Quadratic Configuration Interaction. a General Technique for Determining Electron Correlation Energies John A

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.

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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

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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/, ... are elementary substitution operators and (Brillouin's theorem). the a arrays involve coefficients to be determined. The effect The CISD energy emerging from (2.14) to (2.16) is not of a substitution operator (e.g., 1 'f/) on a determinant where size consistent. This is because the right-hand sides of Eqs. the occupied orbitals (i or j) are not there or where the vir (2.15) and (2.16) are quadratic in the a vectors, while the tualorbitals (a or b) are already there is zero. The operators I~ft-hand sides are linear. Consider, for example, two infi act only on antisymmetric determinants and hence the a ar Dltely separated molecules X and Y. If ij -+ ab is a double rays will be taken to be antisymmetric in both the occupied substitution on X, the corresponding doubles equation (2.16) becomes and virtual suffixes. Thus T2 '110, for example, is a general b linear combination of all determinants obtained by double ('I1ij IHI(1 + Tl + T2 )'I10 ) substitution in '11 ' 0 = af/(X) [EcorreJation (X) + Ecorrelation (Y)]. (2.17) Various types of antisymmetric wave functions can be obtained by applying various functions of the T operators to The term af/(X)EcorreJation (Y) on the right here implies that '110, the value of an a coefficient on X depends on the nature of Y. since no linear term of the left can cancel this. Hence, size~ (2.3) inconsistent results are obtained. and then determining the unknown a coefficients by appro The simplest method ofcorrecting this type of error is to priate projection of the Schrodinger function (H - E) '11, add terms on the left of Eqs. (2.15) and (2.16) which are

J. Chern. Phys .• Vol. 87. No. 10. 15 November 1987

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 5970 Pople, Head-Gordon, and Raghavachari: Quadratic configuration interaction quadratic in T I, T2 and which eliminate the term on the right method and coupled cluster theory. In the space of single involving Ecorrelation' The equations proposed are and double substitutions, coupled cluster theory (CCSD) uses an exponential type of wave function. ('I1oIH IT2'110) = Ecorrelation' (2.18) 'I1CCSD = exp(TI + Ti )'110' (2.25) ('I1~IH I(TI + T2 + TIT2)'I10) = a~Ecorrelation' (2.19) b If we define T2 = Ti + !Ti and expand the exponential in ('I1ij IHI(1 + TI + T2 +! n)'I1o) =aijbEcorrelation' (2.20) the projection equations (2.4)-(2.6), we obtain the CCSD These equations lead to size-consistent energies, as will be equations in the form proved after further development. For a two-electron system, Eqs. (2.18 )-( 2.20) are iden (2.26) tical with the CISD equations (2.14)-(2.16). This is be ('I1~IH I(T I + T2 + TIT2 - jTi )'I10) = a~Ecorrelation' cause the functions TI T2 '11 0 and! T ~ '110 are sums of triple and (2.27) quadruple substitutions, respectively, which vanish if only b ('I1ij IH I (1 + TI + T2 + TIT2 -!Ti two electrons are present. Thus the proposed method then gives correct (FCl) answers, satisfying condition (3) men +!T~ --bTt)'I1o) =a'f/Ecorreiation' (2.28) tioned in the Introduction. We shall demonstrate that this Equations equivalent to these were first implemented in the and related methods are also efficient and accurate [condi CCSD program of Purvis and Bartlett. 13 They demonstrated tions (4) and (5)] later in this article. that size consistency followed from the exponential form of The theoretical method implicit in Eqs. (2.18)-(2.20) the wave function. However, the equations are quartic and might be described as configuration interaction, with qua considerably more complex that the QCISD equations dratic terms added to restore size consistency. We propose (2.18)-(2.20). the term quadratic configuration interaction (QCI) for such If single substitutions are omitted, TI must be removed a procedure. If the space is restricted to single and double from the CCSD equations, leading to the familiar CCD substitutions, as in these equations, we shall denote the equations. The same CCD equations also follow by omitting method by the acronym QCISD. TI in Eqs. (2.18)-(2.20). Thus CCD and QCID are identi The general concept of quadratic configuration interac cal procedures. tion can also be applied to higher levels of substitution. If all The QCISD equations can be obtained by leaving out substitutions up to m-fold are included, the projection equa certain terms in Eqs. (2.26 )-(2.28), so that QCISD could be tions for all levels up to (m - 2) are retained in the linear regarded as a simplified approximate form of CCSD. How form, but quadratic terms T m _ I T2 and T mT2 are added to ever, we prefer to treat the quadratic configuration proce the equations forlevels (m - 1) and m. An additionalfactor dures as an independent approach to the problem of electron 1/2 must be included ifthe product T2T2 appears. Thus the correlation. QCISDT equations are We now return to the QCISD equations (2.18)-(2.20). ('I10IHIT2'110) = Ecorrelation, (2.21) All matrix elements are easily evaluated in terms of two electron integrals, using techniques decribed elsewhere. 19.20 ('I1~IH I(T I + T2 + T3)'I1 ) = a~Ecorrelation' (2.22) 0 The final form of the equations can be written as a simple ('I1ijhIH I (1 + TI + T2 + T3 +!n )'110 = aijbEcorrelation' modification of the CCD equations presented some years (2.23) ago, I I ('I1ijfIH I(T I + T2 + T3 + T2IT3)'I10) = aij%CEcorrelation' (2.24) Ecorrelation =..!.. L (ijllab )aijb, (2.29) 4 (jab The series of methods will terminate with full configuration interaction if m = n. This is because the operators Tn _ I T2 A'fa~ + u'f + v~ = 0, (2.30) and Tn T2 give zero when applied to '11 0' At all levels, the (ab 11ii) + Aijbaijb + Uijb + Vijb = 0, (2.31) quadratic terms introduced serve to cancel the right-hand where (ab /lij) are the usual antisymmetrized two-electron side for the two highest levels of substitution. integrals It should be noted that no TI T2 term appears on the left of the doubles equations (2.20) for the QCISD method. This (pqllrs) = X;(l)X:(2) (1lr ) [Xr(l)Xs(2) means that the equations do not constitute a full set of pro ff 12 jection equations (2.4)-(2.6) with a well-defined wave - Xs (1 )Xr (2)]dT dT2 (2.32) function '11. However, this is not necessarily a disadvantage. I We note that the defining equations lead to a well-defined and energy E, so that specification of a full electronic wave func A~ = Ea - Eo (2.33) tion may not be necessary. We also point out that many Aijb = Ea + Eb - E; - Ej' (2.34) electron correlation methods such as CEPA,17 CPF,I8 CI including Davidson correction,3 perturbation techniques4 u~ and Uijb are the linear arrays used in CISD theory, such as MP2, MP3, or MP4, or several approximate CCSDT Uf= - L (jallib)aJ -..!.. L (jallbc)aZC methods 14 also do not have an associated unique wave func jb 2 jbc tion. Before proceeding with further development of the - ..!.. L (jk Ilib )aj't, (2.35) QCISD equations, we will discuss the relation between this 2 jkb

J. Chern. Phys., Vol. 87, NO.10,15 November 1987

u';/ = I'[ (ab Jlej)a~ - (ab Ilei)a)] The first iteration of Eqs. (2.39) and (2.40), with c u = v = 0, leads to zero values for a'; (X) ,a'; ( y), and aijb(XY), since (ab Jlij) is nonzero only if i,j, a, b are on the + L [ - (kb Jlij)a% + (kaJlij)a~ ] same center. For later iterations, the u and v arrays must be k determined from Eqs. (2.35) to (2.38). This gives nonzero values for a';(X) and a'; ( Y). However, aijh(XY) remains zero. This is because if aijh(XY) is zero in the previous cycle, Uijb(XY) and Vijb(XY) are both zero from Eqs. (2.36) and - L[ (kb Ilje)a';{ + (kaIVe)a~Z kc (2.38). Examination of individual terms shows this result. For example, consider the term - ~kc(kb IVe)a':{ in the (2.36) + (kb Ilie)a%j + (kallie)a%J] , expression for Uijb. If (ia) is on X and (jb) is on Y, then the v'; and v';/ are the quadratic arrays integral is nonvanishing only if the suffixes (ke) are both on Y. But a';{ (XY) is then zero from the previous cycle. Similar v'; =.!. L (jk JIbe) [afaj; + aja~Z + 2aJa,;{] , (2.37) considerations apply to all other terms, so the aijb(XY) re 2 jkbc mains zero at every iteration by induction. As a result, the vijab = -1 £.,~(kllld)[ e aijcdab ak1 - 2( aijacbd+ ak1 aijbdac) ak1 computation of the u and v arrays at each iteration splits into 4 klcd separate computations for the isolated molecules X and Y. - 2( aikajlab cd + aikacd jlab) + 4( aikaac jlbd + aikajlbd ac)] . (238) . This leads to a final T2 which is a sum T2(X) + T2( Y) lead ing to energy additivity from Eq. (2.29). Equations (2.30) and (2.31) may be solved iteratively in the form a'; = - (a';)-I[u'; + vn, (2.39) III. SIMPLIFIED METHOD FOR TRIPLE SUBSTITUTIONS a';/ = - (aijh) -I [(ab IIV) + Uijb + Vijb] , (2.40) The QCISD theory introduced in the previous section still takes inadequate account of the contributions of triple using u and v arrays from the previous iteration, starting substitutions. These are known to be important and could be with u = v = O. The computer program to do this is very incorporated by proceeding to the next level, QCISDT. close to that previously developed for CISD and CCD.II,19 However, implementation of QCISDT for large systems is The only additional work is evaluation of the singles v'; array impractical at present because of handling the large number by Eq. (2.37). This can be done using intermediate arrays oftriple--triple Hamiltonian matrix elements. Some simpler already computed for Vijb. Using a notation explained in Ref. procedure is therefore desirable. 11, v'; is obtained from A useful approximation for triple substitutions is to treat them as a perturbation on a solution already obtained at v'; = -.!. L (aIX2 Ib)af -.!. L (jIX3 Ii )aj 2 b 2 j the singles-doubles level. This means that the Hamiltonian matrix elements Vs, between 'l's (s = single or double) and + L (ijIX4 Iab )aJ. (2.41) '1', (t = triple) are supposed small. At the same time, the jb energy denominator that occurs in a perturbation type ap The most time-consuming term here is the third, which in proach is taken from the eigenvalues of the Fock Hamilto 2 2 volves _n N multiplications. This is quite negligible com nian matrix or the unperturbed matrix of M011er-Plesset 2 4 pared with the - n N steps needed in the evaluation of Uijb theory. If an initial CISD wave function is, for example, by Eq. (2.36). The other terms involving single substitu tions, e.g., the evaluation of u'; by Eq. (2.35), are also (3.1 ) 2 smaller, requiring only - n N 3 computational steps. Hence, the computational effort per cycle for QCISD is close to that then the leading term in the energy lowering due to triples required for QCID or CCD. would be We now give a proof of size consistency for the QCISD T aE = (Eo - E, )-11 ('I'IVI'I',) 12 method. Suppose that we are dealing with two separated T L, molecules X and Y. Then all substitution operators 1': and 1 ijb S+D T which transfer electrons from one molecule to another = L L (Eo-E,)-lasv",V,uau' (3.2) (leading to ionic structures such as X + + Y -) can be ig su , nored, since they lead to wave functions which will have zero Here (E, - Eo) is the triples excitation energy using the matrix elements with '1'0 for any operator. We are left with: Fock Hamiltonian. Such a procedure is close to that intro ( 1) Singles operators localized on one molecule, duced by one of us 15 to compute the contribution of both 1 ';(X) ,1 ';( Y). singles and triples, starting from a converged CCD (or (2) Doubles operators localized on one molecule, QCID) wave function. A similar procedure has also been 1ijb(XX) ,1 ';/( YY). used by Urban, Noga, Cole, and Bartlettl4 to add a triples ( 3) Doubles operators split between the molecules, contribution to a converged CCSD calculation. 1ijb (Xy) with (ia) on X and (jb) on Y. Application of Eq. (3.2) to get a triples correction to Each of these operators will be associated with an a coeffi QCISD cannot be fully justified, since some contribution cient. from triples is already included in the QCISD energy, due to

J. Chern. Phys., Vol. 87, No.10, 15 November 1987

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 5972 Pople, Head-Gordon, and Raghavachari: Quadratic configuration interaction the presence of the TI Tz term in Eq. (2.19). In fact, as dem where as and au are QCISD coefficients. onstrated in the Appendix, the part of (3.2) with sand U Evaluation of the matrix elements in Eq. (3.3) leads to both being single substitutions contains the leading contri the explicit formula bution of this TI Tz quadratic term. We therefore chose to omit the sum l:~u in (3.2) and propose the remaining triples correction AD 1 ~~(Aabc)-1(2-abc -abc)-abc (34) ~T= --~~ ifk Uifk +Uijk Uijk' • 36 ijk abc

(3.3) , where u-array elements are

uijr: = a~(jk IIbe) + at(jk Ilea) + af(jk lIab) + a}(killbe) + aJ(killea)

+ aj(killab) + ak (ijllbe) + at (ijllea) + ak (ijllab) , (3.5)

ilijr: = L [aije(bellek) + ate(eallek) + af/(ab lIek) + ak~(bellej) e + a~ (eallej) + ak~ (ab lIej) + a}: (bellei) + aJ: (eallei) + aj: (ab lIei) ]

+ L[a~!(emllik) +at~(amllik) + af::'(bmllik) + a}! (emllki) +aJ':,(amllki) m (3.6)

This procedure is closely related to the treatment of triple then a~ will be zero, since it is of the type a(Xy) which was substitutions in full fourth-order M0ller-Plesset theory shown to vanish from the QCISD equations. A similar argu (MP4SDTQ). The MP4 result follows if the double coeffi ment can be constructed for all 18 terms in Eq. (3.6). It cients aijb in Eq. (3.6) are replaced by their first-order values, follows that the triples energy correction (3.4) is just the - (Aijb)-I(ab ljij). The term resulting from Eq. (3.5) is sum of XXX and YYY parts, so that size consistency is re closely related to a term in fifth-order perturbation theory tained. which results from the interaction of single and triple substi tutions. The method in which the triples correction aET from Eq. (3.3) is added to the QCISD energy will be denoted by IV. PRELIMINARY APPLICATIONS QCISD(T), as it represents a compromise in which triples In this section, we report the application of the QCISD are only treated in a partial manner. A computer program to and QCISD (T) methods to a set of atoms and molecules for evaluate aET is close to that used in MP4 theory, which which Bauschlicher and co-workers have recently reported already contains code to generate uijr: from Eq. (3.6). This full configuration interaction (FCl) results. I The objective step requires _n3N 4 multiplications and is the only part of is to test whether the new methods are as successful or more the computation proportional to the seventh power of the successful then previous ones in coming close to the correct size of the system. The additional terms ilijte from Eq. (3.5) result (within the space defined by a particular basis set). 3 require only - n N 3 steps and form a relatively insignificant The results test the applicability of these theoretical methods part of the total computation. Since the preceding QCISD to chemically interesting problems such as determination of calculation itself requires a number of iterations, each in the electron affinity of F, the singlet-triplet separation in 2 4 volving _n N steps, the addition of the triples correction to CHz, and excited electronic states of NHz. give the QCISD (T) energy is usually possible with moderate The energies from 24 such QCISD and QCISD(T) cal extra expense. culations are given in Table I, together with the FCI energies It remains to confirm that the triples correction (3.4) is from Ref. 1. In Table II, we give the errors, that is the differ still size consistent. This follows by arguments similar to ences E(approximate) - E(FCl) , together with corre those presented in the previous section. For a compound XY sponding results for other approximate methods. These oth system (X, Y infinitely separated), the ilijte elements may be er methods are MP4 (full MP4SDTQ), reported by Cole divided into XXX, XXY, XYY, and YYY classes. In fact the and Bartlett,9 CISD with the Davidson correction from XXYand XYY terms are zero. To show this, suppose that Bauschlicher et al., I and CCSD and CCSDT-1, also due to (ijab) are located on X and (ke) on Y. Then consider each Cole and Bartlett. 9 At the end of the table, we give the mean term in Eq. (3.6) separately. In the first term, aije will only be absolute errors for each method, using those calculations for nonzero if (e) belongs to X, but then the integral (bellek) which the data is available. Since some of the calculations will be zero since it is spread over X and Y. In the sixth term, refer to molecules with stretched bonds, which are some the integral (ab lIej) will be nonzero if (e) belongs to X, but times subject to large errors, we also list mean absolute er-

J. Chem. Phys., Vol. 87, No. 10, 15 November 1987

TABLE I. Total energies (hartrees).

Moleculel Procedureb I (state, geom." ) basis" QCISD QCISD(T) FCI"

Ne R/4s2pld - 128.700 754 - 128.702617 - 128.702462 R/6s4pld - 128.765 378 - 128.767 929 - 128.767889 F U/4s3pld - 99.545 157 - 99.546 631 - 99.546 620 U/4s3p2d - 99.564233 - 99.566 516 - 99.566483 U/5s4p2d - 99.591780 - 99.594 830 - 99.594 877 F- R/4s3pld - 99.651 109 - 99.653 229 - 99.653 341 R/4s3p2d - 99.674143 - 99.677 591 - 99.677 676 R/5s4p2d - 99.700 755 - 99.707057 - 99.706 690 HFIRe R/DZP - 100.248402 - 100.250727 - 100.250969 1.5Re R/DZP - 100.156257 - 100.159729 - 100.160 393 2Re R/DZP - lOO.D72 713 - 100.079725 - 100.081 108 H20lRe R/DZP -76.252745 -76.256007 -76.256624 1.5Re R/DZP -76.062040 -76.069591 -76.071405 2Re R/DZP -75.930888 - 75.953 530 -75.952269 l.5Re U/DZP -76.057706 -76.065701 -76.071405 2Re U/DZP -75.935541 -75.940310 -75.952269 NH2j2B»Re U/DZP - 55.739 486 - 55.742 110 - 55.742 620 2B,,1.5Re U/DZP - 55.595 224 - 55.601600 - 55.605 209 2B,,2Re U/DZP - 55.494490 - 55.498 131 - 55.505 524 2A"Re U/DZP - 55.685 854 - 55.688 270 - 55.688 762 2A,,1.5Re U/DZP - 55.510 338 - 55.515 648 - 55.517 614 2A»2Re U/DZP - 55.395 207 - 55.405 182 - 55.415 133 3 CH2/ B, U/DZP - 39.044 206 - 39.045 917 - 39.046260 'A, R/DZP - 39.023 661 - 39.026 316 - 39.027183

• Geometries, basis sets, and FCI energies are all giyen in Ref. 1. b Restricted (RHF) reference denoted by R; unrestricted (UHF) reference denoted by U.

TABLE II. Energy differences, E (approx) - E(FCI) (mhartrees).

Moleculel Procedurel CISD (state, geom.) basis MP4 +Dav" CCSD QCISD CCSDT-l QCISD(T)

Ne R/4s2pld - 0.873 + 1.300 + 2.142 + 1.708 -0.227 -0.155 R/6s4pld + 2.562 + 2.030 + 2.875 + 2.511 -0.095 -0.040 F U/4s3pld + 1.830 + 1.830 + 1.463 -0.052 - 0.011 U/4s3p2d + 0.268 + 2.639 + 2.250 -0.074 - 0.033 U/5s4p2d +0.529 + 1.428 + 3.335 + 3.097 + 0.Q35 + 0.047 F- R/4s3pld - 4.748 + 2.962 +4.730 + 2.232 - 1.108 + 0.112 R/4s3p2d - 5.044 + 5.806 + 3.533 - 1.137 + 0.085 R/5s4p2d - 5.398 + 5.518 + 7.715 + 5.935 - 1.207 - 0.367 HFIRe R/DZP - 1.263 + 1.596 + 3.007 + 2.567 + 0.170 +0.242 1.5Re R/DZP + 0.769 + 1.411 + 5.100 + 4.136 +0.490 +0.664 2Re RlDZP +4.841 - 1.291 + 10.181 + 8.395 +0.222 + 1.383 H20lRe R/DZP +0.917 + 2.075 + 4.122 + 3.879 + 0.596 +0.617 l.5Re R/DZP + 5.764 +4.402 + 10.158 + 9.365 + 1.991 + 1.814 2Re R/DZP + 14.860 + 10.012 +21.404 + 21.381 - 2.646 - 1.261 l.5Re U/DZP + 56.150 + 13.676 + 13.699 + 6.252 + 5.704 2Re U/DZP + 25.960 + 17.385 + 16.728 + 10.332 + 11.959 NH2/2B"Re U/DZP + 1.900 + 0.572 + 3.212 + 3.134 + 0.519 +0.510 2B,,1.5Re U/DZP + 39.230 + 1.584 + 9.841 + 9.985 + 3.735 + 3.609 zB,,2Re U/DZP + 18.790 + 9.026 + 11.306 + 11.034 + 6.497 + 7.393 zA"Re U/DZP + 1.615 +0.618 + 2.992 + 2.908 + 0.496 +0.492 zA»1.5Re U/DZP + 7.779 +2.403 +7.670 + 7.276 + 2.116 + 1.966 2A,,2Re U/DZP + 41.297 + 6.886 + 20.168 + 19.926 +9.319 + 9.951 3 CHzI B, U/DZP -0.650 + 2.054 + 0.343 'A, R/DZP -0.039 + 3.522 + 0.867 Mean abs. dey. 11.018 2.937 7.786 6.780 2.242 2.067 Mean abs. dey. (Re only) 2.246 1.708 3.700 2.914 0.476 0.280

• All CISD calculations in this column are based on an RHF reference function.

J. Chern. Phys., Vol. 87, No. 10,15 November 1987

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 5974 Pop Ie, Head-Gordon, and Raghavachari: Quadratic configuration interaction rors for the smaller set of atoms and molecules close to equi the QCISD results. For both methods, the errors are all posi librium geometries. tive and the means are relatively large. The mean absolute The QCISD total energies reported in Table I all lie difference between the two, for the 22 cases where both re above the corresponding full configuration interaction re sults are available, is only 0.658 mhartree. This indicates that sults, even though the method is no longer variational. The the terms that must be omitted in Eq. (2.26)-(2.28) to go errors in this method range from 1.7 mhartree (F atom with from CCSD to QCISD are relatively insignificant (for the the smaller basis) to more than 20 mhartree (RHF-based HF case) . Clearly the principal deficiency of both methods is calculation on H 20 with bonds stretched to twice the equi the inadequate treatment of the triples contributions. librium bond length). The results for molecular states close The most advanced method which has been applied to to equilibrium geometries, however, show only modest er all the systems considered in this study is the approximate rors of 2-4 mhartree, the mean absolute error for this subset treatment CCSDT -1 introduced by Cole and Bartlett. 9 This being 2.914 mhartree. This is consistent with the observation is an approximate implementation of the CCSDT model in T T made in the previous section that QCISD takes only limited which the cluster operator e is replaced by e • + T, + T3 and account of triple substitutions and their contribution to low some further terms are neglected; it involves computational ering the total energy. The effect of triples is known to be steps with summations over seven indices in each iteration.21 much larger for stretched bonds, as already demonstrated by The results of CCSDT-1 calculation are close to those of the perturbation theory.9 full QCISD(T) method proposed in this paper, the mean Inclusion of the triples correction at the QCISD (T) lev absolute difference between the energies being 0.477 mhar elleads to substantial improvement in the calculated total tree. QCISD (T) is slightly superior, particularly for the sys energies. For all of the 24 examples, the QCISD (T) energy is tems close to equilibrium geometries, but the sample is small closer to the correct (FCI) answer than is the QCISD value. for a comparative judgement. Both methods are clearly su In some cases, the QCISD(T) energy is below the FCI re perior to the others listed in Table II. sult, but the largest such overshoot is only about 1.2 mhar Since this work was completed, Noga and Bartlett14 tree. Results on molecules with stretched bonds are im have implemented the full CCSDT scheme and have applied proved but still not really satisfactory, the greatest error this method to 11 of the examples used in this study. This being 12 mhartree, (UHF-based calculation on H20 with consists of Ne, F-, HF, and H 20 (including the stretched twice the equilibrium bond lengths). It is interesting to note bond length cases but only RHF based). For those 11 exam that QCISD (T) performs quite well even for stretched bond ples, the mean absolute deviation for the CCSDT method is lengths when an RHF reference function is used (e.g., HF 0.78 mhartree. For the same 11 examples, the mean devi and H 20 from Table II). The comparatively larger errors for ation for the QCISD(T) method is 0.61 mhartree. More the UHF-based stretched bond lengths appears to be related examples are needed for a rigorous evaluation but the perfor to the very large degree of spin contamination in such wave mance of the QCISD(T) method is clearly excellent. functions, though our preliminary results indicate that the In conclusion, the quadratic configuration interaction method performs very well in cases with moderate amounts method QCISD(T), introduced in this paper, appears to be of spin contamination. It should also be noted that for open simple, efficient, and effective. It should prove valuable in shell cases, our calculations involve frozen UHF Is orbitals, exploring correlation energies for a wide range of molecules. whereas Bauschlicher et al. use frozen RHF Is orbitals and hence the full CI comparisons are not as definitive for open ACKNOWLEDGMENTS shells. For atoms and molecules close to equilibrium geome Part of this work was carried out while one of us (J.A.P) tries, the QCISD(T) results are particularly good, the mean was a Visiting Fellow at the Australian National University, absolute error for the subset being only 0.28 mhartree. Canberra. Support from the National Science Foundation The comparison with other methods displayed in Table (Grant CHE-84-09405) is acknowledged. We are indebted II shows a number of interesting features. Fourth-order to Professor R. J. Bartlett for some valuable comments. M!611er-Plesset energies (MP4), which include contribu tions from triple substitutions, show particularly poor re APPENDIX sults for unrestricted (UMP) cases with stretched bonds. The slow convergence of the UMP series under these cir Here we consider the contribution to the energy that cumstances has been noted before.6 For geometries close to arises from the inclusion of the product term T1T2 in Eq. eqUilibrium, however, MP4 energies show a mean absolute (2.19). This term gives rise to a triple substitution coefficient error of only 2.246 mhartree. The second method document which may be explicitly written as follows: ed is CISD, with the Davidson correction. These energies are quite good, even for the stretched bonds. The Davidson cor (Al) rection, of course, is primarily intended to correct for the effects of unlinked quadruples and no attempt is made to The nine terms in Eq. (AI) can all be derived from the first include contributions from triples. It is likely that the com term by permuting the indices iJ,k and a,b,c and using the parative success of this method arises because this quadru antisymmetric nature of a~. Thus, ples correction is overestimated, thereby effectively compen abc a be + t t' t (A2) sating for the omission of the effects of triples. aijk = ajajk permu a Ion erms. The coupled cluster (CCSD) energies are quite close to In terms of a M!6ller-Plesset perturbation expansion, the

J. Chern. Phys., Vol. 87, No. 10,15 November 1987

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 5975 doubles coefficients are of first order and the singles coeffi cels part ofEq. (A8) and hence the correspondence between cients are of second order and thus the triples coefficients the remaining part of Eq. (A8) and the neglected terms in formed as the product TI T2 are third-order terms. Replacing Eq. (3.2) (those with sand U being single substitutions) is the doubles coefficients by their first-order values, we get not exact. In fact, the full use ofEq. (3.2), though appropri ate for a CI type wave function, is not quite correct in our aij'r: = a~(jk II be )( - aJ'n -I + permutation terms. (A3) case since it yields results which are not size consistent. However, our final correction formula for triple substitu We now use the identity tions which neglects the terms discussed above, viz. Eq. (aJ'D- 1= (aijtC )-I(1 + (af/aJ'n] (A4) (3.4), is fully size consistent as demonstrated earlier. and substitute in Eq. (A3), giving aij'r: = [af(jk IIbe)( - aijtc)-I + permutation terms] + other terms. (AS) I(a) C. A. Bauschlicher, Jr., S. R. Langhoff, P. R. Taylor, and H. Par We note that the other terms in Eq. (AS) result from the tridge, Chern. Phys. Lett. 126, 436 (1986); (b) C. W. Bauschlicher,Jr., S. second part ofEq. (A4) and will not be considered further in R. Langhoff, P. R. Taylor, N. C. Handy, and P. J. Knowles, J. Chern. this discussion. Full details of the contribution of these terms Phys.85, 1469 (1986); (c) C. W. Bauschlicher, Jr. and P. R. Taylor, ibid. 85,2779 (1986); (d) 85, 6510 (1986). will be given in a later paper. Neglecting such terms, Eq. 2J. A. Pople, J. S. Binkley, and R. Seeger, Int. J. Quanturn Chern. Syrnp. 10, (AS) can now be recast using the general notation 1 (1976). s 3S. R. Langhoffand E. R. Davidson, Int. J. Quanturn Chern. 8, 61 (1974). aij'r:=at = L (Eo-Et)-IVtuau' (A6) 4R. Krishnan, M. J. Frisch, and J. A. Pople, J. Chern. Phys. 72, 4244 u (1980). 5C. Mtiller and M. S. Plesset, Phys. Rev. 46, 618 (1934). Evaluating (\IIflH ITIT2\110 ) in Eq. (2.19) now gives 6(a) P. J. Knowles, K. Sornasundararn, N. C. Handy, and K. Hirao, Chern. T S Phys. Lett. 113,8 (1985); (b) W. D. Laidig, G. Fitzgerald, and R. J. wf=ws = LL (Eo-E/)-lV.tVtuau· (A7) Bartlett, ibid. 113, 151 (1985). / u 7J. Cizek, J. Chern. Phys. 45, 4256 (1966). 8R. J. Bartlett, H. Sekino, and G. D. Purvis, III, Chern. Phys. Lett. 98, 66 Ws does not contribute directly to the correlation energy (1983). since the single substitutions do not occur in the energy 9S. J. Cole and R. J. Bartlett, J. Chern. Phys. 86, 873 (1987). expression (2.29). The contribution to the energy occurs as lOp. R. Taylor, G. B. Bacskay, N. S. Huch, and A. C. Hurley, Chern. Phys. follows. The w~ term [which is closely related to the v~ in Eq. Lett. 41, 444 (1976). IIJ. A. Pople, R. Krishnan, H. B. Schlegel, and J. S. Binkley, Int. J. Quan (2.37)] induces a change in the singles coefficients a~ by Eq. turn Chern. 14, 545 (1978). (2.30). The a~ in tum yields a contribution to Uijb [Eq. 12R. J. Bartlett and G. D. Purvis, III, Int. J. Quanturn Chern. 14, 561 (2.36)] and hence changes the doubles coefficients aijb by ( 1978). Eq. (2.31). This finally gives an energy correction via Eq. 13G. D. Purvis, III and R. J. Bartlett, J. Chern. Phys. 76, 1910 (1982). l4(a) M. Urban, J. Noga, S. J. Cole, and R. J. Bartlett, J. Chern. Phys. 83, (2.29). In leading order, this energy contribution may be 4041 (1985); (b) J. Noga and R. J. Bartlett, ibid. 86, 7041 (1987). shown to be equivalent to 15K. Raghavachari, J. Chern. Phys. 82, 4607 (1985). s 16R. J. Bartlett has pointed out that the syrnbols CI, C2, ... are normally used for the substitution operators in CI wave functions, T , T , ... being re 8E= Lasws, (A8) I 2 served for the cluster operators ofcoupled cluster theory. In this paper, we

use the syrnbols TI , T2, ... for substitution operators [defined by Eq. (2.2) J where as are the singles coefficients in second order. Substi throughout, the difference between various correlation rnethods being ac tuting Eq. (A7) into Eq. (A8), we finally have counted for by different values of the amplitudes af,af/, .... S T S 17W. Meyer, Int. J. Quantum Chern. S, 341 (1971); J. Chern. Phys. 58, 1017 8E= LLL (Eo-Et)-lasV./Vtuau· (A9) (1973). s / u 18R. Ahlrichs, P. Scharf, and C. Ehrhardt, J. Chern. Phys. 82, 890 (1985). 19J. A. Pople, R. Seeger, and R. Krishnan, Int. J. Quanturn Chern. Syrnp. This is identical to the term that was neglected in Eq. (3.2). 11, 149 (1977). In the above discussion, the term arising from the right 2~. Krishnan and J. A. Pople, Int. J. Quanturn Chern. 14,91 (1978). hand side ofEq. (2.19) was not considered. This term can- 21y. S. Lee and R. J. Bartlett, J. Chern. Phys. 80, 4371 (1984).

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