26 October 2001

Chemical Physics Letters 347 (2001) 493±498

www.elsevier.com/locate/cplett

Analytical energy gradients of the excited, ionizedand electron-attached states calculated by the
SAC-CI general-R method
Mayumi Ishida, Kazuo Toyota, Masahiro Ehara, Hiroshi Nakatsuji *Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering,
Kyoto University, Sakyou-ku, Kyoto 606-8501, Japan
Received 1 June 2001; in ®nal form 2 July 2001

Abstract
A method of calculating analytical energy gradients of the excited, ionized and electron-attached states calculatedby the symmetry-adapted-cluster con®guration-interaction (SAC-CI) general-R method is formulated and implemented. This method is expected to be a powerful tool in the studies of dynamics and properties of molecules inwhich two- and multi-electron processes are involved. Good performance of this method is shown for the singlet andtriplet excited states of BH and for the doublet ground and excited states of CH. Ó 2001 Published by Elsevier
Science B.V.

1. Introduction
Derivatives of the adiabatic potential energy
surfaces of molecules not only in their groundstates but also in their excited, ionized, and electron-attached states are key fundamental information for investigating molecular geometries,
vibrations, chemical reactions, energy relaxation
processes, and dynamics. Furthermore, the derivatives of energy in these states with respect to external electric and magnetic ®elds provide electricand magnetic properties of molecules in thesedierent electronic states [1]. A purpose of thisLetter is to give a reliable method of calculating*

Corresponding author. Fax: +81-75-753-5910.
E-mail address: hiroshi@subchem.kyoto-u.ac.jp (H. Nakatsuji).

the energy derivatives of molecules based on thesymmetry adapted cluster (SAC)/SAC-CI generalR method for the excited, ionized, and electronattached states.
The SAC/SAC-CI method [2±4] was originallypublished in 1978 for studying ground, excited,
ionized and electron-attached (anion) states ofmolecules and has been successfully applied tovarious chemistries and physics involving manydierent kinds of electronic states [5,6]. The analytical gradients of the SAC/SAC-CI energy were
formulated and implemented in this laboratory at
the single double (SD)-R level [16,17], and applied
to the calculations of equilibrium geometries, dipole moments and vibrational frequencies ofmolecules in various electronic states of molecules. However, as the SAC-CI SD-R method isquantitative only for the electronic states generated by the one-electron process from the ground0009-2614/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V.
PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 9 9 5 - 2

494

M. Ishida et al. / Chemical Physics Letters 347 (2001) 493±498

state, we could not apply this method to themulti-electron processes that sometimes appear in
the excited and ionized states and in the course ofchemical reaction dynamics. Furthermore, when
we study excited states of open-shell systems by
the SAC-CI method, they are often described by
the two-electron processes from the closed-shell
SAC state. Therefore, it was anticipated to develop the analytical energy gradient method thatcan be applicable to such electronic states ofmolecules.
There are two standards in the choice of thelinked operators (R) in the SAC-CI method. In
the SD-R method, only single- and double-excitation operators are adopted for the R-operators,
while in the general-R method [7±10], triple-,quadruple- and higher-excitation operators arefurther included. The SAC-CI SD-R method isreliable for ordinary single-electron excitation andionization processes, while the general-R method
is designed to describe multiple-electron processes
with high accuracy and has been applied successfully, for example, in the calculations of ionization spectra involving shake-up satellite peaks[11±15].
In the present study, the analytical energy gradients of the SAC-CI general-R method were developed and implemented. The targets of themethods are the excited, ionized and electron-attached states of molecules including the multielectron processes. The performance of the method
was examined for the excited states of BH and CH
molecules. More details will be given in a forthcoming paper [18].
2. Theory
The analytical gradients of the SAC-CI correlation energy with respect to the external parameter, a, are most conveniently expressed in terms ofeective density matrices (EDMs) [16,17],
MO
MO
oDE X
ofij Xo ijjkl;cSAC-CI
CSAC-CI
ij
ijkl
oa
oa
oa
ij
ijkl

1

in an MO representation, where fij and ijjkl are
the Fock matrix elements and the two-electron

MO integrals, respectively. Subscripts i, j, k and lrefer to the spatial orbitals. Matrix elements cSAC-CI
ij
and CSAC-CI
are given by
ijkl
cSAC-CI
ij


X
I



(

X
K

XX
M



N

J

dML dMR cMN
ij

XXX
M

X

ZKSAC-CIN

I!
CJ SKJ CI

XX
K

I

)
ZISAC-CI

c0I
ij

ZKSAC-CI CI cKI
ij

dML dMR CI cM;NIij

1 X X X SAC-CIZ
CI CJ cK;IJ
ij
2 K I J K

2and
CSAC-CI
ijkl


X
I



(

X
K

XX
M



N

ZKSAC-CIdML dMR CMN
ijkl

XXX
M

N

I

I

J

J

!

)

CJ SKJ CIXXKI

ZISAC-CI

C0I
ijkl

ZKSAC-CI CI CKI
ijkl

dML dMR CI CM;NI
ijkl

1XXX
2 K

X

ZKSAC-CI CI CJ CK;IJ
ijkl :

3

Here, we adopt the convention that the subscripts
I, J , K, and L refer to the SAC excitation operators
S y , while M and N the SAC-CI excitation operaIJ
tors Ry . cIJ
ij and Cijkl are one-electron (1e) andtwo-electron (2e) coupling constants, respectively,
between linked con®gurations SIy j0i and SJy j0i, cijI;JK
and CI;JK
ijkl are those between linked con®gurationy
SI j0i and the unlinked con®guration SJy SKy j0i, cMN
ij
y
y
M;NI
and CMN
ijkl are those between RM j0i and RN j0i, cijyy yand CM;NIijkl are those between RM j0i and RN SI j0i. CI
LR
is the SAC coecient, and dM and dM are the SACCI left- and right-vectors, respectively. Sij represents the overlap matrix and is independent of theexternal parameters. ZISAC-CI in Eqs. (2) and (3) are
the components of the SAC-CI Z-vectors, and canbe obtained by the SAC-CI Z-vector equation

M. Ishida et al. / Chemical Physics Letters 347 (2001) 493±498

X

(

X

HKI

K

!
CJ SKJ H0I

JDESAC SKI 


XX
M

N

X
J

)

4

CJ HK;IJ ZKSAC-CI

dML dMR HM;NI ;where HIJ , HI;JK and HM;NI are the Hamiltonian
matrices, which are de®ned as h0jSI HSJy j0i,
h0jSI HSJy SKy j0i and h0jRM HRyN SIy j0i. In the SAC-CI
analytical energy gradient method, explicit calculation of the ®rst derivatives of the SAC coecient
oCI =oa is circumvented, introducing the interchange technique [19] (or the so-called Z-vector
method [20]).
For implementing the analytical energy gradients of the SAC-CI general-R method, we haveformulated cSAC-CI
and CSAC-CI
in Eqs. (2) and (3),ijijkl
and the SAC-CI Z-vector equation in Eq. (4) according to the SAC-CI general-R method. The
SAC-CI wave function is written as
WSAC-CI  RWSAC ;

5

where
X
R
dM R 
M;

6

M

and fRM g represents a set of excitation, ionizationand electron-attachment operators. In the SAC-CI
general-R method, fR
M g includes not only singleand double-excitation operators, but also triple-,quadruple-, and higher-excitation operators.
Therefore, the 1e- and 2e-EDMs in the second andfourth terms of Eqs. (2) and (3), and the SAC-CI
Z-vector equation in the right-hand side of Eq. (4),should be formulated to include the contributions
from the higher-excitation operators. Namely, thesummation on M and N in these terms is over the
SAC-CI operators including triple (T)-, quadruple(Q)-, and higher-excitation operators as
!
Q
S
D
T
XX
X
X
X
X



 

M

N

M



M
S
X
N



M

M
D
X
N



T
X
N



Q
X
N

!
  :

7495

Other terms are the same as in the SD-R method.
In the present study, the unlinked terms that are
redundant with the the linked terms are neglected.
The SAC-CI general-R analytical energy gradient code has been implemented into the SAC-CI96
program system [22] and has been incorporated
into the GA U S S I A N 98 suite of programs [23]. The
numerical check was done by con®rming that theanalytical energy gradients gave identical results tothe numerical dierentiations of the energies
within computational accuracy.
3. Applications
The ®rst application of the SAC-CI general-R
analytical energy gradient method is the calculation of the spectroscopic constants of the singlet
and triplet excited states of BH molecule and thoseof the doublet ground and excited states of CH
molecule. BH and CH are good benchmark molecules, since the experimental data of the equilibrium internuclear distance re , the harmonic
vibrational frequency xe , and the adiabatic excitation energy Te have been reported for various
electronic states [21], in which several excited states
are described by the two-electron processes: in theprevious paper of the SAC-CI SD-R energy gradient method [17], we studied only the one-electron
excited states of these molecules.
The basis set we adopted is the double-zeta plus
polarization (DZP) basis set of Huzinaga±Dunning [24,25], (9s5p1d/4s1p)/[4s2p1d/2s1p]. All theMOs were included in the active space and nocon®guration selection was performed. We used
the approximately variational SAC-CI (SAC-CIV) method.
3.1. BH
First, we discuss the spectroscopic properties of
BH: the ground state was calculated by the SAC
method and the excited states by both the SAC-CI
SD-R and general-R methods. In SAC-CI generalR, the R-operators were limited up to triples. Table
1 shows the results of re , xe and Te , compared withexperiments [21]. The excitation level denotes the496

M. Ishida et al. / Chemical Physics Letters 347 (2001) 493±498

Table 1
Excitation level, equilibrium internuclear distance (re ), adiabatic excitation energy (Te ), and harmonic vibrational frequency xe , forthe singlet and triplet states of BH
State

MethodExcitation
level

re

(A)

Te
(eV)

xe
(cm 1 )

X1 R
g

SAC
Exptl.b

0

1.243
1.232

±
±

2362
2356

a3 P

SD-R
General-R
Exptl.b

1
1

1.194
1.195
1.201

1.190
1.163
±

2664
2653
±

0:94 3r 1 1p  0:32 3r 1 2p
0:93 3r 1 1p  0:32 3r 1 2p

A1 P

SD-R
General-R
Exptl.b

1
1

1.236
1.241
1.219

3.074
3.018
2.868

2227
2171
2251

0:94 3r 1 1p
0:93 3r 1 1p

b3 R

SD-R

2

1.206

5.470

2629

General-R

2

1.228

4.616

2405

1.227

±

±

0:86 3r 2 1p2   0:30 3r 2 1p2p
0:30 3r 2 1p2p
0:84 3r 2 1p2   0:30 3r 2 1p2p
0:30 3r 2 1p2p

Exptl.bab

Main con®gurationa

01

C D

SD-R
General-R
Exptl.b

2
2

1.190
1.198
1.196

6.815
6.092
5.701

2776
2673
2610

0:90 3r 2 1p2 
0:89 3r 2 1p2 

C1 R

SD-R
General-R
Exptl.b

2
2

1.191
1.200
1.213

7.550
7.014
6.854

2762
2657
2475

0:63 3r 2 1p2   0:63 3r 2 1p2 
0:63 3r 2 1p2   0:63 3r 2 1p2 

The Hartree±Fock electronic con®guration for the ground state is 1r2 2r2 3r2 .
Ref. [21].

number of electrons involved in the excitation
process and the excitation character is shown by
the SAC-CI coecients of the main con®gurations.
The low-lying A1 P and a3 P states of BH are
described essentially by the one-electron excitations from the ground state, while the C0 1 D, C1 R
and b3 R states are described by the two-electron
excitations from the ground state. For the oneelectron processes, the SAC-CI SD-R and generalR methods give very similar results in good
agreement with experiments: the eects of the tri 0.03 eV, andples are negligible and within 0.005 A,
50 cm 1 for re , Te , and xe , respectively. For theother three states generated by the two-electron
processes, the general-R method gives considerably
better results than the SD-R method, especially for
Te . The eect of including triples amounts up to ca.
0.8 eV for Te of b3 R and C0 1 D states. The calculated re agrees excellently with experiments forthese states; the deviations are within 0.002 A.

Thus, for two-electron processes, we should use
the SAC-CI general-R method for obtaining
quantitatively reliable results. The b3 R state has
not yet been well characterized by experiment andtherefore, the present results of Te and xe should
provide good estimates.
3.2. CH
Next, we examine the performance of the SACCI general-R analytical energy gradient method forthe open-shell molecule CH, for which some of theexcited states are described by the multi-electron
processes relative to the closed-shell SAC (cation)state: we ®rst calculate closed-shell CH by the
SAC method, and then calculate various electronic
states of CH radical by the SAC-CI electron-attachment method. Table 2 summarizes re , xe andTe calculated by the SAC-CI SD-R and general-R
methods in comparison with the experiment [21].
In order to examine the eect of the R-operators inM. Ishida et al. / Chemical Physics Letters 347 (2001) 493±498

497

Table 2
Excitation level, equilibrium internuclear distance (re ), adiabatic excitation energy (Te ), and harmonic vibrational frequency xe , forthe doublet ground and excited states of CH
State

MethodExcitation
level

re

(A)

Te
(eV)

xe
(cm 1 )

Main con®gurationa

X2 P

SD-R
General-R(SDT)
General-R(SDTQ)
Exptl.b

1
1

1.127
1.127

±
±

2906
2889

0.97 1p
0.95 1p

1

1.128

±

2889

0.95 1p

1.120

±

2859

A2 D

2

BR

2

CR



SD-R
General-R(SDT)
General-R(SDTQ)
Exptl.b

2
2

1.074
1.103

4.873
3.075

3488
3045

0:99 3r 1 1p2   0:99 3r 1 1p2 
0:97 3r 1 1p2   0:97 3r 1 1p2 

2

1.105

3.062

3006

0:97 3r 1 1p2   0:97 3r 1 1p2 

1.102

2.875

2931

SD-R
General-R(SDT)
General-R(SDTQ)
Exptl.b

2
2

1.103
1.175

6.083
3.481

3142
2263

2

1.192

3.383

2045

1.198

3.229

1795

SD-R
General-R(SDT)
General-R(SDTQ)
Exptl.b

2
2

1.080
1.111

5.637
4.178

3381
2960

0:70 3r 1 1p2   0:70 3r 1 1p2 
0:69 3r 1 1p2   0:69 3r 1 1p2 

2

1.110

4.168

2966

0:68 3r 1 1p2   0:68 3r 1 1p2 

1.114

3.943

2840

0:57 3r 1 1p2  0:57 3r 1 1p2 
0:55 3r 1 1p2  0:55 3r 1 1p2 
0:20 2r 1 3r 1 1p2 4r
0:54 3r 1 1p2  0:54 3r 1 1p2 
0:22 2r 1 3r 1 1p2 4r

The electronic con®guration and the excitation level are given relative to the closed-shell CH , namely, 1r2 2r2 3r2 .
bRef. [21].
adierent orders, systematic calculations were performed using SDT and SDTQ R-operators.
The ground state, X2 P, is described by the oneelectron process, while the other three doublet
excited states, A2 D, B2 R , and C2 R , are represented by the two-electron processes. For the
ground state, both SAC-CI SD-R and general-R
methods give excellent results in comparison withexperiment. The deviations from the experimental
 and 50 cm 1 for revalues are to within 0.008 A
and xe , respectively. For the excited states, thegeneral-R method drastically improves the resultsof the SD-R method, especially for Te and xe . Theeects of including triples in the R-operators are aslarge as 1.5±2.4 eV and 420±880 cm 1 for Te andxe , respectively. This eect is prominent for theB2 R state. The calculation including up to qua re ,
druples further improves the results by 0.02 A

0.1 eV Te , and 220 cm 1 xe . This is because thethree-electron processes considerably mix to thisstate. The agreement of the general-R results withthe experimental values are satisfactory regardlessof the excitation levels of the states.
In these calculations, the computational cost of
the SAC-CI general-R analytical energy gradient isapproximately twice as much as that for its single
energy calculation. Therefore, this method has theadvantage over the numerical calculations of thederivatives and, in particular, is of great importance for large systems.
4. Summary
In this communication, we reported some results obtained by developing and implementing the498

M. Ishida et al. / Chemical Physics Letters 347 (2001) 493±498

SAC-CI general-R analytical energy gradientmethod for the excited, ionized and electron-attached states of molecules. The reliability andusefulness of the present method are clear from theapplications to the excited states of BH and CH
molecules. The advantages of the SAC-CI generalR analytical energy gradients are summarized asfollows:· The method is quantitatively applicable to theexcited, ionized, and electron-attached states including the multi-electron process.
· In particular, it is useful to study the excitedstates of open-shell systems, since they are often
described by the two-electron process from theclosed-shell SAC state.
· Detailed description can be obtained for thechemical reaction dynamics and energy relaxation processes, in which multi-electron processes sometimes play an important role.
Thus, we think that the present method should be
a powerful tool for investigating the dynamics andproperties in the excited, ionized and electron-attached states involving multi-electron processes.
Acknowledgements
This study has partially been supported by aspecial fund from the Ministry of Education, Science, Culture, and Sports.
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