THE JOURNAL OF CHEMICAL PHYSICS 122, 194108 共2005兲

Iterative CI general singles and doubles „ICIGSD… method for calculating
the exact wave functions of the ground and excited statesof molecules
Hiroshi Nakatsujia兲
Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering, Kyoto
University, Katsura, Nishikyou-ku, Kyoto 615-8510, Japan and Fukui Institute for Fundamental Chemistry,
Kyoto University, 34-4 Takano-Nishihiraki-cho, Sakyo-ku, Kyoto 606-8103, Japan

Masahiro Ehara
Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering,
Kyoto University, Katsura, Nishikyou-ku, Kyoto 615-8510, Japan

共Received 8 February 2005; accepted 7 March 2005; published online 19 May 2005兲
The iterative configuration-interaction general singles and doubles 共ICIGSD兲 method was applied tovarious closed- and open-shell electronic states of molecules within finite basis sets and was shownto give the exact results that are identical to the full CI ones. The structure of the ICIGSD is uniqueamong the ICI formalisms, that is, the singularity problem intrinsic to atomic and molecular
Hamiltonians can be avoided. The convergence of the ICIGSD method was fairly good regardlessof the characters of the electronic states and the qualities of the basis sets; only several iterations
were enough for obtaining microhartree accuracy. These favorable properties are attributed to theunique GSD structure. The present method was shown to be applicable to various spin states and toquasidegenerate states appearing in bond dissociation process. We have also applied the ICIGSD-CI
method to calculate the excited states simultaneously. We have confirmed that the ICIGSD-CI
method is accurate for calculating the excited states the symmetries of which are not only similar to
but also different from that of the ground state. © 2005 American Institute of Physics.
关DOI: 10.1063/1.1898207兴
I. INTRODUCTION

Establishing a method for solving the Schrödinger equation 共SE兲 is a central theme in theoretical chemistry.1 For thispurpose, we have studied the structure of the exact wave
function2–8 and proposed the iterative configurationinteraction 共ICI兲 method2,3 and the simplest extreme
coupled-cluster 共SECC兲 method4 as the methods to generate
the functions having the structure of the exact wave function.
When the variables included in these functions are optimizedby the variational principle, it gives the exact energy and theexact wave function that are the solution of the SE. For
finite-basis approximation, the same method gives the results
identical to the full CI 共FCI兲. Such exactness has been confirmed numerically in both analytical and basis-set expansion
approaches for harmonic oscillator3–5 and atomic and molecular systems.6–8
Historically, Horn and Weinstein, Kosloff and Tal-Ezer,
Cioslowski, and others considered to solve the timedependent SE on the imaginary time axis.9 Huang et al.10
proposed the surplus function method for the variational
Monte Carlo calculations. Nooijen11 considered to solve thecoupled cluster with general singles and doubles variables
共CCGSD兲 with the density equation12 that is equivalent with
the SE. The present author considered independently a possibility of the exactness of the CCGSD and reached a negaFAX: ⫹81-75-383-2739; Electronic mail: hiroshi@sbchem.kyoto-u.ac.jpa兲

0021-9606/2005/122共19兲/194108/7/$22.50

tive conclusion.2 He further considered this subject andshowed that the CCGSD is not guaranteed to be exact, again,
but its possibility was not excluded because of its highly
nonlinear nature.4 Nevertheless, this possibility has received
a lot of interest by many authors.13–18
The true solution of the SE is accessible only with ananalytical method.8 In a finite-basis approximation, FCI isthe best possible solution but it is far from the true solutionof the SE because of the basis-set incompleteness. Nevertheless, we call even the FCI solution as exact, since it is traditionally done so. In the analytical approach, a problem inatomic and molecular calculations was the singularity problem caused by the Coulombic potential involved in the
Hamiltonian.6,8 To circumvent this problem, we proposed theinverse SE6 and the scaled SE.8 In particular, the latter approach was quite successful in developing a general methodof solving the SE in an analytical form.8
In the finite-basis approximation, the singularity problem
becomes a bit vague because when you have a Hamiltonian
matrix H, defined within the basis-set space, you can calculate Hn essentially to an arbitrary order, though this is impossible if the basis is complete because then 共Hn兲ij= 具i兩Hn兩j典 and the integral in the right-hand side can diverge
when n 艌 3.8 When the basis is far from complete, the calculation may proceed without much difficulty,7 but it is due tothe incompleteness of the basis set. We have already shownthat by introducing the inverse Hamiltonian6 and the scaled
Hamiltonian,8 the singularity problem can be eliminated.

122, 194108-1© 2005 American Institute of Physics

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194108-2

J. Chem. Phys. 122, 194108 共2005兲

H. Nakatsuji and M. Ehara

The purpose of this paper is to show a special merit of
the ICIGSD method for calculating the exact wave function.
The merit is twofold: one is the dissolution of the singularity
problem and the other is the rapid convergence. In the
ICIGSD method, we start from the second-quantized Hamiltonian and introduce the GSD number of variables in the ICI
formalism. This makes it possible to formulate the ICIGSD
method to be free from the singularity problem. We haveshown in the previous paper7 that when we perform the simplest ICI 共SICI兲 and ICI3, where the numbers of the variables
are one and three, respectively, the convergence to the exact
wave function is slow; 30–40 iterations for minimal basis
and 40–70 iterations for double-zeta basis for getting theenergy correct to eight decimal figures. In the ICIGSD presented here, the convergence is much faster than in the SICI
and ICI3 given before.
Another purpose of this paper is to show the applicationof the ICIGSD method to molecular excited states, which
was not done in the former calculations with the SICI and
ICI3 methods.7 We have proposed in Paper II3 three methodsof calculating the excited states based on the ICI formalism
and these methods were applied to the harmonic oscillator.5
In this paper, we apply the method B proposed previously3 tomolecular systems and calculate the excited states having thesymmetries not only similar to but also different from that of
the ground states.

II. METHODFirst we briefly review the ICIGSD method.2,3 Originally, the ICI method was introduced in the GSD formalism
given below.2 The Hamiltonian of the system is written in thesecond-quantized form as† †vrpar†a p + 兺 wrs
兺
pqar as aqa p ,
pr
pqrs

共1兲

where ar† and a p are the creation and annihilation operators,
respectively, and p , q , r, s run over all occupied and unoccupied orbitals. The integrals vrp and wrs
pq represent one- andtwo-electron integrals. The energy associated to the wave
function ⌿ is defined by
具⌿兩H − E兩⌿典 = 0.

共2兲2

It was shown previously that the wave function ⌿ is exact
when the following equations are satisfied for all the indicesp , q , r, s;
具⌿兩共H − E兲ar†a p兩⌿典 = 0,

共3兲具⌿兩共H − E兲ar†as†aqa p兩⌿典 = 0.

共4兲

Based on this theorem, the ICIGSD method was introduced in Paper I2 for calculating the exact wave function. It
is defined in a recursion form as⌿n+1 = 共1 + Tn兲⌿n ,where Tn is the GSD operator given by,

共5兲

† †
n r †
C par a p + 兺 nCrs
兺pqar as aqa p ,
pr
pqrs

共6兲

where nCrp and nCrs
pq are the variables associated to thesingles and doubles operators. In Tn, only the totally symmetric one- and two-electron operators were included. The
variables nCrp and nCrs
pq in Tn are determined by solving thesecular equation,
具⌿n+1兩共H − E兲兩⌿n典 = 0,

共7a兲

具⌿n+1兩共H − E兲ar†a p兩⌿n典 = 0,

共7b兲

具⌿n+1兩共H − E兲ar†as†aqa p兩⌿n典 = 0,

共7c兲with the size of the general singles and doubles 共GSD兲.
When the convergence is achieved, the ICIGSD wave function satisfies Eqs. 共3兲 and 共4兲, and therefore it is exact. We
call the above formulation of the GSD method as integralfree algorithm, since no molecular integrals are included in
the Tn operator. Note that ICI satisfies size consistency because it is exact.2
A slightly different formulation of the ICIGSD is possible based on the general definition of the ICI theory.3 In thegeneral context of the ICIND theory,3 where ND stands forthe number of division of the Hamiltonian, we first divide the
Hamiltonian into the GSD parts and define the variable operator S by assigning a variable to each divided element.
Namely, referring to the second-quantized Hamiltonian given
in Eq. 共1兲, we define the variable operator S asSn =A. ICIGSD with integral-free and integralincluding algorithms

H=

Tn =

n r r †
rs † †
c pv par a p + 兺 ncrs
兺pqw pqar as aqa p
pr
pqrs

共8兲

and define the ICIGSD theory by⌿n+1 = 共1 + Sn兲⌿n .

共9兲

The secular equation for this ICIGSD is
具⌿n+1兩共H − E兲兩⌿n典 = 0,

共10a兲

具⌿n+1兩共H − E兲vrpar†a p兩⌿n典 = 0,

共10b兲

† †
具⌿n+1兩共H − E兲wrs
pqar as aqa p兩⌿n典 = 0.

共10c兲

The latter formulation differs from the former one simply inthe presence of the integrals vrp and wrs
pq in the one- andtwo-electron parts. These integrals can be taken off without
affecting the result, simply by redefining the variable operan rs rs
tors as nCrp = ncrpvrp and nCrs
pq = c pqw pq and by dividing Eqs.
rrs
共10b兲 and 共10c兲 by v p and w pq, respectively. The differenceof these two formulations is thus very small, but as will beseen later, the effect is very big numerically in the actual
calculations. We therefore call this latter formulation as
integral-including algorithm, in contrast to the integral-freealgorithm based on Eqs. 共5兲 and 共6兲. Since the integrals given
in Eqs. 共10b兲 and 共10c兲 include three-time products of theintegrals vrp and wrs
pq, respectively, the singularity problem
may appear8 and cause a difficulty in the integral-including
algorithm.
Finally, we note that the integral-free formulation is possible only for the GSD case based on the second-quantized

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194108-3

J. Chem. Phys. 122, 194108 共2005兲

Iterative CI

Hamiltonian given in Eq. 共1兲. In the general ICIND method
based on the ordinary Hamiltonian in the coordinate representation, only the integral-including formulation is possible.

TABLE I. ICIGSD and FCI for C2H4 with the minimal basis set.a
FCI

ICIGSD

Hartree–Fockenergy共a.u.兲

Dim.

Energy共a.u.兲

−77.826602

107952−77.991647

Iter.

Dim.

Energy共a.u.兲

1
2
3
4
5

125
465
532
532
532

−77.978734−77.991152−77.991627−77.991645−77.991647

B. ICIGSD-CI for excited states

Next, we explain the ICI-CI method, which is themethod for calculating the excited states simultaneously withthe ground state. This method was called method B in Paper
II3 and the calculated excited states are not guaranteed to be
exact, though they satisfy some necessary conditions of theexcited states. On the other hand, method A gives the “exact”
excited state, though only one solution is calculated at atime.
When we obtain the exact ground-state wave function⌿g by the ICIGSD method, we have at the same time the
GSD number 共NGSD兲 of good basis functions for the excitedstates, which we designate as 兵⌽K其,
兵⌽K其 = 兵ar†a p兩⌿g典,ar†as†aqa p兩⌿g典其共K = 1,…,NGSD兲,
共11兲

where K represents p , r and p , q , r , s. Eq. 共7兲 implies thatthese excited functions satisfy the Brillouin-orthogonality
with the exact ground state,
具⌿g兩共H − E兲兩⌽K典 = 0,

共12兲

and this relation means that the functions 兵⌽K其 constitute thebasis of the excited states.19 Then, we expand our excitedstate by a linear combination of these functions as
NGSD

⌿e =

兺 d K⌽ K ,

共13兲

K=1

which is written in a more explicit form as
⌿e =

冉兺
pr

drpar†a p +

† †
drs
兺
pqar as aqa p冊 ⌿g .
pqrs

共14兲

This wave function gives a good approximation of the excited states. Comparing Eq. 共14兲 with Eq. 共5兲, you will see,however, that these excited states are nothing but the higher
solutions of the converged ICIGSD solutions.3 Eq. 共14兲 has arealistic meaning when the symmetry of the excited states isdifferent from that of the ground state. For example, we cangenerate triplet excited states using Eq. 共14兲 from the singlet
ground state ⌿g. Similarly, we can generate ionized andelectron-attached states by replacing the GSD excitation operators 兵Ri其 = 兵ar†a p , ar†as†aqa p其 with the ionization operators
兵Ri其 = 兵ar , ar†aqa p其 and the electron-attachment operators 兵Ri其= 兵ar† , ar†as†a p其, respectively.
You find a close similarity of the above formulation withthat of the symmetry-adapted-cluster configurationinteraction 共SAC-CI兲 method for the excited states;19
ICIGSD corresponds to SAC20 and ICIGSD-CI to SAC-CI.19
Further, we can get more accurate wave functions by using
the ICIGSD-CI general-R method,3 in which triple andhigher operators are included in the R operators in addition
to Eq. 共14兲. This approach was called method C in Paper II3
and corresponds to the SAC-CI general-R method.21

a

ICIGSD is due to the integral-free algorithm.

C. Details of calculations

In the ICIGSD and ICIGSD-CI calculations, the basic
wave functions were expanded by the Slater determinants,
and in each iteration, we calculated the functions
兵ar†a p兩␺n典 , ar†as†aqa p兩␺n典其 and the Hamiltonian matrix elements between them. The Knowles–Handy algorithm22 was
useful. The present algorithm may be summarized as follows:
共1兲 An appropriate initial guess 兩␺0典 is chosen. It was
mostly Hartree–Fock.
共2兲 兵ar†a p兩␺n典 , ar†as†aqa p兩␺n典其 are calculated for the operatorswith 兵vrp , wrs
pq其 in the Hamiltonian nonzero.
共3兲 The Hamiltonian and overlap matrices for the basis
functions 兵兩␺n典 , ar†a p兩␺n典 , ar†as†aqa p兩␺n典其 are calculated.
共4兲 The ICIGSD secular equation given by Eq. 共7兲 or 共10兲is diagonalized.
共5兲 The convergence is checked.
After obtaining the ground-state ICIGSD wave function, the
ICIGSD-CI wave functions and the secular equation wereconstructed and solved.
Most calculations were done by the integral-free algorithm, except when remarked. We also carried out the calculations by the integral-including algorithm to see the difference from the calculations with the integral-free algorithm.
The coding was done rather straightforwardly and was not
aimed to be efficient.
III. RESULTS
A. Integral-free versus integral-including algorithms
and the convergence behavior

First, we performed the ICIGSD calculations for the singlet ground state using minimal STO-6G basis23 and doublezeta basis.24 The minimal basis calculations were done for
H2O, BH, N2, HCN, acetylene 共C2H2兲, ethylene
共C2H4兲 , CH3F, HCHO, CO2, and O3 and the double-zeta calculations were done for Be, LiH, BH, 1⌺ and 1⌸ states of
CH+, and HF. In the minimal basis calculations, the 1s orbitals of the second-row atoms were kept as frozen cores. Thesesystems were studied in the previous SICI and ICI3 works7
and so a direct comparison with the previous result is possible. Note, however, that the dimension here is the numberof the Slater determinants, and that of the previous paper7
was the number of the symmetry-adapted configurations.

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194108-4

J. Chem. Phys. 122, 194108 共2005兲

H. Nakatsuji and M. Ehara

TABLE II. ICIGSD with integral-including and integral-free algorithms.
Numbers in parenthesis indicate powers of 10. NA means that the diagonalization of ICIGSD failed.
Molecule iteration

Integral-including algorithm

Integral-free algorithm−1.29共−6兲– − 3.34共4兲
−223.638944−223.677392−223.679863−223.679968−223.679979
NANA−1.04共1兲– − 5.40共2兲−223.638944−223.677406−223.679870−223.679980−223.679983−223.679983−223.679984 共Conv.兲

1.54共1兲– − 5.51共4兲
−37.965931−37.969372−37.969396−37.969396−37.969397 共Conv.兲

−1.63共2兲–2.45共1兲
−37.965931−37.969374−37.969397 共Conv.兲¯¯

7.22共1兲– − 7.07共4兲
NA
NA
NA
NA
NA

−2.23共2兲–4.50共1兲
−100.154141−100.160063−100.160282−100.160287−100.160289 共Conv.兲

−3.27共−6兲– − 2.71共7兲
NANA−3.98共1兲– − 1.64共3兲−397.386385−397.387006 共Conv.兲

−8.06共−6兲– − 1.85共7兲
NA
NA
NA

−3.49共1兲– − 1.44共3兲
−341.357283−341.359218−341.359238 共Conv.兲

O3 共minimal basis兲
Hdiag
1
2
3
4
5
6
7
CH 共double zeta兲
Hdiag
1
2
3
4
5
HF 共double zeta兲
Hdiag
1
2
3
4
5
H2S 共minimal basis兲
Hdiag
1
2
PH3 共minimal basis兲
Hdiag
1
2
3

Table I shows the ICIGSD energies and the dimensions
in each iteration of C2H4 with the minimal basis set. This
calculation is due to the integral-free algorithm. The dimension and the energy at the first iteration are identical to thoseof the conventional SDCI since the deexcitation operators
applied to the Hartree–Fock function vanish identically in

Eq. 共5兲. The energy of the second iteration was slightly
worse by about 5 ⫻ 10−5 a.u. than that of the singles,
doubles, triples, and quadruples CI 共SDTQ-CI兲. After several
iterations, the ICIGSD dimension becomes a constant, i.e.,
the GSD dimension. In the ethylene case, the dimension became 532 after the third iteration. The ICIGSD energy rapidly converges to the FCI one; only five iterations were necessary for obtaining the microhartree 共10−6 a.u.兲 accuracy.
The convergence behaviors of the ICIGSD for other systems
were almost the same as those of this example, as far as we
used the integral-free algorithm.
Now, we compare the ICIGSD calculations due to theintegral-free algorithm and the integral-including algorithm.
We used the Householder bisection and the inverse iteration
method with double-precision accuracy in the diagonalization step of each iteration. In Table II we gave the iteration
processes of these two algorithms for O3, CH, HF, H2S, andPH3. The basis sets are minimal for O3 , H2S, and PH3 anddouble zeta for CH and HF. In the integral-including algo† †rithm, the basis functions 兵兩␺n典 , vrpar†a p兩␺n典 , wrs
pqar as aqa p兩␺n典其were used. As seen from Table II, the diagonal elements ofthe Hamiltonian matrices in the integral-including algorithm
widely range from the order of 10−6 a.u. to the order of
104 a.u. for the molecules including up to first-row atoms
and from the order of 10−6 a.u. to the order of 107 a.u. forthe molecules including up to second-row atoms. For thisreason, the ICIGSD diagonalization of the integral-including
algorithm failed for most of the molecules shown in Table II.
For CH, the convergence of the integral-including algorithm
was reached later than the integral-free algorithm. On theother hand, in the integral-free algorithm, the ranges of the
diagonal elements of the Hamiltonian matrices were muchmoderate; 10⬃ 102 a.u. for the molecules including first-row
atoms and 10⬃ 103 a.u. for the molecules including secondrow atoms. Therefore, no singularity problem occurs in thiscase, and the ICIGSD calculations rapidly converge to the
FCI.
Now that we have demonstrated how important it is to
use the ICIGSD method with the integral-free algorithm, we
will perform hereafter the calculations with only the integralfree algorithm.
Table III summarizes the ICIGSD results, together withTABLE III. ICIGSD and FCI for the ground states of ten molecules with the minimal basis sets.a
FCI

ICIGSD

Molecule

Activespaceocc. ⫻ unocc.

Hartree–Fockenergy共a.u.兲

Dim.

Energy共a.u.兲

Dim.

Iter.

Energy共a.u.兲

H 2O
BH
N2
HCN
C 2H 2C 2H 4
CH3F
HCHO
CO2
O3

4⫻2
3⫻3
5⫻3
5⫻4
5⫻5
6⫻6
7⫻4
6⫻4
8⫻4
9⫻3

−75.676507−25.001486−108.541824−92.573460−76.602406−77.826602−138.472331−113.440285−186.852493−223.415852

37
104
396
4076
8152
107952
54692
11148
30901
12126

−75.727911−25.059317−108.700217−92.741207−76.775867−77.991647−138.570669−113.584518−187.065936−223.679984

37
53
119
345
289
532
993
494
506
884

2
3
4
6
6
5
5
8
9
7

−75.727911−25.059317−108.700217−92.741207−76.775867−77.991647−138.570669−113.584518−187.065936−223.679984

a

ICIGSD is due to the integral-free algorithm.

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194108-5

J. Chem. Phys. 122, 194108 共2005兲

Iterative CI

idly converge to the FCI ones. The numbers of iterations forconvergence were quite small, about two to five iterations to
get a 10−6 a.u. accuracy, and even less than the minimalbasis case, though the GSD dimensions became larger here.
From these results, we may conclude that the convergence of
the ICIGSD calculations are quite fast, irrespective of thequality of the basis set and the size of systems, as far as wetake the integral-free algorithm. This result is very encouraging in comparison with the previous SICI and ICI3 calculations. The iteration numbers for convergence would be related to the numbers of the electrons of the system, since in
the Nth iteration 2N-electron excitations are generated in thepresent method.
B. Open-shell higher-spin states

FIG. 1. Convergence behaviors of ICIGSD calculations with the minimalbasis sets.

the FCI results, for the ten molecules calculated with theminimal basis sets. In the table, the ICIGSD dimension wasgiven for the final iteration, and the FCI dimension shows thenumber of the Slater determinants involved. The dimensionof the present ICIGSD is in the order of 共N / 2兲2 ⫻ 共M
− N / 2兲,2 where N and M are the numbers of electrons andorbitals, respectively, and the difference from the FCI dimension becomes remarkable for large systems. The numbersofiterationsforobtainingtheaccuracyof
10−6 a.u.共0.0006 kcal/ mol兲 were three to nine, which weremuch smaller than those of the SICI and ICI3 reported
previously.7
The convergence behaviors of the ICIGSD method wereshown in Fig. 1 for the minimal basis calculations. Since
each iteration of ICI is variational, the energy decreases
monotonically and converges to the FCI energy from above.
This trend was also observed in the previous SICI and ICI3
calculations.7 It should be noted that the fast convergence
was also obtained for O3 the ground state of which has aquasidegenerate biradical character. In the SICI and ICI3
cases, the convergence of O3 was slow in the initial stage of
iterations.7
Table IV shows the results for the molecules calculatedwith the double-zeta basis. Again the ICIGSD energies rap-

We have applied the ICIGSD method to various spin
multiplicities. Table V shows the results for the doublet toquintet states of N2 with the limited active space 关five occupied and five unoccupied molecular orbitals 共MOs兲兴 using
double-zeta basis set with s-type Rydberg function.25 The
ground state of each spin multiplicity was calculated. Since
the Tn operator given in Eq. 共5兲 is totally symmetric, we
simply used the initial-guess function of the target spin-spacesymmetry for obtaining the ICIGSD wave function of thedesired symmetry. We adopted single dominant spin-adapted
configuration as an initial guess for the doublet to quintet
excited states. Again, the ICIGSD method converged to the
FCI results after four to six iterations. The ICIGSD dimensions of different spin symmetries were almost the same,
since the ICIGSD operators are determined by the structureof the Hamiltonian.
Thus, the ICIGSD method for open-shell and higher-spin
multiplicities are as easy as for the closed-shell ground state.
C. Quasidegenerate states

The performance of the method to the bond dissociationor quasidegenerate states is an important issue. The present
method was applied to the ground state of CO along the bonddissociation process, and the result is given in Table VI. As
seen from the Hartree–Fock weight, the Hartree–Fock dominant character is lost quickly as the CO length increases. The
weight changes as 91.0, 52.1, and 0.8% for r = re , 1.5⫻ re,and 2.0⫻ re, respectively. The ICIGSD calculations were
performed using the same quality of basis sets and active

TABLE IV. ICIGSD and FCI for the double-zeta basis sets.a
FCI

ICIGSD

Molecule

Activespaceocc. ⫻ unocc.

Hartree-Fockenergy共a.u.兲

Dim.

Energy共a.u.兲

Dim.

Iter.

Energy共a.u.兲

Be
LiH
BH
CH+共 1⌺兲
CH+共 1⌸兲HF

2⫻22 ⫻ 10
3⫻9
3⫻9
3⫻9
5⫻7

−14.568534−7.981094−25.113743−37.885843−37.761931−100.021970

36
1428
12936
12936
12064
157984−14.582693−8.008682−25.187657−37.969397−37.853620−100.160289

20
417
1164
1146
1144
1176

2
2
3
3
3
5

−14.582693−8.008682−25.187657−37.969397−37.853620−100.160289

a

ICIGSD is due to the integral-free algorithm.

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194108-6

J. Chem. Phys. 122, 194108 共2005兲

H. Nakatsuji and M. Ehara

TABLE V. ICIGSD and FCI for the doublet ionized, doublet electron-attached, triplet, quartet, and quintet
states of N2 with double-zeta basis set.a
FCI

State

Hartree–Fockenergy共a.u.兲

Doublet 共ionized兲
2 +⌺g−107.992338
Doublet 共electron-attached兲2 −⌺u−108.472335
Triplet3⌸u−108.326615
Quartet4 +⌺u−107.804890
Quintet5⌸u−108.078228
a

ICIGSD

Dim.

Energy共a.u.兲

Dim.

Iter.

Energy共a.u.兲

6716−108.214908

289

6−108.214908

6716−108.641658

289

6−108.641658

5612−108.426040

289

6−108.426040

3120−107.949326

289

5−107.949326

1792−108.140790

288

4−108.140790

ICIGSD is due to the integral-free algorithm.

space as in the calculations of N2, an isoelectronic molecule.
Although the initial guess of the present ICIGSD calculations was the ground-state Hartree–Fock in all cases, six toeight iterations were enough for obtaining the convergence to
FCI. We understand this result because the ICIGSD method
achieves the quality of the SDTQ-CI at the second iteration.
D. ICI-CI calculations of the excited states

Finally, we examine the ICIGSD-CI method for calculating the excited states. Table VII shows the ICIGSD-CI
results for the 1⌺ and 1⌸ excited states of CH+ in comparison with the FCI results. The excitation level denotes thenumber of electrons involved in the excitation process. The
ICI-CI wave functions were calculated using the ICIGSD
ground-state wave function converged with the 10−6-hartree
accuracy. Method B of Paper II explained above wasadopted, and therefore the excited states were described here
within the general singles and doubles operators applied to
the ICIGSD wave function of the ground state. As explainedin the previous section, the 1⌺ excited states, whose symmetry is the same as that of the ground state, correspond to thehigher solutions of the converged GSD diagonalization,
while the 1⌸ excited states were calculated, following the
ICIGSD-CI formalism given in Eq. 共13兲 or 共14兲, because thesymmetry of these excited states is different from that of theground state. Note that only a single diagonalization is necessary in the ICI-CI method; no iteration step is involved asin the SAC-CI method.
As shown in Table VII, the ICIGSD-CI method reproduced the FCI energies in excellent accuracy regardless ofthe excitation levels; the deviations were less than 10−5 aufor one-electron processes, and the errors for two-electron
processes were within 10−4 au at most. It is remarkable that
the ICIGSD method with general singles and doubles operators described well the two-electron processes. Further improvement is, of course, possible by using the ICIGSD-CI
general-R method, in which triples and higher operators areadded to the GSD operators in the ICI-CI stage for describing the excited states. For other systems calculated in thepresent study, the accuracy and the performance of the
ICIGSD-CI method were almost the same as those of thisexample.
IV. SUMMARY

The ICIGSD method lies in a unique position among the
ICI formalisms based on the basis-set expansion algorithm.
In an ordinary ICI method, it is impossible to formulate theintegral-free algorithm; it is possible only in the GSD case is
it possible since the second-quantized Hamiltonian consistsof the GSD terms, and therefore we can eliminate the elements of the integrals of H3 that may lead to a divergence
due to the singularity of the Coulombic potential of the
Hamiltonian. This makes the ICIGSD method free from thesingularity problem that originates from the integrals of H3.
This was demonstrated by performing the integral-including
calculations as a contrast, where the calculations became impossible due to the existence of too large matrix elements
within the double-precision accuracy. Another merit of the
ICIGSD is that the number of the variables is GSD, i.e.,moderately large in contrast to the SICI or ICI3 performed

TABLE VI. ICIGSDa and FCI for the ground state of CO at r = re , 1.5⫻ re, and 2.0⫻ re.
FCI

ICIGSD

RCO

Hartree–Fockenergy共a.u.兲

HF weight
共%兲

Dim.

Energy共a.u.兲

Dim.

Iter.

Energy共a.u.兲

re1.5⫻ re2.0⫻ re−112.334141−111.997532−111.867560

91.0
52.1
0.8

16304
16304
16304

−112.482772−112.295416−112.150999

538
538
538

6
8
6

−112.482772−112.295416−112.150999

a

ICIGSD is due to the integral-free algorithm.

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194108-7

J. Chem. Phys. 122, 194108 共2005兲

Iterative CI

TABLE VII. ICIGSD-CI and FCI for the 1⌺ and 1⌸ excited states of CH+.
ICIGSD-CI

State

Excitation
level

1 ⌺
2 1⌺
3 1⌺
4 1⌺
5 1⌺
1 1⌸
2 1⌸
3 1⌸
4 1⌸

0
2
2
1
2
1
2
1
1

1

a

Character

FCI energy
共a.u.兲

Energy共a.u.兲

⌬a
共a.u.兲

ground␴2 → ␲2
␴2 → ␲2
␴ → ␴*
␴ , 2s → ␲2
␴→␲
␴2 → ␲ , ␴*
2s → ␲
␴→␲

−37.96940−37.70787−37.65990−37.43040−37.32862−37.85262−37.43836−37.31875−37.30099−37.96940−37.70778−37.65989−37.43040−37.32852−37.85262−37.43835−37.31875−37.30099

0.00000
0.00009
0.00001
0.00000
0.000100.00000
0.00001
0.00000
0.00000

Energy difference between FCI and ICIGSD-CI.

previously, where the numbers of the variables were only one
and three, respectively. For this reason, the convergence of
the ICIGSD method was remarkably good in contrast to the
SICI and ICI3 carried out previously. Thus, the two merits of
the ICIGSD originate from the unique GSD structure of theformalism.
The ICIGSD calculations of this paper were mostly donewith the integral-free algorithm. It was applied to the various
molecular electronic states with minimal and double-zeta basis. It was shown that the convergence of the ICIGSD
method is very rapid in contrast to the SICI and ICI3 calculations shown in the previous work,7 where the integral-freealgorithm was impossible. This rapid convergence property
was observed regardless of the character of the electronic
states and the quality of the basis sets. Though the test calculations were done only for small molecules, the numbersof electrons are expected to be closely related with the convergence rate from the structure of the ICIGSD wave function. The ICIGSD method is applicable to various spin multiplicities simply by adopting the initial guess function
belonging to the desired spin symmetry, since the excitation
operators in Tn are totally symmetric. The method is also
shown to be useful for the quasidegenerate states and bonddissociation processes.

The ICIGSD-CI method was also examined for calculating molecular excited states simultaneously with Method B
reported in Paper II.3 We examined both one- and twoelectron excited states and confirmed that the ICIGSD-CI
method reproduces the FCI results in good accuracy.
ACKNOWLEDGMENT

This study has been supported financially by the Grant
for Creative Scientific Research from the Ministry of Education, Science, Culture, and Sports of the Government of Japan.
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