TitleFormulation and implementation of direct algorithm for thesymmetry-adapted cluster and symmetry-adapted cluster-configuration interaction method

Author(s) Fukuda, Ryoichi; Nakatsuji, Hiroshi

Citation JOURNAL OF CHEMICAL PHYSICS (2008), 128(9)

Issue Date 2008-03-07

URL http://hdl.handle.net/2433/84613

Right

Copyright 2008 American Institute of Physics. This article maybe downloaded for personal use only. Any other use requiresprior permission of the author and the American Institute ofPhysics.

Type Journal Article

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Kyoto University

Formulation and implementation of direct algorithm for the symmetry-adapted cluster and symmetry-adapted cluster–configurationinteraction method

Ryoichi Fukuda1,2,a� and Hiroshi Nakatsuji1,2,b�

1Quantum Chemistry Research Institute,c� Kyodai Katsura Venture Plaza, North Building 106,36-1 Goryo Oohara, Nishikyo-ku, Kyoto 615-8245, Japan2Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering,Kyoto University, Kyoto-Daigaku-Kaysura, Nishikyo-ku, Kyoto 615-8510, Japan

�Received 26 July 2007; accepted 14 December 2007; published online 4 March 2008�

We present a new computational algorithm, called direct algorithm, for the symmetry-adaptedcluster �SAC� and SAC–configuration interaction �SAC-CI� methodology for the ground, excited,ionized, and electron-attached states. The perturbation-selection technique and the molecular orbitalindex based direct sigma-vector algorithm were combined efficiently with the use of the sparsenature of the matrices involved. The formal computational cost was reduced to O�N2�M� for asystem with N-active orbitals and M-selected excitation operators. The new direct SAC-CI programhas been applied to several small molecules and free-base porphin and has been shown to be moreefficient than the conventional nondirect SAC-CI program for almost all cases. Particularly, theacceleration was significant for large dimensional computations. The direct SAC-CI algorithm hasachieved an improvement in both accuracy and efficiency. It would open a new possibility in theSAC/SAC-CI methodology for studying various kinds of ground, excited, and ionized states ofmolecules. © 2008 American Institute of Physics. �DOI: 10.1063/1.2832867�

I. INTRODUCTION

The symmetry-adapted cluster �SAC�1 and SAC–configuration interaction �SAC-CI�2 methodology proposedand first coded by one of the authors in 1978 is an electroncorrelation methodology for ground, excited, ionized, andelectron-attached states of molecules. It has been success-fully applied to diverse chemistry, physics, and biology in-volving various kinds of electronic and vibrational states �forrecent reviews, see Ref. 3�. The methodology is based on thecluster expansion formalism combined with the variationalprinciple.2 Later, theoretically identical methodologies, suchas coupled-cluster linear response theory4 and equation-of-motion coupled cluster,5 have been reported. These methodsestablished a highly accurate way to study excited electronicstructures of molecules.

The earlier version of the SAC/SAC-CI program6 waspublished in 1985. A great deal of effort has been made tothe theory and program development, and a lot of new ideashave been implemented. The SAC-CI general-R method,7

multireference version of the SAC/SAC-CI method,8 expo-nentially generated wave function idea,9 extension up to sep-tet spin multiplicity,10 and the analytical energy gradientmethod11 were the representative fruits of these efforts. Re-cently, the giant SAC/SAC-CI theory for giant molecularcrystals has been proposed.12 The SAC/SAC-CI method hasbeen implemented in the GAUSSIAN03 software programpackage

and has been widely used in universities and industries.13 Sofar, the SAC/SAC-CI code has been written with theexcitation-operator driven algorithm1,2 with the use of theperturbation-selection technique.14,15 The integral driven al-gorithm was introduced to the SAC/SAC-CI method byHirao,16 but his algorithm was not combined with theperturbation-selection technique.

Molecular orbital �MO� integral driven direct algorithmfor electron correlation methods was first proposed by Roosfor singles and doubles �SD� CI method.17 In such a proce-dure, the iteration vector �usually termed as sigma vector� isdirectly constructed from MO integrals, without an explicitconstruction of a Hamiltonian matrix. At the same time,when the SAC/SAC-CI SD code was established,1,2 thecoupled-cluster doubles �CCD� method only for ground statewas formulated by Pople et al. in integral driven form.18 Anefficient computation algorithm of CCSD was designed byScuseria et al.,19 who introduced intermediate arrays for ef-ficiency. These MO integral driven approaches include allSD or D excitation operators. We consider that this featureconflicts with the policy of our SAC/SAC-CI program ex-plained below.

A policy of our SAC/SAC-CI program is that we discardminor unimportant terms by introducing some selection pro-cedures with thresholds. Accordingly, depending on the de-sired accuracy, we introduce appropriate thresholds of selec-tions, and less important but time-consuming terms areneglected.20 The perturbation selection of the linked opera-tors is a particularly important technique.14,15 By virtue ofthis technique, we can use a single theory and a single pro-gram for various research subjects;3 from fine theoretical

a�Electronic mail: fukuda@qcri.or.jp.b�Electronic mail: h.nakatsuji@qcri.or.jp.c�URL: http://www.qcri.or.jp/.

THE JOURNAL OF CHEMICAL PHYSICS 128, 094105 �2008�

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spectroscopy of small molecules21–23 to photobiology of bigbiological molecules.24,25 This feature enables us to investi-gate various chemical phenomena from an equal viewpoint.The perturbation-selection technique is particularly impor-tant for excited states. The atomic orbital �AO� basis func-tions are usually optimized for the ground state. We need aflexible AO basis to describe various kinds of excited states.Diffuse functions and/or sizable Rydberg functions are oftenrequired. Therefore, the number of active MOs �N� forexcited-state calculations is much larger than that for groundstate. Order N6 algorithms �the optimal scale of CCSDmethod� will soon face the limit of computer. The limitationsof O�N6� algorithms would be much more severe for excitedstates than for ground states.

A problem in the algorithm of the conventional SAC-CIprogram lies in the calculations of unlinked terms. TheHamiltonian matrix elements between selected excitation op-erators are evaluated within the loops driven by the labels ofexcitation operators. Consequently, the unlinked integral partbecomes an O�M3� step, where M is the number of selectedexcitation operators. This step is so time consuming that theadditional approximation shall be introduced for practicalcalculations, and the approximation was to cut less importantexcitation operators than a given threshold off the loops. Thiscutoff approximation is, however, problematic because theunlinked terms are essentially important in the SAC/SAC-CItheory.

An efficient use of the core memory is also very impor-tant. We use a projective reduction formula26,27 to evaluatethe Hamiltonian matrix elements between selected excitationoperators. The MO integrals are randomly requested fromthe projective reduction routine in the loops of the excitationoperator labels. Therefore, we have to load all the MO elec-tron repulsion integrals �ERIs� on a core memory. If the corememory is not enough to load them on, we have to use amultipass algorithm for separated MO ERIs, but this multi-pass algorithm is inefficient.

In this paper, a solution for these problems is providedby combining the direct algorithm with the perturbation-selection technique. We have developed the direct SAC andSAC-CI algorithm running within the selected excitation op-erators. The MO based direct algorithm that utilizes thesparseness of the Hamiltonian matrices provides an efficientalgorithm that works within the selected excitation operators.This also results in an efficient minimal memory require-ment.

The theoretical framework and the algorithm of the di-rect SAC/SAC-CI method are described in the next section.The implementation of the new direct SAC-CI algorithm isgiven in Sec. III, and applications to several test cases arereported in Sec. IV. The present article focuses on single-point calculations. The analytical energy gradient of theSAC/SAC-CI method has been given in GAUSSIAN 03. Wehave also adapted the direct algorithm for the analytical en-ergy gradient calculations, and it will be reported in thesubsequent paper.

II. THEORY

A. SAC/SAC-CI method

The details of the SAC/SAC-CI methodology have beenreviewed in several articles.1–3 Here, we summarize thepoints pertinent to the present study. The SAC expansion fora totally symmetric singlet ground state is written as

�SAC = exp�S��0� , �1�

where �0� is a closed-shell Hartree-Fock �HF� single determi-nant and

S = �I

cISI†. �2�

SI† is a symmetry-adapted excitation operator. The SAC co-

efficient cI is calculated by solving the nonvariational equa-tions

�0��H − ESAC���SAC� = 0 �3�

and

�0�SL�H − ESAC���SAC� = 0, �4�

where H is the Hamiltonian and ESAC is the SAC energy.The SAC theory provides not only the SAC wave func-

tion �SAC, but also a set of functions that constitute the basisfor the excited states. Namely, the set of functions

�K = PSK† ��SAC� �5�

provides an adequate basis for expanding the excited states.Here, P is a projection operator P=1− ��SAC���SAC�. There-fore, we expand our excited states by linear combinations of�K as

�SAC-CI�p� = �

K

dK�p��K, �6�

which is the SAC-CI wave function for the excited states.Here, the index p labels the excited state. The SAC-CI wavefunction can also be defined for the excited state having dif-ferent symmetries and for the ionized and electron-attachedstates. Including these states, Eq. �5� is generalized as

�K = PRK† ��SAC� , �7�

where RK† denotes a set of excitation, ionization, and

electron-attachment operators. The SAC-CI coefficients andenergies are calculated by solving the nonvariational equa-tion

�0�RL�H − ESAC-CI�p� ���SAC-CI

�p� � = 0. �8�

In the SAC/SAC-CI program, we can use theperturbation-selection method. We select only important ex-citation operators by the second-order perturbation theoryand reduce the labors in the SAC and SAC-CIcalculations.14,15,20 In the singles and doubles method �SAC/SAC-CI SD-R�, we include all singles and apply a perturba-tion selection to doubles. In the ground state SAC calcula-tion, the doubly excitation operator SK

† is included if

094105-2 R. Fukuda and H. Nakatsuji J. Chem. Phys. 128, 094105 �2008�

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�Es�SK† �� � �g, �9�

where Es�SK† � is the second-order energy contribution of SK

†

to the ground state,

Es�SK† � =

�0�SKH�0��0�HSK† �0�

�0�SKHSK† �0� − �0�H�0�

. �10�

For excited, ionized, and electron-attached states, the doublyexcitation operators which satisfy

�Ep�RK† �� � �e �11�

with

Ep�RK† � =

�0�RKH��Ref�p� ���Ref

�p� �HRK† �0�

ERef�p� − �0�RKHRK

† �0��12�

are included in the SAC-CI calculations. Here, �Ref�p� is the

reference wave function which has the energy ERef�p� . The rec-

ommended usage of the perturbation-selection technique isfound in the SAC-CI Guide.20

B. Conventional nondirect SAC/SAC-CI algorithm

The conventional version of the SAC/SAC-CI programwas formulated to be driven with the excitation operator la-bels. For example, the SAC-CI secular equation is written as

�K

�HLK + ULK�dK�p� = ESAC-CI

�p� �K

dK�p�SLK. �13�

Here, HLK, ULK, and SLK denote linked, unlinked, and over-lap matrix elements, respectively,

HLK = �0�RLHRK† �0� , �14�

ULK = �I

�0�RLHRK† SI

†�0�cI, �15�

and

SLK = �0�RLRK† �0� . �16�

For simplicity, we dropped higher-order unlinked terms inEqs. �13� and �15� and the projection operator that will ap-pear in Eq. �16�. The details of this simplification were de-scribed in Ref. 20. The generalized eigenvalue problem ofEq. �13� is solved iteratively with the modified Davidson’sprocedure,28 where the sigma vector,

�L�m� = �

K

�HLK + ULK�bK�m�, �17�

is constructed in each iteration step from the basis vector band the matrix elements. In the conventional code, the matrixelements are evaluated with the projective reduction formulaand are stored on disk. The loops for the integral evaluation�Eqs. �14�–�16�� and the sigma-vector construction �Eq. �17��are driven through the excitation operator labels I, L, and K.We call this algorithm conventional “nondirect” algorithm, incontrast to the “direct” algorithm introduced in this paper.

The important features of the conventional nondirect al-gorithm are as follows:

�1� The perturbation-selection technique is easily intro-duced into the program. The perturbation selection re-

duces the number of the matrix elements to be evalu-ated and the range of the summation in Eqs. �13� and�17�, thus dramatically reducing the total computationtime.

�2� The extensions to include general excitation operatorsare straightforward. This feature was important for thegeneralizations of the SAC-CI code, like the SAC-CIgeneral-R method7 and SAC-CI for high-spinmultiplicities.10 These extensions were done by ex-panding the nature of the excitation operators and couldbe easily implemented in the conventional code by ex-panding the loops of the operators labels I, L, and K.The matrix elements are calculated using the projectivereduction formula, PROJR.27 The PROJR program evalu-ates the matrix elements by comparing the left and rightconfiguration state functions. The algorithm repre-sented by Eqs. �14�–�17� does not depend on the spe-cific form of the R operators.

Because of these attractive features, the conventionalSAC-CI program has been extended to cover a wide range ofchemistry and accuracy3 and has been successfully appliedfrom fine spectroscopy21–23 of rather small molecules tobiospectroscopy24 and photobiology involving moderatelylarge molecules.25 However, the conventional algorithm hasfollowing demerits in comparison with the direct algorithmintroduced in this paper.

�1� The calculation is time consuming, particularly for theunlinked terms. The unlinked terms U in Eq. �15� havethree indices of excitation operators, which result in atriply nested loop structure. However, many of theterms, �RL�H�RK

† SI†�, are identically zero because of

Slater’s rule, but the calculation of these unlinked termscan be done only at the innermost part of the loop.

�2� A whole MO ERI has to be loaded on core memory.This is because the MO ERI is randomly accessed fromthe PROJR subroutine. There is no regulation in the ac-cess of the MO ERI.

In constructing the direct SAC/SAC-CI code, we want toovercome these demerits, keeping the merits of the perturba-tion selection.

C. Direct SAC/SAC-CI algorithm

In the direct SAC method, the excitation operators S†

and the coefficients c are defined by the MO labels instead ofthe excitation operator labels. For singles �S1� and doubles�S2�, they are

S† = S1 + S2 = �i

�a

ciaSi

a† +1

2�ij

�ab

cijabSij

ab†. �18�

Hereafter, we use indices i , j ,k , . . . for occupied MOs,a ,b ,c , . . . for virtual MOs, and p ,q ,r ,s , . . . for generalMO’s. Inserting Eq. �18� into Eqs. �3� and �4�, we obtain theexpressions of the SAC energy, the SAC equations for S1 andS2. In the direct SAC algorithm, we first write down theseworking equations explicitly as summarized in Table X ofthe Appendix, where fq

p= �p�f �q� and vqspr= �pq �rs�= �pr �qs�

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denote the Fock matrix element and the MO ERI, respec-tively. We introduced the permutation operator

Pijab�¯�ij

ab = �¯�ijab + �¯� ji

ba �19�

for convenience. Note that the expressions in Table X are notunique. We will introduce intermediate arrays to make thenumber of operations minimal.19 The expressions in Table Xwere designed to be effective in the perturbation-selectiontechnique.

The SAC-CI secular equation �Eq. �8�� is written in thenonsymmetric eigenvalue problems

Hd�p� = ESAC−CI�p� Sd�p� �20�

and

H*d�p� = ESAC−CI�p� S*d�p�, �21�

where d and d denote the right-hand and left-hand eigenvec-tors, respectively. The superscript � denotes a Hermitiantranspose. In the generalized Davidson procedure,28 we use

m basis vectors bi and bi, which are collected in matrices Band B as

B = �b1,b2, . . . ,bm�, B = �b1,b2, . . . ,bm� . �22�

Here, m is some small number used in generalized David-

son’s procedure.28 The basis vectors bi and bi satisfy thebiorthonormal relation

B*SB = 1 , �23�

where 1 is an m�m unit matrix. Then, we form a smallDavidson matrix as

H = B*HB = B*� = �*B . �24�

The elements of �=HB and �=H*B are the so-called sigmavectors for the right-hand and left-hand basis vectors, respec-tively. Explicitly, they are

�i = Hbi, �i = H*bi. �25�

To impose a biorthonormal relation to the basis, we definetau vectors as

�i = Sbi, �i = S*bi. �26�

In each iteration, the small matrix

C*HC = D �27�

is diagonalized. After convergence, we obtain the SAC-CIvectors by the following expression:

d = BC, d = BC . �28�

In the SAC-CI SD-R method, the sigma and tau vectorshave three blocks: R0 �HF�, R1 �singles�, and R2 �doubles� as

� = ��0

�1

�2� = � �0

�ia

�ijab� . �29�

The MO-indexed representations of the working equation areobtained from Eq. �8� by using the MO-indexed R operators.For singlet excited states, the R operators are

R† = R0 + R1 + R2 = d0 + �i

�a

diaRi

a† +1

2�ij

�ab

dijabRij

ab†.

�30�

The SAC-CI coefficients are obtained from the basis b asfollows:

dijab = �

m

bijab�m�C�m�. �31�

The resultant working equations for the singlet excited statesare summarized in Table XI of the Appendix. For the left-hand projection, we use the following notation for the Her-mite conjugation:

��ijab�* = �ab

ij , �bijab�* = bab

ij . �32�

To simplify our representation, upper bars were dropped forthe left-hand projections in Table XI.

For triplet states, the MO-indexed R operator is

R† = R1 + R2 = �i

�a

diaRi

a† + �ij

�a�b

dijabRij

ab†. �33�

The working equations for the triplet states are summarizedin Table XII of the Appendix. The equations for cation andanion doublet states are obtained from the triplet equations:Cation doublet states are obtained by replacing one of theunoccupied MO indices to infinitely separated orbital, e.g.,Ri

�†. Similarly, the electron-attached �anion doublet� statesare written as an electron transfer from an infinitely sepa-rated orbital to one of the unoccupied orbitals like �→a.Thus, the R operators for the cation doublet and anion dou-blet are written as

R† = �i

di�Ri

�† + �ij

�b

dij�bRij

�b† �34�

and

R† = �a

d�a R�

a† + �i

�a�b

d�jabR�j

ab†, �35�

respectively. With the replacements to infinitely separatedorbitals in Eqs. �34� and �35�, the working equations for thecation doublet and anion doublet are obtained. Note that thematrix elements including indices �, such as f i

�, vij�a, etc., are

zero.

III. IMPLEMENTATION OF THE DIRECT SAC/SAC-CIALGORITHM

In the conventional nondirect program, the loops aredriven with the labels, I, of excitation operators. The MOindices corresponding to the excitation operator are takenfrom the predefined labels of the excitation operators. This

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correspondence may be written as SI†→Sij

ab† and is kept inthe LABEL array, which is a 4�M matrix, where M is thenumber of the selected operators and each operator has fourMO indices.

Because the direct program drives loops with MO indi-ces, we have to refer to the label of the excitation operatorfrom the MO indices such as Sij

ab†→SI†. This correspondence

is done by introducing INDEX arrays. The INDEX arrays aretwo-dimensional integer matrices, whose elements are exci-tation operator labels. We need six types of INDEX arrays:INDEX��ij� , �ab��, INDEX��ab� , �ij��, INDEX��ia� , �jb��,INDEX��jb� , �ia��, INDEX��ib� , �ja��, andINDEX��ja� , �ib��, where the first elements are referred to asleading indices.

When we use the perturbation-selection technique, thelength of the LABEL array is reduced and the INDEX arraysbecome sparse matrices. The INDEX arrays have no ele-ments for the unselected excitation operators because the el-ements are appended only for the selected operators. There-fore, the INDEX arrays are compressed into a one-dimensional vector together with the location array,Loc�LBL, that carries the information of the two-dimensional structure of the original INDEX arrays. Theloops are driven through the element of the INDEX vectorsthat assign the element of the LABEL array, which define theactual form of the excitation operator.

Let us explain the above algorithm using a simple ex-ample. We consider a system with two occupied �i and j� andtwo virtual �a and b� MOs. We assume that within ten pos-sible excitation operators, the following six operators,

S1† = Sii

aa†, S2† = Sii

bb†, S3† = Sjj

aa†,

�36�S4

† = Sijbb†, S5

† = Siiab†, S6

† = Sijab†,

were selected by the perturbation selection, discarding Sijaa†,

Sijba†, Sjj

bb†, and Sjjab†. They actually have symmetry-adapted

forms.1,2 The LABEL array is a 4�6 matrix

S1† S2

† S3† S4

† S5† S6

†

i i j i i i

i i j j i j

a b a b a a

a b a b b b� , �37�

and the original INDEX��ij� , �ab�� array is a sparse matrixgiven by

aa ab ba bb

ii

ij

ji

j j

1 5 0 2

0 6 0 4

0 0 0 0

3 0 0 0� , �38�

where only the selected elements are considered. This IN-DEX array is compressed into a one-dimensional INDEXvector

�1 5 2 6 4 3� , �39�

with the information of the original array stored in theLoc�LBL array

ii ij ji j j j j + 1

�1� 4 6 6 �7�, �40�

where the element indicates the location in the INDEX vec-tor that initiates the elements of the designated MO label,e.g., ij. The elements designated by the MO label ij locatebetween fourth and fifth positions in INDEX. This can beschematized as

�1� 5 2 6 4 �3��ii

�ij

�ji,j j

, �41�

where an arrow, whose location is obtained from theLoc�LBL array, indicates the starting location of the desig-nated MO label. The extra element of j j+1 is used for de-noting the number of elements that belong to row j j. In theoriginal INDEX array given by Eq. �38�, the MO label jimay be discarded from the beginning because of the symme-try of S†, e.g., Sji

ba†=Sijab†. The loop is driven through the

INDEX vector given by Eq. �39� aided with the informationstored in the Loc�LBL array.

We consider the system of the N-active orbitals andM-selected excitation operators. In the modified Davidsonprocedure, the sigma vector is constructed by the multiplica-tion of the basis vector b and the intermediate arrays �V� as,for example, �ij

ab← �V�ijklbkl

ab �see the Appendix�. The MOERIs and intermediate arrays have been sorted in a desired

FIG. 1. Loop structure of direct SAC-CI for �ijab← �V�ij

klbklab.

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order on a disk. Consequently, at most, only the N2 memoryis required for the MO ERIs. A typical loop structure of thedirect SAC-CI algorithm is shown in Fig. 1 for �ij

ab

← �V�ijklbkl

ab. The intermediate array of �V�ijkl for fixed i and j is

loaded within the outer loops for i and j, and these loopsrequire O�N2� computations. The inner loops run only fornonzero elements of b. The computations of the inner loopsare O�M�. The total computational cost of the direct algo-rithm is O�N2�M�. In cases without perturbation selection,M =O�N4�, so that this algorithm becomes O�N6�, whichagrees with the optimal cost of the singles and doublescoupled-cluster theories in the canonical MO basis.

IV. SAMPLE APPLICATIONS

As test applications of the new direct SAC-CI program,we performed calculations for water �H2O�, ethylene �C2H4�,pyrrole �C4H5N�, tetrathiomolybdate anion �MoS4

2−�, andfree-base porphin �C20N4H14�. The molecular geometrieswere taken from the SAC-CI Guide.20 The basis sets were asfollows: D95�d , p� with Rydberg �2s2p� �Ref. 29� on oxygenfor H2O, D95�d , p� with Rydberg �2s2p2d� on carbon for

C2H4, D95�d� for C4H5N, LANL2DZ �Ref. 30� for MoS42−,

and D95 for free-base porphin. The perturbation selectionwith LevelThree threshold ��g=1.0�10−6, �e=1.0�10−7�was used for all molecules since this is the recommendedone. Additionally, LevelOne ��g=1.0�10−5, �e=1.0�10−6�and LevelTwo ��g=5.0�10−6, �e=5.0�10−7� calculationswere carried out for free-base porphin. The levels of theselection can be selected with the GAUSSAIN03 keyword. Thedefault setting was used for other conditions. We used anapproximate variational method for solving the SAC-CIsecular equation, in which the Hamiltonian matrix was sym-

metrized as HIJ= 12 �HIJ+HJI�.

Note that in the conventional nondirect SAC-CI algo-rithm, an additional cutoff threshold was introduced for theunlinked terms to save computer time. The direct SAC-CIalgorithm does not require such approximation, so that thedirect and nondirect SAC-CI computations do not give com-pletely identical results. Theoretically, the direct SAC-CI re-sults should be more accurate than the conventional ones.Both versions of the SAC-CI programs run on the develop-ment version31 of GAUSSIAN03. All computations were car-

TABLE I. Excitation energy, oscillator strength, and ionization potential of water.

State

SAC-CI

Expt.b

Eex �eV�Nature

Direct Nondirect

Eex �eV� Osc �au�a Eex �eV� Osc �au�a

Singlet states1 1B1 �→3s�OH*�, �→4s 7.26 0.054 7.33 0.053 7.4, 7.491 1A2 �→3py, �→4py 9.14 0c 9.21 0c 9.12 1A1 �→3px, n→3s 9.58 0.060 9.67 0.053 9.67, 9.732 1B1 �→3pz 9.72 0.008 9.78 0.008 10.01, 9.9963 1A1 �→3px, n→3s 9.89 0.038 9.96 0.044 10.17, 10.143 1B1 �→4s, �→3s 11.12 0.007 11.20 0.0071 1B2 n→3py, n→4py 11.50 0.014 11.59 0.0142 1A2 �→4py, �→3py 11.57 0c 11.68 0c

4 1B1 n→3px 11.97 0.000 12.04 0.0004 1A1 n→3pz 12.01 0.004 12.09 0.0045 1A1 �→4px 12.80 0.004 12.87 0.004

Triplet states1 3B1 �→3s�OH*�, �→4s 6.86 0c 6.93 0c 7.0, 7.21 3A2 �→3py, �→4py 8.98 0c 9.08 0c 8.9, 9.1, 9.21 3A1 n→3s�OH*�, n→4s 9.25 0c 9.35 0c 9.32 3A1 �→3px, �→4px 9.47 0c 9.54 0c 9.80, 9.812 3B1 �→3pz 9.66 0c 9.74 0c 9.983 3B1 �→4s, �→3s 10.86 0c 10.94 0c

1 3B2 n→3py, n→4py 11.25 0c 11.35 0c

2 3A2 �→4py, �→3py 11.33 0c 11.45 0c

3 3A1 n→3pz 11.75 0c 11.84 0c

4 3B1 n→3px 11.92 0c 12.00 0c

4 3A1 �→4px 12.20 0c 12.26 0c

Cation states1 2B1 ���−1 12.11 0.934 12.15 0.939 12.611 2A1 �n�−1 14.41 0.937 14.48 0.939 14.731 2B2 ���−1 18.78 0.949 18.84 0.952 18.55

aMonopole intensity for ionized states.bReferences 33 and 34.cSymmetry forbidden.

094105-6 R. Fukuda and H. Nakatsuji J. Chem. Phys. 128, 094105 �2008�

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ried out with the Hewlett-Packard Integrity rx2620 serverwith the IA64 architecture.

A. Small molecules

For water and ethylene, the direct SAC-CI results arevery close to the conventional nondirect ones, as seen fromTables I and II, and they reproduced well the experimentalvalues. The average differences between the direct and non-direct results were 0.08 eV for water and 0.12 eV for ethyl-ene. These differences are due to the cutoff of some smallunlinked terms in the conventional program, which is accept-able for small molecules. Compared to the experiments, thedirect results are worse than the nondirect ones, which webelieve to be due to the basis set insufficiency.

Table III shows the singlet and triplet excitation ener-gies, ionization energies, and electron affinities of pyrrole.These test calculations were done without including the Ry-dberg basis, so that we cannot expect to reproduce the ex-

perimental values because the valence-Rydberg mixing wasshown to be strong in some excited states of this molecule.22

Thus, we focus here only on the differences between thedirect and nondirect SAC-CI results. The differences are par-ticularly large for the electron affinities: The direct SAC-CIresults are about 0.25–0.3 eV lower than the nondirect ones,which arose from the accumulated effects of small unlinkedterms cut off in the nondirect program. Since orbital reorga-nizations are expected to be important as well as electroncorrelations in the electron-attached states, the cutoff ap-proximation in the unlinked term may be wrong.

Table IV shows the excitation energies of the tetrathio-molybdate anion. The orbital characters of this molecule areas follows: 1t1 :S�3p� lone pair. 3t2 :Mo�5p�. 2e :Mo�d�+S�p� antibonding. 4t2 :Mo�d�+S�p� antibonding.

The excitation energies of the direct SAC-CI methodwere lower than those of the nondirect method. The gapbetween the two methods depends on the nature of the exci-

TABLE II. Excitation energy, oscillator strength, and ionization potential of ethylene.

State

SAC-CI

Expt.b

Eex �eV�Nature

Direct Nondirect

Eex �eV� Osc �au�a Eex �eV� Osc �au�a

Singlet states1 1Ag �→3p��� 8.18 0c 8.33 0c 8.281 1B1g �→3p��� 7.82 0c 7.94 0c 7.802 1B1g �→�* 8.74 0c 8.94 0c

1 1B2g �→3p��� 7.82 0c 8.00 0c 7.901 1B3g �→3s 9.53 0c 9.58 0c 9.511 1Au �→3d��� 8.84 0c 8.95 0c

1 1B1u �→�* 8.12 0.349 8.21 0.366 �8.02 1B1u �→3d��� 9.14 0.051 9.25 0.048 9.331 1B2u �→3d�� 8.92 0.011 8.94 0.012 8.901 1B3u �→3s 7.15 0.092 7.33 0.093 7.112 1B3u �→3d��� 8.69 0.004 8.86 0.003 8.623 1B3u �→3d�� 8.88 0.028 9.04 0.031 8.90

Triplet states1 3Ag �→3p��� 8.03 0c 8.13 0c 8.151 3B1g �→3p��� 7.79 0c 7.90 0c 7.792 3B1g �→�* 8.42 0c 8.60 0c

1 3B2g �→3p��� 7.76 0c 7.94 0c

1 3B3g �→3s 9.53 0c 9.58 0c

1 3Au �→3d��� 8.84 0c 8.95 0c

1 3B1u �→�* 4.40 0c 4.50 0c 4.362 3B1u �→3d��� 8.95 0c 9.07 0c 8.861 3B2u �→3d�� 8.92 0c 8.94 0c

1 3B3u �→3s 7.02 0c 7.12 0c 6.982 3B3u �→3d��� 8.66 0c 8.83 0c 8.573 3B3u �→3d�� 8.84 0c 8.98 0c

Cation states1 2Ag �2ag�−1 14.56 0.917 14.68 0.924 14.661 2B3g �1b3g�−1 12.87 0.927 13.01 0.933 12.851 2B1u �2b1u�−1 19.33 0.848 19.48 0.852 19.231 2B2u �1b2u�−1 16.00 0.883 16.13 0.885 15.871 2B3u �1b3u�−1 10.34 0.949 10.36 0.952 10.51

aMonopole intensity for ionized states.bReferences 35 and 36.cSymmetry forbidden.

094105-7 Direct algorithm for SAC-CI J. Chem. Phys. 128, 094105 �2008�

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tation. The lower two states are valence-type excitations,from ligand lone pair to metal-ligand antibonding MO. Forthese states, the differences were around 0.2 eV. The nextthree states have a Rydberg nature, i.e., the excitations tomolybdenum 5p orbital. For these states, the differenceswere as large as 0.4 eV, showing that the neglected smallercontributions in the unlinked term could sum up to a largenumber. Such cases seem to occur when the coupling be-tween the orbital reorganization and the electron correlationis large. The direct algorithm includes all such terms, and theresults reproduced the overall experiments better than thenondirect one. When we reduce the cutoff threshold in thenondirect method, the results become close to those of thedirect method, which will be shown in Sec. IV D.

B. Free-base porphin

The results for free-base porphin were summarized inTable V. The results for four threshold levels were given, buthere we discuss only the LevelThree results because it is arecommended level. Except for the 1B2u state, the excitationenergies of the direct SAC-CI method were lower than thoseof the nondirect one. Consequently, the energy gap betweenthe 1B1u and 1B2u states was increased, and the gap betweenthe 1B2u and 2B1u states was reduced. The observed gapbetween Qx and Qy bands is 0.44 eV and that between Qy

and B bands is 0.91 eV. The direct SAC-CI method im-proved the spectral shape by the reduction of the 1B2u and2B1u gap.

TABLE III. Excitation energy, ionization potential, and electron affinity of pyrrole.

State

SAC-CI

Expt.b

Eex �eV�Nature

Direct Nondirect

Eex �eV� Osc. �au�a Eex �eV� Osc. �au�a

Singlet states1 1A1 �2→�

4*, �3→�

5* 6.68 0.004 6.53 0.003

1 1A2 �3→�* 7.62 0c 7.83 0c

1 1B1 �2→�* 8.40 0.011 8.65 0.0111 1B2 �3→�

4* 6.90 0.190 7.00 0.214 6.2–6.5

Triplet states1 3A1 �2→�

4*, �3→�

5* 5.61 0c 5.61 0c 5.10

1 3A2 �3→�* 7.64 0c 7.87 0c

1 3B1 �2→�* 8.12 0c 8.42 0c

1 3B2 �3→�4* 4.47 0c 4.52 0c 4.20

Cation doublet states1 2A1 6a1���−1 12.64 0.908 12.98 0.928 12.60, 12.581 2A2 1a2��3�−1 7.85 0.934 7.93 0.953 8.02, 8.211 2B1 2b1��2�−1 8.69 0.923 8.81 0.938 9.05, 9.201 2B2 4b2���−1 13.12 0.910 13.49 0.929 13.0

Anion doublet states1 2A1 7a1��*�+1 4.90 5.141 2A2 2a2��

5*�+1 4.93 5.23 3.45

1 2B1 3b1��4*�+1 3.76 4.01 2.36

1 2B2 5b2��*�+1 6.43 6.68

aMonopole intensity for ionized states.bReferences 37 and 38.cSymmetry forbidden.

TABLE IV. Excitation energy and oscillator strength of tetrathiomolybudate anion.

State

SAC-CI Expt.a

Main configuration

Direct NondirectEex

�eV� Osc.Eex �eV� Osc. �au� Eex �eV� Osc. �au�

Singlet states1 1T1 1t1→2e 2.11 0b 2.30 0b 2.37 Weak1 1T2 1t1→2e 2.49 0.047 2.71 0.064 2.65 0.11 1E 1t1→3t2 3.17 0b 3.61 0b

2 1T1 1t1→3t2, 1t1→4t2 3.19 0b 3.60 0b

2 1T2 1t1→3t2 3.26 0.016 3.75 0.028 3.22 Weak

aReference 39.bSymmetry allowed states are 1T2.

094105-8 R. Fukuda and H. Nakatsuji J. Chem. Phys. 128, 094105 �2008�

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We also note the differences in the intensities betweenthe two calculations. The most intense peak was the 2B2u

state. In the nondirect results the next intense state was 3B1u,whereas in the direct results the second intense state was1B1u. In comparison with the observed spectrum, the inten-sity of the 3B1u state with the nondirect method seems to betoo strong, so that the direct algorithm improved not only theexcitation energies, but also the absorption intensities. How-ever, our assignments of the basic peaks are the same asthose of the previous results.15,32 Since the basis set of thepresent calculations might be too poor, we postpone detailedarguments in a forthcoming paper.

We examined the convergence of the perturbation-selection techniques. The differences between the LevelTwoand LevelThree results are about 0.15–0.25 eV. To confirm

TABLE V. Excitation energy and oscillator strength of free-base porphin.

State

SAC-CI Expt.a

Nature

Direct NondirectEex

�eV� BandEex �eV� Osc. �au� Eex �eV� Osc. �au�

Level four1 1B1u �→�* 1.92 0.000 2.03 0.000 1.98 Qx

1 1B2u �→�* 2.51 0.000 2.46 0.001 2.42 Qy

2 1B1u �→�* 3.68 1.414 3.95 0.907 3.33 B2 1B2u �→�* 3.79 1.820 4.19 1.726 3.65 N3 1B1u �→�* 4.29 0.660 4.53 1.3073 1B2u �→�* 4.53 0.207 4.70 0.359 4.25–4.67 L4 1B2u �→�* 5.03 0.410 4.96 0.388 5.0–5.5 M4 1B1u �→�* 5.18 0.497 5.36 0.347

Level three1 1B1u �→�* 1.87 0.000 2.00 0.000 1.98 Qx

1 1B2u �→�* 2.47 0.000 2.43 0.001 2.42 Qy

2 1B1u �→�* 3.63 1.385 3.91 0.957 3.33 B2 1B2u �→�* 3.74 1.799 4.13 1.745 3.65 N3 1B1u �→�* 4.24 0.636 4.48 1.2303 1B2u �→�* 4.48 0.204 4.67 0.337 4.25–4.67 L4 1B2u �→�* 4.99 0.389 5.16 0.377 5.0–5.5 M4 1B1u �→�* 5.11 0.490 5.32 0.354

Level two1 1B1u �→�* 1.71 0.001 1.86 0.000 1.98 Qx

1 1B2u �→�* 2.30 0.000 2.31 0.000 2.42 Qy

2 1B1u �→�* 3.43 1.297 3.70 1.140 3.33 B2 1B2u �→�* 3.54 1.646 3.86 1.722 3.65 N3 1B1u �→�* 4.02 0.503 4.27 0.9413 1B2u �→�* 4.24 0.205 4.46 0.305 4.25–4.67 L4 1B2u �→�* 4.79 0.324 4.99 0.324 5.0–5.5 M4 1B1u �→�* 4.85 0.464 5.10 0.378

Level one1 1B1u �→�* 1.58 0.001 1.77 0.000 1.98 Qx

1 1B2u �→�* 2.19 0.000 2.25 0.000 2.42 Qy

2 1B1u �→�* 3.31 1.202 3.58 1.192 3.33 B2 1B2u �→�* 3.40 1.526 3.71 1.690 3.65 N3 1B1u �→�* 3.89 0.431 4.15 0.7743 1B2u �→�* 4.10 0.202 4.33 0.275 4.25–4.67 L4 1B2u �→�* 4.64 0.306 4.86 0.311 5.0–5.5 M4 1B1u �→�* 4.66 0.473 4.93 0.412

aReference 40.

TABLE VI. Computational time �wall clock time� of SAC-CI calculations.

Molecule Direct Nondirect

H2O 52 s 51 sC2H4 9 min 20 s 11 min 52 sMoS4

2− 12 min 12 s 18 min 23 sC4H5N 40 min 15 s 1 h 34 min 17 sFree-base porphin �level one� 1 h 7 min 20 s 56 min 11 sFree-base porphin �level two� 1 h 37 min 54 s 3 h 42 min 28 sFree-base porphin �level three� 7 h 16 min 35 s 48 h 21 min 07 sFree-base porphin �level four� 14 h 14 min 50 s 106 h 18 min 59 s

094105-9 Direct algorithm for SAC-CI J. Chem. Phys. 128, 094105 �2008�

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the convergence of the perturbation-selection techniqueagainst the selection level, we performed additional calcula-tions with tighter thresholds. �LevelFour: �g=5�10−7 and�e=5�10−8� The differences between the LevelThree andLevelFour results are about 0.05 eV, showing the convergingbehavior at higher levels of calculations toward the energieswithout selection.

C. Computational time and space

The computational times were summarized in Table VI.For C2H4 and MoS4

2−, the direct method was slightly faster

than the nondirect method. For C4H5N, the direct methodwas about 2.3 times faster than the nondirect method. Gen-erally, the direct method was more efficient and the accelera-tion was more significant for larger dimensional computa-tions. Actually, for free-base porphin, the acceleration factorwas 6.7 for LevelThree calculations, 2.3 for LevelTwo cal-culations, and comparable to LevelOne calculations. Thus,the efficiency of the direct SAC-CI algorithm would extendthe possibility of the SAC-CI method.

The number of nonzero MO ERIs of the free-base por-

TABLE VII. Sets of cutoff thresholds for the unlinked terms of the nondirect method.

Keyword Set 0 Set 1 Set 2 Set 3 Direct

g CThreULS2G 5.0�10−3 5.0�10−3 1.0�10−3 1.0�10−8 0e �single� CThreULR1 5.0�10−2 2.0�10−2 1.0�10−3 1.0�10−8 0e �double� CThreULR2 5.0�10−2 2.0�10−2 1.0�10−3 1.0�10−8 0

TABLE VIII. Excitation energy and ionization potential of ethylene �in eV� with various levels of approxima-tion.

State

SAC-CI

Expt.Nature

Nondirect

DirectSet 0 Set 1 Set 2 Set 3

Singlet states1 1Ag �→3p��� 8.33 8.18 8.18 8.18 8.18 8.281 1B1g �→3p��� 7.94 7.83 7.82 7.82 7.82 7.802 1B1g �→�* 8.94 8.80 8.74 8.74 8.741 1B2g �→3p��� 8.00 7.84 7.82 7.82 7.82 7.901 1B3g �→3s 9.58 9.50 9.53 9.53 9.53 9.511 1Au �→3d��� 8.95 8.83 8.84 8.84 8.841 1B1u �→�* 8.21 8.13 8.12 8.12 8.12 �8.02 1B1u �→3d��� 9.25 9.13 9.14 9.14 9.14 9.331 1B2u �→3d�� 8.94 8.89 8.92 8.92 8.92 8.901 1B3u �→3s 7.33 7.18 7.15 7.15 7.15 7.112 1B3u �→3d��� 8.86 8.70 8.69 8.69 8.69 8.623 1B3u �→3d�� 9.04 8.96 8.87 8.88 8.88 8.90

Triplet states1 3Ag �→3p��� 8.13 8.03 8.02 8.03 8.03 8.151 3B1g �→3p��� 7.90 7.80 7.79 7.79 7.79 7.792 3B1g �→�* 8.60 8.48 8.42 8.42 8.421 3B2g �→3p��� 7.94 7.79 7.75 7.76 7.761 3B3g �→3s 9.58 9.50 9.53 9.53 9.531 3Au �→3d��� 8.95 8.83 8.84 8.84 8.841 3B1u �→�* 4.50 4.43 4.39 4.40 4.40 4.362 3B1u �→3d��� 9.07 8.94 8.94 8.95 8.95 8.861 3B2u �→3d�� 8.94 8.89 8.92 8.92 8.921 3B3u �→3s 7.12 7.05 7.02 7.02 7.02 6.982 3B3u �→3d��� 8.83 8.68 8.66 8.66 8.66 8.573 3B3u �→3d�� 8.98 8.84 8.84 8.84 8.84

Cation states1 2Ag �2ag�−1 14.68 14.53 14.55 14.56 14.56 14.661 2B3g �1b3g�−1 13.01 12.86 12.86 12.87 12.87 12.851 2B1u �2b1u�−1 19.48 19.34 19.33 19.33 19.33 19.231 2B2u �1b2u�−1 16.13 15.98 16.00 16.00 16.00 15.871 2B3u �1b3u�−1 10.36 10.29 10.34 10.34 10.34 10.51

Computational time 11 min 52 s 21 min 05 s 4 h 21 min 33 s 8 h 47 min 31 s 9 min 20 s

094105-10 R. Fukuda and H. Nakatsuji J. Chem. Phys. 128, 094105 �2008�

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phin was 40 755 181. Therefore, we need approximately41 Mbyte memory for storing all the MO ERIs for the usageof the PROJR program in the nondirect algorithm. We havecompressed the MO ERIs with the use of the point groupsymmetry of free-base porphin. Thus, the demand formemory was not severe for the present examples. On theother hand, the direct method does not require the storage ofthe MO ERIs on memory. This improvement would becomesignificant when we apply the direct method to big biologicalmolecules, which involve hundreds of orbitals without anysymmetry.

D. Unlinked terms

As we have seen in some examples, the approximationsin the unlinked terms used in the nondirect method have ledto some degrees of errors. This error is due to the cutoff ofsmaller unlinked terms, and the amount is dependent on thecutoff threshold. In the present cases, the errors were ap-proximately 0.1–0.3 eV. Considering the errors originatingfrom other sources, the errors less than 0.3 eV ��0.01Eh�might be practically acceptable for most cases. However, forMoS4

2−, the error amounted to 0.4 eV, so that we have toshow the method to reduce this error in the nondirectmethod.

If we reduce the cutoff thresholds in the unlinked terms,the nondirect results become close to those of the directmethod. A severe problem is an increase in the computa-tional time. So, optimal sets of thresholds are necessary todiminish the cutoff errors within permissible computationaltime. To find such optimal threshold, we investigated theeffects of the cutoff of unlinked terms.

The cutoff scheme in the nondirect program is as fol-lows. For the SAC ground state, we cutoff the unlinked termsgenerated by the product of the linked operators whose SDCIcoefficients C is smaller than a threshold g, and this g valuecan be controlled with the CThreULS2G keyword catego-rized as “detailed keywords.”20 The unlinked terms ofSAC-CI are generated by the product of the linked operatorsRK and SI, where we include all the double excitation opera-tors SI but we select important RK operators whose SDCIcoefficients dK satisfy dK�e, where the e values for singleand double excitation operators can be input by the keywords

CThreULR1 and CThreULR2, respectively. The directSAC-CI program does not use this type of cutoff, so that thedirect calculations correspond to g=e=0. To see the effectof this approximation in the nondirect method and to find anoptimal set of thresholds, we examined four sets of thresh-olds for ethylene and tetrathiomolybudate anion. The thresh-olds are summarized in Table VII. Set 0 is the thresholdsadopted as the default.

Table VIII shows the results for ethylene. The nondirectresults for set 0 are the same as those given in Table II. Wesee that the results with set 2 thresholds reproduced the re-sults of the direct method. However, this calculation takestoo much computational time. In this case, set 1 thresholdsseem to be a good compromise. It approximately reproducesthe direct results and the computational time is twice of set 0.If accurate results are necessary, this additional computa-tional cost could be permissible.

Table IX shows the results for tetrathiomolybudate an-ion. The set 2 calculation also well reproduced the directcalculation even in this case. The computational time is,however, too large. It takes 70 times longer for the set 0calculations. In this case, set 1 would be a realistic choiceeven though it still contains some degrees of errors. Thediscrepancy of 0.3 eV in the 2T2 state could be an acceptablerange.

As a result of these test calculations, we recommend touse set 1 thresholds when one wants to get higher accuracywith the nondirect SAC-CI calculations of the released ver-sion. Of course, the direct SAC-CI version will be the bestchoice after the new version of the program is released. Wenote here that the non-direct method is still useful when onewants to use the general-R method7 and the high-spincodes.10

In Table IX, we further showed the results obtained withset 3 thresholds. These results are essentially the same as thedirect ones. This means that the same calculations were per-formed with different algorithms. The efficiency of the directalgorithm is obvious. It is more than 200 times faster thanthe nondirect one for MoS4

2−. This efficiency enables us tochoose better thresholds for the perturbation selection of thelinked operators. When we release the nondirect program, weselected LevelOne, LevelTwo, and LevelThree sets from theexperiences in our previous researches and computationaltimes. Since we now have more freedom due to the present

TABLE IX. Excitation energy of tetrathiomolybudate anion �in eV� with various levels of approximation.

State

SAC-CI

Expt.Main configuration

Nondirect

DirectSet 0 Set 1 Set 2 Set 3

Singlet state1 1T1 1t1→2e 2.30 2.09 2.09 2.11 2.11 2.371 1T2 1t1→2e 2.71 2.51 2.47 2.49 2.49 2.651 1E 1t1→3t2 3.61 3.41 3.17 3.17 3.172 1T1 1t1→3t2, 1t1→4t2 3.60 3.38 3.19 3.19 3.192 1T2 1t1→3t2 3.75 3.56 3.27 3.26 3.26 3.22Computational time 18 min 23 s 38 min 54 s 21 h 30 min 07 s 44 h 56 min 25 s 12 min 12 s

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introduction of the direct method and the general advances incomputer technology, we would be able to set up better setsof recommended thresholds for the perturbation selection oflinked operators adopted in the SAC/SAC-CI program.

V. CONCLUDING REMARKS

We have developed a new computational algorithm,called direct algorithm, for the SAC/SAC-CI program. In thenew direct SAC-CI SD-R program, we could combine theMO-index direct algorithm with the perturbation-selectiontechnique. The sparse nature of the matrices involved wasfully utilized. The key of the present method was an efficientdesign of the index vectors and arrays that map the selectedexcitation operators. For the system of the N-active orbitalsand M-selected excitation operators, the computational costis O�M �N2�, which is optimal for singles and doubles theo-ries with perturbation selection. In addition, an efficient us-age of core memory has been achieved. Because of theachieved efficiency, the computer time became shorter, andthe cutoff approximation in the unlinked terms with a giventhreshold, which was used in the conventional �nondirect�version of the program, became unnecessary.

The direct SAC-CI program has been applied to sometest molecules. The direct SAC-CI algorithm was shown tobe more efficient than the conventional one. Particularly, thecomputational speed was accelerated significantly for largedimensional computations with increasing accuracy of theresults. The errors due to the cutoff approximation in theunlinked terms were minor in small molecules, but wouldbecome non-negligible for the systems where both of thecorrelations and orbital relaxations are important.

Thus, the direct SAC-CI program will provide theoreti-cally more accurate results than before in a shorter compu-tational time. This would extend the applicability of theSAC/SAC-CI methodology with increased accuracy. The

merits of the direct SAC-CI program will be recognizedmuch more clearly for studies of excited-state geometriesand potential energy surfaces. The SAC-CI energy gradientmethod based on the direct algorithm is certainly necessaryfor these studies. The idea of the direct SAC-CI method andthe usage of sparse matrix techniques can be applied to theSAC-CI energy gradient method, which is in progress in ourresearch institute.

TABLE X. Working equations for SAC. �We assume summation over allrepeating indices.�

SAC energy

�ESAC=�2�x�+ �X�

S1 equations

0=�2fai −�2�x�ci

a− �X�cia+ fc

acic− f i

kcka+2�vic

ak− 12vci

ak�ckc

+�2�fckcik

ac+ �vcdak− 1

2vdcak�cik

cd− 12 fc

kckiac− �vic

kl− 12vci

kl�cklac�

S2 equations

2vabij −vba

ij =−�2�x��cijab− 1

2cijba�+ �V�cd

abcijcd+ �V�ij

klcklab

+Pijab�2�f j

bcia+ �vcj

ab− 12v jc

ab�cic�−�2� 1

2 f ibcj

a+ �vijkb− 1

2v jikb�ck

a�+2�V� jc

bkcikac− �V� jc

bkckiac− �V�ic

bkcjkac− �V�ci

bkckjac+ �F�c

a�cijcb− 1

2cijbc�

− �F�ik�ckj

ab− 12cjk

ab�

Intermediate arrays for SAC equations�x�= fc

kckc

�X�=2�vcdkl − 1

2vdckl �ckl

cd

�F�ca= fc

a− �vdckl − 1

2vcdkl �ckl

da

�F�ik= f i

k+ �vcdkl − 1

2vdckl �cil

cd

�V�cdab=vcd

ab− 12vdc

ab

�V�ijkl=vij

kl− 12v ji

kl+ �vefkl − 1

2v fekl �cij

ef

�V�icbk=vic

bk− 12vci

bk+ �vcdkl − 1

2vdckl ��cil

bd− 12cil

db��V�ci

bk=vcibk− 1

2vicbk− 1

2�vcd

kl − 12vdc

kl �cilbd− 1

2�vdc

kl − 12vcd

kl �cildb

TABLE XI. Working equations for SAC-CI singlet excited states. �We as-sume summation over all repeating indices.�

Right-hand projection

�0=�2fckbk

c+2�vcdkl − 1

2vdckl �bkl

cd

�ia=�2f i

ab0+ �X�bia− �F�i

kbka+ �F�c

abic+2�vic

ak− 12vci

ak�bkc

+2�G�ck�cki

ca− 12cki

ac�+�2�fc

k�bkica− 1

2bikca�+ �vka

cd− 12vak

cd�bkicd− �vci

kl− 12vic

kl�bklca�

�ijab=2�vij

ab− 12v ji

ab�b0+ �X��bijab− 1

2bijba�+ �Y��cij

ab− 12cij

ba�+ �V�ij

klbklab+ �V�cd

abbijcd

+Pijab�2��F�i

abjb− 1

2 �F�ibbj

a+ �V�cjbabi

c− �V� jibkbk

a�+ �G�ca�cij

cb

− 12cij

bc�− �G�ik�ckj

ab− 12cjk

ab�+2�V�cj

kb�bkica− 1

2bkiac�− �V�ci

kbbkjca+ �V�ic

kbbkjac+ �F�c

a�bijcb− 1

2bijbc�

− �F�ik�bkj

ab− 12bjk

ab�0=b0

ia=bi

a

ijab=bij

ab− 12bij

ba

Left-hand projection

�0=�2bckfk

c+2bcdkl �vkl

cd− 12vlk

cd��a

i = ��2b0+ �W��fai +ba

i �X�−bak�F�k

i +bci �F�c

a+2bck�vka

ci − 12vak

ci �+2�vac

ik − 12vca

ik ��I�kc

+�2�bcaki − 1

2bcaik ��F�k

c+bcdik �V�ak

cd−bcakl �V�kl

ci− 12 fa

k�I�ki − �val

ik

− 12vla

ik��I�kl − 1

2 fci �I�a

c + �vacid − 1

2vcaid ��I�d

c�ab

ij =2b0�vabij − 1

2vbaij �+ �bab

ij − 12bba

ij ��X�+ �W��vabji − 1

2vbaji �

+babkl �V�kl

ij +bcdij �V�ab

cd

+Pijab�2�ba

i fbj − 1

2baj fb

i +bci �vab

cj − 12vab

jc �−bak�vkb

ij − 12vbk

ij ��+2�bac

ik − 12bac

ki ��V�bkjc −bac

jk �V�bkic

+bcajk �V�kb

ic + �bcbij − 1

2bbcij ��F�a

c − �babkj − 1

2babjk ��F�k

i − �vbcji − 1

2vcbji �

��I�ac − �v jk

ba− 12vkj

ba��I�ki

0=b0

ai =ba

i

abij =bab

ij − 12bba

ij

Intermediate arrays for SAC-CI singlet states

�X�=2�vcdkl − 1

2vdckl �ckl

cd

�Y�=�2fckbk

c+2�vcdkl − 1

2vdckl �bkl

cd

�W�= �bcdkl − 1

2bcdlk �ckl

cd

�F� ji = f j

i − �vcdik − 1

2vcdki �cjk

cd

�F�ba= fb

a− �vcbkl − 1

2vbckl �ckl

ca

�F�ia= f i

a+ fck�cik

ac− 12cik

ca�− �vickl− 1

2vcikl�ckl

ac+ �vcdak− 1

2vcdka�cik

cd

�G� ji =

�22 fc

i bjc+�2�vcj

ki − 12v jc

ki�bkc+ �vcd

ki − 12vcd

ik �bkjcd

�G�ba=−

�22 fb

kbka+�2�vbc

ak− 12vcb

ak�bkc− �vcb

kl − 12vbc

kl �bklca

�G�ai = �vac

ik − 12vca

ik �bkc

�I� ji =bcd

ki �ckjcd− 1

2cjkcd�+bcd

ik �cjkcd− 1

2ckjcd�

�I�ba=bcb

kl �cklca− 1

2cklac�+bbc

kl �cklac− 1

2cklca�

�I�ia=bk

c�ckica− 1

2ckiac�

�V�ciab=vci

ab− 12vic

ab+ �vcdak− 1

2vdcak��cik

bd− 12cik

db�− 12

�vdckb− 1

2vcdkb�cik

ad

− 12

�vcdkb− 1

2vdckb�cik

da+ 12

�vcikl− 1

2vickl�ckl

ab

�V�ijak=vij

ak− 12vij

ka+ �v jckl − 1

2vcjkl��cil

ac− 12cil

ca�− 12

�vickl− 1

2vcikl�cjl

ac

− 12

�vcikl− 1

2vickl�cjl

ca+ 12

�vcdak− 1

2vdcak�cij

cd

�V�ijkl= �vij

kl− 12v ji

kl�+ 12

�vcdkl − 1

2vdckl �cij

cd

�V�cdab= �vcd

ab− 12vcd

ba�+ 12

�vcdkl − 1

2vcdlk �ckl

ab

�V�cika= �vci

ka− 12vci

ak�+ �vcdkl − 1

2vdckl ��cil

ad− 12cil

da��V�ic

ka=−�vicka− 1

2vcika�+ 1

2�vcd

kl − 12vdc

kl �cilad+ 1

2�vdc

kl − 12vcd

kl �cilda

094105-12 R. Fukuda and H. Nakatsuji J. Chem. Phys. 128, 094105 �2008�

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ACKNOWLEDGMENTS

The discussions in the SAC-CI meeting of our laboratorywere very valuable. We thank the members of the SAC-CImeeting, particularly Dr. M. Hada and Dr. M. Ehara. Thiswork has been supported by the grant for creative scientificresearch from the Ministry of Education, Science, Culture,and Sports of Japan.

APPENDIX: WORKING EQUATIONS

We summarize here the working equations for the singletSAC equation in Table X and for the singlet and tripletSAC-CI equations in Tables XI and XII, respectively. Theworking equations for the SAC-CI ionized and electron at-tached doublet states are obtained by replacing the orbitalsby infinitely separated orbitals as in Eqs. �34� and �35�, re-spectively. Note that the matrix elements including indices�, such as f i

�, vij�a, etc., are zero.

1 H. Nakatusji and K. Hirao, J. Chem. Phys. 68, 2053 �1978�.2 H. Nakatsuji, Chem. Phys. Lett. 59, 362 �1978�; 67, 329 �1979�; 67, 334�1979�.

3 H. Nakatsuji, Acta Chim. Hung. 129, 719 �1992�; Computational Chem-istry: Reviews of Current Trends, edited by J. Leszczynski �World Scien-tific, Singapore, 1997�, Vol. 2, pp. 62–124; M. Ehara, M. Ishida, K.Toyota, and H. Nakatsuji, in Reviews of Modern Quantum Chemistry: ACelebration of the Contributions of Robert G. Parr, edited by K. D. Sen�World Scientific, Singapore, 2002�, pp. 293–319; M. Ehara, J. Hase-gawa, and H. Nakatsuji, in Theory and Application of ComputationalChemistry, The First 40 Years, edited by C. E. Dykstra, G. Frenking, K.S. Kim, and G. E. Scuseria �Elsevier, Oxford, 2005�, Chap. 39, pp. 1099–1141.

4 D. Mukherjee and P. K. Mukherjee, Chem. Phys. 39, 325 �1979�; H.Koch and P. Jorgensen, J. Chem. Phys. 93, 3333 �1990�; H. Koch, H. J.Aa. Jansen, T. Helgaker, and P. Jorgensen, ibid. 93, 3345 �1990�.

5 J. Geersten, M. Rittby, and R. J. Bartlett, Chem. Phys. Lett. 164, 57�1989�; J. F. Stanton and R. J. Bartlett, J. Chem. Phys. 98, 7029 �1993�.

6 H. Nakatsuji, Program System for SAC and SAC-CI Calculations, Pro-gram Library No. 146�Y4/SAC�, Data Processing Center of Kyoto Uni-versity, 1985; Program Library SAC85 �No. 1396�, Computer Center ofthe Institute for Molecular Science, Okazaki, Japan, 1986.

7 H. Nakatsuji, Chem. Phys. Lett. 177, 331 �1991�.8 H. Nakatsuji, J. Chem. Phys. 83, 713 �1985�.9 H. Nakatsuji, J. Chem. Phys. 83, 5743 �1985�; Theor. Chim. Acta 71,201 �1987�; J. Chem. Phys. 94, 6716 �1991�; 95, 4296 �1991�.

10 H. Nakatsuji and M. Ehara, J. Chem. Phys. 98, 7179 �1993�; 99, 1952�1993�.

11 T. Nakajima and H. Nakatsuji, Chem. Phys. Lett. 280, 79 �1997�; Chem.Phys. 242, 177 �1998�.

12 H. Nakatsuji, T. Miyahara, and R. Fukuda, J. Chem. Phys. 126, 084104�2007�.

13 M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., GAUSSIAN03, Gaussian,Inc., Pittsburgh, PA, 2003.

14 H. Nakatsuji, Chem. Phys. 75, 425 �1983�.15 H. Nakatsuji, J. Hasegawa, and M. Hada, J. Chem. Phys. 104, 2321

�1996�.16 K. Hirao, J. Chem. Phys. 79, 5000 �1983�.17 B. Roos, Chem. Phys. Lett. 15, 153 �1972�.18 J. A. Pople, R. Krishnan, H. B. Schlegel, and J. S. Binkley, Int. J. Quan-

tum Chem. 14, 545 �1978�; R. J. Bartlett and G. D. Purvis, ibid. 14, 561�1978�.

19 G. E. Scuseria, C. L. Janssen, and H. F. Schaefer III, J. Chem. Phys. 89,7382 �1988�.

20 H. Nakatsuji, M. Hada, M. Ehara et al., SAC-CI Guide, 2005. �a pdf fileis available at http://www.qcri.or.jp/sacci/�.

21 K. Ueda, M. Hoshino, T. Tanaka et al., Phys. Rev. Lett. 94, 243004�2005�; M. Ehara, H. Nakatsuji, M. Matsumoto et al., J. Chem. Phys.124, 124311 �2006�; M. Ehara, K. Kuramoto, H. Nakatsuji, M. Hoshino,T. Tanaka, M. Kitajima, H. Tanaka, Y. Tamenori, and K. Ueda, ibid. 125,114303 �2006�; R. Sankari, M. Ehara, H. Nakatsuji, A. De Fanis, H.

TABLE XII. Working equations for SAC-CI triplet states. �We assume sum-mation over all repeating indices; however, b with identical virtual orbitalsmust be dropped �e.g., vcd

ki bjkcd� k c���dvcd

ki bjkcd�.�

Right-hand projection�i

a= �X�dia− �F�i

kbka+ �F�c

abic−vci

akbkc− �G�c

kckiac

+�2� fck�bki

ca− 12bik

ca− 12bki

ac�+ �vkacd− 1

2vakcd�bki

cd− �vcikl− 1

2vickl�bkl

ca

− 12 �vka

cdbikcd−vci

klbklac��

�ijab= �X��bij

ab− 12bij

ba− 12bji

ab�+�2�F�iabj

b+ �V�cibabj

c− �V� jiakbk

b

+ �V�cjbabi

c− �V� jibkbk

a+�V�ij

lkbklba+ �V�cd

abbijcd+2�V�ci

ka�bkjcb− 1

2bkjbc�− �V�ci

kabjkcb+ �V�ic

kbbkjac

+ �V� jckabik

cb

+�V�ijklbkl

ab+ �V�dcabbji

cd+2�V�cjkb�bik

ac− 12bik

ca�− �V�cjkbbki

ac+ �V� jckabki

bc

+ �V�ickbbjk

ca

+�F�cabij

cb− �F�ikbkj

ab+ �G�cbcij

ac+ �F�cbbij

ac− �F� jkbik

ab− �G� jkcik

ab

−Pijab �2

2 �F�ibbj

a+ �V�cikbbjk

ac+ �V�cikbbkj

ca+ 12 �F�c

a�bijbc+bji

cb�− 1

2 �F�ik�bjk

ab+bkjba�+ 1

2 �G�cacij

bc− 12 �G�i

kcjkab

ia=bi

a

ijab=bij

ab− 12bij

ba− 12bji

ab

Left-hand projection�a

i = fai �W�+ba

i �X�−bak�F�k

i +bci �F�c

a−bckvak

ci −vcaik �I�k

c

+�2 12 �−fa

k�I�ki +vla

ik�I�kl − fc

i �I�ac −vca

id �I�dc�+ �bca

ki − 12bca

ik − 12bac

ki ���F�k

c+bcd

ki �V�akdc+bcd

ik �V�akcd−bca

kl �V�klci−bac

kl �V�lkci

�abij = �bab

ij − 12bba

ij − 12bab

ji ��X�+�2bbj fa

i +bcj �vab

ic − 12vab

ci �− 1

2bci vab

jc −bbk�vak

ij − 12vka

ij �+ 12ba

kvbkij

+bcdij �V�ab

cd +bbakl �V�kl

ji +2�bcbkj − 1

2bcbjk ��V�ak

ic −bbckj �V�ak

ic +backj �V�kb

ic

+bcbik �V�ka

jc

+bcdji �V�ba

cd +babkl �V�kl

ij +2�bacik − 1

2backi ��V�bk

jc −bcaik �V�bk

jc +bbcki �V�ka

jc

+bcajk �V�kb

ic

+bcbij �F�a

c −babkj �F�k

i +bacij �F�b

c −babik �F�k

j −vcaji �I�b

c −vkiba�I�k

j

−Pabij �2

2 baj fb

i +bacjk �V�bk

ic +bcakj �V�bk

ic + 12 �bcb

ji +bbcij ��F�a

c − 12 �bab

jk

+bbakj ��F�k

i − 12vcb

ji �I�ac − 1

2vkjba�I�k

i a

i =bai

abij =bab

ij − 12bba

ij − 12bab

ji

Intermediate arrays for triplet states

�X�= �vcdkl − 1

2vdckl �ckl

cd

�F� ji = f j

i − �vcdik − 1

2vcdki �cjk

cd

�F�ba= fb

a− �vcbkl − 1

2vbckl �ckl

ca

�F�ia= f i

a+ fck�cik

ac− 12cik

ca�− �vickl− 1

2vcikl�ckl

ac+ �vcdak− 1

2vcdka�cik

cd

�G� ji =

�22 fc

i bjc+�2�− 1

2v jcki�bk

c+ �vcdki − 1

2vcdik �bkj

cd+ �− 12vcd

ki �bjkcd

�G�ba=−

�22 fb

kdka+�2�− 1

2vcbak�dk

c− �vcbkl − 1

2vbckl �bkl

ca− �− 12vcb

kl �bklac

�G�ai =− 1

2vcaik dk

c

�I� ji =bcd

ki �ckjcd− 1

2cjkcd�− 1

2bcdik ckj

cd

�I�ba=bcb

kl �cklca− 1

2cklac�− 1

2bbckl ckl

ca

�I�ia=− 1

2bkccki

ac

�V�ciab=vci

ab− 12vic

ab+ 12

�vcikl− 1

2vickl�ckl

ab+ �vcdak− 1

2vdcak��cik

bd− 12cik

db�− 1

2�vdc

kb− 12vcd

kb�cikad− 1

2�vcd

kb− 12vdc

kb�cikda

�V�ijak=vij

ak− 12vij

ka+ 12

�vcdak− 1

2vdcak�cij

cd+ �v jckl − 1

2vcjkl��cil

ac− 12cil

ca�− 1

2�vic

kl− 12vci

kl�cjlac− 1

2�vci

kl− 12vic

kl�cjlca

�V�ijkl= �vij

kl− 12v ji

kl�+ 12

�vcdkl − 1

2vdckl �cij

cd

�V�cdab= �vcd

ab− 12vcd

ba�+ 12

�vcdkl − 1

2vcdlk �ckl

ab

�V�cika= �vci

ka− 12vci

ak�+ �vcdkl − 1

2vdckl ��cil

ad− 12cil

da��V�ic

ka=−�vicka− 1

2vcika�+ 1

2�vcd

kl − 12vdc

kl �cilad+ 1

2�vdc

kl − 12vcd

kl �cilda

�V�ijak=− 1

2vijka− 1

4vdcakcij

cd+ 14vcj

klcilca− 1

2�vic

kl− 12vci

kl�cjlac+ 1

4vicklcjl

ca

�V�ciab=− 1

2vicab− 1

4vicklckl

ab+ 14vdc

akcikdb− 1

2�vdc

kb− 12vcd

kb�cikad+ 1

4vdckbcik

da

�V�ijkl=− 1

2v jikl− 1

4vdckl cij

cd

�V�cdab=− 1

2vcdba− 1

4vcdlk ckl

ab

�V�cika=− 1

2vciak+ 1

4vdckl cil

da

�V�icka= 1

2vcika+ 1

2�vcd

kl − 12vdc

kl �cilad− 1

4vcdkl cil

da

094105-13 Direct algorithm for SAC-CI J. Chem. Phys. 128, 094105 �2008�

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094105-14 R. Fukuda and H. Nakatsuji J. Chem. Phys. 128, 094105 �2008�

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