Quadratic configuration interaction. A general technique for determining electron correlation energies John A. Pople, Martin HeadGordon, and Krishnan Raghavachari Citation: J. Chem. Phys. 87, 5968 (1987); doi: 10.1063/1.453520 View online: http://dx.doi.org/10.1063/1.453520 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v87/i10 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 14 Aug 2012 to 140.123.79.51. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions Quadratic configuration interaction. A general technique for determining electron correlation energies John A. Pople and Martin Head-Gordon Department of Chemistry, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Krishnan Raghavachari AT&T Bell Laboratories, Murray Hill, New Jersey 07974 (Received 1 June 1987; accepted 14 August 1987) A general procedure is introduced for calculation of the electron correlation energy, starting from a single Hartree-Fock determinant. The normal equations of (linear) configuration interaction theory are modified by introducing new terms which are quadratic in the configuration coefficients and which ensure size consistency in the resulting total energy. When used in the truncated configuration space of single and double substitutions, the method, termed QCISD, leads to a tractable set of quadratic equations. The relation of this method to coupled-cluster (CCSD) theory is discussed. A simplified method of adding corrections for triple substitutions is outlined, leading to a method termed QCISD(T). Both of these new procedures are tested (and compared with other procedures) by application to some small systems for which full configuration interaction results are available. I. INTRODUCTION no longer variational. However, this Davidson correction fails to give correct answers for a two-electron system (con­ There is extensive literature on theoretical methods for dition 3). The CISD method also completely omits effects of the electron correlation correction to single-configuration, triple substitutions, which are known to be important.4 Hartree-Fock wave functions and energies. Even with orbi­ A second range of approximate correlation methods is tal expansion in a finite basis, a full solution (full configura­ based on many-body perturbation theory, introduced origin­ tion interaction or FC!) is only possible for small systems. ! ally by M011er and Plesset5 (MP). These treat the correla­ Other approximate methods are subject to various advan­ tion part of the Hamiltonian as a perturbation on the Har­ tages and disadvantages which have been debated at some tree-Fock part and truncate the energy expansion at some length. Desirable features of a theoretical model chemistry order. The theory is fully practical to fourth order (MP4 ), at 2 including electron correlation are : which level it incorporates the effects of single, double, tri­ ( I ) It should be well defined, leading to a unique energy ple, and quadruple substitutions.4 The method is size consis­ for any nuclear configuration and a continuous potential tent at any order but not variational. Its principal deficiency surface. is that the MP series sometimes converges slowly,6 so that a (2) It should be size consistent, so that when applied to truncated expansion may not adequately satisfy condition an ensemble of isolated molecules, calculated energies (5). should be additive. The third set of methods utilize coupled cluster theory, (3) It should be exact (equivalent to FCI) when applied introduced into quantum chemistry by Cizek. 7 These meth­ to a two-electron system. ods introduce substituted configurations in a multiplicative ( 4) It should be efficient, so that application is possible (exponential) rather than an additive manner. The configu­ for large basis sets. ration coefficients are found by requiring that the projection (5) It should be accurate enough to give an adequate of the Schr6dinger function (H - E) 'II onto various deter­ approximation to the FCI result. minants is zero. Such procedures are size consistent but not (6) It should be variational, so that the computed ener­ variational. There are indications that they are still effective gy is an upper bound to the correct energy. under circumstances where perturbation theory conver­ 8 No current method satisfies all of these criteria; most com­ gence is slow. ,9 When limited to double substitutions, the promise by introducing approximations with varying de­ coupled cluster method is usually denoted by CCD. It was grees of success. originally implemented partially by Taylor, Bacskay, Hush, Of current methods, limited (or truncated) configura­ and Hurley,1O and fully by Pople, Krishnan, Schlegel, and tion interaction has been used extensively. If all double-sub­ Binkley!! and by Bartlett and Purvis.!2 The latter authors stitution configurations from a Hartree-Fock reference are later extended their work to include single and double substi­ included, the method is called CID. If single substitutions tutions (CCSD).13 Like CISD, the CCD and CCSD meth­ are added, it becomes CISD. These methods are variational ods take inadequate account of triple substitutions, although (condition 6) but not size consistent (condition 2). In appli­ recently Bartlett and co-workers have implemented the cou­ cations to a range of molecules, size consistency is generally pled cluster theory including triples (CCSDT) and have regarded as being more important than provision of an ener­ also proposed some approximate treatments. !4 In other ap­ gy upper bound, so CISD results are usually modified by a plications, hybrid methods are used in which triple substitu­ simple correction, introduced by Langhoff and Davidson,3 tions are added to CCD or CCSD theory by perturbation which makes the energies approximately size consistent but methods. 14,!5 5968 J. Chem. Phys. 87 (10),15 November 1987 0021-9606/87/225968-08$02.10 © 1987 American Institute of PhySics Downloaded 14 Aug 2012 to 140.123.79.51. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions Pople. Head-Gordon. and Raghavachari: Quadratic configuration interaction 5969 In this paper, we introduce a new series of methods where H is the full Hamiltonian, and E the total energy. which are, in a sense, intermediate between configuration Thus interaction and coupled cluster theory. The idea is to modify ('I1oIH - E 1'11) = 0, (2.4) the configuration interaction equations in a simple manner, so as to restore size consistency [condition (2) ] at the cost of ('I1fIH - E 1'11) = 0, (2.5) loss of variational character [condition (6)]. This requires ('I1f/IH - E 1'11) = 0, (2.6) additional terms which are quadratic in the general substitu­ where 'I1f is the singly substituted determinant 1 f'l1 0 and so tion operators and leads to a treatment that is somewhat forth. The number of such projection equations can be cho­ simpler then coupled cluster methods, at least for higher sen to be equal to the number of unknowns. For example, if levels of substitution. The theory and its implementation for only single and double substitution operators are used (SD single and double substitutions is outlined in Sec. II. This is methods), Eqs. (2.4) to (2.6) suffice to determine the un­ followed in Sec. III by an approximate treatment of the triple knowns E, af,a'f/. substitution contributions. Finally, Sec. IV presents some The simplest choice for the configuration function f is comparisons between the new method and some of the ear­ linear, leading to the configuration interaction wave func­ lier techniques described above. tions16 II. FORMULATION OF QUADRATIC CONFIGURATION 'I1CID = (1 + T2 )'I10' (2.7) INTERACTION 'I1C1SD = (1 + Tl + T2)'I10 ' (2.8) All of the theoretical techniques discussed here begin It should be noted that we use the intermediate normaliza­ with the Hartree-Fock single-determinant wave function tion convention ('1101'11) = I throughout. The projection '110= (n!)-1/2det{xl"'Xn}' (2.1) equations (2.4) to (2.6) are then identical with those ob­ tained by minimization of the expectation energy where Xi (i = 1,' . ,n) are occupied spin orbitals. We will consider only the spin-unrestricted (UHF) form of (2.1), E = ('I1IH 1'11)/('111'11), so that the CID and CISD energies for which each Xi is an eigenfunction of the one-electron are variational upper bounds to the exact (FCI) result. If we define Fock operator F with eigenvalue Ei • The remaining eigen­ functions and eigenvalues ofF (actual spin orbitals and ener­ H=F+ V, (2.9) gies) are Xa,Ea (a = n l,n 2, ... ,N), where N is the di­ + + EHF = ('I1oIH 1'110), (2.10) mension of the basis. Following a widespread convention, we will use symbols i,j, k, ... for occupied and a, b, c, ... for virtual E = EHF + Ecorrelation, (2.11 ) spin orbitals without further identification. H=H-EHF , (2.12) Other determinantal wave functions are derived from v= V - ('110 IV 1'110), (2.13 ) '110 by substitution of occupied spin orbitals by virtual spin orbitals. Following general practice, we define complete sin­ where F is the Fock Hamiltonian, then the CISD projection gle, double, triple, ... substitution operators. equations can be written ('I1oIH IT2 '11 0) = EcorreJation' (2.14 ) Tl = L a':1~, ia ('I1fIH I(TI + T2)'I10 ) =afEcorreJation' (2.15) b (2.2) ('I1ij IHI(1 + TJ + T2)'I10 ) =aijbEcorreJation' (2.16) In deriving these equations from the projection conditions (2.4)-(2.6), we have replaced ('I1fIH 1'11 ) by zero for all i,a. T 3--- 1 ~ a-",-I--k'abc<>abe 0 36 ij be IJ IJ This is because '110 is the optimized Hartree-Fock function where 1 f,l f/, ..