Herzberg [5], and the first high resolution continuum spectrum was thatof Dehmer and Chupka [6]. These results, improving on previous resolution by asmuch as two orders of magnitude, not only successfully resolved the rotational andvibrational separations of the molecular spectrum with precision better than a fractionof a wavenumber, but also were capable of accurately defining line widths and shapes
to an extent that prompted the development of new theory describing strong energy
dependence (i.e., resonant effects) of the photoexcitation cross-sections with quantitative
rigor.
The emergence of even better experimental methodology in the last twenty years

6
has continued to improve the quality of Rydberg spectroscopy. Noteworthy examples
include the popularization of high-intensity synchrotron radiation sources, permitting
even weak features in the spectrum to contribute an observable signal, and the introduction of narrow bandwidth laser sources extending into the far ultraviolet throughthe use of tunable lasers, frequency doubling, and higher harmonic generation. Laser
technology has not only superseded the use of dispersive monochromators, but has also
opened the door to exquisitely fine control over the phase and coherence properties of
the incident light. Among the many delicate effects now accessible by experimental
techniques are the tunneling between the vibrational potential wells of highly excited
double-well adiabatic H2 potential curves [7], the breaking of g-u symmetry in HD, [8],singlet-triplet mixing near the H(n=1)+H(n=2) dissociation limit [9], and competition
between dissociation and ionization decay dynamics in the regions of the H 2 spectrum inan energy regime where multiple dissociative fragmentation channels are simultaneously
open [10].

1.2Emerging applications for Rydberg state theory
Much of the appeal of Rydberg states lies in their potential to serve as a bridge

between classical and quantum mechanics. Traditionally, the former has been associatedparadigmatically with macroscopic systems, and the latter with microscopic systems.
In theory, of course, classical mechanics is merely the expression of certain limiting
procedures necessary to extend quantum theory to systems with arbitrarily large particle
numbers, energies, and state densities. With respect to the boundary between theclassical and quantum regimes, one may identify two areas of burgeoning recent researchinterest: First, techniques to demonstrate explicitly quantum mechanical properties ona macroscopic (or at least mesoscopic) scale, and second, new methods to control orselectively influence the evolution of quantum systems.
Much of the renewed interest in atomic physics generally may be attributed to7
rapid improvements in laser trapping and cooling technology. From the standpoint ofRydberg spectroscopy, the ability to cool atoms to temperatures to fractions of a Kelvinis especially appealing, since highly-excited states can survive for long times under suchconditions, allowing their evolution over time to be systematically manipulated and externally directed toward a controlled outcome. Possibilities include arranging Rydberg
atoms in an orderly spatial geometry, entangling them with one another, constructing
Rydberg electron wavepackets that mimic classical particles or display long-time recurrence effects, and exploring the controlled or spontaneous transition of Rydberg atoms
into molecules or plasmas.
The range and diversity of such work has been extensive enough that it can only be
surveyed here by a selected subset of representative examples. Rydberg atoms have been
proposed as a pathway to quantum information processing, either via a dipole blockade
effect [11, 12], entanglement in superconducting cavities using microwave photons [13], or
half-cycle pulses shaped using optimal control theory [14, 15]. A system of cold trapped
atoms excited to high Rydberg levels has been observed to evolve spontaneously into acold plasma, at temperatures four orders of magnitude lower than any other method ofcold plasma generation, in a phase transition postulated to be analogous to the Mott
transition in semiconductors [16, 17, 18, 19]. The angular momentum composition of
Rydberg wavepackets has been selectively controlled by phase-locked laser pulses [20].
The coherent control of a four-wave mixing signal has been observed as a manifestationof the interference between Rydberg excitation pathways [21]. The collision potentials
between Rydberg atoms have been examined in detail, revealing curve crossings at
thousands of a.u. capable of supporting bound vibrational levels [22], which may alreadyhave been experimentally observed [23]. Finally, Rydberg states have been proposed as asensitive probe of small electric fields and gas phase ion concentrations, as well as various
other measurements of fundamental molecular constants like ionization potentials andionic energy level structure [24].

8Less attention has been paid to what might be termed “Rydberg chemistry”,
the study of bond formation and dissociation processes facilitated or influenced by thepresence of Rydberg electrons. Such applications would define a new field of conceptual
overlap between the increasing interest in quantum control of chemical reaction products, in the physical chemistry community, and the atomic Rydberg investigations by
the AMO community discussed above.

1.3

Outline of presented topics
In this thesis I explore a number of the properties of molecular Rydberg statesolutions, and the calculation and analysis of spectroscopic observables. In the development of new techniques and predictions, one logically begins by working from thesimplest cases toward the more complex, and the simplest of all molecules are homonuclear diatomics. I shall consider two varieties of diatomic Rydberg bound states, one
being the more conventional case in which the molecular ion is bound by the core electrons, with the Rydberg electron serving as a probe of the core structure and dynamics,
and the other being long-range molecules with two distantly separated centers in which
the bonding is controlled by the Rydberg electron itself.
In Chapter 2, I introduce the basic concepts and vocabulary of scattering theory
in an informal overview. The significance and origin of resonances is considered, withparticular attention to their relationship to the different classes of motion that characterize the degrees of freedom that create resonant effects. I specialize to the example
of molecular hydrogen, and move on to the consideration of multichannel spectroscopy,
and how it can be described from the standpoint of a quantum defect formalism. The
spectrum of the molecular hydrogen isotopomer HD is calculated by way of quantum
defect theory in conjunction with application of Fano’s frame transformation procedure,
followed by a critical assessment of the limitations of the approach.
Chapter 3 pursues the separate topic of the perturbed spectrum of a Rydberg

9atom in the presence of a ground state atom or molecule. The interaction between the
Rydberg electron and the perturbing atom or molecule is first considered at the level of azero-range interaction approximation, where the particle is described by a delta function
potential with a strength proportional to the energy-dependent generalization of thescattering length. This approximation yields an exact analytical result in degenerate
perturbation theory for a hydrogenic Rydberg atom, if spin-orbit and other degeneracybreaking terms can be neglected. The results of this approximation are tested by meansof a more detailed finite-range model pseudopotential defined in a way that separates
out scattered partial waves non-locally. Properties of the bound state potentials thatarise from such interactions are noted and discussed.
Chapter 4 returns to the same problem, but instead treats it using a Green’s
function formalism. I demonstrate the ability of this method to reproduce the othermethods of the previous chapter, both for the hydrogenic and finite quantum defectcases. The case of perturbation of a Rydberg molecule by a weak external electric field isconsidered, with example calculations. Finally, some relationships between semiclassicalclosed-orbit theory and the nodal pattern of the quantum wavefunction.
Chapter 5 revisits the diatomic photoionization spectrum, and introduces thecomplication of competing dissociative processes. The history of methods that treat
competition with the dissociative continuum is reviewed, and the advantages and limitations of various methods are observed. As an alternative to these, I propose a new representation of coupling to the continuum, using a discretized pseudostate basis obeying
Siegert boundary conditions. The properties of Siegert pseudostates are briefly summarized, and a recently developed method for their efficient generation is described. Using
these pseudostates as a finite basis representation for the MQDT-frame transformation
technique, it is shown that scattering into the ionization and dissociation continua canbe treated simultaneously within a unified formalism.

Chapter 2General review of molecular MQDTFor over a century, spectroscopy has served as a faithful midwife for the birth ofcountless new discoveries in modern chemistry and physics. Bohr’s theory of the atom
was designed to accommodate classical stoichiometric concepts of valency to explain the
spectra of hydrogen and alkali atoms. Organic chemists depend on infrared spectroscopy
to detect the presence of certain bond types, and the study photoelectron spectra in
the visible and ultraviolet regions to draw inferences about the distribution of electrons
within those bonds. Biologists utilize the sensitivity of fluorescence decay lifetimes toperturbation by analyte atoms and molecules in order to perform image mapping ofcellular structure. The reflectance and emission spectra of rocks and minerals provide
clues to geological history, whether of the upheaval of the Earth’s crust or of the flowof ancient rivers on Mars. Astronomers rely on spectroscopic observations to determine
the composition of stars, as well as their distance, their age, their motion, their temperature, the influence of any extra-solar planets they might support, and the density andcomposition of the diffuse interstellar matter through which their light subsequently
travels. The most accurate clocks in the world depend on spectroscopic standards, as
does the definition of the SI unit of length, the meter. The spectroscopic study of distantgalaxies and quasars is essential to the pursuit of fundamental cosmological questions
regarding the distribution of matter in the universe, its age and early history, and thepossible variation of fundamental physical constants over cosmic time scales.

11
The richness of the information available from spectroscopic analysis is belied byits complexity. The farther removed one becomes from the simple picture of transitions
between discrete hydrogenic levels, the more difficult it becomes to make sense of thedense forest of spectroscopic features that encode the details of an atom or molecule’s
nanoscale structure and dynamics.

1

This onset of complexity may be related toseveral successive complications of the primitive hydrogenic picture: first, the transition
to systems with multiple interacting electrons; second, the passage beyond bound statetransitions to the consideration of excitations into the scattering continuum; third, theexistence of multiple internal degrees of freedom that may be simultaneously excited,and subsequently exchange energy with one another.
This thesis will be concerned with the resonant continuum spectroscopy of atoms
and molecules, for which all three complications are potentially present and important.
By the “continuum”, we refer to any region of the spectrum that describes complete separation of two fragment particles, either of an electron from its parent atom, molecule, orion, or of two dissociative fragments that result from the cleavage of a bond in a molecule
or molecular ion. Since the potential energy of separating fragments is not subject toquantization, the states of the continuum region are continuously distributed. The spectrum of such a region, however, will in general display quite complex and irregular peakand valley structures that signify the presence of resonant scattering effects. The distribution, shape, width, and intensity of these features reflect the geometric configurationand dynamic evolution of the system during the brief period in which fragmentation
occurs. 2
Throughout this chapter, as well as in chapter 5, molecular hydrogen is used forillustrative purposes, as well as for various example calculations. Molecular hydrogen
1

The British biologist and Nobel Laureate Francis Crick famously described the interpretation of acomplicated biomolecular spectra as being “like trying to determine the structure of a piano by listening
to the sound it made while being dropped down a flight of stairs.”
2
The material in this introduction, and the two sections following, is discussed in most introductory
graduate level atomic/molecular physics textbooks. See, for example, chapters 9-12 of [25] or chapters
1-3 of [26].

12
is a homonuclear diatomic, and as such it is characterized by greater symmetry thanheteronuclear diatomic or polyatomic molecules. This greatly simplifies the analyticalform of many expressions, sacrificing generality for the sake of improved conceptual
clarity. For sake of comparison, some discussion and computational results for the
more complicated diatomics N2 and NO can be found in [27], and a recent example ofthe successful application of quantum defect theory to a simple polyatomic system ispresented in [28].

2.1

Degrees of freedom in molecular physics
In a molecular system, there are four classes of motion: electronic, vibrational,

rotational, and translational. Since all inertial frames of reference are physically equivalent, the translational motion can be eliminated by a transformation to the rest frameof the molecule, with no effect on the observed spectrum other than an overall Doppler
shift. (For a macroscopic sample, of course, translational motion is randomized andthese effects are averaged over many atoms, imposing a limit on experimental resolution.) All other motions, however, are fully quantized, potentially spectroscopically
active (subject to the relevant symmetry-dependent selection rules), and in general define degrees of freedom in which energy can be stored and potentially transferred bycoupling to the other degrees of freedom.
The fact that the motion can be resolved in this way is a consequence of the differing energy scales with respect to which these motions occur. The separation between
electronic energy levels is commonly on the order of a few tenths of an atomic unit, atleast for the lowest levels, and the associated transitions appear as lines in the visible
or ultraviolet. The separation between vibrational levels is on the order of thousandthsof atomic units (102 to 104 cm−1 ), and appears in the infrared. The separation between
rotational levels is on the order of hundreds of thousandths of an atomic unit (10 0 to
102 cm−1 ), extending into the far infrared or microwave. These figures are all approx-

13
imate, of course, and the vibrational and rotational level splittings depend specificallyon the masses of the nuclei; molecular hydrogen and its isotopomers have light nuclei(protons or deuterons), for example, and thus their splittings are increased by an orderof magnitude relative to a heavier molecule like NO.
Table 2.1 shows the level spacings for molecular hydrogen relative to the groundstate. This data is representative of the literature references used to check the accuracyof the rovibronic energy levels in my own computations. For extended discussion of
the techniques used to calculate these and similar quantities, including the absolute+
dissociation threshold energy of the H +2 and HD molecular ions that are used to fix

the spectrum relative to threshold in the example calculations, see [29, 30, 31, 32, 33, 34].
Table 2.1: Tabulation of the lowest rovibrational levels for molecular hydrogen, in cm −1 ,based on the ab initio data of [35]. These values reflect the inclusion of both adiabaticand nonadiabatic correction terms beyond the Born-Oppenheimer approximation.
v
0
1
2
3
4
5

J=0
0.0
4161.28087.011782.515250.5
18492.1J=1
118.5
4273.88193.811883.6
15345.9
18581.9

J=2354.4
4497.98406.412084.8
15535.8
18760.5

The time scale associated with motion in a particular degree of freedom is inversely
proportional to its energy splittings, and thus the differences in energy scale can also
be interpreted as defining a hierarchy of time scales. Rotational motion is much slower
than vibrational motion, and both are slow relative to electronic motion. Conceptually
this means that an electron experiences the nuclear structure of a molecule as “fixed inspace”; the nuclei, inversely, experience the electron only as a highly averaged spatial
distribution of its motion. The result is an approximate separability of the equationsof motion governing these two disparate degrees of freedom; when this assumption istaken to be exact, the result is the familiar Born-Oppenheimer approximation. To find14
a Born-Oppenheimer solution, the nuclei are first frozen in space, and the electronic
Hamiltonian diagonalized subject to the potential for that configuration. The electronic
calculation is repeated for a systematically varied range of nuclear coordinate values,
mapping out variation in energy levels to define associated potential surfaces. The
potential surface is then used to define a Hamiltonian controlling the nuclear motion,
which is diagonalized for the vibrational energy spectrum and vibrational wavefunctions.
Figure 2.1 shows the lowest electronic singlet ungerade potential curves for molecular hydrogen and the Rydberg series limiting curve of the molecular ion, based on datafrom the ab initio calculations of Kolos and Wolniewicz. Notice the avoided crossingin the 41 Σu curve, resulting from the crossing of a repulsive electronically autoionizing (doubly excited) state potential curve that descends through this region, creating asimilar series of avoided crossings with all of the singly excited electronic states higher
than n=4.
At the level of the adiabatic approximation, the quantum defect may be definedin the molecular body frame as a function of the internuclear separation R,

1+UnΛ (R) = U1sσ(R) − [n − µΛ (R)]−2 .
2

(2.1)

In practice, the quantum defect function µ Λ (R) has a weak energy dependence, although
it rapidly approaches an energy-independent value for increasing principal quantum
number n. For energy-independent quantum defect calculations, the quantum defectmust be defined either relative to one particular potential curve, or else through some
limiting procedure in the energy. As an additional approximation, the curves are here
diabatically continued through the avoided crossings, since the the relation in Eq. 2.1
assumes there are no avoided crossings with other electronic states. (This is a reasonableapproximation for the ungerade states reached in ground state photoionization of H 2 ,since the avoided crossings appear outside the Franck-Condon region and have little

−0.4

Potential energy (a.u.)

−0.5

−0.6

−0.7

−0.8

0

2

4

6

8

10

R (a.u.)Figure 2.1: Potential energy curves for the Σ (solid) and Π (dashed) symmetries of H 2 , together with the ionic potential curve (dotted),
based on the ab initio calculations of Kolos and Wolniewicz [36].
15

16
influence on the preionization spectrum.) For the sake of all remaining calculations inthis section, we will use the 5Σ and 6Π potential curves to define the energy-independent
quantum defects, as shown in Figure 2.2. The potential curves associated with these
defects are shown in Figure 2.3, giving some visual estimation of the extent of theapproximation involved in neglecting energy dependence. The most obvious error isthe catastrophic failure of the 2Σ state to reach the correct asymptotic limit. The
higher Rydberg states that dominate the continuum spectrum, however, are quite welldescribed by the energy independent approximation. Corrections to this approximation
are discussed in [37, 38, 39].
The rotation and vibration of a molecule are coupled more strongly than the
nuclear and electronic degrees of freedom, by a combination of centrifugal and Coriolislike terms of the Hamiltonian. In general, one should not speak of separable vibrationaland rotational states, but only a fully coupled “rovibrational” state that obeys theentire nuclear Hamiltonian. To a good approximation, however, the energy of such astate is still recognizable as the combination of a rotational excitation and a vibrational
excitation, provided the rotational excitation is small.
The coupling between rotation and electronic angular momentum is particularly
sensitive to the excitation of the Rydberg electron. The low temperature spectrumof molecular hydrogen is strongly influenced by an l-uncoupling process, whereby theelectron undergoes a gradual transition from the low n case of being strongly coupled
to the internuclear axis (Hund’s case [b], where the approximately “good” quantum
number Λ is the projection of the total orbital angular momentum onto the internuclearaxis) to the high n case where the electronic angular momentum decouples from thesymmetry axis and is instead quantized along the ionic core’s axis of rotation (Hund’scase [d], where the approximately “good” quantum number N + is the rotational angularmomentum of the core). In the range n=6-10, the coupling is mixed, and both Λ andN + are only meaningful in an approximate sense.

17

Quantum defect

1

0.5

0

−0.5

0

2

4
R (a.u.)6

8

Figure 2.2: Quantum defects for the Σ (solid) and Π (dashed) symmetries of H 2 , basedon the fittings of Jungen and Atabek [40].

−0.4Potential energy (a.u.)

−0.5

−0.6

−0.7

−0.8

0

2

4

6

8

10

R (a.u.)Figure 2.3: Effective energy-independent potential energy curves for the Σ (solid) and Π (dashed) symmetries of H 2 , based on the quantum
defects in Figure 2.2, together with the ionic potential curve.
18

19

2.2

Resonances
In phenomenological terms, a resonance in the continuum scattering behavior of a

dynamical system appears as a strong and localized energy dependence. This may manifest itself as either an enhancement or a suppression of the background cross-section.
Mathematically speaking, a resonance is associated with the existence of a nearly boundstate of the Hamiltonian, an approximate eigensolution that would become a true boundstate if the Hamiltonian were modified in some intentional way. Resonances in atomic
and molecular physics result in continuum wavefunctions that accumulate probability
density in some restricted region of configuration space for which the magnitude of theseparation between the scattering particles is “small”, such that it resembles the distribution of a bound state wavefunction of the two-particle system. If a state at a resonant
energy is initially prepared in a superposition that gives it a spatial distribution localized in this way, it will generally remain localized for a long period of time before slowly
decaying into the exterior continuum region; in time dependent terms, a resonance canbe described as a decaying nearly-bound state, with an associated lifetime inversely
proportional to the strength of its coupling to the surrounding continuum.
Resonances commonly arise from one of two situations. A shape resonance occurs
when the single-channel potential describing the interaction of two particles has a local
maximum separating two regions, and the scattering energy is below the energy of thismaximum such that it lies at what would become a bound state in the inner region
if the potential beyond the local maximum were replaced by an infinite wall. In thetime-dependent picture, a particle must travel back and forth between these regions
by tunneling through the potential barrier; since tunneling probability is exponentiallysensitive to the height of the barrier, the particle can be effectively trapped inside theinner region for a duration sufficient to complete many cycles of the orbital period
appropriate for classical motion within the local potential well. Shape resonances in20electron-atom or atom-atom interactions are commonly encountered as a result of thecentrifugal correction to the effective potential arising from the (l+1)l2r 2 repulsive term;
attractive interactions are typically shorter-range, and thus the cross-over between thestrong but shorter-range attractive potential region and the weaker but longer-range
repulsive potential region creates a local maximum close to threshold for scattering
states with a finite angular momentum.
A Feshbach resonance is the result of coupling between channels associated withthe excitation of additional degrees of freedom in a system. For example, the energy
associated with two electrons being simultaneously excited to hydrogenic Rydberg states
would, in the absence of electron correlation, be a stable doubly excited state. Since
one of the two electrons may relax back into the ground state and give up its energy
to promote the other into the continuum, the doubly excited state is unstable withrespect to autoionization, and the two-particle bound state solution instead becomes aresonance. In molecular systems, autoionizing resonances can also result from coupling
to the nuclear degrees of freedom. Because of small terms in the Hamiltonian that violate
the Born-Oppenheimer assumptions and introduce coupling between the nuclear andelectronic motion, a system excited in both a nuclear and an electron degree of freedom
is unstable with respect to transfer of energy from the nuclear excitation into the electron(if it has a total energy greater than the ionization threshold). As a result, the electron
is able to escape into the continuum. At the energy of the hypothetically uncoupled
doubly-excited state, the continuum will show a preionization resonance, a resonancethat reflects the existence of a short-lived metastable state that eventually undergoes
ionization. Similarly, if the hypothetical doubly-excited system is above the energy
required for dissociation, the electron can relax to a lower energy level, transferring its
energy to the nuclear motion and yielding a predissociation resonance. If both typesof continua are energetically accessible at once, then the processes will compete, withrelative intensities of the preionization and predissociation spectra at that energy that21
correspond to the probabilistic branching ratio between the two processes. (we shall set
aside the question of predissociation until Chapter 5, and for the moment limit attention
to the preionization spectrum.)
Bound states are characterized by a single intensity parameter, governed (in thelimit of the dipole approximation) by the dipole oscillator strength of that transition.
The width of a bound state transition line is generally controlled by spectral broadening
effects that arise from the finite thermal energy of the sample; as the temperature isdecreased, the lines become sharper. Bound state transition lines cannot be made arbitrarily narrow, however, due to the natural linewidth arising from their finite lifetime fordecay back to the ground state. Resonances, on the other hand, have a natural width
associated with the much faster time scale for energy interchange between coupled degrees of freedom; they can correspondingly display much larger linewidths. The shapeof a bound state transition line is usually a symmetric Gaussian, in accordance with theexponential Maxwell distribution of kinetic energies at finite temperature. Resonances,by contrast, have a (potentially asymmetrical) Fano line shape. The asymmetry canbe explained by recognizing that the excitation to the continuum can occur either by adirect pathway (just as at non-resonant energies), or by the creation of the intermediate
autoionizing state; these two pathways must be summed coherently, and thus could giverise to either destructive interference (suppressing the intensity) or constructive interference (enhancing the intensity). Depending on the value of the lineshape parameter,
the peak may even be converted into a window resonance, a nearly symmetric dip in
the cross-section.
In the limit of extremely weak coupling (and thus extremely long decay lifetime),
resonances strongly resemble bound states. Indeed, as the coupling vanishes, a resonance must pass over smoothly into a true bound state embedded in the continuum.
Although the continuum itself is uniformly and densely packed with states, it is possible
to imagine taking a carefully arranged linear combination of continuum wavefunctions

22(with, in general, complex coefficients) that behaves very much like a discrete eigenstate[41, 42]. In particular, it will display time dependence proportional to e iEa t , where Ea
is a complex value; the real part of E a describes periodic motion within the local quasibinding potential, and the imaginary part describes the slow leak of electron probability
density into the rest of the continuum. This complex energy state is in some sense theoptimally discretized eigenstate of the Hamiltonian for representing that particular resonant process; it is a true pole of the scattering Green’s function, analytically continued
into the complex plane. One can imagine performing the same transformation in theother direction; taking a finite basis of such scattering eigenstates, and using them as arepresentation of a portion of the continuum. we shall revisit this idea in chapter 5.
For the Coulomb potential of an electron in a neutral molecule, there exists an
infinite number of excited electronic Rydberg states. The ionizing electron leaves behinda molecular ion which may be in either the ground state or any energetically accessible
rotationally or vibrationally excited state. The residual state, in conjunction with allother quantum numbers needed to describe the escaping electron, is termed a channel, and labeled by the appropriate rotational or vibrational quantum numbers of themolecular ion. (The use of asymptotic states to define the channel labels provides anatural connection with the scattering matrix (or S-matrix) representation of the scattered continuum wavefunction– the indices of the S-matrix correspond to the asymptotic
channels.) The energy at which a channel becomes energetically accessible, or open, is
simply the energy of the molecular ion in that rotational and vibrational state. At lower
energies, the channel is closed, but still influences the continuum by creating an infinite
series of Rydberg resonances. Unlike bound state transitions, which have experimental
widths that reflect an incoherent sum over many particles in a sample, resonances can(and frequently do) overlap with one another and produce complicated interference effects. In particular, resonances belonging to a Rydberg series attached to one channel
threshold may interact with resonances belonging to a different channel threshold. If the

23
two series have distinct properties– say, one has broad widths and decays primarily byionization, and the other is narrow and decays primarily by dissociation– the composite
effect on the (coherent) total spectra can be quite complex and difficult to untangle.
Surmounting this difficulty is the principal task of theoretical multichannel Rydberg
spectroscopy, known more commonly in the literature as multichannel quantum defecttheory (MQDT).

2.3

Channel elimination: Traditional and energy-smoothed approaches
Quantum defect theory begins from the assumption that the N o independent

solutions of an No -channel Schrödinger equation can be written as an expansion withterms consisting of a product of channel functions (describing the state of the core andall degrees of freedom of the outer electron except the fragmentation coordinate) anda linear combination of radial Coulomb functions. These long range solutions, valid
beyond the region where channel coupling remains significant, can be written in morethan one representation, but a convenient choice for conceptual purposes is the outgoingwave formΨphys(E) = A
i0

No
X
1
i

r

Φi (ω)[fi+ (r)Sii0

phys− fi− (r)δii0 ], r → ∞,(2.2)

where the scattering matrix is labeled with the superscript phys to denote that only
channels which are energetically accessible (open) will contribute to the sum. The
Coulomb functions fi± (r) are those that obey incoming and outgoing wave boundary
conditions [43], and the operator A denotes a formal antisymmetrization procedure.
In this expression, Φi (ω) includes the total wavefunction of the core associated withionic threshold energy Ei , and the spherical harmonics Ylm (θ, φ) and spin wavefunctionof the outer electron. For the example system of molecular hydrogen, we adopt theapproximation that only p-wave scattering contributes to the state probed in total

24
absorption– even parity states are symmetry forbidden transitions, and f and higher
partial waves are unable to penetrate through the centrifugal barrier into the couplingregion– and thus the index i describes only the vibrational and rotational states v + , N +of the core. (In fact, for the calculation for HD in the following section the breakdownof gerade-ungerade symmetry compromises the accuracy of this approximation, but thisis expected to only be significant in the vicinity of the dissociative thresholds of highly
excited vibrational states.)
The critical point realized by Seaton, Fano, and other MQDT pioneers is that theasymptotic boundary conditions implied by the above expression need not be enforced
until the final stage of the calculation. The short range interactions involved in thecreation of auto-ionizing resonances can be fully included by the use of an alternate
expansion that incorporates an additional N c accessible excitation channels, whose Rydberg electron channel energies are below the threshold for ionization. This modified
expression is still only accurate beyond some value r 0 bounding the small-r region where
coupling occurs, and can be written in the similar form

Ψi0 (E) = A

N
X
1
i

1
Φi (ω) √ [fi+ (r)Sii0 − fi− (r)δii0 ], r > r0 .
r
i 2

(2.3)

Here Sii0 indicates a quantity familiar in multichannel quantum defect studies, thesmooth, short-range N-channel scattering matrix.
Roughly speaking, the i0 -th independent solution in Eq. (2.3) describes a simple
one-step scattering process in which the outermost electron encounters the ionic core
while moving inward in channel i0 , and scatters outward into all channels i. Some of thei channels may be energetically closed at r → ∞, but the finiteness boundary condition
has not yet been imposed.
In order to remove unphysical closed-channel divergences, the usual MQDT “channelelimination” procedure is employed, which amounts to forming an appropriate linear

25
combination of the solutions in Eq. (2.3) to construct the physical scattering matrix. We
write this expression in the form used in Ref. [44], in terms of the Hermitian conjugate
of S, partitioned into open and closed sub-blocks:

S†phys = S†oo − S†oc [S†cc − e2iβ ]−1 S†co .

(2.4)

The four matrices in the right-hand side of (2.4) are physically open and closed partitionsof the smooth, short-range scattering matrix (Hermitian conjugated) from Eq. (2.3),


†
 Soo

S† = 
S†co

S†oc 
S†cc

.


(2.5)

The matrix β is diagonal and of dimension N c × Nc ; its elements βi = π[−2(E −Ei )]−1/2 are the Coulomb phase parameters in each closed channel. These quantities
vary strongly with energy near their respective thresholds, and account for nearly all
the energy dependence of the physical scattering process. This allows the expression
to correctly describe resonant spectra even while the energy dependence of the smooth,short-range scattering matrix is entirely neglected.
The physical significance of the eigenvalues of the scattering matrix, as first realized by Fano [45], centers on the recognition that S will be diagonal in the representation
where the short-range Hamiltonian is diagonal. That is, there exist N = N c + No solutions, the eigenchannel solutions Φ α , each of which have a common phase shift πµ α
in all physical channels. The scattering matrix can be recovered from the eigenchannel
defects µα by means of a transformation,

Sii0 =X

Uiα e2iπµα (UT )αi0 .

(2.6)

α

The matrix U is known as the frame transformation matrix as a consequence of itsrole in transforming from the short-range (“body frame”) diagonal representation to
the long-range (“lab frame”) electron-plus-core representation. In the inner region, the26outer electron is moving fast enough that a definite value of the internuclear separation
R and orbital angular momentum along the axis Λ can be defined; in the outer region,
the electron is moving slowly on the time scale of nuclear motion, and the system ismore appropriately described by the ionization channel indices. The rotational andvibrational components of the frame transformation matrix can be separated to give+Uiα = hi|αi = hv + |Ri(N ) hN + |Λi(lJ) .

(2.7)

The first factor is simply the vibrational wavefunction for the v + state, and the second
factor is a rotational frame transformation matrix element defined in [27]. Note that
this expression assumes that a single value of the orbital angular momentum l is present.
If additional partial waves contribute significantly in the outer region, then the form of
this transformation becomes more complicated.
The eigenchannel quantum defects, since they are labeled by quantities that aremeaningful in the Born-Oppenheimer limit, can be extracted from the bound state
potential energy curves of H2 through their definition given in Eq. 2.1. The actual use of
these quantum defects in a numerical frame transformation is problematic owing to thecontinuous variable R, which effectively necessitates an infinite number of channels to
be included in the scattering matrix. In practice, however, a reasonable approximation
to the exact frame transformation can be accomplished by truncating the sum overalpha after a relatively small number of vibrational states (usually of the order of one or
two dozen per ionic rotational quantum number in this work). The approximate frametransformation matrix is formed from the eigenvectors of the quantum defect matrix,
defined by(J 0 )µv+ N + ,v+0 N +0 =XΛ

0hN + |Λi(lJ )

Z

+

0

dRhv + |Ri(N ) µΛ (R)hR|v + iN

+0



0

0

hΛ|N + i(lJ ) ,
(2.8)

27
The numerical diagonalization of Eq. 2.8 is generally more convenient than direct
construction and diagonalization of the scattering matrix itself.
Because of the weak energy dependence of the quantum defect (see Ref. [37]) asingle function for each body frame symmetry, µ Λ (R), suffices to describe all bound state
potential energy curves of the π symmetry, and provides a reasonable approximation tomost higher n curves of the σ symmetry. We follow Raoult and Jungen [46] in selecting
an eigenquantum defect function derived from fitting to the bound states of the n=6
curve for the σ case, and n=5 for π. Since the 3pσ state deviates substantially fromthis simple approximation, and the 2pσ state also differs qualitatively, peaks associatedwith resonances in these low-n channels are not expected to be located precisely by
this method. Gao and Greene [39] have shown how the frame transformation may bereformulated to account for such energy dependent effects.
The total photoionization cross section is directly proportional to the differential
oscillator strength, and is given by the expression

σ(ω) =

4π 2 ωα X X (−)phys 2|Dj|
3(2J0 + 1) J j(2.9)

where J0 is the initial total angular momentum state, J is the final angular momentum(−)physstate, Dj

is the reduced dipole matrix element between the j-th physical incoming-

wave solution and the molecular ground state, ω is the frequency of the absorbed photon,(−)physand α is the fine structure constant. D j

can be calculated by a channel eliminationtransformation analogous to that for the physical scattering matrix above,− †2iβ −1 †D(−)phys = D−
] Sco ,o − Dc [Scc − e(2.10)(−)

using the reduced dipole matrix elements for the long-range solutions D j , which canthemselves be found from the body-frame dipole matrix elements by application of theframe transformation coefficients [47, 27]. Note that in a real physical system at finite

28
temperature, more than one J0 will typically be present, and a final average must be
carried out over the initial statistical distribution.
In traditional MQDT, an analogue of the scattering matrix channel elimination
procedure is performed to determine the final solutions used in the dipole matrix elements. As before, this allows the N o physical Djphys to be factored into the matrix
product of a vector consisting of N weakly energy dependent short-range D j and an No
by N matrix containing all of the resonant structure. The cross section may then be
convolved over some finite width Γ using (for example) a Lorentzian kernel

1
σconv (ω0 ) =2πZdω σ(ω)Γ(ω − ω0 )2 + (Γ/2)2(2.11)

to produce a spectrum that approximates the finite resolution achieved in any actual
experiment.
Robicheaux [48] showed that this analytical form for the convolution can be
rewritten in the suggestive form (suppressing the sum over J)iΓ X
4π 2 αIm (ω0 + )σconv (ω0 ) = −3(2J0 + 1)
2 j

Z

(−)

dE |Dj |2iΓ
Eg + ω 0 +
−E
2

−1




(2.12)

where E = Eg +ω is the energy of the final state and E g is the ground state energy. The
sum and integration can be formally carried out to yield a new formula which involves
the Coulomb functions defined at complex electron energies E = E r + iΓ
2 . Robicheaux’s
original expression for this preconvolved cross section, after rearrangement and theapplication of a matrix identity, can be written asiΓ X (−)∗ X  †4π 2 α(−)(S − e2iβ )(−1) (S † + e2iβ )D Im (ω0 + )Djσconv (ω0 ) = −j,k k
3(2J0 + 1)
2 jk

(2.13)

29
Unlike the partitioning scheme for channel elimination (as in Eq. (2.4), thisformula treats open and closed channels on an equal footing. In essence Eq. (2.13)
relies on the exponential to naturally close a channel every time its argument β (acomplex generalization of the phase parameter) switches from being almost real atenergies below the channel threshold to having a large imaginary component at energies
above the threshold. Although the exact expression derived by Robicheaux for β iscomplicated to evaluate, the approximation

βj = πκj ,Er − Ej < 0(2.14)

βj = i∞,E r − Ej > 0(2.15)

qcan be used in its place. Here κj = 1/ −2(E − Ej ) is a complex generalization of theeffective quantum number in channel j, with the branch chosen such that Re β j > 0
when Er − Ej < 0.

2.4

Application: HD photoionization spectrum
The experimental photoionization spectrum of HD was first observed at high res-olution by Dehmer and Chupka [49], and theoretically modeled to a high degree of
accuracy by Du and Greene [50]. In this section, we revisit the HD problem using Robicheaux’s faster, more direct reformulation of MQDT to generate a much broader rangeof the photoabsorption spectrum. The decision to return to this system is motivated byseveral factors. First, high resolution experimental data exists for both the photoionization and photoabsorption cross-sections, and it provides a qualitative measure of theinfluence of competing dissociation channels or other decay pathways. This allows us
to study the anticipated failure of the Ref. [50] implementation of energy independent
MQDT. Discrepancies are expected in the neighborhood of strong coupling to the dis-30
sociative regions of the potential energy surface, where g-u symmetries become mixed,
and where the 1sσ potential curve may no longer adequately suffice to describe the coreelectronic state. Second, most of the peaks in the experimental spectrum of HD haveremained unassigned, or have only tentative spectroscopic labels, and the present study
can identify them with minimal ambiguity. Third, this comparatively straightforward
application offers a reasonable opportunity to assess the speed and robustness of thepreconvolution variant of MQDT, with the intent of considering it for more generalapplication to other molecular problems in the future.
Below the first ionization threshold, located at 124568.5 cm −1 , the spectrum is
discrete, consisting of isolated bound states. The preconvolution algorithm describes thisregion equally well, but owing to our primary interest in the photoionizaton spectrum,
discrete photoabsorption will not be discussed further here. At higher energies, thespectrum consists of sets of Rydberg series that converge to thresholds associated withrovibrational states of the residual ion. Because the vibrational levels are more widely
spaced than the rotational levels by approximately an order of magnitude (∼2000 cm −1
vs. ∼200 cm−1 ), ionization thresholds appear as neighboring pairs, with all allowedtransitions corresponding to an ionic rotational state of either J + 1 or J − 1 (since
the Rydberg electron is restricted to the p-wave solutions excited when an HD groundstate electron absorbs a single VUV photon). To simulate the results of a spectrum
taken at 78 K, the contributions from the three energetically accessible initial rotational
states, labeled by their total angular momentum as J 0 =0, 1, and 2 are weighted by the
Boltzmann statistical factors 61.9%, 35.8%, and 2.2%, respectively. This results in four
distinct thresholds, each with up to three attached series (R, Q, and P branches), which
originate from each vibrational state of the core.
Much of the complexity of the photoionization spectrum arises in regions where
peaks in one series are perturbed by the lower-n resonances of a higher threshold thatlie at roughly the same energy. In such cases, spectral features can become so strongly

31
mixed as that any formal spectroscopic assignments would be somewhat arbitrary. A
common example occurs when a lower-n resonance appears just slightly below a highern threshold, such that its width is less than the energy separations of the near-thresholdseries. In this instance, the low-n resonance will be distributed over many adjacentpeaks in the series, causing the total number of such peaks to be increased by one anduniformly enhancing their intensity, yet no one peak can be unambiguously assigned tothe perturber.
To a limited extent, the spectrum exhibits repeated patterns attributable to thenearly uniform separation between the first few vibrational levels. As a result, a subsection of the total spectrum that spans one vibrational energy spacing serves to provide
a qualitatively representative measure of accuracy over a broader range. We have chosen to focus on the spectrum in the energy regime near the second (v + = 1) set ofrovibrational thresholds, from 125500 cm −1 to 127300 cm−1 .
Figures 2.4-2.7 show the theoretical results superimposed on the experimental
data points. Since the experiment does not provide an absolute normalization for theintensity, we have normalized this data to match the theoretical cross-section at 127000
cm−1 . The peaks in Figure 1 have been labeled through the application of a techniquedetailed in [37] which involves searching for the roots of a determinantal equation andexamining the magnitude of the coefficients in each closed channel. As n becomes large(≥10), the coupling (as noted earlier) undergoes a transition from Hund’s case b (where
the projection of the angular momentum of the Rydberg electron along the internuclear
axis Λ is a “good” quantum number) to Hund’s case d (where the preferred quantum
number is the rotational state of the core). Although this decoupling occurs gradually,
we adopt here the convention of describing resonances of n<10 according to the firstcase, and n>10 in terms of the second.
In Figures 2.4 - 2.7 several peaks are labeled that exhibit some significant discrepancies between theory and with experiment. These discrepancies likely arise from our

32

Figure 2.4: Experimental[49] and preconvolved theoretical oscillator strength [51] forHD photoionization between 125500-126000 cm −1 photon energy.

33

Figure 2.5: Experimental[49] and preconvolved theoretical oscillator strength [51] forHD photoionization between 126000-126500 cm −1 photon energy.

34

Figure 2.6: Experimental[49] and preconvolved theoretical oscillator strength [51] forHD photoionization between 126500-127000 cm −1 photon energy.

35

Figure 2.7: Experimental[49] and preconvolved theoretical oscillator strength [51] forHD photoionization between 126000-127500 cm −1 photon energy.

36neglect of competing predissociation processes in the MQDT calculations. As previously
shown by Du and Greene[50], departures from linear absorption in the experiments fromphoton energies ∼12600 to greater than 12700 cm −1 are also a potential problem. Regarding predissociation, for example, the resonances labeled R(0), R(1), and Q(1) thatare assigned to perturbers associated with v = 3 − 8 ionic cores are likely to be affectedby coupling with either the repulsive, dissociative inner wall of the potential energy
surface via the Coriolis interaction (for Λ 6= Λ 0 ), or the outer potential regions where
nλn’λ0 doubly-excited potential curves are known to intersect and strongly perturb the
Rydberg potential surface. Such examples of channel interactions are well-documented
in H2 (e.g. the 3pσ-3pπ Coriolis-type predissociation discussed in [52], and predissociation of the 3pπ, v = 8 and 4pπ, v = 5 states discussed in [53] and [54]). Although these
two mechanisms appear to be distinct, MQDT in principle allows a unified descriptionof joint Coriolis interactions and electronic core excitations.
Comparison of our results with experiment in Figure 2.4 shows that many of theobserved resonances are in good agreement in terms of position and intensity. Most ofthese resonances are assigned to those classified in the Hund’s case d limit and theyalso have low vibrational excitation. Discrepancies in intensities, however are apparent
for the strong v=3 resonances predicted at 125750 and 125830cm −1 , as well as the v=8
set between 125860 and 125900 cm−1 . These resonances are associated with 5pπ, v = 3
interlopers. For HD it is somewhat surprising (but not unprecedented) that Rydberg
states attached to ionic cores with vibrational quanta as low as v = 3 excitation willstrongly sample the dissociative regions of the potential energy surfaces, resulting in asignificant competition with the autoionization channel. The increase in the effective,
non-adiabatic couplings in HD may therefore be related to the breakdown of the g-u
inversion symmetry, particularly near its dissociative limits; for the corresponding peaksin the H2 spectrum the predissociation is only moderate [55].
Another class of discrepancy in Figure 2.4 is typified by the predicted R(0), v = 2,

37
7pσ interloper at 125970 cm−1 , which, while in good agreement in terms of position,is too intense. This may be due to the breakdown of symmetry, or else to the nonlinearity of the intensity dependence on the cross section arising from saturation effects[50]. (The results of Glass-Maujean [55] imply that ionization prevails for the 7pσ seriesin H2 except in cases where cases where the peaks are strongly mixed with anotherseries.)Figures 2.5 - 2.7 show a comparison of the present theory with the remaining
range of experiment considered here, leading up to 127500 cm −1 . Figure 2.5 shows
an increasing number of low-n, high-v interlopers predicted by theory with minimalcorrelation to resonance observed in the experiment. It is also apparent that shifting ofmany predicted resonances ∼3-5 cm−1 compared to experiment occurs near the v = 1
threshold at 126600 cm−1 (shifts of this magnitude are also noticeable for some of theresonances in Figure 2.4 as well). The origin of these shifts is not certain, althoughRef. [50] pointed out that they are likely due to our neglect of energy dependence inthe quantum defects, or else associated with random slippage errors in the experimental
wavelength calibration.
The discrepancies seen in Figure 2.6 are most apparent for the very strong resonances predicted at 126610, 126680, and 126700 cm −1 , respectively. These resonances
are associated with 4pπ, v = 5 interlopers which can predissociate. This resonance andintensity pattern parallels the three predicted 5pπ, v = 3 interlopers shown in Figure
2.4, perhaps strengthening the conclusion that those are strongly predissociated as well.
In Figure 2.7 one observes a complete breakdown of agreement between the theoretical predictions and experiment beyond ∼127200 cm −1 . This must be attributed tomy neglect of predissociation processes, particularly since the predicted resonances areassociated with interlopers which have very high vibrational excitation, for example,
states with 3pλ, v = 8 − 9. These states would predissociate primarily through Coriolis
coupling with nσ states of comparable total energy.

38
In summary, disagreement between the MQDT calculations and experiment isprobably partly attributable to the neglect of predissociation via the Coriolis interaction
and channel-coupling with two-electron excited core states. In limited spectral regions,departures from linearity in the original experimental conditions could also be a sourceof discrepancy. In chapter 5, approaches to the treatment of predissociation will bereconsidered in greater detail.
Within regions of the spectrum dominated by high-n resonances, however, theframe transformation-based MQDT method remains impressively accurate. The very
recent work of Merkt and coworkers [56] provides a persuasive example of the capabilitiesof the frame transformation to describe even densely spaced resonances in the nearthreshold regime. Figure 2.8 shows the experimental photoionization spectrum plottedin a narrow region near the v = 1, np0 threshold, where the Rydberg series has many
sharp and narrow resonances. Just above the threshold is a complex resonance arisingfrom the perturbation of the v = 1, np2 series (converging to a threshold at 789.847
Å) by an interloper attached to the v = 2 threshold. Each of the modulated groupingsof peaks below the threshold at 790.630 Å can similarly be understood as a complex
resonance arising from the mixing of the overlapping v = 1, np0 and v = 1, np2 series.
The theory (b) in 2.8 is a reproduction of the result in [50], calculated at too coarse amesh to identify any of these high-n structures. Figure 2.9 shows the same spectrum
recalculated on a much finer mesh; the positions as well as the intensity modulationand shifting line profiles across the complex resonance sets are all well-described by thetheory, aside from a few very weak and broad features in the experimental spectrum.
(The disagreement in intensity for the Q(1), v = 2 8pπ peak at 790.07 Å is presumably
due to predissociation.)

39

Figure 2.8: Ultra high resolution experimental (a) [56] and theoretical (b) oscillator
strength for HD photoionization near the v=1 threshold, using theory from the mid1980s by Du and Greene [50].

Figure 2.9: Recent recalculation of the theoretical oscillator strength by the same
method, on a much finer grid and at a rotational temperature (25 K) appropriate forthe newer experiment.

Chapter 3

Long-range Rydberg state: Approximate and model approaches

In the previous chapter, we considered a system consisting of a Rydberg electron
attached to a single positively-charged core. Within a small region of space definedby the dimensionality of the core, the electron would experience a complex potential,
but outside that region, the behavior of the electron would be well-characterized bysome linear combination of bound or continuum hydrogenic solutions. The method ofmultichannel quantum defect theory was explicitly designed to exploit this natural partitioning of the Hamiltonian into a complicated form in one tightly localized region of
space, and a far simpler form everywhere outside of it. Suppose that, instead of a single
core, the Rydberg electron had two such regions of “hard physics”, well separated fromone another, within the volume of space where its probability density was non-vanishing.
The philosophy of quantum defect theory would still be highly appropriate for this classof nuclear geometries. The vast majority of the volume in which the electron resided
would still be simple Coulombic physics, and the interaction with the isolated and localized regions where the electron was governed by a more complex Hamiltonian could
still be conceptualized as scattering centers embedded in a region of far smoother trajectories, like divots in a fairway of a golf course, or bumpers in the playfield of a pinballmachine. It would still be fantastically wasteful, from a computational standpoint, toexpend effort describing the short-range and long-range interactions using the same
approach; any method that would be adequate to describe the difficult many-particle

41
interactions of the inner regions would entail accounting for all manner of correlations,(anti)symmetrizations, and excitations, and once the method was outside that region,
all of this specialized machinery would no longer be essential.

3.1

The Fermi pseudopotential approximation
When an atom in a low-lying excited state or a Rydberg state is brought neara ground state atom or molecule, much of the interaction derives from simple electron
scattering off of the ground state species. This picture was developed in a classic paper
by Fermi [57], to describe pressure shifts and broadening of atomic Rydberg spectral
lines, as well as for the scattering of neutrons by protons in a hydrogen-rich sample [58].
Fermi’s approximation, although conceptually crude, was surprisingly accurate for its
time, in part because certain effects neglected by his analysis (the “excluded volume” and“wave distortion” corrections) tend to cancel one another [59]. 1 Presnyakov, however,noted that the Fermi model would be difficult to extend to alkali-alkali interactions, due
to the electron affinity of alkali atoms [62]; most of them have either true ionic boundstates, or else near-threshold resonances.
The outer electron of a Rydberg atom is distributed over a region that extends
far from the ionic core. At large radii the Coloumb potential is weak, and varies slowly
as a function of position; consequently, the valence electron has low momentum andmay be viewed as occupying a ’quasi-free’ state. This allows the interaction between anearby ground state particle (the ’perturber’) and the Rydberg atom to be describedin terms of isolated scattering events between the electron and perturber and between
the electron and the ionic core. In effect, the three-body problem (electron, ion, andperturber) is treated as a combination of two separate two-body interactions.
Ordinarily, the scattering of a low energy electron (such as a Rydberg electron
1

For detailed reviews of the theory of pressure shifts and broadening, including comparison of the
Fermi model with several other methods, see Allard and Kielkopf [60] or Beigman and Lebedev [61].

42
far from the core) can be well-approximated by including only S-wave scattering. Thisis a familiar consequence of the Wigner threshold law. When the angular momentumof the scattered electron relative to the perturbing particle is greater than zero, however, the short-range potential may support quasibound negative ion states, i.e., shaperesonances. Since the scattering parameters vary rapidly as a function of energy in thevicinity of a shape resonance, such resonance channels can result in the formation ofmolecular bound states. This situation will be considered later in further detail, andfor now we restrict attention to the S-wave case.
The Fermi pseudo-potential appropriate for S-wave scattering is given by~ = 2πaT [k(R)]δ(~r − R)~
VS (~r, R)(3.1)~ is the position of the perturber, both from thewhere ~r is the position of the electron, R
ionic core, k(R) is the valence electron wavenumber, and we define the triplet scattering
length for electron-perturber collisions as a T = − tan δ0T /k. The energy variation of
the triplet S-wave phase shift δ0T , comes from its implicit dependence on R, according
to 12 k 2 (R) = − 2n1 2 + R1 , for a Rydberg electron with principal quantum number n. In
general, it is necessary to account for singlet scattering as well, but for Rubidium, thesinglet scattering interaction is much weaker and does not give rise to the same boundstates, so to a good approximation it may be neglected.
Alternatively, a more general zero-range pseudopotential form appropriate to partial waves beyond l = 0 can be formulated using a Green’s-function technique that treats
the perturber as an additional boundary condition on the atomic solution [63]. Suchmethods will be considered in further detail in the following chapter. The Green’s
function approach has been applied fruitfully to the study of electron transfer effects
in atomic collisions [64], particularly those in which the collision is assisted by a laser
field that excites one of the atoms into a Rydberg state [65, 66]. A second, and fully

43
equivalent, method applies the zero-range potential as an additional term in the molecular Hamiltonian, consisting of a sum of separable projections for each l-channel onto
angular momentum eigenstates [67].
Even before the work considered in this thesis, two particularly noteworthy andrelevant predictions have been based on the Fermi approximation. First, Ivanov, in his
study of the perturbation of strongly degenerate Rydberg levels, observed that the interaction potential associated with S-wave scattering of the Rydberg electron from theperturber would experience a sign reversal as it passed through the separation radius associated with the Ramsauer minimum of the electron-perturber scattering cross-section[68]. Since the potential vanishes asymptotically, and is attractive at intermediate distances, Ivanov concluded that the potential was sufficiently broad that it must support
a bonded complex at large distances. Second, Du and Greene [69, 70] applied a combination of the Fermi method and MQDT to rare gas Rydberg dimers in order to explain
features of their photoionization spectra. Even at the level of the Fermi approximation,
their potential curves displayed long-range oscillations comparable to those witnessedby de Prunelé.

3.2

Degenerate perturbation theory for the hydrogenic caseIn a high Rydberg state, the electron kinetic energy is so low that the electron-

perturber scattering is primarily S-wave, and such that the scattering properties of theelectron at non-resonant energies are dominated by the S-wave scattering length. Analysis by Greene, Dickinson, and Sadeghpour [71] showed that, in an ultracold gas, thenegative 3 S-wave scattering length can produce a new type of ultra-long-range moleculewith unusual properties. This model has already been used to predict the existenceof two classes of ultra-long-range (i.e. of size 10 2 -104 a.u.) Rydberg molecular boundstates in an ultracold Rb gas; these molecules consist of one excited and one ground state
atom [71]. In addition to being much larger than any previously known diatomic ground44
state, 2 these molecules are expected to have permanent electric dipole moments in thekdebye range, and lifetimes on the order of tens or hundreds of microseconds (for n in
the range 30-70). The Born-Oppenheimer potential curves of these states are oscillatory,
with many local minima, each supporting multiple vibrational levels. The second class,the perturbed hydrogenic states [69, 70], also have an unusual valence electron probability distribution resembling a trilobite fossil, which can be viewed as a semiclassicalelectron interference pattern [73]. Both classes of molecular states, however, are boundonly weakly, by just a small fraction of the spacing between adjacent n-manifolds. For
n = 30, for example, the minimum of the first type of state is approximately 100 MHz,
whereas the minimum of the second type is roughly 10 GHz, compared to the 250 GHz
separation between the n = 30 and n = 29 manifolds. All further discussion will befocused entirely on the perturbed hydrogenic states.
For a completely degenerate hydrogenic n 2 manifold, the Fermi S-wave pseudopotential effectively recombines the manifold into n 2 -1 states that remain unperturbed
and degenerate (and thus have zero amplitude at the perturber position) and one nondegenerate state that maximizes the amplitude at this position. Using first-order degenerate perturbation theory for the delta-function perturbation of Eq. 3.1, the resulting
non-degenerate internuclear potential may be written in closed form,

Un (R) = −n−1
X 2l + 11+
2πa(k(R))Rnl (R)2 ,
T
2n2
4π
l=l

(3.2)

min

where Rnl is the radial hydrogenic wavefunction and the states below l min are possibly
non-degenerate levels that are sufficiently distant from the manifold that their contribution may be reasonably neglected. This formula is exact for a Coulomb potential, andremains reasonably accurate for quasi-degenerate approximations to the level structureof an alkali atom. (For short internuclear distances, the potential must also be ad2
Since the first publication of this prediction, the observation of other classes of diatomics withcomparably large internuclear separations, generated by various dipole-dipole interaction effects, hasbeen reported [23, 72].

45
justed to include the polarization interaction between the ion and the perturber, since
the highly excited Rydberg electron no longer shields the ionic core.) A plot of themolecular bound state defined by Eq. 3.2 is shown in Figure 3.1.
This class of potentials supports many vibrationally bound states. The lowest
states are confined to individual local potential wells associated with the oscillation of
the perturbed hydrogenic potential, while higher states are distributed over multiple
minima, or over the entire width of the broad well formed by the energy-dependence ofthe scattering length. For n=30, the potential supports about 70 vibrational wavefunctions. All of these states are characterized by strongly asymmetrical charge distribution,with the electron density accumulated heavily on top of the perturber in a pronounced
peak. This charge separation is the origin of the unusually large dipole moments associated with this class of states, on the order of D = R − 12 n2 a.u.; for states in thevicinity of n=30, the degenerate perturbation theory predicts electric dipole momentson the order of a kdebye (see [74] for a recent calculation of the dependence of the dipole
moment on R).
It may seem somewhat peculiar to suggest that a homonuclear diatomic bonded
complex could possess a permanent dipole moment. In principle, the solutions should
appear in parity eigenstate pairs (gerade/ungerade) symmetrized in such a way as tomake the nuclei indistinguishable. In any practical application, however, the parity
eigenstates are so nearly degenerate as to make such symmetrization irrelevant; eventhe smallest external perturbation (by stray electric field, say, or a nearby Rydbergatom) is sufficient to break degeneracy and form the charge-separated states calculated
above by hybridization of adjacent J states of opposite parity. Equivalently, one could
think of preparing the system in a charge-localized state, and estimating the tunnelingtime necessary for the ground state valence electron on the perturbing alkali to penetrate
to the location of the ionic core; even a rough estimate of the parity splitting (< 10 −100
a.u., from [71]) gives a localization half-life many orders of magnitude longer than the

46

Figure 3.1: A surface plot of the Rydberg electron probability density ρ|Ψ(ρ, z)| 2 in cylindrical coordinates for the n = 30 3 Σ S-wave
scattered (“trilobite”) state [71]. The perturber is located at the position of the lowest minimum in the potential energy curve (at R = 1234
a.u.), and corresponds to the largest peaks in electron density along the axis of symmetry.

47
entire history of the universe.

3.3

Omont’s generalized delta-function approximation
For electron-perturber scattering in partial waves beyond the S-wave case, thestrongest contributions are expected to occur when some higher partial wave is resonant.
Our treatment of this problem follows a method developed by Omont [75], based on
approximating the zero-range pseudopotential with an l-expansion of the R-matrix. For
the present results, we shall be content to examine only systems for which the scattering
is dominated by the S- and P-wave partial waves, although the same approach should
also apply to perturbers with D-wave or F-wave shape resonances (such as N 2 or SF6 .)we will here assume the 87 Rb Rydberg atom, whose large ionic core destroys the orbital
angular momentum degeneracy and whose heavy mass ensures the existence of manyclosely-spaced vibrational levels. Rubidium, of course, is a favorite atom utilized in the
study of ultracold atomic gases. I shall further assume that experimental ability exists
to prepare a Rydberg state containing high angular-momentum components, either by amultiphoton process or by the imposition of a weak electric field that breaks the dipole
selection rule.
Following Omont, the matrix element associated with P-wave electron-perturber
scattering can be written ashΨ1 |Vp |Ψ2 i = −6π tan δ1T ~~ 0 Ψ2 (~r0 ) | 0∇Ψ1 (~r) · ∇~r=~r =Rk 3 (R)(3.3)

where δ1T is now the triplet P-wave scattering phase shift. we adopt the ab initio calculations of Bahrim et al [76, 77, 78], for the S- and P-wave triplet e − -Rb(5s) scattering
phase shifts.
In order to generate a bound state, the perturbation should ideally result in a negative energy shift, though bound states might in some cases result from repulsive scat-

48
tering lengths. For S-wave scattering, a negative shift occurs when the scattering lengthaT = − tan δ0T (k)/k is negative. For P-wave scattering, the corresponding parameterthat must have a negative value is the effective scattering volume, a T = − tan δ1T (k)/k 3 .
(Again, in general both singlet and triplet scattering must be included, but for Rb only
the triplet P-wave scattering is resonant.) Qualitatively, these pseudopotentials may
be viewed as selecting a linear combination of atomic states that maximizes either thevalue of the wavefunction (S-wave) or of its derivative (P-wave) at the position of theperturber. Note that for the P-wave correction, the derivative acts in all three spatial
directions, giving rise to two possible sets of states: those that maximize the gradientparallel to the internuclear axis, and those that maximize the gradient perpendicular toit. The former have a nodal plane perpendicular to the internuclear axis, and thus a Σmolecular symmetry (m = 0), and the latter place the nodal plane along the axis, andhence have a Π molecular symmetry (m = 1).

3.4

Resonance scattering effects
At the position of a resonance, the tangent of the phase shift diverges, resulting

in an unphysical form of the interaction potential. In practice, however, the energy
eigenstate is bounded by manifolds corresponding to n + 1 and n − 1, and is subject tolevel repulsion by states of identical symmetry attached to the adjacent manifolds. Thispermits a diagonalization even at energies close to the resonance energy, circumventing
the need to resort to explicit renormalization of the potential. The number of manifolds
retained above and below the n-value of interest can be varied to test the eigenvalue
stability. In a typical calculation, adequate convergence is obtained with three to seventotal manifolds.
The Born-Oppenheimer potential curves (both S- and P-waves) associated withthe Σ molecular symmetry are shown in Figure 3.2, and the curve for the Π symmetry
state (possible for P-wave only) is shown in Figure 3.3. These results are found by

49
diagonalization of the Omont psedopotential within a basis of Rb atomic states associated with the included manifolds, including both the high-l quasidegenerate states
and the lower-l states with finite quantum defect. The associated S- and P-wave phaseshifts, as functions of energy, are shown in Figure 3.4. Recall that the phase shifts
at the perturber R are implicit functions of R as a result of the change in the local
kinetic energy of the scattered electron. The most prominent qualitative features ofthe potentials are directly controlled by the energy dependence of the phase shift. For
example, the point at which the Σ S-wave potential passes through zero corresponds to
the Ramsauer-Townsend zero in the e − -Rb(5s) phase shift at 0.042 eV (around R = 450
a.u.) [71].
For the present context, the most notable characteristic of the P-wave potentials
is their broad and comparatively deep minimum. As R decreases through the vicinityof the shape resonance around R = 700, the associated P-wave potentials detach fromthe n = 30 manifold, pass through a point where the slope of the potential becomeslarge, and run close to the n = 29 manifold for smaller R. In essence, a rise by π inthe scattering phase shift due to the P-wave resonance results in a promotion of theassociated molecular curve to the next lower manifold. We emphasize that this effect isquite general; it will occur between any two adjacent n-manifolds, for any partial-wave
electron-perturber scattering channel that contains a shape resonance. This behavior
should consistently define the existence of a global minimum in the potential curve, andthus a set of bound vibrational levels. In the inset of figure 3.2 we show the first ten
vibrational levels for the P-wave scattered Σ state.
Both sets of potential curves exhibit multiple avoided crossings, with both thehydrogenic energy levels and (in the case of identical molecular symmetry) with eachother. Depending on the Landau-Zener parameters for each crossing, sufficiently highvibrational levels may dissociate on a time scale shorter than the natural lifetime of the
Rydberg state. Even for states lower in energy than the crossing energy, the possibility

50

Figure 3.2: 3 Σ Born-Oppenheimer potential curves for states arising from both S-wave
and P-wave scattering [79]. Several of the lowest vibrational levels, along with theirassociated wavefunctions, are shown in the inset. The zero of the energy axis is taken
to lie at the position of the n = 30 manifold.

51

Figure 3.3: 3 Π Born-Oppenheimer potential curve arising from P-wave scattering [79].

52

Figure 3.4: Phase shifts for low energy S- and P-wave scattering from 87 Rb as a functionof energy [79]. The Ramsauer-Townsend zero in the S-wave curve at E = 0.05 eVcorresponds to the point at R = 450 where the S-wave potential crosses through then = 30 manifold. The resonance in the P-wave phase between 0.02 and 0.04 eV controls
the rapid rise of the P-wave potential in the range between R = 500 and R = 1000 a.u.

53
of tunneling cannot be excluded; this is of particular concern for the crossing between Sand P-wave 3 Σ states, which occurs close to the S-wave minimum and could potentiallydestabilize even the lowest vibrational level. Based on this and other anticipated decay
mechanisms, including a significant contribution from blackbody radiation, we predictthat the lifetime of such states should scale as n 2 [80] and be of the order of 50µs forn = 30. This is longer than the vibrational period of low-lying levels, suggesting thepossibility of resolving vibrational substructure in the absorption spectrum.
Figure 3.5 contains a wavefunction of the Σ P-wave scattered molecular state in
the vicinity of the minimum of the potential curve. Rather than being distributed overthe entire classically allowed region, the electron density is confined to an envelope withthe approximate shape of a butterfly. The nodal pattern features two large “wings”
of electron density extending to the usual spatial boundary of the atomic Rydberg
state, but along the internuclear axis the density accumulates near the position of theperturber.
Like the trilobite states controlled by pure S-wave scattering [71], the P-wave”butterfly” states develop large electric dipole moments, despite the fact that the electron density vanishes at the perturber. The behavior of the dipole moment at the equilibrium separation with n scales roughly linearly with n, and its value for the n = 30
states is approximately 1.05 kdebye, rising to 3.91 kdebye for n = 70. The Π symmetry
states have similarly large dipole moments, but negative, with a value of -1.53 kdebye
at n = 30. Such large permanent dipole moments can be manipulated by external
electromagnetic fields or by dipole-dipole interactions.

3.5

Finite range nonlocal pseudopotential
In order to confirm the accuracy of the zero-range model, we have adopted a short-

range nonlocal approximation of the form developed by Pascale [81]. In particular, thisallows the model to represent longer-range polarization terms arising from the dipole and54

Figure 3.5: A surface plot of the Rydberg electron probability density ρ|Ψ(ρ, z)| 2 in cylindrical coordinates for the n = 30 3 Σ P-wave
scattered (“butterfly”) state [79]. The perturber is located at the position of the lowest minimum in the potential energy curve (at R = 308
a.u.), and corresponds to the largest peaks in electron density along the axis of symmetry.

55
quadrapole polarizability of the perturber. This technique has been used successfully,
for example, in the calculation of adiabatic potential curves for helium-barium collisions[82]. To verify the accuracy of the Omont theory, we performed a full diagonalization
on a two-dimensional spline basis set using a nonlocal model pseudopotential with freeparameters that could be varied to reproduce the observed phase shifts. (See [83] for a
recent example of the application of this method.) As a further test, we also implemented
the zero-range potential approximation using a Green’s-function technique, as presented
in Chapter 4. A comparison of these three methods is shown in Figure 3.6. In eachcalculation, the overall shape of the P-wave bound state confirms the validity of theextended Fermi model, with quantitative agreement to within a few percent. Notethat for the sake of this comparison, the core quantum defects have been set to zero(i.e., an H + Rb system), allowing the use of the exact analytically-known Coulomb
Green’s-function.
These conclusions can be extended to molecular perturbers, through a generalizedversion of the same zero-range potential [66]. The simplest such system, a diatomic
molecular perturber, introduces a second axis of symmetry into the problem, defining theorientation of the molecule, with the result that states of differing molecular symmetry(i.e., projection of the electronic angular momentum onto the ion-perturber axis) can
become coupled. Note, however, that if the perturber has a permanent dipole moment,
the longer-range electron-dipole interaction controls the scattering physics, invalidating
the assumptions of the zero-range approximation. The large number of shape resonances
known in electron-molecule scattering (N 2 , SF6 , BF3 , CO2 , for instance) suggests that
this class of Born-Oppenheimer potential curves should be observable in many different
contexts, ranging from the conventional quantum chemistry of valence states to theRydberg states of an ultracold gas.

56

Figure 3.6: Comparison of the Omont generalization of the Fermi pseudopotential(dashed) with the finite-range Pascale-type pseudopotential (points), and the Green’s
function method (solid) [79].

Chapter 4Long-range Rydberg states: Green’s function method

If requested to derive a method more rigorous than the perturbative or model
potential methods discussed in the last chapter, the first inclination of a researcher
trained in the tradition of quantum chemistry would be to turn to a variational method.
This would be done with the expectation that improving both the volume of the basis
set space and the description of electron correlation (by CI or some other method)
would result in a solution set converging with arbitrary accuracy to the true energy
spectrum. From a computational standpoint, however, applying the standard machineryof quantum chemistry to the system of a separated neutral atom and Rydberg atom
entails the acceptance of considerable inefficiencies. The Rydberg electron is distributed
over an enormous volume of space, only a small fraction of which is occupied by either
the ionic core or the perturbing atom. Apart from these isolated regions of space,
the electron experiences only the residual Coulombic potential of the core, completely
screened by the core electrons to yield an effective long-range charge of unity. Thus,
the system is identical to atomic hydrogen in this region, and must yield a solution
assembled by linear superposition of the hydrogenic solutions.
Conceptually, this amounts to envisioning the system as a series of scattering
events, with the electron being tossed back and forth from one center to the other, likea ping-pong ball that receives an impulse “kick” and “spin” from the paddle of eachplayer on repeated transits back and forth across the table. If we envision enclosing

58
all short-range modifications to the asymptotic Coulomb potential within small spheres
surrounding the separated centers, the eigenenergies of the system can be found bysolving for general solutions locally within each of the three partitioned regions of theelectron’s motion, and then matching them suitably at the surface of the spheres, toselect out appropriate linear combinations of the solutions in each region.
In practice, we still have no interest in actually describing the form of the wavefunction in regions where it interacts strongly with the other electrons. The electron
spends only a small fraction of its orbit in these complicated regions, and the net effectof such interactions on the final spectrum is merely to generate appropriate boundary
conditions for the “external” Coulombic solutions. In the philosophy of multichannel
quantum defect theory, the consequence of such localized boundary conditions is to define a phase shift of the usual hydrogenic solution, which must be accommodated by theformation of a linear combination of the regular and irregular solutions that reproduces
that phase shift. The same idea applies in the current context— except that because
the system has a boundary condition with symmetry dissimilar to that of the Coulombic potential, the solution will mix not only regular and irregular solutions, but also
solutions with different values of angular momenta relative to the core.
Because the solution will be controlled entirely by boundary conditions at a surface, the most computationally advantageous implementation of the solution will involve
writing a formal solution in integral form, and then evaluating the effect of phase shifts
by means of surface integral corrections to that solution. Surface integrals have reduced
dimensionality relative to any method that would involve diagonalization of functions inthree-dimensional configuration space, potentially improving the speed of calculations
by several orders of magnitude. Moreover, the surface integrals are expected to be overextremely small and localized surfaces, permitting certain natural approximations. The
most familiar way of reducing a volume integral solution to surface integral conditions
is the application of Green’s theorem (variantly known as the “second Green’s identity”

59
in some references), and so we now turn to the task of formulating a Green’s function
technique for construction of the solution.

4.1

The Coulomb Green’s function
The Coulomb Green’s function, fortunately, is known since the work of Hostler

and Pratt in the 1960s [84, 85, 86] to have an analytic closed form, derived from formallycompleting the sum over eigenstates in its eigenfunction expansion. To fix the normalization convention, which tends to vary from reference to reference in the literature,
we explicitly define the Green’s function at energy E, G c (~r, ~r 0), as the solution to the
Coulomb Schrödinger equation



1
1 2 1
∇ + − 22r 2νGc (~r, ~r 0; ν) = −δ(~r − ~r 0 )

(4.1)

1
where δ is the Dirac delta function and ν = √−2E
is the effective wave number associatedwith this energy. Then the Green’s function can be written in terms of the Whittaker
functions Mν,1/2 (r) and Wν,1/2 (r) as

Gc (~r, ~r 0; ν) =i
Γ(1 − ν) h 0
0
M
(η)W
(ξ)
−
M
(η)W
(ξ)
.
ν,1/2
ν,1/2
ν,1/2
ν,1/2
2π|~r − ~r 0 |(4.2)

The primed functions denote derivatives with respect to the total arguments, which are
given by
ξ = (r + r 0 + |~r − ~r 0 |)/νand
η = (r + r 0 − |~r − ~r 0 |)/ν.
Note that these two parameters depend only on the orientation of r̂ and r̂ 0 relative to
one another, and not the absolute orientation of the entire system, with the result thatthe entire expression is invariant with respect to rotation, just as one would expect forthe solution to a spherically symmetric potential. The fact that the solution can be60
reduced to only two dynamical variables, r + r 0 and |~r − ~r|, instead of three as might
be anticipated purely from conservation of angular momentum, is a consequence of thehigher symmetry associated with the invariance of the Runge-Lenz vector.
Alternatively, the Green’s function can be written as an expansion in sphericalharmonics Ylm (θ, φ):Gc (~r, ~r 0; ν) =X

gl (r, r 0 )Ylm (θ, φ)Ylm (θ 0 , φ0 ).

(4.3)

lm

The one-dimensional radial Green’s function for a particular angular momentum l isthen

ν Γ(l + 1 − ν)gl (r, r ) = 0
Mν,l+1/2
rr Γ(2l + 2)
0



2r<
ν



Wν,l+1/2



2r>
ν



,

(4.4)

with r< defined as the lesser of r and r 0 , and r> as the greater of r and r 0 . In termsof the regular and irregular Coulomb functions f l (r) and gl (r), the same function is
written as

gl (r, r 0 ) = −π1
fl (r< ) (fl (r> ) cos π(ν − l) + gl (r> ) sin π(ν − l)) .
0
rr sin π(ν − l)

(4.5)

The second factor in the product should be recognized as the linear combination of solutions that vanishes at infinity. (Here and elsewhere, the function g l with one argument
denotes the irregular Coulomb function, whereas g l with two arguments is the radial
Green’s function.)
Finally, the most general form of the Green’s function (from which equation 4.2
can be derived, and of which 4.3 is a special case) is the eigenfunction expansion,

Gc (~r, ~r 0; ν) =with E(ν) = − 2ν1 2

X Φnlm (r, θ, φ)Φnlm (r 0 , θ, φ)

En − E(ν)

,

(4.6)

nlm
1
and En = − 2n2 . In principle, the sum runs over both the infinityof bound states and extends as an integral into the continuum, although truncation

61
to a finite number of bound states often provides a reasonable approximation to theconverged sum. The eigenstates are the familiar hydrogenic wavefunctions2
Φnlm (r, θ, φ) = 2ns(n − l − 1)!
(n + l)!



2r
n

l

e

−r/n

L2l+1
n−l−1



2r
n



Ylm (θ, φ),(4.7)

defined in terms of the generalized Laguerre polynomial L ab (x).

4.2

Hydrogenic solution for S- and P-wave scattering
The use of Green’s function-based scattering formalism to derive the perturbationof a Rydberg spectrum has been an object of periodic interest, primarily among Russian researchers, since the mid-1960s. In their classic studies of the zero-range potential,
Demkov and Drukarev [87, 88, 63] demonstrated that the effect of a short-range potential well on a weakly bound electron in an external electric (or magnetic) field could beapproximated as a zero-range boundary condition on the Green’s function solution enforced at a single point. Twenty years later, they revisited the same problem for higher
angular momenta [89], replacing the zero-radius potential approximation with a boundary condition on a small sphere. A more powerful implementation of the same ideas
was proposed by Andreev and coworkers [90, 91], based on Landau and Lifshitz’s classic
treatment [92] of the scattering of slow particles by a short-range center. Tkachenko[93] explicitly applied this generalized theory to the Coulombic system in the context
of studying charge-transfer complexes (i.e., systems in which the short-range potential
has at least one bound state that perturbs the spectrum). The charge transfer process
has continued to attract attention, for example, in the consideration of laser-assisted
collision complex formation [65, 66], or certain fundamental summation relations [94].
One of the more interesting applications of the Green’s function method during this period was that of de Prunelé [67], who observed small oscillations in the potential curvesof alkali-He systems at separations as large as hundreds of Bohr radii. Most recently,

62
the work of Fabrikant, Chibisov, and Khuskivadze [95, 74] applied a Kirchoff integral
technique involving the matching of the Green’s function solution to a short-range solution calculated by ab initio methods. The approach of this section differs somewhat
from any of these earlier methods in the detailed derivation, but overlaps importantlywith them on the conceptual level, and predicts similar results in the appropriate limits.
In particular, we follow the same basic procedure as Fabrikant, but we rely here only
on phase shift data, without any need to find a short-range electron perturber solution
wavefunction.
To find the eigenspectrum of a perturbed hydrogenic Rydberg atom, we begin byassuming the same geometry and coordinate system as in the previous chapter, with the
Rydberg atom nucleus at the origin and the perturber located at a distance R along thez-axis. The dimension of the perturber is again taken to be small on the scale of boththe electron wavelength, and the variation of the electron wavelength. This permits
the use of the same simple quasi-classical relationship between R and the scattering
wavenumber k that appeared in equation 3.1, as a parameter for the energy-dependent
phase-shift δl (k).
For the hydrogenic case, we consider a solution Ψ(~r) to the Coulomb Schrödinger
equation, valid everywhere outside of a small sphere of radius y around the perturber
~
at R:



1
1 2 1
∇ + − 22r 2νΨ(~r) = 0.

(4.8)

By multiplying 4.1 from the left on both sides of the equation by Ψ(~r), 4.8 by G c (~r, ~r 0; ν),taking the difference, and integrating over the outer region, we find an integral form forthe formal solution,Ψ(~r) = −

1
2

Z n

o

Ψ(~r 0 )∇2 Gc (~r, ~r 0 ) − Gc (~r, ~r 0 )∇2 Ψ(~r 0 ) d3 r 0 .

(4.9)

63
By application of Green’s theorem, this then reduces to the desired surface integral:Ψ(~r) = −
=

1
2

1
2

I n

I 

o

~ c (~r, ~r 0 ) − Gc (~r, ~r 0 )∇Ψ(~~ r 0 ) · d~a 0
Ψ(~r 0 )∇G∂Ψ(~r 0 )∂Gc (~r, ~r 0 )
0
G
(~
r
,
~
r
)
−
Ψ(~r 0 ) da 0 .
c
∂n0
∂n0


(4.10)The derivative with respect to n’ denotes differentiation along a surface normal vector
directed outward from the original integration volume, that is, pointing into the sphericalsurface of integration of radius y surrounding the perturber. After defining a local
coordinate system centered on the perturber,~
rx = |~r − R|,θx = arctan

r sin θ,r cos θ − R

φx = φ,r0 sin θ,0
r cos θ − R

φy = φ0 ,and for the primed coordinate,~ry = |~r 0 − R|,θy = arctan

the outer solution becomes1
Ψ(~r) =2I ()∂Ψ(~r 0 )∂Gc (~r, ~r 0 )Ψ(~r 0 ) −Gc (~r, ~r 0 ) da 0 .
∂ry∂rx(4.11)The surface of integration is defined as the spherical set of points for which r y = y,where y  1 is a constant.
The inner region solution, defined in terms of the same local coordinate system,
can be expanded in terms of a set of spherical harmonics centered on the perturber.
The sphere enclosing the inner region is small, and the potential due to the ionic core isvarying slowly at this distance. To a good approximation, the solution at the bounding
surface (where we also assume that the short-range atom-electron potential is negligibly
small) can be written in terms of the radial wavefunctions appropriate to free-electron
partial-wave scattering: the regular and irregular spherical Bessel functions, j l and ηl .
In conjunction with the previously discussed quasi-classical approximation that fixes the64
local wavenumber as k =at the boundary asΨ(~r) =

Xq

− ν12 + R2 , this permits us to write the inner region solution(k)Alm (jl (kx) cos δl

lm

(k)

where the k-dependent parameter δl(k)− ηl (kx) sin δl )Ylm (θx , φx ),(4.12)

is the phase shift for free-electron scattering in

the lth partial wave channel. In analogy with y above, x is defined as a parameter fixing
the value of rx , this time with y < x < 1. (That is to say, not only is this solution valid
on the boundary at y, but we have further assumed that it remains approximately valid
even on a larger boundary at x where the solutions will be matched. This assumption
can be made without loss of generality, since the r and r 0 coordinates are symmetric in
the Green’s function, and will be convenient for use in a future Taylor expansion. The
approximation only becomes exact, of course, in the limit as both x and y approach 0,
but that limit may still be taken in a way that preserves the ratio between x and y.)~ axis, andSince the current system displays rotational symmetry around the R
the magnetic quantum number m is defined relative to this axis in both coordinate
systems, the final solution wavefunction will still have m as a good quantum number.
In particular, we expect to find molecular wavefunctions of Σ symmetry for m = 0,
Π symmetry for m = 1, and so forth. For present applications, we will assume thatelectron-perturber scattering for angular momenta beyond l = 1 is negligible, although
the method may be naturally extended for scattering states with a d-wave resonance.
Thus, just as for the previous methods, we anticipate three types of solutions: a Σ statearising from S-wave scattering, a Σ state arising from P-wave scattering with m = 0,
and a Π state arising from P-wave scattering with m = 1. For geometries in this chapter
where this symmetry is preserved, the solution will be written with a subscript, Ψ m (~r),
to specify this symmetry.
In order to reduce the formal solution in terms of a linear system of integral

65
equations into a numerical problem in linear algebra, the solution must be resolved
into components defined in terms of some discrete basis. For the hydrogenic case, themost logical choice for a finite basis is the set of spherical harmonics centered on theperturber, for several reasons. First, the inner region solution is already defined as
an expansion in this basis, such that the projection onto spherical harmonics simply
picks out a single term in the expansion. Since all but the first few l-values will havevanishing phase shifts (reflecting the absence of any interaction with the core), thematrix elements arising from these higher terms decouple entirely from the molecularstates generated by lower angular momentum scattering, and including them in the
projection basis gives only trivial solutions that correspond to non-interaction. Thus,
the summation in equation 4.12 may be effectively truncated after only the low-l terms.
Second, the Green’s function varies slowly enough over this distance that it may be
expanded in a Taylor series with rapidly diminishing higher-order terms. Third, because
the symmetries are controlled by the local behavior of the wavefunction at the perturber,an expansion local to that center will naturally result in a matrix equation with diagonal
elements that can be associated with states of the appropriate symmetries, and offdiagonal elements that reflect the coupling between them.
To reduce the integral equation to a matrix equation, we first insert the solution
from 4.12 into 4.11, and then project both sides onto spherical harmonics:< lm|Ψm > =I∗
dΩx Ylm(θx , φx )Ψm(k)= Alm (jl (kx) cos δl=1X
Al 0 m
2 l0

I−Ilm (x, ~y )(4.13)(k)− ηl (kx) sin δl )
h



(k)

0
y 2 dΩy Ilm(x, ~y ) jl (ky) cos δl(k)− ηl (ky) sin δl


∂ 
(k)
(k)
jl (ky) cos δl − ηl (ky) sin δl
Yl0 m (θy , φy ),∂y∗ (θ , φ ),where the function Ilm (x, ~y ) is the projection of the Green’s function onto Y lm
x x

66

Ilm (x, ~y ) =I∗dΩx Ylm(θx , φx )Gc (~r, ~r 0; ν),(4.14)

0 (x, ~and Ilmy ) is the partial derivative of this function with respect to x.

The resulting matrix equation may be written in its most general form as


















a00 (x, y; k)
..
.

···
..

a0,l0 (x, y; k)···.

al,0 (x, y; k)..
.

al,l0 (x, y; k)..

.

a0,lmax (x, y; k)..
.
al,lmax (x, y; k)..
.

almax ,0 (x, y; k) · · · almax ,l0 (x, y; k) · · · almax ,lmax (x, y; k)


















A0m
..
.
Alm
..
.
Almax m










 = 0.








(4.15)For the case lmax = 1, the resulting matrix equation has dimensionality 2 × 2 for m = 0,
with the form

 a00 (x, y; k)







a01 (x, y; k)   A00 

a10 (x, y; k) a11 (x, y; k)and 1 × 1 for m = 1, with the form

a11 (x, y; k)A11 = 0.

A10

 = 0,
(4.16)(4.17)

Eigenvalues of the Hamiltonian are thereby identified as zeroes in the determinant of
the matrix a, as a function of the wavenumber k. For a known eigenvalue k, the linear
system 4.14 may be solved, giving the relative values of the unknown coefficients A lm
(with their absolute values fixed by the additional constraint of normalization). This
process, naturally, must be repeated for varying values of the internuclear separation R
in order to map out the internuclear potentials.
One important test of the accuracy of a numerical solution of this equation is thatthe resulting determinant must be independent of y. This is a consequence of the y-

67
independence of the inner region wavefunction at the boundary x, as seen in 4.12. Thisimplies that, in taking a Taylor series expansion of Green’s function used in the outerregion wavefunction expression 4.11, only the leading order terms in y (after accounting
for any possible cancellation between the two integrand terms) need be retained.
To evaluate the matrix elements(k)all0 (x, y; k) = δll0 (jl (kx) cos δl
−

1
2

I

h

(k)

− ηl (kx) sin δl )


(4.18)(k)0(x, ~y ) jl (ky) cos δly 2 dΩy Ilm−Ilm (x, ~y )(k)− ηl (ky) sin δl




∂ 
(k)
(k)
jl (ky) cos δl − ηl (ky) sin δl
Yl0 m (θy , φy ),∂y~ As noted previwe expand the Green’s function in a Taylor series around the point R.
ously, the Coulomb Green’s function, although defined in a six dimensional configuration
space, depends on only two independent variables as a result of its high symmetry. Although these variables have already been defined as ξ and η above for the sake of acompact notation, it will be more convenient here to define them as
u = |~r − ~r 0 | = ν (ξ−η)
2andv = r + r 0 = ν (ξ+η)
2 .
~ the variable u is close to 2R, and the variable v is
For ~r and r~0 both in the vicinity of R,
close to 0. After separating out the 1/v factor containing the divergence at ~r → ~r 0 , theTaylor expansion of the analytical Green’s function, to third order in these variables,
yields1
Φ(u, v; ν)
2πv
1
1
1
=
+ Φv + Φuv (u − 2R) + Φvv v + Φuuv (u − 2R)2
v
2
2
1
1
+ Φuvv (u − 2R)v + Φvvv v 2 .
2
6

Gc (u, v; ν) =(4.19)

68
The subscript notation signifies a partial derivative with respect to that variable; Φ uv
designates the partial derivative of Φ = 2vG c (u, v; ν) with respect to both u and v, i.e.
∂ 2 Φ/∂u∂v, always evaluated at u = 2R and v = 0. The expansion lacks terms that
correspond to first and higher partial derivatives with respect to u only, since all suchderivatives of Φ identically vanish.
The expressions for the Φ terms often have a surprisingly simple form, due to theability to convert higher derivatives into lower derivatives using the Whittaker equation
identityMν00 (

2R1
ν2
2R
ν 2 k22R) = −(− +)Mν ( ) =Mν ( ).
ν
4 2R
ν
4
ν

(4.20)The second derivative M 00 here is with respect to the total argument, and the wavenumber remains defined as before according to the quasi-classical approximation k =q− ν12 + R2 .

A series of such identities can be defined recursively to reduce any expression of higher
order derivatives to an expression written using only M and W , and their first derivatives. The first few cases required for the p-wave case are given here explicitly forreference:Φvv = −k 2νΓ(1 − ν)
Mν Wν
2R2 (
ν
k4 ν
2k 2 0 0= Γ(1 − ν) ( 3 −)Mν Wν −Mν Wν +
2R
2
ν

1
0
0
Mν Wν + M ν Wν
2R2
2
=
R2


ν
1
0
0
= Γ(1 − ν)
.
M
W
+
M
W
M
W
−
ν ν
ν ν
ν ν
2R3
2R2

(4.21)

Φuv = −

(4.22)

Φvvv(4.23)

Φuvv
Φuuv(4.24)(4.25)

It is implied that all Whittaker functions and derivatives appearing above are evaluatedat the position of the perturber, that is, with the argument 2Rν .

69
The integrals in Eq. 4.14 may be simplified by noting that the projections onto
∗ (θ , φ ) will eliminate all terms of the Taylor expansion except those with the proper
Ylm
x x

symmetry, based on the orthogonality of spherical harmonics. To make the symmetry
explicit, it is advantageous to expand the variables u and v in the spherical polar
coordinate system centered on the perturber:1
u = 2R + (x cos θx + y cos θy ) + O( )
R
v =



(4.26)
x2 + y 2 − 2xy(sin θx sin θy cos(φx − φy ) + cos θx cos θy ) .

(4.27)

Then the Taylor series terms are regrouped according to their angular symmetry withrespect to the perturber center:

Gc (u, v; ν) =1
1
1
+ Φv + Φvv x + Φvvv x2 )(4.28)
x2
6
1+(Φuv x + Φuvv x2 ) cos θx2
1+(Φuv y + Φuvv xy) cos θy2
y1
1+( 2 − Φvv y − Φvvv xy)(sin θx sin θy cos(φx − φy ) + cos θx cos θy ) } .
x
2
3
1
2π



(

Additional terms proportional to higher spherical harmonics are not shown, but would
be relevant if Λ took on values greater than 1 (i.e. ∆ states and higher). The φdependence controls which terms survive in the Σ and Π molecular symmetries. For
example, in the Σ symmetry, the terms proportional to cos(φ x − φy ) will vanish in the
projection integral, whereas for the Π symmetry the reverse is true, and all terms except
for these will vanish. The spherical Bessel functions are already relative to the perturber
center, so it is trivial to expand them in powers of x or y.
Once all of these expansions in the perturber-centered coordinate system areinserted into the determinantal equation 4.15, and the relevant integrals over the Green’s
function are carried out, the elements of the determinant will be written as expansions in

70
x or y. For the zero-range approximation, where x and y are assumed to be vanishingly
small, only the leading order term is significant, and all higher powers can be neglected.
Since the inner region solution 4.12 is independent of y, the determinant matrix elements
must also be y-independent to leading order; this provides a consistency check on various
cancellations that occur in analytical simplification of the matrix elements, and has also
been confirmed numerically. The dependence on x is generally finite (since the sphericalBessels do display x dependence), but should cancel on either side of the equation to
give a final equation in ν and R only. (Factors of k may also need to be divided out of
the entire equation in order to clarify the identity of the scattering length, although thisis a purely cosmetic modification.) For the Σ symmetry (m = 0), the solution condition
from the determinantal equation is

0 =

(

(k)

(k)

1 − Φvtan δ0k+ (Φvvv − 3Φuuv ))tan δ1
−
k3

(k)
(k)
tan δ0 tan δ1(Φv Φvvv + 3Φ2uv − 3Φv Φuuv )
3

k

k

)

(4.29).

Note that the k dependence could have been entirely rewritten in terms of ν and R,
but has been left explicit in order to reveal the energy- dependent triplet scattering(k)length, aT = − tan δ0 /k, that appears in the Fermi pseudopotential treatment. A
logical factorization of this expression yields the equivalent form((k)

tan δ1{1 + aT Φv } 1 + (Φvvv − 3Φuuv )k3(k)tan δ1aT+(3Φ2uv)
1 + aTk3)= 0.

(4.30)

The final term in the second bracketed factor is numerically “small” under most circumstances, and so this amounts to a near-separation of the total quantization condition
into two independent solutions. In fact, the first bracketed factor is immediately recognizable as the result of Demkov [63] for S-wave scattering of a Rydberg electron off azero-range potential, confirming that the method recovers the predicted S-wave solution

71
in the limit that P-wave scattering is negligible, as first derived by Dalidchick and Ivanov[96]. The capability to describe higher partial waves in our theory is comparable to thetheories of Tkachenko [93] and Dubov and Rabitz [66], both derived by the method ofAndreev [90], and the results here agree for the special cases considered in those papers.
(To our knowledge, however, no previous study ever applied these results to a systemwith a scattering resonance in the manner of the current work.) Thus, Eq. 4.30 may
be understood as a generalization of the results of [63] and [96], but with the additional
capability of describing avoided crossings between the two classes of solution. For theΠ symmetry, the equivalent expression is simply(k)1 − Φvvv

tan δ1k3= 0.

(4.31)

The form of the Taylor expansion coefficients Φ xy provides a source of analyticalinformation about the behavior of the solution potential, albeit without much accompanying physical insight. For example, one can observe that both Φ v , Φuv , and Φuuv
all oscillate as a function of R such that they may vanish at particular R values, whileΦvv , Φuvv , and Φvvv do not. This implies that the Σ solution potential curves willshow oscillations, while the Π curve will be smooth. It also suggests that the coupling
between the S-wave and P-wave scattered bound states (which is proportional to Φ uv )will potentially vanish for certain values of R, such that the avoided crossing between
the potential curves may become nearly diabatic.
Extension of the above results to a non-hydrogenic case (i.e., for finite quantum
defects, as with an alkali Rydberg atom) can be accomplished in one of two ways. First,
the Rydberg core can be treated as a second scattering center for the electron, and an
equivalent set of surface integrals performed local to its center. Of course, the inner
region solution around the ionic center will need to be expanded in terms of Coulomb
functions, rather than spherical Bessel functions, but in all other respects the derivation

72
is identical. Each non-zero quantum defect creates an additional linear equation thatmust be solved, and increases the dimensionality of 4.14 by one. Alternatively, theGreen’s function can be expanded in spherical harmonics relative to the ionic core, with
the quantum defect explicitly inserted into the radial solution of Eq. 4.5:

glRb (r, r 0 ) = −1π{fl (r< ) cos πµl − gl (r< ) sin πµl }0rr sin π(ν + µl − l)(4.32){fl (r> ) cos π(ν − l) + gl (r> ) sin π(ν − l)} .
If the Coulomb Green’s function given by the expansion for vanishing quantum defect(i.e., using Eq. 4.5) is subtracted from this expression, the result is a closed form for thecorrection term, first derived (based on a somewhat different argument) by Davydkin
et al. [97],

Gq.d. corr (~r, ~r 0; ν) =X
sin πµlπ∗W(r)Wν,l+1/2 (r 0 )Ylm (θ, φ)Ylm(θ 0 , φ0 ),rr 0 sin πν lm sin π(µl + ν) ν,l+1/2(4.33)

where the notation W denotes the Whittaker function normalized according to the same
convention as the standard Coulomb f and g, i.e. W = f cos π(ν − l) + g sin π(ν − l).
The integrals over this correction term are identical in form to those of the Coulomb
Green’s function itself, and carried out numerically. (Since the sum cannot be performed
analytically, there is no advantage to attempting a Taylor series expansion as before.)
In the actual evaluation of these integrals, numerical stability of the root-findprocedure is greatly improved by the analytic removal of singularities. In the Coulomb
Green’s function, the prefactor Γ(1 − ν) is singular at each integer ν, and diverges as(sin πν)−1 ; this is expected, since the Green’s function has a pole at the hydrogenic
solutions. Similarly, the terms in the summation form for the correction for finite
quantum defects each diverge whenever sin π(µ l + ν) vanishes; in the absence of theperturber, this would define the set of non-degenerate poles at ν = n − µ l .

73
A plot of the Σ potential curves for the Rb ∗ -Rb long-range molecular Rydberg
state in the vicinity of the n=30 manifold are shown in Figure 4.1.

4.3

Solution in the presence of an external field
Because of the unusually large dipole moments of these molecules, even a weakexternal electric field will be sufficient to align them. For this reason, it is desirable tohave some estimate of the effect of an electric field on the level structure of the boundstate potential curves. Furthermore, the functional form of the field dependence of thelevel shifts may prove useful in the experimental distinguishing of Rydberg-ground stateinteractions like the trilobite state, and Rydberg-Rydberg long-range dipole interactions
such as those discussed in [22]. The theory of Rydberg-perturber interactions was
already the subject of study by de Prunelé [67], in the context of non-resonant states
formed for Rydberg-rare gas dimers (which do not support bound states, but do yieldoscillatory long-range potential curves).
In introducing a weak external electric field, we follow the approach of de Prunelé,and consider the form of the correction term from the standpoint of an eigenfunction
expansion in the Stark eigenstates. This requires that the Stark spectrum first be expanded in terms of the zero-field spectrum by a full diagonalization. In the hydrogenic
case, the results of this transformation are known exactly; the Stark Hamiltonian isexactly separable in parabolic coordinates, and the transformation coefficients for arbitrary field were first derived by Fano and Harmin [98, 99], with strong analogies to the
MQDT frame transformation [100]. For an alkali system, the diagonalization must be
performed numerically by first-order perturbation theory (see, for example, [101]). The
Stark map for several manifolds of Rb in the vicinity of n=15 is shown in Figure 4.2.
Since both the Coulomb Green’s function and the Coulomb-Stark Green’s function
can be expanded in eigenstates, the correction due to the external field is [67]0Energy (GHz)

−50

−100

−150

−200

−250

0

500

1000
R (a.u.)

1500

Figure 4.1: 3 Σ potential curves for Rb∗ -Rb long-range interaction, relative to the energy of the n=30 hydrogenic manifold. The nearlyvertical line of zeroes occurring beyond R=1750 is an artifact of the root finder at the classical turning point k=0.
74

1000

n=18, l=1
n=15 manifold
0

Energy (GHz)n=18, l=0
n=16, l=2−1000

n=17, l=1
n=14 manifold−2000

n=17, l=0
−3000

0

100

200
300
Field (V/cm)

400

500

Figure 4.2: Stark splitting for Rb in the vicinity of the n=15 manifold.
75

76∆Gs (~r, ~r 0 ; E) = hr|

X |αmihαm|α,mE − Eα−X |νlmihνlm|

ν,l,m

E − Eνl



 |~
r 0i

(4.34)

In principle, the sum runs over the entire eigenspectrum, including the continuum. For
most practical purposes, however, the correction is only important when the terms inthe denominator become large, restricting the sum to levels close to the bound stateenergy of interest.
If this correction term is added to the Green’s function in the outer region solution,
the matching condition will find solutions for a finite-field Coulomb-Stark Hamiltonian.
We will initially assume that the external field is aligned with the internuclear axis.
The level structure may be investigated as a function of both the dynamical parameters
E and R. Holding E constant and varying R gives the Born-Oppenheimer potential
curves for the molecular state. Figures 4.4 and 4.5 show the energy level spectrum at a
fixed field strength, as a function of the internuclear separation R.
Alternatively, R may be held constant and the field strength varied, in order tostudy the Stark effect itself. Figure 4.5 shows the Stark maps for several energy levels
at a fixed internuclear separation, as a function of the field strength.

4.4

Semiclassical interpretation
Rydberg states are often celebrated in the literature for their potential to bridge

the conceptual gap between the quantum and classical worlds. The motion of a Rydberg electron may be described accurately using semiclassical techniques, which relate
the classical closed-orbit trajectories of the spatial potential to the underlying nodal
structure of the true quantum mechanical solution. In this section, we discussion theorigin of the nodal pattern for the 3 Σ-scattered state. This is a condensed version of thesemiclassical interpretation presented in [73], which may referenced for a more extended
discussion of these results in terms of the EKB quantization.

Energy (GHz)

0

−1000

−2000

0

200
R (a.u.)400

77

Figure 4.3: Adiabatic energy level spectrum for states of Σ molecular symmetry in the vicinity of then = 15 manifold, for a field strengthof 51.42 V/cm. Note that the degenerate manifold has been split in accordance with the linear Stark effect. The avoided crossings between
the P-wave scattered molecular state and the atomic P state are partially obscured by artifacts that reflect insufficient sensitivity of theroot-finding algorithm.

30

Energy (GHz)

10

−10

−30

−50

0

100

200
R (a.u.)300

78

Figure 4.4: Magnified region of the previous plot showing the energy spectrum near the degenerate manifold, with a series of avoided
crossings between the trilobite state and high-angular momentum atomic states. Occasional spurious roots appear at the position of theatomic levels, reflecting imperfect cancellation between the Stark correction term and the Coulomb Green’s function.

−1282−258

Energy (GHz)−258.5
−1283
−259
−1284

n=18, l=0
atomic state−259.5n=17, l=1
atomic state

−1285

−260
−2000

Energy (GHz)

−1600
−2005

−1620
−2010
trilobite (s−wave)
molecular state
−2015

0

100
200
Field (V/cm)

butterfly (p−wave)
molecular state
300

−1640

0

100
200
Field (V/cm)

300

Figure 4.5: Energy dependence of selected atomic and molecular levels with respect to electric field, at an internuclear separation of
R = 157.
79

80
The robust, elliptic nodal structure of the electronic states observed in Figure 3.1
suggests that the problem will simplify in elliptic coordinates. Elliptic coordinates havebeen used to treat a number of other two center problems such as H +2 [102] and atomic
hydrogen perturbed by a point dipole [103]. From this standpoint, it is no surprisethat elliptic coordinates are useful for the current system. However, it is desirable tounderstand both why elliptic coordinates seem to be preferred in our system, and exactly
how the description is simplified when elliptic coordinates are used.
Guided by the elliptically shaped nodal patterns, we introduce elliptic coordinates,
defined with the foci of the elliptical coordinate system placed on the ion (at the origin)and the perturbing Rb atom (ρ, z) = (0, R). If r 1 and r2 are the distances between the
Rydberg electron and the two foci of our coordinate system, the elliptic coordinates aredefined as

ξ =
η =

r1 + r 2R
r1 − r 2.
R(4.35)

These coordinates are constrained to the ranges 1 ≤ ξ < ∞ and |η| ≤ 1 . Next, ourunperturbed elliptic eigenstates are introduced.
The elliptic eigenstates we construct here are stationary eigenstates of the hydrogen atom. These states differ from the traditional states of hydrogen (eigenstates of~ 2 , Lz ) through the replacement of L
~ 2 by a constant of the motion that emerges out
H, Lof elliptic coordinates. The Schrödinger equation for an electron (in a molecular L z = 0
Σ state) in a Coulomb potential in elliptic coordinates reads [104, 105, 106]

"

#

 ∂
 ∂

1 ∂  21 ∂ 
RR2  2ξ −1
1 − η2−−− (ξ − η) −E ξ − η 2 Ψ(ξ, η; R) = 0.
2 ∂ξ∂ξ 2 ∂η∂η
2
4(4.36)

81
This equation separates into two one-dimensional eigenvalue problems for theelliptic separation constant B(R)""#

(4.37)#(4.38)

 ∂R
R2 2
1 ∂  2− ξ−
Eξ F (ξ, R) = B(R)F (ξ, R)
−
ξ −1
2 ∂ξ
∂ξ
2
4

 ∂
R
R2 2
1 ∂ 
+ η+
Eη Φ(η, R) = −B(R)Φ(η, R),−
1 − η2
2 ∂η
∂η
2
4

−1
where Ψ(ξ, η; R) = F (ξ, R)Φ(η, R). Given a total energy E = 2n2 and a distance R, theseparation constant B(R) becomes quantized when boundary conditions appropriate
for bound states are imposed. We label these states by the number of nodes in the ξ
direction, nξ = (0, . . . , n − 1). The number of nodes in the angular η direction for thism = 0 state is then given by the constraintnη = n − 1 − n ξ .

(4.39)

Although analytical power series solutions of these Equations (4.37, 4.38) havebeen found [104, 105, 106], an efficient way of calculating the solutions is to diagonalize
the one-dimensional Hamiltonians in a b-spline basis set. This procedure produces theelliptic Coulomb states Ψnξ (ξ, η; R) along with the corresponding values of the separation constant Bnξ (R). In all of the preceding equations, the internuclear separation
R appears as a continuous parameter. However, these equations do not yet include
the effect of the perturbing Rb atom. So far, the parameter R only gives the distance
between the foci of our elliptic coordinate system.
At least back to Sommerfeld [107], it has been known that the hydrogen atom
is separable in elliptic coordinates. Although the separability of the hydrogen atom
in elliptic coordinates is not widely known, a number of authors have investigated theproperties of the elliptic eigenstates. Erikson and Hill first [108] showed that the elliptic
separation constant B(R) was related to the orbital angular momentum about the two

82foci of the coordinate system (see also [102, 109]). Figure 4.6 shows contour plots ofthe electron probability density for four of the degenerate elliptic states in the n = 30
manifold. The value of R for the elliptic coordinates underlying these states is 1232 a.u.
To our knowledge, the unusual nodal patterns of these states, which range from elliptical(0, 29) to semicircular (14, 15) to wedge (29, 0) shaped, have not been reported in theliterature thus far. Next, these states are used as zero order eigenstates in perturbation
theory.
To include the effect of the perturbing Rb atom, the perturbation is diagonalized
in the basis of elliptic eigenstates. The relevant perturbation matrix isDE~ n0ξ .
Vnξ n0ξ (R) = 2πAT [k(R)] nξ δ(~r − R)

(4.40)

Because this matrix is separable, only one state splits away from the degenerate nmanifold when it is diagonalized. The total energy of the state can be found analytically,
and is given by the expression:

En (R) =

X
2
2
−1 X
−1
+
Vnξ nξ (R) = 2 + 2πAT [k(R)]Ψnξ (1, 1; R) ,
2
2n
2n
nξ
nξ

(4.41)

which is used in Figure 4.7 to calculate the Born-Oppenheimer potential curve. The
wavefunction is then a linear combination of the elliptic eigenstates Ψ nξ (ξ, η; R):Ψn (ξ, η; R) =X

Ψnξ (1, 1; R)Ψnξ (ξ, η; R).

(4.42)nξ

It is clear that both the Born-Oppenheimer potential curve (4.41) and wavefunction (4.42) are determined by the values of the primitive elliptic eigenstates Ψ nξ (1, 1; R)at the position of the perturbing Rb atom (ξ, η) = (1, 1). In general, the wavefunctions
and eigenvalues include contributions from all n elliptic eigenstates in the degenerate
manifold. That the molecular states shows a strong elliptic character is seen whenthe values Ψnξ (1, 1; R) are shown as a function of internuclear separation R. Figure

83

Figure 4.6: Examples of elliptic eigenstates of the n = 30 hydrogen atom defined with anelliptic coordinate system where the distance between the foci of the coordinate system
is 1232 a.u., and plotted in cylindrical polar coordinates [73]. The foci of the elliptic
coordinate system have been placed at the origin and at (ρ, z)=(0,1232) (shown by asmall circle).

84

0

POTENTIAL ENERGY (GHz)

n=30

−5

−10
(0,29)(4,25)(3,26)−15500(2,27)(1,28)

1000
1500
INTERNUCLEAR SEPARATION R (a.u.)

2000

Figure 4.7: Born-Oppenheimer potential curve for a highly polar n = 30 Rydberg stateof Rb2 [73]. The approximate numbers of nodes in the electronic wavefunction (ν ξ , νη )are shown below each minima. Note that the effective quantum numbers (ν ξ , νη ) evolve
continuously as a function of internuclear separation, but are integer valued at theminima.

85
4.9 shows a plot of the values of Ψnξ (1, 1; R)

2(nξ = 0, 1 . . . , 10) for an unperturbed

n = 30 state as a function of R. At certain internuclear separations, a single primitive elliptic states dominates the molecular wavefunction (4.42) and Born-Oppenheimer
potential curve (4.41). A comparison between the coefficients in Figure 4.9 and theBorn-Oppenheimer potential curve in Figure 4.7 shows that the minima in the potential curve occur precisely where a single elliptic state is dominating the molecular
wavefunction. This “quasi-separability” of the full wavefunction means that at certain
internuclear separations the full Hamiltonian (including the perturbation) is nearly diagonal is the basis of primitive elliptic eigenstates. In some sense, the potential energy
is minimized (or maximized, for AT > 0) when the amount a non-separability in elliptic
coordinates is the least.
Furthermore, as the internuclear separation is changed, the specific elliptic statethat dominates the sums in Equations (4.42) and (4.41) changes. This accounts for the
redistribution of the nodes in the molecular wavefunction from the η to the ξ direction as the internuclear separation decreases (see Figures 4.7 and 4.8). Thus, at theoutermost minimum (R = 1575 a.u.), Figure 4.9 shows a peak in the contribution of
the nξ = 0 state. At the next minimum (R = 1382 a.u.), the n ξ = 1 state has cometo dominate. As the character of the molecular state evolves from one elliptic state toanother the numbers of nodes in the electronic wavefunction (ν ξ , νη ) evolve according
to the constraintνξ (R) + νη (R) = n − 1.

(4.43)

The characteristic elliptically shaped nodal pattern is preserved as the nuclei get closer
together, but the number of nodes in the two elliptic directions get redistributed from theangular η direction to the radial ξ direction as the internuclear separation is decreased.
Thus we have shown how an unexpected “quasi-separability” in elliptic coordi-

86

Figure 4.8: Contour plots of ρ |Ψ(ρ, z)| 2 of the Born-Oppenheimer wavefunctions of along-range Rb2 molecule at the energy of an n = 30 Rydberg state [73]. Four internuclear
separations are shown (R = 1575, 1382, 1234, 1110) corresponding to the outermost
minima in the oscillating potential curve (see Figure 4.7). These states are labeled by
the approximate numbers of nodes (ν ξ , νη ) in the two elliptic directions.

87

−8

EXPANSION COEFFICIENTS (10 )86(4,25)
4(3,26)(2,27)(1,28)(0,29)

2

0
500

1000
1500
INTERNUCLEAR SEPARATION R (a.u.)
2

2000

Figure 4.9: The values of the expansion coefficients Ψnξ (1, 1; R) for a n = 30 state[73]. These coefficients determine which elliptic eigenstate of hydrogen (see Figure 4.6)
dominates the molecular wavefunction and Born-Oppenheimer potential curve (Figures
4.7 and 4.8) at a given radius. Each peak in the graph is labeled by the elliptic quantum
numbers (nξ , nη ) of the state the peak represents.

88nates emerges at certain internuclear separations. These special internuclear separations
are precisely the minima in the Born-Oppenheimer potential curve. That the BornOppenheimer Hamiltonian is nearly diagonal in elliptic coordinates at a sequence ofinternuclear radii is an unexpected simplification that accounts for the elliptical shapeof the nodal structure of the molecular states. The evolution of the molecular statewith internuclear separation can be viewed as an change in the contributions of theprimitive elliptic states that comprise the wavefunction. Now we turn to a semiclassicaldescription of these states.
Semiclassical methods permit a simple interpretation of multidimensional quantum systems [110]. In the absence of the perturbing Rb atom, the long range dynamics
of the Rydberg electron are purely Coulombic. Our main challenge is therefore to include the effect of the perturbing Rb atom in a semiclassical treatment. In principle, itwould be possible to include the perturbing potential using classical perturbation theory[111]. Rather than this approach, we focus on a more qualitative viewpoint. To include
the effect of the perturbing Rb atom semiclassically, we replace its delta-function Fermi
potential by an inhomogeneous delta-function source at the perturbing atom. While thisis clearly an approximation, we show that this approximation reproduces the solutions
from degenerate perturbation (Equation 4.42) theory extremely well. The resulting
object of interest is then the Coulomb Green’s function with the source placed at theposition of the ground state Rb atom. The semiclassical Green’s function can be written[110] as a sum over classical trajectories that propagate from ~x 0 to ~x with energy E:
G(~x, ~x0 , E) ≈X

0

AeiS(~x,~x ,E)−iµπ/2 .

traj

For our purposes the most important quantities in this expression are the classicalaction S(~x, ~x0 , E) and the Maslov index µ of each trajectory, which counts the numbersof sign changes of the amplitude A. The amplitude A is a measure of the stability ofeach classical path. If the energy E is fixed and the source coordinate of the Green’s

89~ there are only fourfunction is set to be at the position of the perturbing atom (~x 0 = R)
~ E). These fourclassical trajectories that contribute to the Green’s function G(~x, R,~ and have apaths always lie on two Kepler ellipses that intersect the points ~x and R
Coulomb potential at one focus. An example of these four paths is seen in Figure 4.10.
The foci of the elliptic coordinate system (the Coulomb singularity and the perturbing
Rb atom) are shown as solid circles. The Green’s function is then a coherent sum of
the four trajectories that propagate from the ground state Rb atom to the observation
point marked by a hollow circle.
Figure 4.11 shows contour plots of both the quantum and semiclassical Coulomb
Green’s function with the source point placed at the ground state Rb atom (ρ, z) =(0, 1232). The semiclassical Green’s function has been constructed as described above,and shows strong agreement with the molecular wavefunction found using degenerate
perturbation theory (Figure 4.8) and the quantum Green’s function also pictured here.
The quantum Green’s function shown in Figure 4.11 is based on an analytical expression
first derived by Hostler [85, 86]. The good agreement of these three methods (quantum and semiclassical Green’s function, and degenerate perturbation theory) show thatthe inclusion of the perturbing Rb atom through an inhomogeneous source term in the
Schrödinger equation is a good approximation for the problem at hand. Additionally,
from a semiclassical perspective, then, the nodal structure of these electronic wavefunctions is governed by two things. First, the long range Coulomb physics provides the~ selects
majority of the dynamical evolution. Second, the ground state Rb atom at R
only Coulomb orbits that pass through this point. From this perspective, constraining
the trajectories to pass through the ground state Rb atom is a perfectly sensible way of
including the effect of the perturbation in a semiclassical treatment.

90

Figure 4.10: An example of the four classical paths lying along the two possible closed~ E), at
orbit trajectories that contribute to the semiclassical Green’s function G(~x, R,
the energy of an n = 30 Rydberg state [73]. The trajectories lie on two Kepler ellipses~ (right solid circle) and the observation point ~x (hollow
intersecting the perturber at R
circle). These ellipses are unique when one focus is constrained to be at the Coulomb
singularity at the origin (left solid circle).

91

Figure 4.11: Contour plot of the semiclassical (upper) and quantum (lower) Coulomb
Green’s function with the source coordinate placed at the ground state Rb atom (ρ,z)=(0, 1234) [73]. The semiclassical Green’s function has only been shown up to theclassical turning point where it diverges unphysically.

Chapter 5

Siegert states

Progress toward a truly general theory of molecular systems excited at continuum
energies depends on the ability to describe all accessible continua concurrently. Modern
experiments routinely probe energy regimes where multiple channels of both ionizing
and dissociative types are in competition [10]. The philosophy of multichannel quantum
defect theory outlined in Chapter 2 is naturally suited to the description of physics inthe asymptotic region, where channel indices correspond to the quantum numbers of
the separated system, and complicated short-range scattering dynamics are reduced toa minimal set of interaction parameters. In principle, the problems of predissociationand preionization are based on exactly the same interaction parameters, since they bothhinge on the coupling between the electronic excitation of the Rydberg electron and therovibronic excitation of the core. In preionization, sufficient energy is transferred fromthe core to the electron to promote it into the continuum of the separated electron-ion
system; in predissociation the reverse process occurs, with energy transferring from theelectron into vibrational excitation, dissociating the molecule.

5.1

Dissociative channels in molecular MQDT: General considerations
The extension of quantum defect theory to include a nuclear continuum is notnaturally suggested by the form of the channel expansion on which the theory relies,

93
because the usual theory has only ionization channels represented explicitly (see Eq. 2.2
in Chapter 2). The time-independent Schrödinger wavefunction is conventionally expanded using a solution ansatz form that distinguishes the channel states (discrete, andusually calculated numerically) from the asymptotic states of the scattering coordinate(continuous, and usually known analytically at large separations). The necessity of selecting one coordinate set to serve as the “channels” inevitably entails a preimposed bias
as to which coordinate should be considered to correspond to the reactive dynamics ofthe fragmenting particles. From that perspective, the very notion that MQDT could beof utility in double continuum problems may appear on the surface to be questionable.
At the same time, much of the conceptual appeal of quantum defect methods
for single continuum studies is retained or even enhanced by the computational challenges of working within a two-continuum problem. The physics of the system is still
dominated by a finite set of energetically open channels, and the selection of an expansion form that incorporates those channel states directly will still greatly reduce
computational labor. Some coordinates and asymptotically “good” quantum numbers
refer to coordinates that obey periodic rather than asymptotic boundary conditions,such as the angular electronic coordinates or the rotational eigenvalues; it would bea great waste of computational resources to solve for wavefunctions already known inadvance, such as when the ionization occurs via a few partial waves or into a small
number of product rotational states. Moreover, the transition from single continuasystems to multiple continua only complicates the necessity of efficiently describing thecoupling between metastable resonant states and the continua in which they are embedded. Since the short-range interaction physics to which resonant processes are sensitiveis relatively indifferent to the number of different continua or types of continua that are
simultaneously open, the same quantum defects and phase shifts that characterize thesingle-continuum case still ought to contain all necessary information for handling any
associated multiple-continuum generalizations of the same system.

94

5.2

Established methods for handling dissociative channels
The extension of quantum-defect-style techniques to the description of dissociative

channel physics was initially pursued in the late 1970s by Schneider, Burke, and other coworkers [112, 113, 114, 115], in the context of electron scattering from neutral molecules.
Their calculations were based on the Born-Oppenheimer approximation, and neglected
all non-adiabatic coupling effects. This allowed for an adequate handling of dissociative
attachment and vibrational excitation processes of neutrals, but could not be extended
to electron scattering from positive ions (or the associated half-scattering processes ofneutral molecule photoionization and photodissociation), which involve an essentiallynon-adiabatic coupling interaction with electronic Rydberg states. Still, Schneider et al.
were able to demonstrate that their R-matrix method gave results formally equivalent tothat of the Fano frame transformation formulas (see the concluding discussion of [115]).
Other applications of quantum defect theory to the dissociative spectra of diatomic
molecules were later developed by Giusti [116] and [117] . These methods are analogous
to the standard one-dimensional formulation of quantum defect theory, but with theelectronic coordinate replaced by the nuclear degree of freedom. (For an extended
discussion of the extension of MQDT to non-Coulombic long-range potentials, see [43,118, 119].) These methods are rigorous, but naturally involve no reference to electronic
physics except at the level of relying on adiabatic or diabatic potential curves.

5.2.1

Jungen eigenphase method
One of the earliest attempts to apply multichannel quantum defect theory to the

problem of competing dissociation and ionization processes was carried out by ChristianJungen [53]. Jungen adopted an approach inspired by the utility of Fano’s approach [45]in treating the “recoupling” transformation between the short-range interaction of the
Rydberg electron with the core, and the long-range Coulombic forces of the Rydberg

95
states. For escape into an electronic continuum only, the result of this procedure is
the orthogonal frame transformation already discussed in Chapter 2. To represent the
dissociative physics, the vibrational basis must be augmented in some way that permits
a finite value of the wavefunction on the fixed-R boundary between the inner region(where the wavefunction is unknown) and the outer region (where it can be expanded
in dissociative channels). R, as before, refers to the internuclear separation, as opposed
to r, the radial coordinate of the electron. The assumption is made that the potential in
the outer region vanishes except for a constant that defines the channel energy threshold.
Note that while the boundary condition of the electronic coordinate r is formally applied
at infinity, the boundary condition for the nuclear coordinate R is necessarily imposed atsome finite boundary R0 that delineates the range over which the inner region vibrational
basis is defined. Outside the region the solution is a linear combination of the regularand irregular vibrational continuum functions, F nΛ cos δ − GnΛ sin δ, multiplied by thechannel function that describes the state of the dissociating atoms or molecular fragment
pair, with the Rydberg electron in a low-n electronic state.
Figure 5.1 shows the inner and outer regions, the boundaries between them, andthe form of the solutions in each region, as found in [53]. Solutions with finite amplitude
in region II correspond to ionization, and solutions with finite amplitude in region III
correspond to dissociation. Solutions in region IV, which would describe dissociative
ionization, are not possible in the current version of Jungen’s method.
If a vibrational basis for the molecular ion H +2 is generated in accordance with afixed logarithmic derivative boundary condition at R 0 , which all functions in the basis
must satisfy, then the (discrete) basis consists a ladder of energy levels that begin toresemble the eigenstates of a particle in a one-dimensional box as soon as they risehigh enough in energy to exceed the dissociation threshold; they shall be referred toas the “box states”, in contrast to the ordinary vibrationally bound states that vanish as R → ∞. For an arbitrary choice of the logarithmic derivative, it is unlikely

96

Figure 5.1: Partition of configuration space into inner and outer regions, taken from thework of Ch. Jungen [53].

that the wavefunction expansion constructed using the box states (which all share thesame boundary condition) will have an energy at precisely the right value to correspondto the true (energy-conserving) continuum state. The selection of a single logarithmic derivative discretizes the continuum values available for continuum-like H +2 channel
states, and this also restricts the continuum H 2 states to a finite range of discrete values. Thus, the system must be solved iteratively, with a systematic variation of thelogarithmic derivative from −∞ to ∞ that causes the locations of the energy levels toscan across the position of the true dissociative state energy. This is accomplished bycasting the eigenequation system normally as for the ionization-only R-matrix approach[120] approach, but including the box state channels as “strongly closed” channels withphase parameters that depend implicitly on the logarithmic derivative. These additional strongly closed channels produce exactly one extra solution, corresponding to thedissociative continuum state of proper energy. This and the other eigenphase solutions
are then projected back onto the channel basis, defining the usual smooth, short-range
reaction matrix (or scattering matrix). From this point, channel elimination can be
performed, usually to close off the “weakly closed” channels associated with the true
vibrationally bound states of the molecular ion potential that give rise to the ionization
Rydberg series structure.

97
As originally developed, this method is encumbered by the need to recalculate
the eigensolution coupling matrix elements at each new logarithmic derivative. Since
the values at which the linear eigensystem possesses an eigensolution are dependenton energy, this means that the system must be solved repeatedly at each energy until
the result has iteratively converged to the desired accuracy. Moreover, the method, as
originally conceived, functions only for a single dissociative channel per Λ symmetry, andonly in the energy regime below dissociative ionization. The reaction matrix contains an
explicit channel for dissociation, reflecting the constraint that the solution must vanish
either at infinity in all ionization channels, or else at the boundary in the dissociation
channel.
More recently [121], Jungen and Ross generalized their method to allow for multiple dissociative channels, and to remove the necessity for iteration. The standard
frame transformation calculation is done first for the normal basis set (vanishing at
R0 , and it is then repeated with a different logarithmic derivative for each dissociation
channel. The eigenchannel solutions for all the different boundary condition choices
are combined into a single generalized basis, and the asymptotic boundary conditions
then give a generalized eigenvalue problem for the full two-dimensional eigenphases andeigensolution vectors. As before, this gives a short-range reaction matrix with “weakly
closed” channels, which may be then eliminated with the usual algebra. This method
has provided the most accurate theoretical results currently available, and is in the process of being extended to other systems [122]. It does depend, however, on a physicallymotivated choice of the “additional” logarithmic derivatives; the accuracy of the results
reflects the extent to which an energy in the vibrational basis appears close to the energy at which the calculation is being performed. In effect, this optimization condition
amounts to a semi-iterative procedure that must be adjusted “by hand” in different energy ranges. This is easily accomplished for the study of a single peak at some specified
energy, but could be potentially quite inconvenient for the calculation of a continuously

98energy-dependent spectrum with many peaks over a wide energy range. Moreover, itremains unclear how to extend this method to energies above the dissociative ionizationthreshold.

5.2.2

Stephens-Greene box averaging method
A more conceptually direct approach to the representation of coupling to a dis-sociative continuum is that of box size averaging [54]. As in the Jungen method, thedissociative continuum is discretized by allowing the vibrational basis to extend far
above the ionic dissociation threshold, to include many box states. Instead of looking
at the variation of these state energies with logarithmic derivative, one may instead consider their sensitivity to variations in the box radius R 0 . As the box increases in size,
the box state levels shift to lower energies, passing smoothly through all continuum
energies. The coupling between these fictitious channels and the physical ionization
channels provides a measure of the extent to which flux can be expected to pass into
those dissociation channels. The positions of resonances with strong dissociative character will be perturbed significantly by the introduction of these additional box states, andthe variation in this perturbation as a function of R 0 is thus a measure of the couplingstrength between the dissociative continuum and the ionization channel associated withthat resonance.
In order to quantitatively interpret this effect, the standard MQDT frame transformation procedure must be performed sequentially for a series of closely spaced box
radius parameter values. The cross-sections arising from each individual value of thebox radius are then averaged together, giving a cross-section in which the linewidthsof dissociation-sensitive peaks have been broadened. The range of R 0 values used in
the calculation is determined by the separation between box state energy levels in thevicinity of the energy for which coupling to the dissociative continuum is anticipatedto be important; roughly speaking, it is necessary to change R 0 by exactly enough to99shift all the levels down by one cycle, such that the n + 1-th level is lowered to theposition initially occupied by the n-th level. Since the eigenvalues of a particle in a box
have a relative spacing itself proportional to n, the selection of an appropriate reference value for n is essential for producing an evenly weighted average. This is a trivial
choice at low total energies, where only one dissociative channel is accessible; if there
are multiple competing dissociative product channels open at once, however, the criterion becomes more difficult to specify, and if the cycle periodicities are fundamentally
incommensurate, potentially impossible.

5.2.3

Two-dimensional R-matrix method
R-matrix techniques derive their name from the original formulation of Wigner

and Eisenbud [123], who first recognized that the matrix of generalized logarithmic
derivatives on the surface bounding a continuum system of interacting particles could
be written as an expansion over resonant terms located at the eigenenergies of thebounded system1 . This approach demonstrated good utility for the semi-empiricalcharacterization of small collections of known nuclear resonances in terms of a minimal
parameter set, but it was not applied widely for ab initio computation in its originalform, owing to its poor convergence properties. Although it is possible to improve
convergence through the inclusion of certain correction schemes, such as the Buttle
correction, this can only be accomplished at the expense of some amount of conceptual
clarity and computational efficiency.
An alternate version of the R-matrix theory was introduced by Fano and Lee [120],
based on transformation to the eigenchannel representation. Rather than selecting asingle arbitrary value for the surface logarithmic derivative and solving the Schrödinger
equation subject to that condition, the eigenchannel representation provides a set ofsolutions, each of which has a different logarithmic derivative that is common to all of its
1

For a review of early applications of R-matrix theory to problems in nuclear physics, see [124].

100
physical channel components. Since the boundary condition is itself the undeterminedeigenvalue parameter, it is permissible to express the matrix equation in a basis set
that uses basis functions with a range of different logarithmic derivatives, improving
the convergence properties of the final R-matrix expression. This comes at the expense,
however, of requiring an iterative solution procedure. For an application of this approachto photoionization, see [125].
The eigenchannel approach may be recast in a noniterative form by means of avariational principle first discovered by Kohn [126]. In contrast to the more traditionalbound state applications of the calculus of variations to quantum systems, which apply
the Rayleigh-Ritz method to determine a stationary value of the energy for a given basis
set, Kohn’s principle gives a stationary value for the logarithmic derivative in terms ofa fixed scattering energy. For continuum systems this is conceptually advantageous,in that it reflects an awareness that the total energy is typically a controllable input
parameter for scattering experiments. This simplification was first proposed for a fieldperturbed atomic system by Greene [127], and shortly thereafter extended to molecular
photoionization [128]. Although this approach successfully combines the stable convergence of the eigenchannel method with the non-iterative efficiency of the traditional
Wigner-Eisenbud theory, it still suffers from the disadvantage of requiring the eigensystem to be solved again at each new energy. This disadvantage may be reduced by use ofa modified “streamlined” solution technique for the generalized eigenequation, although
for larger basis sets (e.g., such as those commonly required for the two-dimensional Rmatrix method), the computational advantage is limited by the inability to efficiently
diagonize the Γ matrix in the closed-closed subspace [129, 44]. Additionally, the energy
dependence of the R-matrix elements over small energy ranges is frequently smoothenough that they may be interpolated from a limited number of data points, provided
sufficient care is taken with respect to the handling of divergences in the logarithmic
derivative eigenvalues.

101
A detailed review of the variational R-matrix method may be found in [129].
Here, we will focus specifically on the specialization of this method to the problem ofphotoionization and photodissociation in diatomics. For a system with two dissimilar
modes of fragmentation, as is the case for competing dissociation and ionization, thevariational principle must be adjusted to account for the nonequivalence of the kinetic
energy terms of the Hamiltonian. Assume the potential within a two-dimensional box
bounded by surfaces at the axes and at r = r 0 and R = R0 may be written as V (r, R).
Then the time independent Schrödinger equation requires

!

1 ∂2
1 ∂2
−
−
+ V (r, R) ψ(r, R) = Eψ(r, R).
2µr ∂r 2 2µR ∂R2(5.1)

The mass factors µr and µR denote the reduced masses for electronic and nuclear motion,
respectively. The angular degrees of freedom have already been separated in sphericalcoordinates, such that the full solution in the inner region is given by~ =Ψ(~r, R)1
ψ(r, R)Φ(Ω).
rR

(5.2)

The notation Ω represents all angular degrees of freedom. In general, the possibility ofadditional electronic symmetries and partial wave components would require a sum overmultiple Φ channels, although for the ungerade photodissociation spectrum of molecularhydrogen it suffices to assume that only the p-wave channel is active; the R coordinate,however, is strictly confined to s-wave scattering by conservation of momentum. The
partition of configuration space that defines the inner region is identical to that ofJungen’s work as previously shown in Figure 5.1, with Jungen’s radius r 2 corresponding
to the box boundary labeled here as r 0 .
The mass factors may be temporarily absorbed by a change of variables, x =and X =√

µr r√
µR R, to recast the Hamiltonian in a more symmetric form. Then the

differential equation is transformed into the usual matricial form by the taking of an102
inner product over the enclosed box volume,

E

Z

∗

dx dX ψ ψ =Z

dx dXψ

!

(5.3)



(5.4)

1 ∂2
1 ∂2
−
−
+ Ṽ (x, X) ψ2 ∂x2 2 ∂X 2∗

followed by the application of Green’s theorem,EZ

dx dX ψ ∗ ψ =

Z

+

dx dX
Z



1 ∂ψ ∗ ∂ψ ∗ 1 ∂ψ ∗ ∂ψ ∗+2 ∂x ∂x
2 ∂X ∂X

dx dXψ ∗ Ṽ (x, X)ψ

(5.5)

Z x0

(5.6)

1
∂ψ
|x=x0
dx ψ ∗ (x, X0 )
2 0
∂X
Z
∂ψ
1 X0
dx ψ ∗ (x0 , X)
|X=X0 .
−
2 0
∂X

−

(5.7)

The logarithmic derivative is defined based on an outward normal with respect to theappropriate scaled variable along each surface,∂ψ|X=X0 = −bψ,∂X∂ψ
|x=x0 = −bψ.
∂xAfter rearrangement and restoration of the original variables, the variational quantity
b may finally be expressed as

b=

√



1 ∂ψ 22
2µr µR dr dR 2E − µ1r | ∂ψ∂r | − µR | ∂R | − 2V (r, R)|ψ|√ R√ R r0µr 0 dr |ψ|2 + µR 0R0 dR |ψ|2

R



(5.8)

Note that the boundary parameter b is not the logarithmic derivative itself, but rather
the mass-rescaled logarithmic derivative; to recover the usual logarithmic derivative, bmust be multiplied by the appropriate√µ factor.

Proceeding from this relation, the solution is expanded in a two-dimensional basis,
ψ=

XX
i

ci,j ui (r)vj (R)j,to produce the usual generalized eigenvalue equation,

103

Γ~c = bΛ~c,(5.9)

where Γ is the matrix representation of the numerator of 5.8 and Λ is the matrix representation of the denominator of 5.8 (i.e., the overlap matrix for that basis). Eacheigensolution will have a common boundary parameter value b β on both surfaces. Outside the box boundary, the scattered eigenchannel solutions can be written as a channel
decomposition of the regular and irregular solutions [44],ψβ (r, ω) =

X1
i

r

Φi (ω)(fi (r)Iiβ − gi (r)Jiβ ).

(5.10)The symbol ω includes all degrees of freedom necessary to describe the channel, including R and all angular coordinates Ω. This wavefunction requires a formal antisymmetrization, but for sufficiently large r 0 this can usually be neglected. For the solutions
asymptotic in R, the variable r is replaced by R wherever it appears, and the Coulomb
functions fi and gi are replaced by the regular and irregular vibrational continuum functions, which can be found by the Milne method [118, 130] or else by similar numerical
techniques. The matching condition of continuity for the wavefunction and its derivative
permits the determination of I and J in terms of Wronskians,
Iiβ = −

W (gi , ψiβ )W (fi , gi )Jiβ = −W (fi , ψiβ ),W (fi , gi )andand this suffices to define either the reaction matrix K = IJ −1 or the scattering matrix
S = (I + iJ)(I − iJ)−1 . The notation ψiβ refers to the projection of the solution ψ β onto
the i-th channel function at the appropriate matching surface.
The variational R-matrix method requires an explicit form for the potential energyfunction within the two-dimensional box. In principle, the potential energy of molecular104hydrogen as a bivariant function of r and R is quite complicated, reflecting short-range
electron correlation effects that are challenging to define for the highly excited pseudocontinuum states that characterize the R-matrix eigenspectrum. Instead, we have opted
to work with a simplified two-dimensional model potential that reproduces the featuresof the exact H2 intermolecular potential to reasonable quantitative accuracy. A separate
potential form is utilized for each Λ-space symmetry, with several parameters that may
be optimized through a one-dimensional R-matrix procedure to reproduce the known
body-frame quantum defects µΛ (R). For the Σ symmetry, this procedure yields (inatomic units)

V (r, R) = −α1

α2 − R − α 3 R 4
1 − tanh
7!

tanh (R/α4 )4 e−r 2 /3r(5.11)

where α1 = 1.6435, α2 = 6.2, α3 = 0.0125, and α4 = 1.15. For the Π symmetry, theoptimization gives e−r2 /3√ 
2
V (r, R) = α1 R e−(R−α2 ) /α3r(5.12)

with α1 = 0.480, α2 = 3.35, and α3 = 6.5. The resulting model potential curves areshown in 5.2. Note that the model potential begins to break down for Σ states of
low n, owing to the neglected energy dependence of the Σ quantum defect which is notdescribed by this fitted potential. The shape of the 2Σ potential is not even qualitatively
correct, a reflection of the inability of a 1 Σ+u state to dissociate into two ground statehydrogen atoms [40]. This is a problem common to all QDT techniques that use anenergy-independent quantum defect, but since the n=2 state lies so low in energy as
to be inactive for scattering or photoabsorption processes near the ionization threshold,this does not constitute a serious defect for the purpose of using the model as a test
system.
As mentioned above, the R-matrix procedure is computationally expensive to re-105

Figure 5.2: Potential curves generated from the model Hamiltonian using Eqs. 5.11
and 5.12. Potentials drawn with a dashed curve are of 1 Σu symmetry, and those witha solid curve are of 1 Πu symmetry. (These may be compared with the exact curves forH2 shown in Figure 2.1.)

106peat at each new energy (unless the basis set is small enough to use the streamlined
version). Fortunately, the quantities I and J are only weakly and smoothly energy
dependent, and can be interpolated from data computed on a coarser energy grid. Because they incorporate relative phase information, and because the phase is ambiguouswith respect to a shift of π, the eigenvector solutions are susceptible to branch discontinuities associated with the underlying trigonometric functions; these must be repaired
before the interpolation is applied. Given the interpolated reaction matrix values, theonly remaining step that must be performed at each energy is the elimination of closed
channels according to the conventional channel elimination procedure. Note that thismethod results in the appearance of channels in the S-matrix that are explicitly labeled
as either ionization or dissociation channels.
Results of the R-matrix calculation, and a comparison with experimental data,
are summarized in Table 5.1 (aside from the v + = 2 resonance series, for which thepeaks are exclusively ionized, and occur within a wavenumber or two of the correct
position). The photoionization, photodissociation, and total cross-sections are shownin Figures 5.3, 5.4, and 5.5. The photoionization and photodissociation portions of thecross-section are separated by summation over only the appropriate physical channels.
Note that since this calculation is based on a model potential, and resonance features
are often highly sensitive to small variations in the form of the short-range potential,
the comparison is not expected to be quantitative. A comparison with the Siegert
method (below) using the same model potential, however, suggests that only errors in
the positions (as opposed to widths or relative yields) of the peaks are likely to be aconsequence of the failure of the model approximations. The inability of the relative
yields to reproduce experiment in anything beyond a general qualitative correspondence
is disappointing, and may reflect either a fundamental weakness of the technique, orsome undetected error in the implementation. Moreover, the dissociation spectrum
reveals an unassignable peak at 786 Å; this may be an artifact related to the anomalous

107
shape of the 2pσ and 3pσ potential curves. This method probably merits further study
before any definite conclusions are drawn concerning its capabilities and limitations.
Table 5.1: Positions and fractional yields for select ungerade J=0 resonances in H 2 .
3pπ, v = 85pσ, v = 44pπ, v = 5

Position (cm−1 )Width (cm−1 )% Ionization% Dissociation
Position (cm−1 )% Ionization% Dissociation
Position (cm−1 )% Ionization% Dissociation

Model
127295.7
3.9
0.1
99.9
127622.00.42
0.58
127701.4
0.998
0.002

Obs. (H2 )127248.2
3.4
≤2
≥ 98
127599.4
0.77
0.10 ± 0.10
127667.6
0.85
0.18 ± 0.05

The two-dimensional R-matrix method, unlike the MQDT approaches, works directly from the potential energy function, rather than incorporating that information
through intermediate parameters such as the quantum defects. Its advantage is that it
does not rely on the vibrational frame transformation method, and thus serves well as abenchmark calculation. In situations where the short-range potential can be describedin full quantitative detail, it provides a “brute force” alternative to the more approximate quantum defect methods, with a controlled convergence behavior relative to thevariation of geometric parameters like the number of channels and the box dimensions.
This can be of potential utility for testing other methods, or studying representativemodel systems. For experimental spectra, however, the reproduction of accurate highresolution spectra at the sub-wavenumber level is much more easily attained by the
inclusion of semi-empirical parameter adjustments that naturally account for small correction effects.

108

Figure 5.3: Infinite resolution photoionization cross-section calculated using the twodimensional R-matrix technique.

Figure 5.4: Infinite resolution photodissociation cross-section calculated using the twodimensional R-matrix technique.

Figure 5.5: Infinite resolution total photoabsorption cross-section, taken from the summation of Figures 5.3 and 5.4.

109

5.3

Siegert pseudostates: Basic concepts
The difficulties of the Stephens-Greene box-averaging stems from its inability torepresent a true outgoing wave solution. With the box averaging method, for example,
the inner region solutions are fixed by normalization in such a way as to guarantee
conservation of the probability within the box radius; the solution cannot have a finite
flux at the boundary, but can only “bounce”. As such, the shape of an averaged peakcan be broadened, but the total integrated area can never be reduced. This is a necessary consequence of flux conservation. For methods that incorporate the dissociative
channel as if it were an additional ionic channel, on the other hand, the solution forces
flux to be conserved despite distribution over all channels; this neglects the possibility
that the true solution may be coupled to both types of continua at once, implying a
failure when the first threshold for dissociative ionization is crossed; it is simply notpossible in these methods for flux to be outgoing in both channel types at once. Despite the ability of MQDT to provide, in principle, the true two-dimensional solution
within a finite region of space, the use of a conventional basis fails to provide sufficient
flexibility to characterize a wavefunction with outgoing wave form in both the asymptotic electronic coordinate (controlled by the assumed asymptotic form of the outerregion solutions) and the finite-bounded nuclear coordinate simply in terms of a single
set of channel states. Thus, methods such as the treatment of Ross-Jungen [121] must
explicitly supplement the basis using additional sets of basis functions obeying nonDirichlet boundary conditions, with the logarithmic derivative of each new set selected
in a somewhat arbitrary and ad hoc manner.
The construction of outgoing-wave solutions for a given Hamiltonian can be accomplished by analytically generalizing the energy eigenvalue spectrum of that Hamiltonian into the complex plane. Because outgoing wave states behave asymptotically as
exponential functions of a complex argument, they cannot be described in terms of the

110
usual boundary conditions encountered in Sturm-Liouville theory, since neither theirvalues nor their first derivatives vanish at any point in space; thus, the hermiticity of
the Hamiltonian operator is no longer sufficient to guarantee that the eigenvalues are
confined to real axis. The physical interpretation of complex eigenenergies was first
explored by Gamow [131], who recognized that the lifetime τ of a resonance state wasinversely proportional to the imaginary part of its complex energy. Subsequent work by
Wiesskopf and Wigner [132] related the lifetime τ to the linewidth parameter Γ = 1/τ ,at least at the level of the approximation of their theory.
The definition of resonance states in terms of the solutions of the Schrödinger
equation may be approached from two somewhat different perspectives. The first theory,developed by Kapur and Peierls [133], was based on the partitioning of the Hamiltonian
into an outer region (with a continuous spectrum) and and inner region (with a discrete
spectrum), treated as two interacting coupled channels. The Kapur-Peierls eigenstates
are defined as the eigenfunctions of an effective Hamiltonian defined by inverting the
projection of the resolvent operator onto the inner channel. Since this is an energydependent operator, with the wavenumber of the outer channel continuum state as a freeparameter of the Hamiltonian, the eigensolution set itself has a parametric dependence
on the scattering energy. Although the Kapur-Peierls functions have the advantage ofwell-behaved orthogonality and completeness properties, they are inconvenient for use
in any calculation which involves a range of scattering energies; because they have sucha complicated parametric dependence on the energy, they are quite difficult to evaluate
except by rediagonalizing the effective Hamiltonian at each new energy.
A alternate, parameter-free version of the resonance states was developed bySiegert [134]. Siegert states are defined, in a much more intuitive way, as the eigensolutions of the Hamiltonian that vanish at the origin and have pure outgoing wave
character in the asymptotic limit R → ∞. Siegert’s derivation, notably, allowed foroverlapping resonances of arbitrary width, giving a smooth background term in the111cross-section. Siegert eigenstates formally correspond to S-matrix poles in the complex
plane. Sharply resonant features associated with bound states can be identified with
poles lying on the real axis, while broad background scattering can be described byclosely spaced eigenstates with finite imaginary parts that serve as a discretized approximation to the true continuum. The Siegert states offer several advantages over the
Kapur-Peierls theory. First, they correctly describe long-lived resonances in the sensethat they pass smoothly over into bound states; any states that lie lower in energy
than the continuum threshold of a potential are guaranteed to correspond to the usual(real-valued) bound states. For the Kapur-Peierls formulation, this only occurs for a
fortuitous selection of the partitioning radius. Moreover, the Siegert bound states areguaranteed to be correctly normalized and orthogonal. Second, the Siegert states are
uniquely defined for a given Hamiltonian at any scattering energy. This allows themto serve as an expansion basis for other functions, including the Green’s function. (Amore detailed survey of the relations between the Siegert and Kapur-Peierls theories
may be found in [135].) In addition to their identification as poles of the scatteringmatrix, one may note from the Siegert boundary conditions at 0 and ∞ that the Siegert
eigensolutions are proportional to the Jost solution f ± (k, R) [136], and the eigenvalues
are equivalent to the zeroes of the Jost function J ± (k) = f± (k, 0) in the complex plane[137].
A renewed interest in Siegert functions during the 1970s was stimulated by therealization that they could be applied to the direct calculation of total [138] or partial[139, 140] linewidth parameters in many-electron multichannel systems. These methods
do not calculate cross-sections, but extract the resonance eigenvalues directly as poles ofthe scattering matrix; they are closely related to the various complex coordinate scaling
methods [141]. The utility of Siegert states for numerical calculations is compromisedby their rapid exponential divergence for increasing r. One of several approaches forcircumventing divergence-related instabilities was developed by Meyer and Walter [137],

112
who confined the range of the Siegert states to a finite radius, and solved for theireigenspectrum within an L2 primitive basis set using variational calculus. That is, they
replaced the exact Siegert boundary conditiond− ik φ(R)
=0
dR
R=∞


(5.13)

with the finite range boundary conditiond− ik φ(R)
= 0.
dR
R=R0


(5.14)(As a historical note, it is the latter definition that actually corresponds to that ofSiegert’s original paper, which was only concerned with very short-range nuclear interaction potentials.) This method allowed a single formalism to yield all the bound,virtual, and resonance states of the potential, a significant accomplishment.
A complication of the method involves the necessity of differentiating the ‘’cutoff” resonances, a string of closely spaced poles in the complex plane arising from thetruncation of the potential at a finite cut-off radius, from the true physical resonances.
These are not artifacts, but rather the direct scattering states associated with thatmodified potential. They are highly sensitive to the shape of the potential tail andthe location of the cut-off discontinuity, even if the tail is exponentially small in that
region. This is a general peculiarity of the analytical properties of Siegert states in any
infinite-range potential, even one that decays exponentially rapidly– they invariably
display a pathological sensitivity to the asymptotic potential behavior, preventing all
but a finite subset of the Siegert states for cut-off potentials (SSCPs) from converging
to exact Siegert states. From the standpoint of rigorous mathematical formalism, it isimportant to consider all further discussion in light of this recognition. Since all realistic
physical potentials have a tail behavior that decays more slowly than exponentially, theultimate test of any Siegert method, including the elegant work of [142], is whether or not

113
the derived observable quantities (cross-sections and S-matrices) are numerically stablewith respect to variation of the box radius parameters. Since scattering observables inatomic physics are in general not sensitive to weak long-range potential terms (except insufficiently low energy scattering regimes that such terms can no longer be consideredweak), however, the pathological behavior of the SSCPs is expected not to have physicalsignificance for any realistic system; even with two very different eigenvalue spectrums,
an expansion-based expression for an observable should converge to exactly the samevalue in both cases.
Even methods based on the SSCPs still suffer from the serious drawback of being
computationally dependent on iterative procedures, either in diagonalizing the full matrix [138], or the secular equation resulting from a variational principle [137], or mostrecently, a partitioned submatrix reduced to the dimension of the open channels [143].
The solution must be iterated many times to converge on every individual eigenvalue.
While this is potentially practical for “direct” methods, in which the only objective isthe determination of a finite number of true resonant states and perhaps a representative topology of the surrounding scattering (cut-off) resonances, it would be impossible
to calculate all the wavefunctions of the infinite series of SSCPs, which are necessarycomponents of any complex basis vector expansion, such as would be used (for example)
to construct the Green’s function by means of the Mittag-Leffler expansion theorem.
In principle, all of the information describing the continuum resonance features andstate densities should be contained in these eigenvectors, thus bypassing the need forapproximating any integrals over the continuum. Turning that promise into a computational reality requires a careful study of the relationship between the SSCPs, and theirrepresentation in any finite basis, as well as a way of finding all of the eigensolutions defined by that basis selection without the need for cumbersome and potentially unstable
iterative searches for eigenvalues in the complex plane.
Both of these difficulties were resolved recently by Tolstikhin et al. in their study

114
of the Siegert pseudostates (SPSs) [144, 142], the set of N eigenstates the result fromsolving the Hamiltonian with Siegert boundary conditions on the interval [0,R 0 ] for an
N -dimensional basis set. Tolstikhin et al. assume orthonormality of the basis functions,but most of their expressions can be straightforwardly generalized to a non-orthogonalbasis, and we will lift that assumption except where otherwise noted. Regardless oforthogonality, the basis is in general not assumed to be complete except in the limitof N → ∞. The SPS basis representation allows the derivation of completeness andnormalization properties of the Siegert state functions to be replaced by linear algebra
operations, which are then related to the true properties of the SSCPs by investigationof the N → ∞ limiting behavior.
We seek a solution expanded in terms of some primitive basis set

φ(R) =NX

cj yj (R),j=1

0 ≤ R ≤ R0 .

(5.15)

Here N is the dimension of our basis, and we have selected a non-orthogonal b-spline
basis [145, 146] for the yj (R). Inserting this into the Schrödinger equation, premultiplying by yj 0 , integrating with an integration by parts, and employing the boundary value
5.14, we find a matrix equation for the coefficients c j

PN

j=1

 R
1 R0 dyj 0 dyj
2 0

ik 0
dR dR dR − 2 yj (R0 )yj (R0 )+ 0R0 yj 0 (R)µ[V (r) − E]yj (R)dR cj = 0.
R

(5.16)(5.17)(5.18)

Note that we have used a Green’s theorem identity before substituting in the boundary
condition, and that the Hamiltonian has been multiplied through by the reduced massµ. Written more concisely in matrix notation, we have a system of the form(H̃ − ikL − k 2 O)~c = 0,(5.19)

115d
yj 0 (R0 ),where Lj,j 0 is the surface matrix yj (R0 )yj 0 (R0 ), H̃j,j 0 is the matrix 2µHj,j 0 +yj (R0 ) dr

and O is the overlap matrix for the spline basis set.
This equation is manifestly nonlinear, but the method of Tolstikhin et al. allows it
to be “linearized” by recasting it as a new eigensystem in a basis of doubled dimension.
([147, 148] discuss related techniques for solving differential equations where the eigenvalue appears in a boundary condition.) We continue by defining d i = ikci , yielding a~ Substituting this into the original eigenequation
trivial second equation ikO~c = Od.
now gives a linear equation in the doubled basis space


 H̃



0

0
−O







  ~c  = ik 




d~

L
−O





−O   ~c 
0




d~

.


(5.20)

This is an equation for the eigenvalue λ = ik, giving 2N solutions lying either on the(Re λ)-axis or in conjugate pairs in the right half of the complex λ-plane. The solutions(plotted here in the k plane, rather than the λ-plane) are shown in Figures 5.6 and 5.7.
An atypical feature of the SPS eigensystem should be noted here– the eigenvalue
parameter is proportional to the wavenumber k, rather than the energy. Since the energy
has a k 2 functional dependence, the energy spectrum itself will need to be mapped onto
a two-sheet Riemann surface in order to become single-valued, with a branch point at
the origin and the cut following the positive real semiaxis. States with k in the upper
half-k-plane will be mapped onto the first sheet of the Riemann surface on which theenergy eigenspectrum is defined, and those with k in the lower half-plane will map to
the second sheet. Since the bound state eigenvalues (that is, those which exponentiallydecay as R → R0 ) lie on the first sheet, this will be termed the “physical” sheet. The
only states that can lie on the physical sheet are bound states (although if a discrete
state is embedded in the continuum, it will be located on top of the cut line). Thus,
all resonance states, whether they are near the real energy axis and related to real
observable resonances, or deeper in the complex plane and related to smooth background−15

Im(k) (a.u.)

−5

5

15
−100

−50

0
Re(k) (a.u.)

50

100

116

Figure 5.6: Siegert pseudostate eigenspectrum resulting from the solution of of Equation 5.20. Note that the eigenstates have beentransformed from λ to k, corresponding to a rotation in the complex plane by 90 degrees. States above the real axis all lie on theimaginary axis, and are identified as the bound states of the potential. States on the complex axis below the real axis are unphysicalantibound states.

−0.2

Im(k) (a.u.)

0

0.2

0.4

0.6

0.8
−100

−50

0
Re(k) (a.u.)

50

100

117

Figure 5.7: Enhanced detail of the spectrum in the vicinity of the real axis, showing states that correspond to the Seigert discretized
continuum. Note that the “kink” in the spectrum occurs for values of k that oscillate too quickly to be represented in the primitive spline
basis.

118
scattering, are exclusively on the unphysical sheet of the multisheet Riemann surface.
The Siegert states are known to obey unconventional orthogonality, normalization, and completeness relations, and this has historically been a source of controversy in the literature, with several competing schemes for regularizing the non-squarenormalizable continuum-like states. In the SPSs, however, the origin of these questions
become simpler to understand by working within the doubled dimension space. If theeigenvalues of Equation 5.20 are nondegenerate, then it may be easily shown (notingthat the matrices are symmetric, not Hermitian) that the eigenvectors are symmetricwith respect to the weighting matrix that appears on the right hand side,






~c(n)T d~(n)T 




L
−O

−O  
0




~c(m)
d~(m)




 = 0,
∀n 6= m.

(5.21)The normalization condition may be derived by examining the behavior of the exponentially decaying bound states, and imposing unit normalization on them in the limit that
R0 is made sufficiently large that their amplitude on that boundary vanishes. We begin
from the orthonormality expression with an undetermined normalization constant,






~c(n)T d~(n)T 


L



−O  ~c(m) = αδnm .


(5.22)−(λn + λm )~c(n)T O~c(m)T + ~c(n)T L~c(m)T = αδnm ,(5.23)

−O

0




d~(m)

~ this condition reduces to an equivalent version in the original
By the definition of d,N-dimensional basis,or in the position-space representation,−(λn + λm )Z R00

φn (R)φm (R)dR + φn (R0 )φm (R0 ) = αδnm .

(5.24)

119
For bound states in the limit R0 → ∞, where the surface term vanishes, we must recover
the conventional unit normalization

Z ∞0

φn (R)φm (R)dR = δnm .

(5.25)

This requirement uniquely determines the normalization constant α = −(λ n +λm ) for allstates. Note that since all the states on the unphysical sheet, the eigenfunction diverges
exponentially for increasing R, and thus the normalization condition 5.24 amounts to acancellation of two exponentially growing terms; this is a potential source of numericalinstability as the box size increases.
The orthonormality relation 5.22 implies an associated completeness relation in
the doubled dimension space,


2N
X
1 

2λ 

n=1

n

~c(n)
d~(n)

−1





  (n)T (n)T   −L O 

 ~c
d~
=




O

,

(5.26)

0which yields a set of multiple completeness relations upon reduction to the originalN -dimensional basis:

2N
X
1

λ
n=1 n
2N
X

n=1
2N
X

c(n) c(n)T= 0(5.27)

c(n) c(n)T= 2 O−1(5.28)

c(n) c(n)T= 2 O−1 LO−1

(5.29)

n=1

In fact, an infinite list of such completeness relations, generated by changing the powerof λn in the weighted sum, may be derived on the basis of a recursion relation based on
the Schroödinger equation 5.19, as shown in [142].

120

5.4

Siegert pseudostates: Single channel Green’s function method
For a single-channel system with a finite-range potential, the SPS eigenstates

serve as a complete expansion set for the Mittag-Leffler expansion theorem. Unlike thetrue Siegert states, for which convergence of this representation is problematic (see Moreand Gerjuoy [135] for an extended discussion), the SPS Green’s function is guaranteed
to converge, owing to the completeness relations given above. The derivation proceeds
entirely by linear algebra [142], and requires no assumptions about the analytical properties of the exact Green’s function in the limit N → ∞.
The form of the Green’s function depends on the choice of boundary conditions.
The Green’s function is required in all cases to vanish at the origin,

G(0, R0 ; k) = 0.

(5.30)For a conventional (i.e. eigenvalue-independent) outer boundary condition, the Schrödinger
Green’s function equation(H − E)G(R, R0 ; k) = δ(R − R0 )(5.31)

yields the solution

0

G(R, R ; k) =2NX
φn (r)φn (r 0 )n=1

kn2 − k 2.

(5.32)

This function is identifiable as the principle value Green’s function; it displays neither
incoming nor outgoing wave character on the boundary. On the other hand, the Siegert
boundary conditiond=0− ik G(R, R0 ; k)
dR
R=R0


(5.33)

has the same quadratic matrix equation representation as 5.19, and gives the alternate
solution form

121

G(R, R0 ; k) =2NX
φn (r)φn (r 0 )kn (kn − k)n=1.

(5.34)

This is the outgoing wave Green’s function. Both forms are solutions to the same
Hamiltonian, and differ only by a homogeneous solution of the Schrödinger equation.
These Green’s functions require all 2N SPS solutions to be included in the summation; the omission of even one state (or pair of states) will give a totally incorrect
solution. Since some of the pseudostates lie very far from the axis, their exponential
growth at the outer boundary may be extremely rapid. This is a potential cause ofserious numerical instability, as noted first in [149]; because the normalization of theSiegert pseudostates 5.24 involves a cancellation of an exponentially large integral withan exponentially large surface term, an exponential growth factor that exceeds the inverse of the machine precision will effectively destroy any orthonormality of the basis
set, and thus the completeness relations on which the derivation of the Green’s function
depends.
Fortunately, the scattering matrix may be analytically derived directly from theGreen’s function expansion, giving either a “sum formula”

S(k) = e

2ikR0"

1 + ik2nX[φn (R0 ]2

k (k − k)n=1 n n#(5.35)or the “product formula”

S(k) = −e2ikR0

2NY
kn + kn=1

kn − k.

(5.36)The second result, rather remarkably, allows the S-matrix to be written entirely interms of the eigenvalues; this completely bypasses the difficulty of numerical instability
associated with the rapid exponential breakdown of the orthonormality relation forincreasing value of R0 , as discussed in [149].
The partitioning of configuration space into inner and outer regions for the SPS

122
eigenvalue and the subsequent symmetrization of the Hamiltonian by use of the Blochoperator are both reminiscent of the eigenvalue R-matrix method. Further, the derivation of the sum and product formulas shown above involves the surface value of theGreen’s function, G(R0 , R0 ; k), which is known to be proportional to the R-function(the one-dimensional R-matrix) [124]. This suggests that there should be some natural
relationship between the SPS formulation and the R-matrix formulation. In fact, Bayeet al. [149] have shown that for the same finite basis set representation, the methods areformally equivalent. 2 This homology provides an alternate expression for the S-matrix
based on R-matrix theory, working directly from the primitive basis and the matrices
in its representation without any need to find the complete set of SPS eigenstates. Of
course, it may still be useful to find SPSs that correspond to the true physical Siegert
states, for the sake of extracting their width and shape parameters, but for only a few
states this can be easily accomplished by using the iterative approach. For the taskof constructing the scattering matrix, the usefulness of the SPS formalism appears tobe subsumed by the more general applicability of the R-matrix formalism; the latter is
not susceptible to the Siegert states’ noted susceptibility to long-range potential tails,
and the multichannel version of R-matrix theory is far simpler than that for the Siegert
state case.
Although the Green’s function SPS theory is completely rigorous, it is severely restricted in utility. The limitations discussed in the last two sections of this chapter may
be summarized: First, the SPS technique is best suited for single channel problems. An
extension to even two-channel problems presents considerable challenge, and depends
on the motivated selection of a fortuitously simple uniformization transformation [150].
Such a transformation is necessary to map the multi-valued function defined on the entire multisheet Riemann surface into a single-valued (but potentially quite complicated)
2This is already anticipated in the discussion of Lane and Thomas, who note that the WignerEisenbud theory is “absolutely equivalent” to the Kapur-Peierls resonance theory [124], which in turnis related to the Siegert theory by the renormalization procedure suggested by More and Gerjuoy [135].

123
function of some new variable. For a two-channel system, the Riemann surface has four
sheets (arising from two branch cuts, with branch points at each of the threshold energies) and the uniformized matrix eigenequation is quartic in the transformed variable.
More generally, a N -channel problem will have a 2 N -sheet Riemann surface, and there
is no known method for uniformization beyond the two-channel case.
Second, the derivation of the SPS Green’s function expressions shown above isconsidered only for one dimensional model systems in the work of Tolstikhin et al. It
remains a matter of untested conjecture whether or not there exists a multidimensional
generalization of the requisite completeness relations, or whether the derivation yields
a comparably simple closed form. At the approximate level, one might imagine an expansion in hyperspherical coordinates [151]; this treatment is only of utility, however, if
the fragmentation modes are qualitatively similar. For the solution of competing ionization and dissociation channels, involving the coupling between fundamentally distinct
degrees of freedom, an entirely different approach is necessary. This problem is further
compounded by the possibility that one set of channels (e.g., the ionization channels
in the hydrogenic system) may have Coulombic long-range interaction or a centrifugal
effective potential term associated with finite angular momentum. In this case, the SPS
formalism is not simply non-rigorous, but involves increasingly complicated corrections
to the boundary condition that must be expressed at the level of a Taylor expansion.
In the conclusion of their work, Tolstikhin et al. hinted at the possibility of an
entirely different philosophy for application of the Siegert states to scattering physics:“The SPS formulation also has advantages as a method of discretization of the continuum.” Their suggestion called for the use of the SPSs as a basis for a time-dependent
close-coupling calculation, with the recognition that this would “enable one to distinguish between the excitation of a resonance state and the underlying continuum scattering.” This idea was implemented shortly thereafter by Yoshida and coworkers [152, 153]
to describe the reflectionless loss of wavepacket probability by flux across a bounding

124
surface.
In fact, the ability of Siegert states to serve as a finite basis approximation to thecontinuum had already been demonstrated some time earlier in the work of Seideman
and Miller [154], in the context of exploring the semiclassical analysis of transition state
theory in nonperturbative regimes. If the wavefunction describing reaction dynamics inthe vicinity of a harmonic transition state are expanded in terms of the normal mode
frequencies, with the reaction coordinate naturally acquiring an imaginary frequency,
then the Hamiltonian expanded in normal mode coordinates becomes complex symmetric. If this Hamiltonian is solved by direct diagonalization, instead of perturbative
corrections to the semiclassical solution, then the spectrum contains a set of complex
eigenvalues identifiable as Siegert states. These states do not represent resonant complex formation, but are instead the discretization of the reaction coordinate continuum,
and give smooth and structureless contributions to the transition probability. Ryaboy
and Moiseyev [155] showed that this set of states could be equivalently generated by acomplex coordinate scaling in which the “white” (non-resonant) continuum was rotatedoff into the complex plane to uncover the Siegert eigenvalues and make their statessquare integrable in the scaled coordinates.
In the next section, we follow the approach of Siedeman and Miller, rather thanthe more rigorous theory of Tolstikhin et al., and begin from the hypothesis that the
Siegert pseudostates can be usefully appropriated as a basis set for representing thediscretized continuum of a channel expansion. In effect, this creates an additional(potentially infinite) set of complex-energy channels that extend arbitrarily high into
the vibrational continuum. As Ryaboy and Moiseyev note, so long as these channels
are associated with poles of the scattering matrix that are separated in the complex
plane by “distances” (i.e., the difference between the real parts of the eigenvalues) thatare smaller than the corresponding widths (i.e., the sum of the imaginary parts ofthe eigenvalues), the contribution to the observable scattering parameters (phase shift,125cross-section, etc.) can be sufficiently smooth to completely represent the backgroundelastic scattering. In effect, the coupling to the continuum has been broken up into asequence of broad, overlapping Lorentzian resonances which may be summed together
to give a slowly varying baseline that reflects direct coupling to the continuum. If any
physical resonance states are present, they will normally have poles much closer to thereal axis, and thus their Siegert eigenstate widths will be much sharper and narrower,
and their contribution to the spectrum will be manifestly resonant.

5.5

Extending MQDT to a Siegert pseudostate basis: Theory
Since the Siegert pseudostate basis consists of two branches of narrowly-spaced

continuum-like complex eigenstates, in addition to the same rovibrational eigenstates
they share with a more conventional basis with fixed logarithmic derivatives on thebox boundary, they appear at first glance to include exactly the basis set contribution
needed to give the wavefunction outgoing flux across that boundary. In principle, wewould like to have a solution state located at precisely the energy of the vibrational continuum state into which the molecule is dissociating at that energy, with an imaginary
part corresponding to the width of the predissociating resonance. In practice, if we havean artificial discretized continuum of sufficiently dense pseudostates with the property
of being complete (or at least, complete with respect to the subset of continuum states
one is attempting to represent), then the lack of a state at exactly the right energy andwidth is immaterial, and accommodated at the level of a complex basis expansion. Thisis quite different from the original method Jungen described above, where producing a
pseudo-continuum state at precisely the correct energy (by varying a parameter iteratively in [53]) was of paramount importance for the success of the solution. (It is moresimilar to the later method of Ross and Jungen, which augments the basis by including
additional sets that do not vanish, although even in this case the choice of additional
sets must be optimized by the choice of a boundary condition resembling the boundary

126
condition of the real solution wavefunction. This cannot typically be known in advance,
but must be optimized by hand for different resonances for best results, whereas theSiegert method avoids the need to readjust any parameters throughout an entire energy
region.) There are, to be sure, numerical concerns to be addressed before this approachshould be naively trusted, especially in light of the known peculiarities of the Siegert
states, including their exponential divergence for increasing argument (on the “unphysical” sheet of the Riemann surface) and their innate overcompleteness. For the sake ofdevising a useful basis, however, it suffices that these concerns be resolved merely tothe level of numerical accuracy (i.e., convergence of the calculated cross-section) undercertain specified parameters that reflect relevant real-world conditions. This is a much
weaker requirement than the one necessary for implementation by the more rigorous
approach of Tolstikin et al..
For resonance series corresponding to high electronically excited intermediates(Rydberg states) of diatomic molecules, the most natural description of the system is
one with quantum defect parameters defined in terms of a fixed internuclear distance
R and a well-defined projection of the orbital angular momentum Λ onto the axis ofsymmetry. This is because the electron spends most of its time far from the nuclear core,
and when it does penetrate into the core, it gains enough speed from falling through the
Coulomb potential that the nuclei are essentially frozen on the time scale of its motion.
The quantum defect functions µΛ (R) in this representation, the so-called “body-frame”,
may either be calculated from highly accurate ab initio techniques, or extracted from asemi-empirical fitting of experimental data [121]. In order to connect them with the true
asymptotic ionization channels defined in terms of Siegert pseudostates of the residual
core, j = {v + , N + }, a frame transformation must be performed [45, 37], where N +is the ionic rotational momentum, and v + is the vibrational quantum number of thepseudostates. In the Siegert MQDT procedure, it is necessary to directly evaluate theS-matrix by the frame transformation integral

127

Sj,j 0 =

P

R R0

Λ hN

+ |Λi

+i

Λ hN

P

0

0

φj (R)e2iπµΛ (R) φj 0 (R) dR hΛ|N + i+ |Λi φj (R0 )e2iπµΛ (R0 ) φ

kj +kj 0

j 0 (R0 )(5.37)

0hΛ|N + i.

The surface term in 5.38 is new, but it is included because also arises in the orthonormality relation [142]. A similar transformation converts the body-frame transition dipole
elements DΛ (R) into reduced dipole matrix elements in the same S-matrix representation,

DjS =(2J + 1)PΛ hΛ|J0 i(J) hΛ|N + i(5.38)

× 0R0 φ0 (R)DΛ (R)eiπµΛ (R) φj (R) dR.
R

Here φ0 (R) is the initial vibrational wavefunction, and J 0 and J are the total angularmomenta of the initial and final states of the system, respectively. (Note that the surface
term is omitted here because φ0 (R) is assumed to be negative on the surface.)
Note that the Siegert pseudostates are never conjugated in these expressions,even when they formally belong to the dual (“bra”) space. In particular, this means~ S† below is calculated by conjugating only e iπµΛ (R) inthat the quantity labeled as D
the definition above, and not the dipole matrix elements directly. The unconventional
nature of the inner product in a Siegert basis may be understood from a somewhat
more intuitive perspective by means of an appeal to the symmetry properties of theGreen’s function in the complex k plane, as first presented by More and Gerjuoy [135].
To briefly sketch this argument, we begin by noting that the Green’s operator formedby eigenvector expansion over a resonance state basis such as the Siegert pseudostatesobeys the identityG† (k) = G(−k ∗ ).

(5.39)

128
This relation follows rigorously from the fact that the Green’s operator is real for negative energies (i.e., for pure imaginary values of k), via the Schwartz reflection principle.
(This is in contrast to the more usual case of the reflection principle, defined for functions which are real when their argument is real.) This nonstandard symmetry also
applies to the left eigenvector states φ̃j and right eigenvector states φjφ̃j (R; k) = φj (R; −k ∗ ) = φ∗j (R; k).

(5.40)

In other words, if hφ̃j | is a left eigenvalue of the Green’s operator G(k), then | φ̃j i is a righteigenvalue of G(−k ∗ ). Since both the transformation between left and right eigenvectors
and the transformation between the bra and ket space representations both involve a
conjugation, the net effect is that the functions of the dual space are unconjugated.
At this stage of the calculation no information about the long-range behavior of
the channels has yet been included, and since the body-frame quantum defects are nearlyenergy independent, the resulting S-matrix is typically a smooth and fairly weak functionof energy. The method of channel elimination [156, 44] systematically eliminates flux
in all electronic channels below the energy threshold for electron escape (the “closedchannel subspace”) to form a “physical” S-matrix S phys , by taking the proper linear
combination of short-range solutions that ensures exponential decay at infinity. For along-range Coulomb potential, this procedure gives

Sphys = Soo − Soc (Scc − e−2iβ )−1 Sco .

(5.41)

Here, β is a diagonal matrix of the usual Coulomb long-range phase parameter πν jwhere νj is the (possibly complex) effective quantum number in the jth channel, S is
the scattering matrix, and the subscripts indicate partitions of the matrices into closed
and open subspaces [44].
For a Siegert state basis, this physical scattering matrix is in general not unitary,

129
but rather subunitary, reflecting the loss of flux at the boundary R 0 via coupling to theSiegert pseudo-continuum states. It can be used to calculate the total cross-section for
dissociative recombination by means of conventional formulas, but with the departurefrom unitarity, 1 −Pphys 2
phys 2
j |Sj,j 0 | , identified as the probability |S d,j 0 | for scattering into

the dissociative continuum. This method also provides all quantities necessary to findthe partial photoionization cross-section into any open channel, σ j ; see Eq. 2.59 of [44]for further details. The contributions from all open channels can then be summed togive the total cross-section for photoionization.
Alternatively, the total photoabsorption cross-section may be found directly froma “preconvolution” formula first derived by Robicheaux to handle the energy smoothing of densely spaced resonances [48], but expressed here in the equivalent but more
symmetrical form of Granger [157, 158],

σtotal (E) =

h
i−1 h
i
4π 2 αω~ S† 1 − Se2iβ~S
Re D
1 + Se2iβ D
3(2J0 + 1)

(5.42)

where Re signifies taking the real part of everything that follows, and the † here conjugates only the operator, not the entire matrix element. The diagonal matrix written
as e−2iβ has a nontrivial definition in terms of the quantum defect parameters, it may
be approximated quite well by taking β j = πνj for “closed” channels with E < Re Ej ,and βi = i∞ for “open” channels with E > Re Ej . Here E is the total energy of theqsystem, Ej is the threshold energy for channel j, and ν j = 1/ 2(Ej − E) on the branchwhere Im ν > 0. The utility of this expression lies in recognizing that the value of thecross-section at a complex energy in the above formula is equivalent to the cross-section
at a real energy, smoothed over a channel-dependent width Γ j = 2 Im j . Within the
Siegert state formulation, the electron energy  j = E − Ej will naturally take on acomplex value in any channel where the channel eigenenergy E j is itself complex, while
E remains real.

130
Given S and j , either of the two cross-section formulas above can be evaluated,with appropriate allowances for the possibility of complex energy eigenvalues. Note thatthe first procedure simply gives a sum over the flux into specific ionization channels,
while the second gives a single value for the total photoabsorption cross-section. Thismeans that the latter will contain information about the solution wavefunction along
the R = R0 boundary not contained in any of the open ionization channels. In general,the value of σtotal will be equal to or greater than the sum over the individual σ j , andany difference may be attributed to the effect of coupling to high-lying Siegert states in
the continuum. Thus, the difference between these two formulae at any energy provides
the photodissociation cross-section.
In order to test the validity of this hypothesis, we will start by defining a set ofSiegert pseudostates for the H2+ internuclear potential. The eigensolutions fall into threeclasses, as shown in Figure 5.8. Those lying along the positive (Im k)-axis are associatedwith negative eigenenergies on the physical sheet of the E-plane, the bound states of thepotential. These are the channel thresholds to which the Rydberg autoionization seriesof the ionization spectrum converge, and so we include all of their states. The solutions
along the negative (Im k)-axis lie on the unphysical energy sheet, and we reject them as
antibound states arising from the doubling of the dimension space. The remainder ofthe solutions fall above and below the (Re E)-axis, corresponding to conjugate solution
pairs of the eigenvalue parameter λ = ik. We select only those with negative Im E j ,a decision that can be justified conceptually on the grounds that these states display
a time dependence in which total probability decreases over time, corresponding todecay into the continuum. (For a much more sophisticated discussion of this topic,
see the extensive discussion of [159, 160, 161], which arrives at the same conclusion forcomplex expansions of the Green’s function using far more rigorous formal arguments.)
For MQDT matrix elements it is also acceptable to reject states lying very high in thecontinuum, since their Franck-Condon overlap with the bound states is negligible.

131

Figure 5.8: Distribution of H+
2 vibrational Siegert pseudostate energies in the complex
energy plane for angular momentum N + =1. Only the circled states are included aschannels in the scattering matrix.

132
Tolstikhin et al. discuss the unusual completeness relation obeyed by the full setof Siegert pseudostates, which has an additional factor of 2. Our restricted subset ofSiegert pseudostates does not, of course, obey that doubled completeness relation. We
have confirmed through numerical tests, however, that this restricted subset behaves
like a complete set, to at least 10−12 accuracy, for representing either L 2 functions
confined within the boundary or functions with purely outgoing wave character at theboundary. The ability to represent L 2 functions diminishes somewhat for extremely narrow functions, for which the primary contribution must come from shorter wavelength
pseudocontinuum functions. For example, when expanding in terms of a truncated set
with a maximum Re(k) of about 50 a.u., only features broader than /1 k=0.02 a.u. inwidth can be well- represented. When using this truncated set numerically, a Gaussian2of the form e−α(R−Req ) can be only represented to 10−6 accuracy when α=100 (i.e., a
Gaussian of width 0.14 a.u.), has a relative error of over 10% for α=1000 (i.e., a Gaussian of width 0.045 a.u.), and fails entirely for α=10000 (i.e., a Gaussian of width 0.014
a.u., smaller than the wavelength of the highest excited state included in the basis).
For an impressive demonstration (in a somewhat different context) of the convergenceproperties of a similarly truncated Siegert basis also used to describe smooth continuum
physics, see [154].

5.6

Extending MQDT to a Siegert pseudostate basis: Results anddiscussion
In the region of the ungerade H2 spectrum between 127200 and 127800 cm −1

there are several strongly predissociated resonances, members of the npπ, v + = 8 and
npπ, v + = 5 series. In each case, our calculated spectrum correctly reproduces them inthe total absorption cross-section, but shows them as weak or absent in the ionization
cross-section. Comparisons of our results with other theoretical and experimental values[55, 121] for the relative yields of selected resonances appear in Table 5.2. Note particu-

133
larly that this method is able to correctly describe the strong rotational dependence ofthe 4pπ, v + = 5 branching ratio, a nontrivial consequence of subtle channel interactions.
Table 5.2: Photoionization and photodissociation yields for select ungerade resonancesin H2 for which the relative yields have been experimentally observed [55].
State
3pπ, v = 8, R(0)5pσ, v = 4, R(0)4pπ, v = 5, R(0)4pπ, v = 5, R(1)SourceObserved
Theory[121]Present
Observed
Theory[121]Present
Observed
Theory[121]Present
Observed
Theory[121]Present

Energy127248.2
127246.9
127242.2
127599.4
127602.2
127606.8
127667.6
127665.4
127666.6
127599.4
127758.4
127759.5% Ion.
10(5)1190(10)
88
76
82(5)
93
97
30(10)
17
29% Diss.
95(5)9999
10(10)
12
24
18(5)
7
3
70(10)
83
71

As a test of the method in an entirely different energy regime we considered the
problem of dissociative photoionization, a three-body breakup channel accessible only
at much higher energies. Experimental measurements of the ratio between pure ionization and dissociative ionization have been performed since the 1970s by a number of
researchers [162, 163, 164], along with at least one early theoretical calculation [165].
Since our ionization spectrum is a sum over individual channels, one can easily distinguish between contributions from channels above and below the dissociative threshold.
Our results, plotted against those of past experiment and theory, are presented in 5.9.
Finally, we have performed a model calculation demonstrating the utility of our
method for treating dissociative recombination, particularly in systems where indirectchannels (those involving scattering into intermediate autodissociating Rydberg states)
play an important role. 5.10 shows the dissociative recombination spectrum of a simplified H2 model potential (neglecting rotation and with R-independent quantum defects),
compared with the familiar approximation of O’Malley for smooth background scatter-

134

Figure 5.9: Dissociative photoionization cross-section, as a ratio to the total photoionization cross-section. The solid line is the current theory, the dotted line is an earlier theory[165], and the points are data from various experimental measurements [162, 163, 164].

135
ing by direct processes [166]. This spectrum accurately reproduces the background, andalso describes complex interference effects from the series of resonances converging toeach Rydberg threshold.

Figure 5.10: Dissociative recombination cross section for the model potential, unconvolved (solid) and convolved with a Lorentzian of width 0.1 eV (dashed), compared tothat resulting from the O’Malley formula (dotted).

The extension of the Siegert MQDT method to polyatomic systems might be
attempted in several ways. First, by transforming the nuclear coordinates to a hyperspherical coordinate system, a multidimensional problem is effectively reduced to a single scattering coordinate (the hyperradius), allowing the formulas above to be adaptedwith minimal alteration. This has already been demonstrated for the dissociative recombination of H+
3 [28], with results that appear to correctly reproduce broad featuresin the experimental spectrum, although the theory appears not to be sufficiently precise
to describe detailed resonance structures. Alternatively, it might be possible to recast
the Siegert eigenproblem in a generalized form that admits arbitrary dimension. Even
if the completeness relations and matrix inversion identities required for the derivation

136
of the Green’s function (as in the Tolstikhin papers) were no longer possible to writeout analytically in multiple dimensions, the ability to construct a “locally complete”
representation of the multidimensional continuum would still be sufficient to provide
appropriate channel states for the MQDT channel expansion.

Chapter 6

Conclusion

On a practical level, this dissertation has considered the physical properties of two
highly excited diatomic systems, one commonplace, and naturally occurring throughout
the universe, and one exotic, and likely to be observed under only carefully controlledlaboratory conditions. On a conceptual level, however, these studies share a common
philosophical framework with respect to the way in which they approach the process of
scattering for Rydberg electrons in quantum mechanical systems.
From the standpoint of classical mechanics, scattering is understood as a discrete
event or series of events associated with the changing trajectories of a set of particle
or bodies or bodies. A pair of scattering billiards, for example, will move along aflat surface in straight line until they hit one another, experience an instantaneous
redistribution of their total energy and momentum (perhaps with some dissipation due
to inelasticity), and then continue away along altered trajectories. A satellite undergoinga close encounter with a planet will slingshot rapidly through its gravitational field in a
hyperbolic orbit, emerging along a new asymptotic trajectory with a modified direction
and velocity. Scattering, by nature, is a localized process. It takes a strong interaction
the occurs during a short period of time and is confined to a limited region of space,and reduces it to an overall effect in terms of the alteration of a system’s trajectories infree space.
In quantum mechanics, the picture of particles traveling along trajectories and138
undergoing isolated events must typically be abandoned. The solution to tunnelingthrough a barrier, the first quantum scattering calculation that any young physicist is
likely to encounter, is solved entirely in terms of wavefunctions and matching conditions;the solution in each region is solved separately, and the solutions are joined together
by the conditions of continuity and smoothness across the boundaries. This picture isentirely delocalized. The solutions are continuum solutions that extend from negative
infinity to positive infinity, and there is no “event” associated with the calculation. A
sufficiently large number of solutions may be superposed, of course, to give a localized wavepacket, and propagating that wavepacket according to the time-dependent
Schrödinger equation will recover a dynamical behavior recognizably similar to the classical case. But the preparation of such a packet will necessarily involve states with arange of energies, whereas many interesting quantum mechanical measurements are associated with experiments for which the preparation of such wavepackets is difficult or
impossible. A common situation is that of experimental continuous-wave spectroscopy,
where a monoenergetic photon source (like a laser) excites an atomic or molecular system with a single, well-defined energy.
The challenge of highly excited quantum mechanical systems, such as Rydberg
atoms and molecules, is that they occupy an intermediate status between the quantum
and classical understandings of scattering. On one hand, they are clearly scattered by
localized interactions. The total volume through which a Rydberg electron moves is typically quite large compared to the volume in which it has a complicated interaction witha scattering body. One is logically motivated to find some way of exploiting the spatialconfinement of this interaction in order to simplify the task of solving the Schrödinger
equation. On the other hand, the interactions themselves are clearly quantum mechanical in nature, such that it would be disastrous to attempt to treat an electron scatteringfrom a molecule as if it were something like a marble scattering from a set of bowlingballs connected by springs. The exchange of energy and momentum between a scattering

139
electron and a molecule is a function of inherently quantum mechanical properties suchas phase, wavelength, spin, quantized orbital angular momentum, and quantized molecular excitation modes. The solution must therefore be quantum mechanical in form.
The possibility of successfully combining advantageous features of both perspectives isrealized by quantum defect theory, which interprets complex short-range interactionsin terms of their net effect on long-range parameters like the quantum defects or phaseshifts.
In Chapter 2, quantum defect theory was introduced as a paradigm for thinking
about the effect of core scattering on a Rydberg state. The electron moves mostly in aregion far separated from the core, under the influence of a long-range Coulomb potential, and therefore has a solution that resembles the hydrogenic wavefunctions; the only
effect of the core is to shift the energy of these wavefunctions in a way that is dependent
on the orbital angular momentum, but at least to a good approximation is independentof the electron’s long-range behavior. In the wavefunction itself, this manifests as aphase parameter that mixes the regular and irregular Coulomb functions. In effect, thequantum defect formalism divides the problem into a short-range solution region anda long-range solution region, and describes the effect of the short-range physics on thelong-range solution in terms of the minimum number of necessary parameters. For aproblem with multiple scattering channels, the scattering matrix suffices to specify theprobability amplitude for inelastic scattering from one type of channel to another. Although this is fully quantum mechanical, the classical picture of scattering as a localized
event that modifies particle trajectories has been restored. In semiclassical terms, theelectron occupies a closed orbital that, once per orbital period, scatters from the corein such a way as to accumulate an additional phase shift, and transfer incoming probability flux in one channel to outgoing flux in the others. The efficacy of this approach
was demonstrated for the diatomic system HD, but the method generalizes naturally to
any Rydberg system, atomic, diatomic, or polyatomic. With respect to the problem of140
describing overlapping Rydberg series of resonances in the electronic continuum, there
is presently no other method that rivals the capabilities of quantum defect methodology
to efficiently reproduce resonant features in the scattering cross-section.
In Chapter 3, a single long-lived Rydberg stationary state is created, and theeffect of perturbation by the presence of a second particle that physically impinges on
the Rydberg wavefunction is considered. This situation is similar to that of Rydberg
excitation in the previous chapter, in the respect that electron-core interaction may
still be described most efficiently by quantum defects, but the second scattering centercreates an additional set of boundary conditions that must be satisfied. Just as themodification of a Coulomb potential could be expressed in terms of phase-like quantum
defect parameters, so also the short-range scattering of the electron by a neutral perturber can be fully described in terms of an overall phase shift in each channel. These
phase shifts provide the basis for a zero-range pseudopotential approximation that performs with accuracy comparable to the more conventional approach of diagonalizing the
Hamiltonian in an L2 basis. In classical terms, one might think about the difference
between a detailed modeling of the local interaction between two colliding billiard balls,
with infinitesimal deformations during the moment of contact accounting for the exchange of impulse, and an hard sphere approximation that treats the collision as truly
instantaneous. The first level of detailed analysis might be necessary to account forslight inelasticity of the collision, or the transfer of translational motion into rotationfor off-center collisions, but once the functional dependence of these effects was known,
they could be incorporated directly into the hard-sphere model as various corrections
and coefficients without the need to repeated the detailed calculation. The parameterization of Rydberg-neutral scattering in terms of phase shifts that depend only on theenergy and angular momentum of the electron accomplishes a similar reduction of thedetailed short-range behavior in terms of its collective asymptotic effect.
Chapter 4 implements the separation between the scattering region and the exter-141
nal free-electron region in an even more literal sense, by using the Coulomb Green’s function to write an exact integral equation solution in the outer region, and match it onto
the phase-shift adjusted short-range solution close to the perturber. Since the Green’s
function serves as a propagator in the energy representation, this may be viewed quite
literally as repeatedly colliding the electron with the perturber, and using the boundary
matching condition to select out the stable closed orbit trajectories of the motion. In
fact, modern semiclassical theory allows this picture to be implemented directly, witha coherent summation over the two closed Coulomb orbits that pass through a given
point and begin and end on a small sphere bounding the perturber. The constructive
interference between the paths is responsible for the distinctive nodal pattern of the
wavefunction that accounts for its resemblance to the ridged shell of a trilobite.
The Rydberg state localized-scattering picture is certainly powerful with respect
to the task of describing highly excited and continuum electronic states. This raises thequestion of whether on can retain the advantages of this philosophy, while at the sametime including the nuclear continuum states that arise in the context of dissociation.
Chapter 5 provides a demonstration of several approaches to extending multichannel
quantum defect theory or R-matrix theory to systems with such a double continuum,
including one that is entirely original, the method of Siegert channel states.
This dissertation has refined and extended a set of existing tools from the fieldsof resonance theory, quantum defect theory, and multichannel spectroscopic analysis inorder to describe two rather dissimilar diatomic systems that arise from Rydberg electron excitation. From this foundation, there are many directions that could be pursued
with respect to verifying, improving, or applying these ideas. In some cases, this may
simply mean coming to a better understanding of the limitations of these methods. The
machinery of quantum defect theory, for example, as powerful as it often proves to be,leaves no obvious procedure for improvement of accuracy. Unlike perturbation theory,
there are no “higher level terms” available in the theory to systematically improve the142
convergence of a solution, and unlike variational methods, there is no guarantee of alower or upper bounding of the calculated values. The incorporation of effects arisingfrom energy dependence of the quantum defect parameters, for example, or symmetry
breaking effects like the loss of gerade-ungerade symmetry in HD and Jahn-Teller couplings in triatomics, must usually be added to the model on a somewhat ad hoc basis.
This increases the importance of confirming the results of such calculations, either bymore rigorous computational methods, or by direct experimental observation.
For the long-range Rydberg bound states, the first crucial test of the theory would
involve experimentally confirming their existence by means of some spectroscopic signature. At present, the most difficult aspect of preparing these states lies in the creationof a high-enough density cold dilute gas sample. Most modern magneto-optical traps
are capable of generating densities in the range of 10 10 -1011 atoms per cm−3 ; the population of nearest-neighbor pairs with the correct separation to form resonance-induced
bound states (e.g., at 500-1000 Bohr radii for n=30-70) would be greatly enhanced atdensities only a few orders of magnitude larger than this. As a possible alternative,the existence and structure of molecular potential curves might be deduced from theobservation of satellite lines even at temperatures above those favorable for the formation of bound states. On the theoretical end, further refinement of the theory in thispaper (particularly with regard to the handling of fine and hyperfine structure) would
be advantageous.
If the existence of such states is confirmed, other opportunities for applicationoriented theoretical study could be investigated more thoroughly. The large permanent
dipole moments attached to these states makes the prospect for alignment-based applications particularly appealing. For example, if a sample of Rydberg molecules wereprepared and oriented along a common axis, and a beam of slow electrons passed throughthe sample in the direction of that axis, one might imagine the electrons displaying adiffraction pattern suggestive of the intricate nodal structure of the electronic wavefunc-

143tion. Another possibility, prompted by the observation that (to a good approximation)
the effect of the perturber is simply to mix atomic states, might be the creation of asimilarly shaped electronic wavefunction in the absence of any perturber at all, perhaps
by the use of pulsed multipole electric fields [167]. A third idea might involve the studyof Rydberg-state induced dissociation of diatomic or polyatomic perturbers via dissociative attachment, as initially proposed in the work of Dubov and Rabitz [65, 66] inthe context of considering laser-assisted exchange reactions.
As noted in the conclusion to Chapter 5, the utility of the Siegert state method
has already been demonstrated beyond the case of molecular hydrogen. The most direct route to generalize the method to polyatomic molecules is probably that used in[28], based on a transformation of the nuclear motion to hyperspherical coordinates.
One might also envision finding a technique for calculating multidimensional Siegert
pseudostates, with the possibility of outgoing flux on several different surfaces or hypersurfaces in a multidimensional volume. In the two-dimensional case, for example, it
could provide a new approach to the handling of two-electron systems with an accessible
double ionization continuum, perhaps in conjunction with R-matrix methodology. Even
more complicated molecules might be analyzed by isolating a subset of particularly “active” normal modes, and defining the Siegert spectrum in the normal mode coordinates
directly.
The study of resonant effects in near-threshold electron-molecule scattering processes may provide the first step toward a greater understanding of resonant effects inlarger molecules with similar local structure. The work of Burrow and coworkers, for example, has described resonant scattering properties of electrons from both chloroalkanes
and DNA bases using a model that assumes a single electronically active bond, with subsequent vibrational coupling to the other bonds of the molecule [168, 169]. And the workof Gianturco and Lucchese has suggested that one of the most important contributions
to DNA radiation-induced decay channels involves the creation of photoionized elec-

144
trons that undergo low-energy dissociative attachment, a process that depends vitallyon energy redistribution between electronic and nuclear excitation modes [170, 171].

Bibliography[1] J. J. Balmer, Ann. Phys. Chem. 25, 80 (1885).
[2] J. R. Rydberg, Kgl. Svenska Akad. Handl. 23, 1 (1889).
[3] G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 (2002).
[4] J. C. Boyce, Rev. Mod. Phys. 13, 1 (1941).
[5] G. Herzberg, J. Mol. Spectrosc. 33, 147 (1970).
[6] P. M. Dehmer and W. A. Chupka, J. Chem. Phys. 65, 2243 (1976).
[7] E. Reinhold, W. Hogervorst, and W. Ubachs, Phys. Rev. Lett. 78, 2543 (1997).
[8] E. Reinhold, W. Hogervorst, and W. Ubachs, Chem. Phys. Lett. 296, 411 (1998).
[9] E. McCormack and E. Eyler, Phys. Rev. Lett. 66, 1042 (1991).
[10] E. McCormack, S. T. Pratt, P. M. Dehmer, and J. L. Dehmer, J. Chem. Phys.
98, 8370 (1993).
[11] M. D. Lukin et al., Phys. Rev. Lett. 87, 037901 (2001).
[12] M. D. Lukin, Rev. Mod. Phys. 75, 457 (2003).
[13] S. Haroche, Fortschr. Phys. 51, 388 (2003).
[14] C. Rangan, J. Ahn, D. N. Hutchinson, and P. H. Bucksbaum, Phys. Rev. A 66,
022312 (2002).
[15] J. Ahn, T. C. Weinacht, and P. H. Bucksbaum, Science 287, 463 (2000).
[16] T. Gallagher, P. Pillet, M. P. Robinson, B. Laburthe-Tolra, and M. W. Noel, J.
Opt. Soc. Am. B 20, 1091 (2003).
[17] T. C. Killian, M. J. Lin, S.Kulin, R. Dumke, S. D. Bergeson, and S. L. Ralston,Phys. Rev. Lett. 86, 3759 (2001).
[18] S. Kulin, T. C. Killian, and S. D. Bergeson, Phys. Rev. Lett. 85, 318 (2000).
[19] T. C. Killian, S. Kulin, S. D. Bergeson, L. A. Orozco, C. Orzel, and S. L. Ralston,Phys. Rev. Lett. 83, 4776 (1999).

146[20] J. R. R. Verlet, V. G. Stavros, R. S. Minns, and H. H. Fielding, Phys. Rev. Lett.
89, 263004 (2003).
[21] W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, Phys. Rev. A
63, 063406 (2001).
[22] C. Boisseau, I. Simbotin, and R. Côté, Phys. Rev. Lett. 88, 133004 (2002).
[23] A. L. de Oliveira, M. W. Mancini, V. S. Bagnato, and L. G. Marcassa, Phys. Rev.
Lett. 90, 143002 (2003).
[24] F. Merkt, Chimia 54, 89 (2000).
[25] B. H. Bransden and C. J. Jochain, Physics of Atoms and Molecules (Longman
Group Limited, Essex, 1983).
[26] H. Friedrich, Theoretical Atomic Physics (Springer-Verlag, Berlin, 1990).
[27] C. H. Greene and C. Jungen, Adv. At. Mol. Phys. 21, 51 (1985).
[28] V. Kokoouline and C. H. Greene, Phys. Rev. A 68, 012703 (2003).
[29] W. Kolos and L. Wolniewicz, J. Chem. Phys. 41, 3674 (1964).
[30] W. Kolos and L. Wolniewicz, J. Chem. Phys. 49, 404 (1968).
[31] D. M. Bishop and L. M. Cheung, Phys. Rev. A 16, 640 (1977).
[32] L. Wolniewicz and J. D. Poll, J. Chem. Phys. 73, 6225 (1980).
[33] W. Kolos and K. Szalewicz, J. Chem. Phys. 84, 3278 (1986).
[34] B. D. Esry and H. R. Sadeghpour, Phys. Rev. A 60, 3604 (1999).
[35] L. Wolniewicz, J. Chem. Phys. 78, 6173 (1983).
[36] W. Kolos and L. Wolniewicz, J. Chem. Phys. 48, 3672 (1968).
[37] C. H. Greene and C. Jungen, Phys. Rev. Lett. 55, 1066 (1985).
[38] H. Gao and C. H. Greene, J. Chem. Phys. 91, 3988 (1989).
[39] H. Gao and C. H. Greene, Phys. Rev. A 42, 6946 (1990).
[40] C. Jungen and O. Atabek, J. Chem. Phys. 66, 5584 (1977).
[41] R. M. More, Phys. Rev. A 3, 1217 (1971).
[42] R. M. More, Phys. Rev. A 4, 1782 (1971).
[43] C. Greene, U. Fano, and G. Strinati, Phys. Rev. A 19, 1485 (1979).
[44] M. Aymar, C. H. Greene, and E. Luc-Koenig, Rev. Mod. Phys. 68, 1015 (1996).
[45] U. Fano, Phys. Rev. A 2, 353 (1970).

147[46] M. Raoult and C. Jungen, J. Chem. Phys. 74, 3388 (1980).
[47] C. Jungen and D. Dill, J. Chem. Phys. 73, 3338 (1980).
[48] F. Robicheaux, Phys. Rev. A 48, 4162 (1993).
[49] P. M. Dehmer and W. A. Chupka, J. Chem. Phys. 79, 1569 (1983).
[50] N. Y. Du and C. H. Greene, J. Chem. Phys. 85, 5430 (1986).
[51] E. L. Hamilton and C. H. Greene, Phys. Essays 13, 265 (2000).
[52] H. Gao, C. Jungen, and C. H. Greene, Phys. Rev. A 47, 4877 (1993).
[53] C. Jungen, Phys. Rev. Lett. 53, 2394 (1984).
[54] J. A. Stephens and C. H. Greene, J. Chem. Phys. 103, 5470 (1995).
[55] M. Glass-Maujean, J. Breton, and P. M. Guyon, Z. Phys. D 5, 189 (1987).
[56] G. M. Greetham, U. Hollenstein, R. Seiler, W. Ubachs, and F. Merkt, Phys. Chem.
Chem. Phys. 5, 2528 (2003).
[57] E. Fermi, Nuovo Cimento 11, 159 (1934).
[58] E. Fermi, Ric. Scientifica 7, 13 (1936).
[59] M. H. Mittleman, Phys. Rev. 162, 81 (1967).
[60] N. Allard and J. Kielkopf, Rev. Mod. Phys. 54, 1103 (1982).
[61] I. L. Beigman and V. S. Lebedev, Phys. Rep. 20, 95 (1995).
[62] L. P. Presnyakov, Phys. Rev. A 2, 1720 (1970).
[63] Y. N. Demkov and V. N. Ostrovskii, Zero-Range Potentials and Their Applicationsin Atomic Physics (Plenum Press, New York, 1988 (USSR, 1975)).
[64] G. K. Ivanov, Opt. Spectrosc. (USSR) 37, 361 (1974).
[65] V. S. Dubov, J. Chem. Phys. 97, 7342 (1992).
[66] V. Dubov and H. Rabitz, J. Chem. Phys. 104, 551 (1996).
[67] E. de Prunelé, Phys. Rev. A 35, 496 (1987).
[68] G. K. Ivanov, Opt. Spectrosc. (USSR) 43, 617 (1977).
[69] N. Y. Du and C. H. Greene, Phys. Rev. A 36, 971 (1987).
[70] N. Y. Du and C. H. Greene, J. Chem. Phys. 390, 6347 (1989).
[71] C. H. Greene, A. S. Dickinson, and H. R. Sadeghpour, Phys. Rev. Lett. 85, 2458
(2000).

148[72] J. Léonard, M. Walhout, A. P. Mosk, T. Muller, M. Leduc, and C. CohenTannoudji, Phys. Rev. Lett. 91, 073203 (2003).
[73] B. E. Granger, E. L. Hamilton, and C. H. Greene, Phys. Rev. A 64, 042508 (2001).
[74] A. A. Khuskivadze, M. I. Chibisov, and I. I. Fabrikant, Phys. Rev. A 66, 042709(2002).
[75] A. Omont, J. Phys. (Paris) 38, 1343 (1977).
[76] C. Bahrim, U. Thumm, and I. I. Fabrikan, Phys. Rev. A 63, 042710 (2001).
[77] C. Bahrim and U. Thumm, Phys. Rev. A 64, 022716 (2001).
[78] C. Bahrim, U. Thumm, and I. I. Fabrikant, Phys. Rev. A 61, 022722 (2000).
[79] E. L. Hamilton, C. H. Greene, and H. R. Sadeghpour, J. Phys. B: At. Mol. Opt.
Phys. 35, 199 (2002).
[80] W. E. Cooke and T. F. Gallagher, Phys. Rev. A 21, 588 (1980).
[81] J. Pascale, Phys. Rev. A 28, 1632 (1983).
[82] J. Brust and C. H. Greene, Phys. Rev. A 56, 2005 (1997).
[83] D. B. Khrebtukov and I. I. Fabrikant, Phys. Rev. A 51, 4675 (1995).
[84] L. Hostler and R. H. Pratt, Phys. Rev. Lett. 10, 469 (1963).
[85] L. Hostler, J. Math. Phys. 5, 591 (1964).
[86] L. Hostler, J. Math. Phys. 8, 642 (1967).
[87] Y. N. Demkov and G. F. Drukarev, Sov. Phys. JETP 20, 614 (1964).
[88] Y. N. Demkov and G. F. Drukarev, Sov. Phys. JETP 22, 271 (1965).
[89] Y. N. Demkov and G. F. Drukarev, Sov. Phys. JETP 54, 650 (1981).
[90] S. P. Andreev, B. M. Karnakov, V. D. Mur, and V. A. Polunin, Sov. Phys. JETP
59, 506 (1984).
[91] B. M. Karnadov and V. D. Mur, Sov. Phys. JETP 60, 657 (1984).
[92] L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics (Pergamon Press,
London, 1959).
[93] S. V. Tkachenko, Sov. Phys. JETP 61, 1149 (1985).
[94] M. I. Chibisov, A. M. Ermolaev, F. Brouillard, and M. H. Cherkani, Phys. Rev.
Lett. 84, 451 (2000).
[95] M. I. Chibisov, A. A. Khuskivadze, and I. I. Fabrikant, J. Phys. B: At. Mol. Opt.
Phys. 35, L193 (2002).

149[96] F. I. Dalidchik and G. K. Ivanov, Teor. Eksp. Khim. 8, 9 (1972).
[97] V. A. Davydkin, B. A. Zon, N. L. Manakov, and L. P. Rapoport, Sov. Phys. JETP
133, 70 (1971).
[98] U. Fano, Phys. Rev. A 24, 619 (1981).
[99] D. Harmin, in Atoms in Strong Fields, NATO ASI, edited by C. Nicolaides(Plenum Press, New York, NY, 1988).
[100] D. J. Armstrong and C. H. Greene, Phys. Rev. A 50, 4956 (1994).
[101] E. Luc-Koenig, S. Liberman, and J. Pinard, Phys. Rev. A 20, 519 (1979).
[102] M. P. Strand and W. P. Reinhardt, J. Chem. Phys. 70, 3812 (1979).
[103] P. D. Robinson, Proc. Phys. Soc. London 71, 828 (1958).
[104] C. A. Coulson and P. D. Robinson, Proc. Phys. Soc. London 71, 815 (1958).
[105] B. R. Judd, Angular Momentum Theory for Diatomic Molecules (Academic Press,
New York, 1975).
[106] S. M. Sung and D. R. Herschbach, J. Chem. Phys. 95, 7437 (1991).
[107] A. Sommerfeld, Atomic Structure and Spectral Lines (Methuen and Co. Ltd.,
London, 1934).
[108] H. A. Erikson and E. L. Hill, Phys. Rev. 75, 29 (1949).
[109] Y. Duan and J. M. Yuan, Eur. Phys. J. D 6, 319 (1999).
[110] M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag,
New York, 1990).
[111] A. M. O. de Almeida, Hamiltonian Systems: Chaos and Quantization (Cambridge
University Press, Cambridge, 1988).
[112] B. I. Schneider, Phys. Rev. A 11, 1957 (1975).
[113] B. I. Schneider and P. J. Hay, Phys. Rev. A 13, 1923 (1976).
[114] P. G. Burke, I. Mackey, and I. Shimamura, J. Phys. B: At. Mol. Opt. Phys. 10,2497 (1977).
[115] B. I. Schneider, M. LeDourneuf, and P. G. Burke, J. Phys. B: At. Mol. Opt. Phys.
12, L365 (1979).
[116] A. Giusti, J. Phys. B: At. Mol. Opt. Phys. 13, 3867 (1980).
[117] F. H. Mies, J. Chem. Phys. 80, 2514 (1984).
[118] C. H. Greene, A. R. P. Rau, and U. Fano, Phys. Rev. A 22, 149 (1982).
[119] J. Burke, C. H. Greene, and J. L. Bohn, Phys. Rev. A 81, 3355 (1998).

150[120] U. Fano and C. M. Lee, Phys. Rev. Lett. 31, 1573 (1973).
[121] C. Jungen and S. C. Ross, Phys. Rev. A 55, R2503 (1997).
[122] S. C. Ross, C. Jungen, and A. Matzkin, Can. J. Phys. 79, 561 (2001).
[123] E. P. Wigner and L. Eisenbud, Phys. Rev. 72, 29 (1947).
[124] A. M. Lane and R. G. Thomas, Rev. Mod. Phys. 30, 257 (1958).
[125] G. Raşeev and H. L. Rouzo, Phys. Rev. A 27, 268 (1983).
[126] W. Kohn, Phys. Rev. 74, 1763 (1948).
[127] C. H. Greene, Phys. Rev. A 28, 2209 (1983).
[128] H. L. Rouzo and G. Raşeev, Phys. Rev. A 29, 1214 (1984).
[129] C. H. Greene, in Fundamental Processes of Atomic Dynamics, edited by J. S.
Briggs, H. Kleinpoppen, and H. O. Lutz (Plenum Publishing Corporation, Essex,
1988).
[130] F. Robicheaux, U. Fano, M. Cavegnero, and D. A. Harmin, Phys. Rev. A 35, 3619
(1987).
[131] G. Gamow, Z. Phys. 51, 204 (1928).
[132] V. Weisskopf and E. P. Wigner, Z. Phys. 63, 54 (1930).
[133] P. Kapur and R. Peierls, Proc. R. Soc. London A 166, 277 (1938).
[134] A. J. F. Siegert, Phys. Rev. 56, 750 (1939).
[135] R. M. More and E. Gerjuoy, Phys. Rev. A 7, 1288 (1973).
[136] R. G. Newton, Scattering Theory of Waves and Particles (Academic, New York,1966).
[137] H.-D. Meyer and O. Walter, J. Phys. B: At. Mol. Opt. Phys. 15, 3647 (1982).
[138] J. N. Bardsley and R. R. Junker, J. Phys. B: At. Mol. Opt. Phys. 5, L178 (1972).
[139] R. Yaris and H. S. Taylor, Chem. Phys. Lett. 66, 505 (1979).
[140] C. W. McCurdy and T. N. Rescigno, Phys. Rev. A 20, 2346 (1979).
[141] W. P. Reinhardt, Ann. Rev. Phys. Chem. 33, 223 (1982).
[142] O. I. Tolstikhin, V. N. Ostrovsky, and H. Nakamura, Phys. Rev. A 58, 2077
(1998).
[143] H. W. Jang and J. C. Light, Phys. Rev. A 51, 1277 (1995).
[144] O. I. Tolstikhin, V. N. Ostrovsky, and H. Nakamura, Phys. Rev. Lett. 79, 2026
(1997).

151[145] H. Bachau, E. Cormier, P. Decleva, J. E. Hansen, and F. Martı́n, Rep. Prog. Phys.
64, 1815 (2001).
[146] C. de Boor, A Practical Guide to Splines (Springer, New York, 1978).
[147] B. Friedman, Principles and Techniques of Applied Mathematics (John Wiley &Sons, Inc., New York, 1956).
[148] D. L. Huestis, J. Math. Phys. (N.Y.) 16, 2148 (1975).
[149] D. Baye, J. Goldbeter, and J.-M. Sparenberg, Phys. Rev. A 65, 052710 (2002).
[150] G. V. Sitnikov and O. I. Tolstikhin, Phys. Rev. A 67, 032714 (2003).
[151] C. D. Lin, Phys. Rep. 257, 1 (1995).
[152] S. Yoshida, S. Watanabe, C. O. Reinhold, and J. Burgdorfer, Phys. Rev. A 60,
1113 (1999).
[153] S. Tanabe et al., Phys. Rev. A 63, 052721 (2001).
[154] T. Seideman and W. H. Miller, J. Chem. Phys. 95, 1768 (1991).
[155] V. Ryaboy and N. Moiseyev, J. Chem. Phys. 98, 9618 (1993).
[156] M. Seaton, Rep. Prog. Phys. 46, 167 (1983).
[157] B. E. Granger and C. H. Greene, Phys. Rev. A 62, 012511 (2000).
[158] B. E. Granger, Ph.D. thesis, University of Colorado at Boulder, 2001.
[159] A. Bohm, S. Maxson, M. Loewe, and M. Gadella, Physica A 236, 485 (1997).
[160] R. de la Madrid, A. Bohm, and M. Gadella, Fortschr. Phys. 50, 185 (2002).
[161] A. Bohm, N. L. Harshman, and H. Walther, Phys. Rev. A 66, 012107 (2002).
[162] R. Browning and J. Fryar, J. Phys. B: At. Mol. Opt. Phys. 6, 364 (1973).
[163] C. Backx, G. R. Wright, and M. J. van der Wiel, J. Phys. B: At. Mol. Opt. Phys.
9, 315 (1976).
[164] Y. M. Chung, E.-M. Lee, and J. A. R. Samson, J. Chem. Phys. 99, 885 (1993).
[165] A. L. Ford, K. K. Docken, and A. Dalgarno, Asrophys. J. 195, 819 (1975).
[166] T. F. O’Malley, Phys. Rev. 150, 14 (1966).
[167] K. B. MacAdam and C. S. Hwang, Rev. Sci. Instr. 74, 2267 (2003).
[168] K. Aflatooni, G. A. Gallup, and P. D. Burrow, J. Phys. Chem. A 102, 6205 (1998).
[169] P. D. Burrow et al., J. Phys. B: At. Mol. Opt. Phys. 31, L1009 (1998).
[170] F. A. Gianturco and R. R. Lucchese, Int. Rev. Phys. Chem. 15, 429 (1996).
[171] F. A. Gianturco and R. R. Lucchese, Phys. Rev. A 64, 32706 (2001).