Size consistent multireference single and double excitation configuration interaction calculations. The multireference coupled electron-pair approximation P. J. A. Ruttink, J. H. van Lenthe, and R. Zwaans Theoretical Chemistry Group, State University of Utrecht, Padualaan 14, 3584 CH Utrecht, The Netherlands G. C. Groenenboom Department of Organic Chemistry, Eindhoven University of Technology, P. 0. Box 513, 5600 MB Eindhoven, The Netherlands (Received 2 1 November 1990; accepted 22 February 199 1) A new size consistent extension to the multi reference configuration interaction method is described. The method termed multireference coupled electron pair approximation (MRCEPA) is akin to a multireference CEPA( 0) approach, though nonlinear terms do receive separate attention. We show the performance of the approach in some model systems as well as in an application to calculation of ground and excited TV* states of ethylene. I. INTRODUCTION The results obtained by this method will be compared to results of the averaged coupled pair functional ( ACPF) the- The coupled electron pair approximation (CEPA) ory, as introduced recently by Gdanitz and Ahlrichs’3 and method has the advantage over the method for configuration which may also be considered as a generalization of the interaction with all single and double replacements (SDCI) CEPA (or CPF) philosophy to MC reference configuration that it yields size consistent results with only a slight increase sets. in computational effort. Here, size consistency means that In Sec. II, the method used for modifying the MRSDCI the energy of two subsystems is additive if the subsystems do equations will be discussed. Applications to some model sys- not interact. It may be successfully applied to both closed’ tems, including comparison with full CI and MRACPFi3 and open* shell systems, provided that a single configuration results, are reported in Sec. III and the method is applied to state function (CSF), e.g., the restricted Hartree-Fock de- the calculation of ground and excited states of ethylene in terminant, is dominant in the correlated wave function. Sec. IV. Finally, conclusions are drawn in Sec. V. However, if there are low lying excitations (in case of multi- bond dissociation, open shell systems with high symmetry), II. THEORY more than one CSF is needed for a proper zeroth order de- scription of the wave function.3*4 In this section the size consistency corrections for sec- Fairly accurate results may be obtained using the multi- ond order wave functions will be discussed in terms of the reference single double configuration interaction spin orbital formalism. Our reasoning is analogous to the (MRSDCI) method. However, since any restricted CI cal- derivation of the coupled pair approximation (CPA) equa- culation suffers from size consistency defects, an extension tions by Hurley. l4 of the CEPA method to multiconfiguration (MC) reference sets is called for. In the following, a method is proposed, A. Wave functions which includes size consistency corrections in MRSDCI cal- culations in a manner analogous to the CEPA(0) method, The second order wave function to be used is designated provided that the reference configuration set is complete in by the active orbital space. Thus, our starting wave function is a second order CI function and the resulting function will ac- IyC)=~cRIR)+~ciIAi)9 (1) cordingly be called the second order CEPA function. where the refetence conhgiration set { IR )} is assumed to be In practice, our method amounts to shifting the diag- complete within some (small) orbital subset (the active spin onal H-matrix elements of the single and double excitations, orbitals) and the set {[Ai)} contains all single and double as in the CI-type implementation of the CEPA method.’ excitations with respect to any (R ). However, our method dithers from the straightforward gen- The size consistency corrections to be applied to the CI eralization of the CEPA(0) method to MC reference sets, secular equations for this function will be calculated in the which should be identical to the linearized version of the MC form of diagonal H-matrix element shifts. These shifts are reference coupled cluster (CC) development.“‘2 Since the calculated by assuming that the second order CEPA func- nonlinear terms in our modified MRSDCI equations are po- tion may be approximated by the corresponding MC refer- tentially important, they should receive proper attention ence coupled cluster function”’ with internal contraction:i5 when formulating size consistency corrections for wave functions based on a MC reference configuration set. IYcc> = exp(T) IO), (2) 7212 J. Chem. Phys. 94 (1 l), 1 June 1991 0021-9606/Q? /I 17212.09$03.00 @ 1991 American Institute of Physics Downloaded 12 Sep 2003 to 131.174.188.149. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp Ruttink eta/: Configuration interaction calculations 7213 where the reference function 10) is given by which is given by Pb Y,, ) do not interact with any reference configuration ]R ), Eq. (9) is identical to the first secular IO>=pI$w c’“7 (3) equation for the corresponding CI problem: R Al-- E = (OIH IO> + (OJHT IO> = E, + IL,,,. (11) with the same coefficients c, as in Eq. ( 1). T generates all single and double excitations with respect to any ]R ), Using Eqs. (2)) (7)) and (8), Eq. ( 10) may be rewritten as T= c 2 t,lPrPP + c c c c L,P~~~v-Po~ (4) (Olr+(H - E) IY,) + (0]7+HP, exp(T) IO) = 0. (12) P P P v P 0 where r is an excitation operator, t is the corresponding con- The second term in Eq. ( 12)) which is missing in the secular nected cluster amplitude, p and Y are inactive or active spin equations for the SDCI function, will be used to obtain the orbital indices, andp and u are active or external spin orbital size-consistency corrections wanted. To this end we note indices. According to Eq. (3)) the reference function IO) is that Eq. (12) may be rewritten as: normalized. The excitation operators are defined such that (OJr+(H+ AH, - E)IY,) = 0, (13) they yield normalized excited functions r]O), i.e., we assume provided that the diagonal shift AH, is given by (olr+rlo) = 1. (5) AH, = (OldHP, exp( T) ]O)/(O]rt]Yc). (14) We do not need the explicit form of T, as given by Eq. In the following, Eqs. ( 13) and ( 14) will be taken as the (4). Instead, we define excitation classes as follows; The ex- set of equations to be satisfied by the second order CEPA citation class (k,(3 contains all excitations with k holes in function Yc. These equations lead to a pseudo eigenvalue the inactive orbitals and /particles in the external orbitals. problem since the shifts AH, depend on the connected clus- Since the number of electrons in the active orbitals is not ter amplitudes in T, or, equivalently, on the configuration fixed, k and kmay differ from each other. Eq. (4) is thus coefficients ci in Y c. equivalent to Note that the coefficients ci in Eq. ( 1) are not identical T= C C T(k,d O<k,&Z, except k= F=O. (6) to the connected cluster amplitudes t in Eq. (4) since the ci k / will contain contributions from disconnected clusters.‘6”7 We also use the corresponding projection operators In the following, Eq. (14) is simplified using two ap- P( k,f’J and the projection operators P,, Pa, and Pb defined proximations which are analogous to the approximations by: used in the CEPA(0) method. PO = P(O,O), C. The direct term approximation This approximation was introduced by Kelly and P, =~~P(k,e), k / Sessler18 and Kelly. l9 In our formalism it takes the follow- O<k,&Z, except k = /= 0 ing form. In the expansion of exp( ZJ IO) in Eq. ( 14), only those terms are retained which contain 7. The selection of Pt,=~~PWl, terms thus depends on the excitation r for which the shift is k / being calculated. In order to determine which terms in k>2 or f52. (7) exp( 2’) IO) contain the excitation 710) and also which terms Thus, P,, projects to the reference configuration space, P, belong to the interacting space of T]O), we divide the excita- projects to the interacting space of the reference configura- tion operators in Tin two groups. The first group T, contains tions spanned by the set {]Ai)] and Pb projects to the space r and all excitation operators r’ which only contain spin spanned by all higher excitations. In practice, Pb may be orbital indices which are also present in 7. The complement considered to project to the interacting space of the set of T, is denoted by T,“. It contains all other excitation opera- tors. Assuming that all excitation operators mutually com- CIAi)I* In the following, we assume that the second order mute, we then have CEPA function may be written as exp(T)lO) = exp(T’,)exp(T,)lO). (15) I*,) = (PC +P,)IY,,) = [I +P, exp(T)]lO) (8) The direct term approximation is obtained by projecting This function will be used for calculating the size consistency the second factor in Eq. ( 15 ) to Q-IO),after projection by Pb corrections to be applied to the second order SDCI function [cf. Eq. ( 14) 1. This yields in order to obtain the second order CEPA function. P,, exp( T) 10) z=Pb exp( T’, )rlO) B. The diagonal H-matrix element shift X (ol~teWT,)lO). (16) Because of the Pauli exclusion principle, we have #r/O) = 0. The coupled cluster function is assumed to satisfy the Therefore, T’, in Eq.