JOURNAL OF CHEMICAL PHYSICSVOLUME 117, NUMBER 1

1 JULY 2002

Structure of the exact wave function. V. Iterative configuration interaction
method for molecular systems within finite basis
Hiroshi Nakatsujia) and Masahiro Ehara
Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering,
Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan

共Received 28 December 2001; accepted 30 April 2002兲
The iterative configuration interaction 共ICI兲 method is applied to molecular systems within finite
basis using only few 共1–3兲 variables and shown to give the exact results that are identical to the full
CI 共FCI兲 ones. Since each iteration step of ICI is variational, the ICI converges monotonically to theexact solution from above. The diagonalization in ICI is so slight as the number of variables is sosmall, in contrast to the huge number of variables of FCI. We calculated the molecular ground statesof various spin-space symmetries using minimal basis and double zeta basis. The number of
iterations for convergence was small for minimal basis but moderate for double zeta basis,considering that only 1–3 variables are optimized in each iteration step. © 2002 American Instituteof Physics. 关DOI: 10.1063/1.1487830兴

I. INTRODUCTION

We may further divide the nuclear attraction operator into
those due to nucleus A, nucleus B, etc., and then, N D is
N N⫹2 where N N is the number of the nuclei involved in themolecule. When we use the Hamiltonian in the second quantized form, it may be divided into singles and doubles operators and then N D is the general singles and doubles 共GSD兲
number. If necessary, the singles part may be further divided
into kinetic part and nuclear attraction part. In this paper, we
use the division associated with Eq. 共1.1兲 and the nuclear
attraction part is not divided, so that N D is three for all themolecules, independent of the quality of the basis set. We
have also performed the simplest ICI 共SICI兲, where the
Hamiltonian is not divided, namely N D ⫽1. The number ofvariables in the present ICI is, therefore, only one or three, incontrast to the huge ones of the FCI.

Solving the Schrödinger equation 共SE兲 is a central themeof theoretical chemistry, since it constitutes a basic principleof chemistry.1 In this series of studies,2– 6 we have investigated the structure of the exact wave function, and proposed
the iterative configuration interaction 共ICI兲 method2,3 and theextended coupled cluster 共ECC兲 method4 as the methods tocalculate the exact wave function. This has been confirmed
numerically in the applications to harmonic oscillator4,5 andhydrogen atom,6 both involving only one-particle operator.
Horn and Weinstein, Kosloff and Tal-Ezer, Cioslowski, andothers considered to solve the time-dependent SE on theimaginary time axis.7 Huang et al.8 proposed the surplus
function method for the variational Monte Carlo calculations. Nooijen9 considered to solve the generalized coupledcluster wave function by the density equation10 that isequivalent with the SE. Van Voorhis and Head-Gordon11
showed numerically that the generalized coupled cluster certainly reproduces the full CI energy for Ne and N2 .
In a finite basis set space, the full CI 共FCI兲 method gives
the best possible approximation of the exact solution of theSE within the linear expansion treatment. The ICI and ECC
methods should therefore give the results identical to the
FCI, when applied using the same basis set. We show in thispaper that this is certainly the case for the ICI method.
The number of the variables of the ICI method, N D , isrelated to the way how we define the Hamiltonian operator:
N D is the dividing number of the Hamiltonian operator.
When the Hamiltonian is given byH⫽

1

1/r i j ,
兺i ⫺ 2 ⌬ i ⫺ 兺i 兺A Z A /r Ai ⫹ 兺i⬎ jII. ICI METHOD

We briefly explain the ICI method.2,3 We divide the
Hamiltonian into N D parts as
ND

H⫽

共2.1兲

and corresponding to this division, we define the variable
operator S byNDS⫽兺 C IH I .

共2.2兲

I⫽1

The ICI wave function is defined by the recurrence,␺ n ⫽ 共 1⫹S n 兲 ␺ n⫺1 ,

共1.1兲

共2.3兲

where n is the iteration number. The variable coefficients C I
in the ith iteration are written as C I,n and are calculated by
the variational principle using the secular equation,

we may divide it into the kinetic operator, nuclear attraction
operator, and electron repulsion operator. Then, N D is three.
Fax: ⫹81-75-753-5910. Electronic mail: hiroshi@sbchem.kyoto-u.ac.jpa兲

0021-9606/2002/117(1)/9/4/$19.00

兺 HI ,I⫽1

9

具 ␺ n 兩 H⫺E n 兩 ␺ n⫺1 典 ⫽0,

共2.4兲

具 ␺ n 兩 共 H⫺E n 兲 H I 兩 ␺ n⫺1 典 ⫽0, 共 I⫽1,...,N D 兲 .

共2.5兲

© 2002 American Institute of Physics

Downloaded 02 Sep 2002 to 130.54.33.130. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

10

J. Chem. Phys., Vol. 117, No. 1, 1 July 2002

H. Nakatsuji and M. Ehara

Note that the energy of the nth iteration, E n , satisfies

具 ␺ n 兩 H⫺E n 兩 ␺ n 典 ⫽0

共2.6兲

as well as Eq. 共2.4兲.
It is convenient to introduce a unity operator by H 0(H 0 ⫽1), and assign C 0 to this operator instead of unity in
Eq. 共2.3兲. Then, the ICI wave function is written as
ND␺ n ⫽ 兺 C I,n H I ␺ n⫺1 ,

共2.7兲

I⫽0

and the above secular equation is rewritten as
ND兺 具 ␺ n⫺1兩 H I共 H⫺E n 兲 H J兩 ␺ n⫺1 典 C J,n ⫽0,

J⫽0

共 I⫽0,...,N D兲 .

共2.8兲Note that this is a very small secular equation of the dimension of 2 and 4 in the present calculations. All that is necessary for the ICI calculation is the evaluation of the few integrals involved in Eq. 共2.8兲, namely
H IJ ⫽ 具 ␺ n⫺1 兩 H I HH J 兩 ␺ n⫺1 典

共2.9兲

S IJ ⫽ 具 ␺ n⫺1 兩 H I H J 兩 ␺ n⫺1 典 .

共2.10兲

andThe essence of the finite basis approximation adopted in
FCI is that these integrals are calculated within the space of
this finite basis set. We denote the full configuration space of
this finite space by 兵 兩 k 典 其 , which is complete within this finite
space, namely兺k 兩 k 典具 k 兩 ⫽1.

共2.11兲

Then, the integrals H IJ and S IJ within this finite space are
evaluated by using the so-called resolution of identity 共RI兲procedure as

具 ␺ n⫺1 兩 H I 兩 k 典具 k 兩 H 兩 l 典具 l 兩 H J 兩 ␺ n⫺1 典
兺k,l共2.12兲

兺k 具 ␺ n⫺1兩 H I兩 k 典具 k 兩 H J兩 ␺ n⫺1 典 .

共2.13兲

H IJ ⫽andS IJ ⫽

An important notice here is that if we calculate the integrals H IJ and S IJ by analytical procedure or some equivalentmethod, the solution of the ICI will be the true solution of
the SE. In general, Eq. 共2.11兲 does not hold, as long as thebasis 兵 兩 k 典 其 is not truly complete. The finite space approximation adopted above is a crude approximation and its accuracy is limited by the quality of the basis set initiallyadopted. A dilemma of FCI is that to improve the quality, we
have to use very large basis set, but it is impractical since thedimension of FCI soon reaches an astronomical figure.
The ICI calculation of FCI is very simple: it was performed by modifying the related parts of the GAMESS
program.12 All we have to do is an evaluation of the ten or
three integrals of H IJ and S IJ , depending on ICI3 (N D ⫽3)
or SICI (N D ⫽1). In the RI formula, the configuration 兩 k 典

FIG. 1. Convergence of SICI and ICI3 calculations with the minimal
STO-6G basis sets. Full and open symbols are SICI and ICI3 results, respectively.

runs over the full configurations. However, since the operator
H I is one or two electron operator, the integral 具 ␺ n⫺1 兩 H I 兩 k 典
is zero if 兩 k 典 is different from the elements of 兩 ␺ n⫺1 典 bymore than two orbital indices. Using such a criterion, we canmake the evaluation of the integrals H IJ and S IJ efficient.
Since each step of ICI is variational, we can calculate theground state of each spin-space symmetry by starting froman appropriate initial guess function 兩 ␺ 0 典 . The higher excitedstates of the same symmetry as the ground state can be calculated by Method A described in Paper II.3 For this purpose,
all the lower states of the desired excited states must exist
within the eigenvalues of the small matrix of the dimension
N D ⫹1 in the iteration process, even though the ground andother lower excited states are approximate except for theexcited state under consideration. In the present very small
N D case, this requirement was difficult to be satisfied evenfor the lowest excited state, so that we calculated only theground state of each spin-space symmetry. The calculationsof the excited states by Method B,3 which is ICI–CI in thenotation of Paper IV,5 were also not performed by the same
reason.
III. RESULTS

We performed ICI calculations using minimal STO-6G
basis13 and double-zeta basis.14 Minimal basis calculations
were done for CO, H2 O, HCN, acetylene (C2 H2 ), ethylene(C2 H4 ), CH3 F, HCHO, CO2 , O3 and HNCO. Double zeta
calculations were done for Be, LiH, CH⫹ , HF, 1 A 1 and 3 B 1states of CH2 , and H2 O in various spin-space symmetries
共1 A 1 , 1 A 2 , 1 B 1 , doublet ionized 2 A 1 , 2 B 1 , 2 B 2 , and triplet
3
A 1 , 3 A 2 , 3 B 1 states兲. In the minimal basis calculations, the1s orbitals of the second-row atoms were kept as frozen
cores.
Figure 1 shows the convergence behavior of the SICI

Downloaded 02 Sep 2002 to 130.54.33.130. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

J. Chem. Phys., Vol. 117, No. 1, 1 July 2002

Exact wave function

11

TABLE I. FCI, SICI, and ICI3 results for the minimal basis sets. The initial guess is Hartree–Fock.
FCI

Hartree–FockMolecule occ.⫻unocc. Energy 共au兲 Dimension

Energy共au兲

Active space

H2 O
HCN
C2 H2
C2 H4
CH3 F
HCHO
CO2
O3
HNCO

4⫻2
5⫻4
5⫻5
6⫻6
7⫻4
6⫻4
8⫻4
9⫻3
8⫻5

⫺75.676 51⫺92.573 46⫺76.602 41⫺77.826 60⫺138.472 33⫺113.440 29⫺186.852 49⫺223.415 85⫺167.105 76

SICI⫺75.727 91⫺92.741 21⫺76.775 87⫺77.991 65⫺138.570 67⫺113.584 52⫺187.065 94⫺223.679 98⫺167.323 97

37
1436
2640
29248
16542
3644
9189
4067
216249Iter.
10⫺5 au
Dim. accuracy
2

10

2
2
2
2
2
2
2
2

29
30
34
23
34
46
67
68

Energy共au兲

⫺75.727 91⫺92.741 21⫺76.775 87⫺77.991 65⫺138.570 67⫺113.584 52⫺187.065 94⫺223.679 98⫺167.323 97

and ICI3 with the minimal basis. The initial guess is
Hartree–Fock. The energy decreases monotonically and converges to the FCI energy from above as theoretically predicted since each iteration process of ICI is variational. The
ICI3 gives initially more rapid convergence than the SICI,
because the number of variables is larger in ICI3. However,since the difference is only two, the overall behaviors of ICI3
and SICI are similar. Generally speaking, the convergence of
ICI is very rapid in the initial stage; for example, at 5 iterations, 99.66% of the correlation energy is achieved for
CH3 F. The convergence of O3 is slower initially than theother molecules, since the ground state of O3 has quasidegenerate biradical character and so the Hartree–Fock initial guess is poor. For the ICI with the double-zeta basis,
similar convergence behavior was observed, though the convergence was slower than the minimal basis case.
Table I shows the SICI and ICI3 results, together with
the FCI results, for nine molecules calculated with the minimal basis. The ICI converges exactly to the FCI energy without any difficulty. The dimension of the present ICI is only 2
or 4 and is remarkably small in comparison with the FCI
dimension 共29 248 for ethylene and 216 249 for HNCO兲. We
show the number of iterations for obtaining the accuracy of
10⫺3 au 共0.6 kcal/mol兲, 10⫺4 au 共0.06 kcal/mol兲, and10⫺5 au 共0.006 kcal/mol兲. For ordinary calculations in chemistry, 10⫺3 au accuracy would be sufficient. Comparing theiteration times necessary for different accuracies, we see that
the ICI converges quite nicely at the initial stage and then, to

ICI3
Iter.
Iter.
Iter.
10⫺3 au 10⫺4 au
10⫺5 auaccuracy accuracy Dim. accuracy
2

5

4

7

8
9
11
4
9
16
29
18

18
19
21
10
21
30
48
39

4
4
4
4
4
4
4
4

27
28
33
20
33
44
68
69

Energy共au兲

⫺75.727 91⫺92.741 21⫺76.775 87⫺77.991 65⫺138.570 67⫺113.584 52⫺187.065 94⫺223.679 98⫺167.323 97

Iter.
Iter.
10⫺3 au 10⫺4 au
accuracy accuracy
2

4

8
9
10
4
9
15
29
19

17
18
21
9
21
29
48
40

improve only the order of 0.1–0.01 kcal/mol, many iterations
are done. For example, for HNCO, the molecule having largest FCI dimension, the iteration time for the 10⫺3 au accuracy is 19, that for 10⫺4 au accuracy is 40 and that for the
10⫺5 au accuracy is 69. Similar behavior is also observed forother molecules. For obtaining millihartree of accuracy,
10–20 iterations are enough, except for ozone.
Table II shows the results for the molecules calculatedwith the double-zeta basis. Again, we confirm that the SICI
and ICI3 energies converge to the FCI energy from above.
Here, we need larger number of iterations than the minimalbasis case, but the convergence was very smooth and monotonic, never oscillating, for all the cases. For obtaining achemical accuracy, i.e., 10⫺3 au accuracy, we need about 60
共average value兲 iterations. The iteration numbers show againthat the convergence is very slow in the final stage to improve only 0.01 kcal/mol. For example, for singlet methylene, it took 260 times of iterations to improve only about
10⫺5 au 共0.006 kcal/mol兲. A reason is clearly a very small
number of variables in the present calculations, which was
kept to 2 to 4 irrespective of the quality of the basis set andthe size of the molecules.
Table III shows the ICI3 results for water with the
double zeta basis. We calculated the ground states of manydifferent spin-space symmetries. Since every step of thepresent ICI is totally symmetric, we just used the initial
guess function of appropriate spin-space symmetry. We used

TABLE II. FCI, SICI, and ICI3 results for the double zeta basis sets. The initial guess is Hartree–Fock except for CH2 ( 3 B 1 ), for which it is a single dominant
spin-adapted configuration.
FCI

SICI

ICI3

Hartree–Fockor single
Active space

config.

Molecule

occ.⫻unocc.

Energy 共au兲

Be
LiH
CH⫹
HF
CH2 共1A 1兲
CH2 共3B 1兲

2⫻2
2⫻10
3⫻9
5⫻7
4⫻10
4⫻10

⫺38.874 92

Iter.
10⫺5 au
Dim. accuracy

Energy 共au兲

Iter.
Iter.
Iter.
10⫺3 au 10⫺4 au
10⫺5 auaccuracy accuracy Dim. accuracy

Energy 共au兲

Iter.
Iter.
10⫺3 au 10⫺4 au
accuracy accuracy

Dimension

Energy 共au兲

⫺14.568 53

20⫺14.582 69

2

24⫺14.582 69

4

13

4

24

⫺14.582 69

3⫺7.981 09⫺37.885 84⫺100.021 97⫺38.861 52

620

2

89

42

4

86

38

286

89

181

4

329

101

208

2

198

59

112

4

201

2

618

145

374

4

679

⫺8.008 68⫺37.969 40⫺100.160 29⫺38.962 44

92⫺8.008 68⫺37.969 40⫺100.160 29⫺38.962 44

11

71 881⫺8.008 68⫺37.969 40⫺100.160 29⫺38.962 44

111 600⫺38.998 142

251⫺38.998 14

107

178

4

288

⫺38.998 144582
44 388

12

63

119

162

419

120

203

Downloaded 02 Sep 2002 to 130.54.33.130. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

12

J. Chem. Phys., Vol. 117, No. 1, 1 July 2002

H. Nakatsuji and M. Ehara

TABLE III. FCI and ICI3 results for the singlet, triplet, and ionized states of H2 O with double zeta basis set.a
FCI

State
Singlet
A1
1
A2
1
B1
Triplet
3
A1
3
A2
3
B1
Ionized
2
A1
2
B1
2
B2
1

Hartree–Fockor single config.
Energy 共au兲

Dimension

Energy 共au兲⫺76.009 56⫺75.567 61⫺75.643 27

256 473245 000
245 776⫺75.613 83⫺75.584 53⫺75.672 20⫺75.456 87⫺75.507 36⫺75.292 61

ICI3

Energy 共au兲Iter.
10⫺3 auaccuracyIter.
10⫺4 auaccuracy

214
362
334

⫺76.157 87⫺75.761 02⫺75.838 26

67
130
117

125
241
221

4
4
4

825
421
330

⫺75.797 15⫺75.779 89⫺75.867 48

169
144
111

480
276
215

4
4
4

319
290
356

⫺75.655 68⫺75.720 23⫺75.460 08

92
85
78

197
175
175

Dim.

Iter.
10⫺5 auaccuracy⫺76.157 87⫺75.761 02⫺75.838 26444

440 475
437 640
437 520⫺75.797 15⫺75.779 89⫺75.867 48

232 968
223 282
232 068⫺75.655 68⫺75.720 23⫺75.460 08a

For inital guess, see the text.

Hartree–Fock for the 1 A 1 ground state, single dominant spinadapted configuration for the singlet and triplet excitedstates, and Koopmans state for the doublet ionized states.
Again, the ICI of only 2 to 4 dimensions has reproduced the
FCI results of the dimensions as large as 223 282– 440 475.
The convergence rate is different from state to state but the
average iteration number for obtaining 10⫺3 accuracy is 110.

ACKNOWLEDGMENT

This study has been supported financially by the special
fund from the Ministry of Education, Science, Culture, andSports of the Government of Japan.

P. A. M. Dirac, Proc. R. Soc. London, Ser. A 123, 714 共1929兲.
H. Nakatsuji, J. Chem. Phys. 113, 2949 共2000兲.
3
H. Nakatsuji and E. R. Davidson, J. Chem. Phys. 115, 2000 共2001兲.
4
H. Nakatsuji, J. Chem. Phys. 115, 2465 共2001兲.
5
H. Nakatsuji, J. Chem. Phys. 116, 1811 共2002兲.
6
H. Nakatsuji, Phys. Rev. A 65, 052122 共2002兲
7
D. Horn and M. Weinsten, Phys. Rev. D 30, 1256 共1984兲; R. Kosloff and
H. Tal-Ezer, Chem. Phys. Lett. 127, 223 共1986兲; T. J. Park and J. C. Light,
J. Chem. Phys. 85, 5870 共1986兲; J. Cioslowski, Phys. Rev. Lett. 58, 83
共1987兲.
8
H. Huang, Q. Xie, Z. Cao, Z. Li, Z. Yue, and L. Ming, J. Chem. Phys. 110,
3703 共1999兲; H. Huang, X. Zeng, and L. Ming, ibid. 112, 5257 共2000兲.
9M. Nooijen, Phys. Rev. Lett. 84, 2108 共2000兲.
10
H. Nakatsuji, Phys. Rev. A 14, 41 共1976兲.
11
T. Van Voorhis and M. Head-Gordon, J. Chem. Phys. 115, 5033 共2001兲.
12
M. W. Schmidt, K. K. Baldridge, J. A. Boatz et al., J. Comput. Chem. 14,1347 共1993兲.
13
W. J. Hehre, R. F. Stewart, and J. A. Pople, J. Chem. Phys. 51, 2657
共1969兲.
14
S. Huzinaga, J. Chem. Phys. 42, 1293 共1965兲; T. H. Dunning, Jr., ibid. 53,
2823 共1970兲.
1
2

IV. CONCLUDING REMARKSThis paper gives a numerical proof that the ICI method
certainly gives the exact wave function for molecular systems within finite basis. The exact nature of the ICI is thus
confirmed, together with the theoretical proof given in Papers I and II.
It is necessary to study the computational algorithm best
suited to the ICI method for practical utilities in molecular
calculations. As this method is different from the conventional FCI, the suitable algorithm may be different from the
FCI ones. Use of the general singles and doubles 共GSD兲
number of variables is also interesting, because it would result in better convergence and simultaneous calculations of
excited states with the ICI–CI method5 共Method B3兲. Suchstudy is in progress in our laboratory.

Downloaded 02 Sep 2002 to 130.54.33.130. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp