28 November 1997

Chemical Physics Letters 280 Ž1997. 79–84

Analytical energy gradient of the ground, excited, ionized andelectron-attached states calculated by the SACrSAC-CI method
Takahito Nakajima a , Hiroshi Nakatsuji a,b,c,)a

Department of Synthetic Chemistry and Biological Chemistry, Faculty of Engineering, Kyoto UniÕersity, Sakyo-ku, Kyoto 606-01, Japanb
Department of Applied Chemistry, Graduate School of Engineering, the UniÕersity of Tokyo, Hongo, Tokyo 113, Japanc
The Institute for Fundamental Chemistry, 34-4 Takano Nishi-Hiraki-cho, Sakyo-ku, Kyoto 606, Japan
Received 30 July 1997; in final form 11 September 1997

Abstract
The method of calculating the analytical energy gradients for the ground, excited, ionized and electron-attached states
calculated by the SAC Žsymmetry adapted cluster.rSAC-CI Žconfiguration interaction. method is formulated and implemented. It is adapted with the configuration selection procedure used in the SACrSAC-CI method. The performance of themethod is investigated for the Li 2 and CO molecules. q 1997 Elsevier Science B.V.

1. Introduction
The derivatives of the adiabatic potential energies
with respect to the nuclear coordinates of a molecule
give fundamental information for investigating
molecular geometries, vibrations, chemical reactions,etc., and the derivative of the energy with respect tothe external electric or magnetic field provides themolecular properties such as the dipole, quadrupole,and magnetic moments w1x. Thus, the method of
calculating the energy derivatives provides a usefuland powerful tool in the field of quantum chemistry.
The SAC Žsymmetry adapted cluster.rSAC-CI
method w2–4x was originally published in 1978 forstudying the ground, excited, ionized and electron
attached Žanion. states of molecules and applied tovarious chemistries and physics involving many different electronic states w5,6x. It has also been applied
to multi-electron processes w7x and the high-spin)

Corresponding author. E-mail: hiroshi@sbchem.kyoto-u.ac.jp

states from quartet to septet spin multiplicities w8x. It
has been applied to the spectroscopies and chemical
reactions of various molecular systems including surface–molecule interacting systems w9x and biochemical systems w10x. Due to these accumulated studies,
the SACrSAC-CI method has now been well established as a powerful, computationally efficient, andaccurate method for calculating the ground and various excited states of molecules and molecular systems w5,6x.
It is thus desirable to develop a practical methodof calculating the gradients of the SACrSAC-CI
energies. This will provide a useful and efficientmethod for studying the dynamics and the propertiesof molecules in the ground and various excited andionized states.
Two approaches have been developed for calculating the energy gradient; one calculates the direct
analytical energy gradient w1x, and the other uses the
Hellmann–Feynman ŽH–F. theorem w11,12x. The useof the H–F theorem is efficient for a basis such as

0009-2614r97r$17.00 q 1997 Elsevier Science B.V. All rights reserved.
PII S 0 0 0 9 - 2 6 1 4 Ž 9 7 . 0 1 0 9 7 - X

T. Nakajima, H. Nakatsujir Chemical Physics Letters 280 (1997) 79–84

80

plane waves Že.g., the Car–Parrinello method w13x.
and provides physically and chemically pictorial force
concepts w14,15x in chemistry. Though the H–F theorem holds for the exact and stable wavefunctions, it
is not satisfied for most approximate wavefunctions
due to the use of the linear combination of atomic
orbital ŽLCAO. –molecular orbital ŽMO. approximation. Nakatsuji and co-workers w16,17x proved a theorem which shows that there is a unique and systematic way of improving the SCF and multi-configuration SCF ŽMC-SCF. wavefunctions to satisfy the
H–F theorem; a sufficient condition for the H–F
theorem to be satisfied is that the basis set includes
the derivative of any basis function included. However, the H–F theorem is not satisfied for the SAC-CI
method and therefore, we have to calculate the analytical energy gradients of the SACrSAC-CI energies.
In this Letter, we briefly summarize the
SACrSAC-CI energy gradient method for the
ground, excited, ionized and electron attached statesof a molecule. It assumes the usage of the configuration selection technique w18x. More details will begiven in a forthcoming paper w19x. To the best of ourknowledge, this is the first time that the energy
gradient method has been implemented for the
method using the configuration selection technique.
The EOM-CC energy gradient, which is equivalent
to the SAC-CI energy gradient, has been developedby Stanton and Gauss w20,21x, but the method wasstraightforward since the configuration selection
technique was not used in their formulation. We
apply the present SACrSAC-CI energy gradientmethod to the various electronic states of the Li 2 andCO molecules.

SAC-CI correlation energy, f i j the Fock matrix element and Ž ij < kl . the two-electron MO integral. g i j
and Gi jk l are the SAC or SAC-CI one- and two-electron Ž1e and 2e. effective density matrices ŽEDMs.,
respectively, in the MO representation, and are given
by

½ žÝÝ

g iSAC
'Ý
j

1q

I

K

ZKSAC C J SK J

J

/ 5C

I

y ZISAC g i0j I y Ý Ý ZKSAC CI g iKj I
K

y

1
2

ÝÝÝ
K

I

I

ZKSAC CI C J g iKj , I J

Ž 2.

J

and

½ žÝÝ

Gi SACjk l ' Ý

1q

I

K

ZKSAC C J SK J

J

/ 5C

I

y ZISAC Gi 0jkIl y Ý Ý ZKSAC CI Gi Kjk Il
K

y

1
2

I

Ý Ý Ý ZKSAC CI C J Gi Kjk,l I J
K

I

Ž 3.

J

for the SAC method and- CI
g iSAC
j

'Ý
I

½žÝ Ý
K

5

ZKSAC - CI C J SK J CI y ZISAC - CI g i0j I/Jq Ý Ý d ML d NRg iMj N y Ý Ý ZKSAC - CI CI g iKj I
MNKqÝ Ý Ý
M

y

1
2

N

I

d ML d NR CI g iMj , N I

IÝ Ý Ý ZKSAC - CI CI C J g iKj , I J
K

I

Ž 4.

J

and
- CI
Gi SACjk l

2. SAC r SAC-CI energy gradient method
The gradients of the SAC and SAC-CI non-variational ŽSAC-CI-NV. energies with respect to theexternal parameter a are written as
ED E
Ea

MO

s Ý gi j
ij

E fi jEaMOq Ý Gi jk lijkl

E Ž ij < kl .
Ea

Ž 1.

'Ý
I

K

5

ZKSAC - CI C J SK J CI y ZISAC - CI Gi 0jkIl/

J

N
SAC - CIq Ý Ý d ML d NRGi M
CI Gi Kjk Il
jk l y Ý Ý Z K

M

N

K

qÝ Ý Ý
Myin a MO representation, where D E is the SAC or

½žÝ Ý
1
2

N

I

, NI
d ML d NR CI Gi M
jk lIÝ Ý Ý ZKSAC - CI CI C J Gi Kjk,l I J
K

I

J

Ž 5.

T. Nakajima, H. Nakatsujir Chemical Physics Letters 280 (1997) 79–84

for the SAC-CI method. Here, g iIjJ and Gi IjkJ l are theone- and two-electron coupling constants between
the configurations F I and F J , respectively, CI the
SAC coefficients, and d ML and d NR are the left- andright-vector coefficients, respectively, in the SAC-CI
method. SI J represents the overlap matrix between
the configurations F I and F J , which is a constant
independent of the external parameters. ZI in Eqs.
Ž2. – Ž5. are obtained by the interchange techniquew22x Žor the so-called Z-vector method w23x., so thatthe explicit calculation of the first derivatives of the
SAC coefficient ECIrEa, is circumvented. Otherwise, it is a costly computational procedure involving 3 N sets of unknowns with N being the numberof atoms. Note that the formulae given by Eqs.
Ž2. – Ž5. are valid for the SAC SD-S and SAC-CI
SD-R methods not including the unlinked terms as
products of the single excitation operators. Inclusionof such terms is straightforward.
The MO representation of the first derivatives of
the SAC and SAC-CI energies is not necessarily
practical since it involves the derivatives of the 2e
MO integrals. However, the derivatives of the 2e
atomic orbital ŽAO. integrals are mostly zero since
only such derivative integrals which include the AO
whose center is involved in the derivative calculation
are non-zero, while the MO derivative integrals are
generally non-zero. Therefore, instead of transforming the derivative integrals, we back-transform theeffective density matrix from the MO to the AO
representation w24x. The detailed discussion is given
in Ref. w19x.
We have also implemented the first derivative of
the approximately variational SAC-CI ŽSAC-CI-V.
energy w3x. This implementation is straightforward
since the SAC-CI-V method is obtained simply bysymmetrizing the SAC-CI-NV matrices. The
SACrSAC-CI energy gradient code, in addition to
the CIS Žsingles., CISD Žsingles and doubles., MP2Žsecond-order Møller–Plesset perturbation. andSACD energy gradient, has been implemented into
the SAC85 and SAC-CI96 program system w25–27x
in the SD-R approximation. The SCF calculation is
performed by the GAUSSIAN94 w28x or HONDO8
w29x program systems.
A numerical check for the implemented program
was carried out by a comparison with the result of
the numerical differentiations.

81

3. GSUM method
A feature of the present SACrSAC-CI energy
gradient method is that it is adapted with the configuration selection scheme used in the SACrSAC-CI
calculations. This is advantageous for the study ofdynamics and properties of large molecules. However, for example in the geometry optimization procedure, an independent selection of operators forindependent geometries may lead to a discontinuity
in the potential surface. To avoid this discontinuity,
we adopt the GSUM method proposed previously
w30x: we use the group sum of the operators selected
at different points in the geometrical domain ofinterest. We applied it in the calculations of thepotential energy curves of the Li 2 and Liq
2 molecules
w30x.
The geometry optimization using the GSUM
method is summarized as follows. Before the optimization, we take the group sum of the linked andunlinked operators selected for all the representative
points in the nuclear configuration space which covers the reaction under consideration. The geometry
optimization is performed within this configuration
space. The usefulness of this method is confirmed in
the applications given below.

4. Applications
4.1. Li 2
As an example of applications of the present
SACrSAC-CI energy gradient method, we calculate
the spectroscopic constants of the ground, singlet andtriplet excited and ionized states of the Li 2 molecule.
Li 2 is a good benchmark molecule, since the experimental results, data for the equilibrium internuclear
distance re , the harmonic vibrational frequency veand the adiabatic transition energy Te are reported
for various states w31x, and in a previous paper wenumerically calculated these spectroscopic constants
w xof Li 2 and Liq
2 by the SACrSAC-CI method 30 .
We here calculate these properties by the
SACrSAC-CI energy gradient method; the singlet
ground state is evaluated by the SAC energy gradientmethod and the other excited states by the SAC-CI
energy gradient method. The doublet ionized state

T. Nakajima, H. Nakatsujir Chemical Physics Letters 280 (1997) 79–84

82

was obtained by the electron attachment to the corresponding di-cationic closed-shell Li 2 molecule. The
SAC-CI calculations are performed by both the nonvariational and approximate variational methods,
SAC-CI-NV and SAC-CI-V, respectively. The basis
set is 6-311 q q G ) ) w32,33x. No configuration selection was done for this molecule.
The calculated results are shown in Table 1 andcompared with the experiment w31x. The results calculated by the SACrSAC-CI energy gradient method
generally give good agreement with experiment. The
differences between the SACrSAC-CI and experi˚ for the equilibmental results are to within 0.03 A
rium distance, to within 10 cmy1 for the harmonic
vibrational frequency, and within 0.15 eV for the
adiabatic transition energy.
The SAC-CI-V results are close to the SAC-CINV results. The adiabatic transition energies calcu-

lated by the SAC-CI-V method are generally a little
smaller than those by the SAC-CI-NV method. The
differences in the equilibrium distance between
SAC-CI-NV and SAC-CI-V are to within a few
˚ for the states for which the vibrational fre10y3 A
quencies are larger than 200 cmy1 . Thus, the accuracies of the SAC-CI-NV and SAC-CI-V methods are
similar.
4.2. CO — with the GSUM method
In the next example, we show the performance of
the GSUM method used in the configuration selection technique. We calculate the equilibrium internuclear distance re and the adiabatic transition energy
Te of the ground, singlet and triplet states of the COmolecule. In a previous paper w34x, we numericallycalculated these spectroscopic constants of CO by
the SACrSAC-CI method.

Table 1Equilibrium internuclear distances, harmonic vibrational frequencies and adiabatic transition energies of Li 2 in the singlet, triplet andionized states
State

Equilibrium internuclear distance, re Harmonic vibrational frequency v e Žcmy1 . Adiabatic transition energy, Te
˚.
ŽA
ŽeV.

singlet state
X 1 SqSACg
exptl. a
A 1 Sq
SAC-CI-NV
u
SAC-CI-V
exptl. a1
B P u SAC-CI-NV
SAC-CI-V
exptl. a
2 1 Sq
SAC-CI-NV
g
SAC-CI-V
1 1 P g SAC-CI-NV
SAC-CI-V

2.698
2.673
3.093
3.094
3.108
2.968
2.968
2.935
3.366
3.343
3.693
3.718

347
351
258
254
255
269
272
270
173
165
149
145

0.0
0.0
1.844
1.843
1.744
2.725
2.725
2.534
2.718
2.718
2.725
2.725

triplet statea 3 Sq
SAC-CI-NV
u
SAC-CI-V
1 3 P u SAC-CI-NV
SAC-CI-V
b 3 Sq
SAC-CI-NV
g
SAC-CI-V

4.225
4.215
2.589
2.586
3.082
3.083

74
76
348
344
251
244

0.978
0.977
1.428
1.428
2.014
2.014

ionized state
1 2 Sq
SAC-CI-NV 3.134
g
SAC-CI-V
3.131

184
183

5.161 b5.161 babRef. w31x.
Ionization potential ŽIP. calculated as the difference energy from the Li 2 ground state. Experimental IP is 5.15 or 4.86 eV from Ref. w31x.

T. Nakajima, H. Nakatsujir Chemical Physics Letters 280 (1997) 79–84

83

Table 2Equilibrium internuclear distances, adiabatic transition energies and dipole moments of CO in the singlet and triplet states
State
X 1 Sq
a 3P
X3

a Sq
d 3D
A1 P
D1D

SAC
exptl. b
SAC-CI
exptl. b
SAC-CI
exptl. b
SAC-CI
exptl. b
SAC-CI
exptl. b
SAC-CI
exptl. b

Equiliblium internuclear distance, re˚.
ŽAAdiabatic transition energy, TeŽeV.

Dipole moment a , meŽdebye.

1.122
1.128
1.195
1.206
1.336
1.352
1.351
1.370
1.231
1.235
1.368
1.399

0.0
0.0
6.207
6.036
6.889
6.922
7.728
7.578
8.442
8.068
8.411
8.174

q0.079q0.122
y1.763y1.374
q1.099q1.0 6
q0.746–y0.625y0.335 c
q0.520–aPolarity CyOq.
Ref. w31x.
cRef. w35x.
b

The basis set is 6-311 q q G ) ) w32,33x. The
singlet ground state was calculated by the SAC
energy gradient method and the excited singlet andtriplet states by the SAC-CI energy gradient method.
The SAC-CI calculations are performed by the SACCI-NV method. The energy thresholds of 1 = 10y5
and 1 = 10y7 hartree are used in the configuration
selection scheme for the ground and excited states,
respectively. The geometry optimization is performed by using the GSUM method. The configuration space in the experimental geometry is preliminarily determined by using the configuration selection technique and the equilibrium distance is calculated by the geometry optimization method keeping
this configuration space. Though the experimental
geometries are used as the initial starting geometries
in the present calculations, other adequate geometries
such as the optimized ones at the HFrCIS level canbe used, as we have checked that the dependence on
the initial geometry is small if a proper geometry one
is used.
The calculated results are shown in Table 2 andcompared with experiment w31,35x. We also calculate
the dipole moments using the energy derivative
method. The results calculated by the SACrSAC-CI
energy gradient method generally give good agreement with experiment. This shows that the
SACrSAC-CI energy gradient method is effective

and also useful when the configuration selection
scheme is used.

5. Concluding remarks
In this Letter, we briefly summarize the
SACrSAC-CI analytical energy gradient method forthe ground, excited, ionized and electron attachedstates. The configuration selection technique can beapplied to our SACrSAC-CI energy gradient method.
The reliability and the usefulness of the present
method are confirmed, based on applications to the
Li 2 and CO molecules. We expect that the present
method will be used efficiently for studies of the
dynamics and properties of molecules in the ground,
excited, ionized and electron attached states.

Acknowledgements
Part of the calculations have been carried out withthe use of the computers at the Computer Center of
the Institute for Molecular Science. This study has
partially been supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Sci-

84

T. Nakajima, H. Nakatsujir Chemical Physics Letters 280 (1997) 79–84

ence and Culture and by New Energy and Industrial
Technology Development Organization ŽNEDO..

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