From Configuration Interaction to Coupled Cluster Theory: the Quadratic Configuration Interaction Approach Dieter Cremer*

Advanced Review From configuration interaction to coupled cluster theory: The quadratic configuration interaction approach Dieter Cremer*

Configuration interaction (CI) theory has dominated the first 50 years of quantum chemistry before it was replaced by many-body perturbation theory and coupled cluster theory. However, even today it plays an important role in the education of everybody who wants to enter the realm of quantum chemistry. Apart from this, full CI is the method of choice for getting exact energies for a given basis set. The development of CI theory from the early days of quantum chemistry up to our time is described with special emphasis on the size-extensivity problem, which after its discovery has reduced the use of CI methods considerably. It led to the development of the quadratic CI (QCI) approach as a special form of size- extensive Cl. Intimately linked with QCI is the scientific dispute between QCI developers and their opponents, who argued that the QCI approach in its original form does not lead to a set of size-extensive CI methods. This dispute was settled when it was shown that QCI in its original form can be converted into a generally defined series of size-extensive methods, which however have to be viewed as a series of simplified coupled cluster methods rather than a series of size-extensive CI methods. © 2013 John Wiley & Sons, Ltd.

How to cite this article: WIREs Comput Mol Sci 2013, 3:482-503 doi: 10.1002/wcms.1131

INTRODUCTION AND BASIC TERMS Theory edited by Schaefer. In the same book, Roos 2 3 onfiguration interaction (CI) theory was the and Siegbahn and Meyer reported on related devel- first post-HF (Hartree-Fock) [also called post- opments in CI theory thus complementing Shavitt's C 4 SCF (self-consistent field)] method used in the early review. In 1998, Carsky summarized in an excellent days of quantum chemistry to correct the mean-field review developments during 70 years of CI theory approach (MFA) of HF for a better description of and 1 year later Sherill and Schaefer gave a detailed 5 electron-electron interactions. In the early literature, account on highly correlated CI theory. one used the term superposition of configurations, Since CI theory has played such a fundamental which as an acronym would be misleading nowadays role in the development, understanding, and applica- because it denotes spin orbit coupling. Therefore, we tion of electron correlation methods, it is discussed in will avoid the latter term in this article. literally all textbooks on quantum chemistry of which 6 A number of review articles have focused on only the book by Szabo and Ostlund is mentioned CI theory. A state-of-the-art account was given by here. In the following, some basic terms will be intro- Shavitt1 in 1977 in Methods of Electronic Structure duced that facilitate the reading of this review. HF theory simplifies the description of the interactions of N electrons by considering just one *Correspondence to: [email protected] electron interacting with the mean field of the N - Computational and Theoretical Chemistry Group (CATCO), De- partment of Chemistry, Southern Methodist University, Dallas, 1 remaining electrons thus reducing the many-body TX,USA (many-electron) problem to an effective one-body DOl: 10.1002/wcms.1131 (one-electron) problem. This is the essence of the

MFA, which implies a drastic simplification of the cal meaning from the electron attachment process.8 description of electron-electron interactions. The HF (ii) The nodal properties of the virtual HF orbitals space orbitals ¢i or spin orbitals lj!i (subscripts i, j, k, (with increasing orbital energies, the number of nodal 1, ... denote orbitals with occupation numbers ni = surfaces of the virtual orbitals increases) can be ex- 2, 1, or 0 in the case of space orbitals or ni = 1 or 0 in ploited to improve the HF description of the electron the case of spin orbitals) are calculated variationally interaction. For this purpose, excited configurations as a linear combination of M basis functions x 11 : are generated by moving electrons from occupied to M virtual orbitals (denoted by subscripts a, b, c, etc.; ¢i = L C11 iX11 with fL = 1, 2, ... , M (1) for orbitals with unspecified electron occupation sub- f1 scripts p, q, r, s, etc., are used). In the case of the wa- where the expansion coefficients c11 i are variationally ter molecule, doubly (D) excited configurations are optimized. Hence, the HF energy E(HF) based on for example: (2222011000000), (2222020000000), these orbitals is an upper bound to the exact energy or (2222002000000). The virtual orbitals possess ad- E(exact). The difference between the exact, nonrela- ditional nodal surfaces, which separate two electrons, tivistic Schrodinger energy (obtained for a clamped and therefore a better description of the correlated nuclei situation, i.e., with the Born-Oppenheimer ap- movements of the electrons is achieved. Depending proximation) and the HF limit energy (obtained for a on the nodal properties of originally and newly oc- 7 complete basis set) is called correlation energy : cupied orbital, one speaks of in-out (ns --+ ks with k > n, etc.), angular (a --+ n*), or left-right (a --+ l'lE(corr) = E(exact)- E(HF, limit) (2) a*) excitations, which describe in-out, angular, and It accounts for the energy lowering due to a corre- left-right electron correlation, respectively. The correlated movement of the electrons in a many-electron sponding Slater determinants are written in the short molecule thus reducing the destabilizing electron in- form lfl), which indicates that the occupied orbitals teractions and correcting in this way the MFA of HF. ¢i and ¢i as well as the virtual orbitals ¢a and ¢b Since HF is carried out with a finite set of M basis are involved in the excitation process. In this way, a functions, the correlation energy depends on the size CID (CI with all D excitations) wavefunction can be of the basis set. It also depends on how the HF method generated: is improved. This can be done in a methodologically a>b 1 simple way by employing CI theory. l\llcw) = coio) + l..:cfllfl) (3) A set of occupation numbers (nt, n2, n3, .. . , nM) i>j is called an electron configuration. When using space orbitals and describing a closed shell molecule in its where co is the coefficient of the HF reference, which ground state, m = N/2 orbitals with the lowest ener- can be one or close to one depending on what normal- gies are doubly occupied. For example, in the case of a ization is used. The cfl are the mixing(= interaction) VDZ (valence double-zeta) basis set description of the coefficients of the excited configurations. Hence, the water molecule (N = 10, M = 13), the ground state CID wavefunction is expressed as a linear combina- (GS) configuration is given by the occupation vector tion of the HF Slater determinant and all possible (2222200000000) because there are five doubly oc- D-excited Slater determinants, which can be gener- cupied (ni = 2) molecular (space) orbitals (MOs) and ated for a set of 13 MOs and 10 electrons. The CID eight virtual orbitals with na = 0 (subscripts i and weighting coefficients cfl are determined by a varia- a denote occupied and virtual orbitals, respectively). tional approach during which the HF orbitals are kept The corresponding wavefunction is the HF Slater de- frozen. The resulting CID energy is lower than the HF terminant I

Volume 3, September/October 2013 © 2013 John Wiley & Sons, Ltd. 483 Advanced Review wires.wiley.com/wcms or quasi-degenerate GS, linear combinations of a few that substitute the internal orbitals in the CI expan- Slater determinants rather than single Slater determi- sion are called external orbitals, where it is generally nants are eigenfunctions of the spin operators, and assumed that the external orbitals are orthogonal to therefore these spin-adapted configurations are used, the internal orbitals. As internal orbitals, one can use which are called configuration state functions (CSFs). HF canonical orbitals, natural orbitals, Brueckner or- For the purpose of simplifying the notation, we keep bitals, localized orbitals, etc. External orbitals can be the symbol lff{) to denote a CSF, which in the the virtual orbitals of a HF calculations or any type simplest case is identical to one Slater determinant. of improved virtual orbitals. A full CI (FCI) expansion for an N-electron The number of CSF terms in the FCI wavefunc- system contains all possible P-excitations (P = tion for a system with N electrons, i.e., the size of the S, D, T, Q, ... , X), where IS) = If) denotes FCI space d(N,S,M), can be calculated using Weyl's 1 12 13 singly(S)-excited CSFs, ID) = lffl D-excited CSFs, formula • • IT) = Iffkl triply(T)-excited CSFs, IQ) = Iffkjd) d 25 + 1 ( M + 1 ) ( M + 1 ) quadruply(Q)-excited CSFs, and IX) all X-fold ex- ( N, S' M) = M + 1 Nj2 - S Nj2 + S + 1 cited CSFs. I\I!FcJ) = Io) + L cplp) (4) (8) all P where S is the total spin of the N electron system, It is convenient to choose the CSFs lp) to be or- 25 + 1 gives its multiplicity, and the symmetry of the thonormal so that the metric S becomes the identity system is not considered. Hence for the GS of wa- matrix I, and the FCI eigenvalue problem takes the ter calculated with a VDZ basis set (M = 13 ), the simple form dimension of the FCI problem expressed in CSFs is He= ESc= Ec (5) 429,429 and increases to 30,046,752 when a valence where the elements of the hermitian matrix H are triple-zeta basis set (M = 19) is used. The number of given by Slater determinants is larger by a factor of 3-4, but (6) symmetry can be exploited to reduce this number. In addition, one can carry out a frozen core (only the or- and the expansion coefficients cp are collected in the bitals of the valence shell are used for the generation vector c. The FCI energy of the GS, E(FCI), is defined of CSFs) and/or a deleted virtual orbital calculation for the basis set of size M used to obtain the HF (a certain number of high-lying virtual orbitals are reference function Io) as the lowest eigenvalue of excluded from the formation of CSF, i.e., they cannot the diagonal eigenvalue matrix E. be occupied by electrons) thus reducing the computa- The HF energy represents a minimum with re- tional load. gard to the mixing of occupied and virtual spin or- Because of the large computational costs, a FCI bitals, Therefore, S-excited CSFs do not interact with calculation can only be carried out for relatively small the HF GS wavefunction, which is expressed in the electronic systems. Accordingly, one has used simpler 9 Brillouin theorem : CI expansions that truncate the number of excitations at a given level thus leading to truncated CI (7) expansions of the CID, CISD, CI with all S, D, and However, as soon as pair correlation or higher n- T excitations (CISDT), or CI with all S, D, T, and Q electron correlation effects are included into the CI excitations (CISDTQ) type. If M basis functions (lead- expansion, the HF orbitals are no longer optimal. For ing to M orbitals) are used, these methods formally 6 8 example, in-out pair correlation requires more dif- scale with O(M ) (CID and CISD), O(M ) (CISDT), 10 fuse orbitals. By accounting for some of these orbital and O(M ) (CISDTQ) where however for large ra- improvements, the S excitations make a significant tios of MIN the computational costs can be reduced 4 6 contribution in a CISD (CI with all S and D exci- to O(M ) for CISD and even O(M ) for CISDTQ. tations) or any more sophisticated CI wavefunction. Truncated CI in the form of CISD was once The calculation of the matrix elements HpQ follows the method of choice, which had to do with its vari- 10 11 the Slater rules. • One important consequence of ational character and the upper bound property of the Slater rules is that CSFs differing by more than the CI energy. For almost 50 years, CI was the most two spin orbitals do not interact because the Hamil- often used post-SCF method. In the mid-1970s, first 14 15 tonian contains just one- and two-particle operators. Moller-Plesset perturbation theory, • then coupled 16 17 Often the orbitals occupied in the HF reference cluster theory, • and finally density functional the- function are called internal orbitals. Those orbitals ory (DFT) 18 replaced CI theory as a basic tool of

484 © 2013 John Wiley & Sons, Ltd. Volume 3, September/October 2013 WIREs Computational Molecular Science Configuration interaction and quadratic configuration interaction a quantum chemist. The reasons for the decreasing are obtained by diagonalizing the one-electron den- application of CI methods will be discussed in the sity matrix of the CI wavefunction. They are orbitals following. with fractional occupation numbers ni (0 _:::: n 1 _:::: 1.0). Lowdin could prove that by using NSOs the CI expansion of most rapid convergence was obtained, which THE EARLY DAYS OF CI THEORY seemed to be of little value since the NSOs have to be determined first by a CI calculation. However, the The first CI calculation was carried out by Hylleraas l9 essence of this finding was already in the same year in 1928 for the helium atom in its ground state. 19 He demonstrated by Shull and Lowdin.24 These authors investigated what one would call today the radial (in- recalculated the He atom with a three-orbital basis out) correlation of electrons by exciting one or both set including all S- and D-substituted configurations of the helium electrons into higher lying ns atomic (in total six configurations, which corresponds for the orbitals (AOs) thus generating a CI expansion, which two-electron system of He to a FCI calculation), de- led to a significantly lower energy than obtained by termined the NSOs, and then repeated the calculation the HF approach. Hylleraas also checked the angular by using instead of AOs the calculated NSOs. In this correlation and found it to be less important where he way, it was shown that the number of configurations was misled by the fact that he could not readjust the could be reduced from six to three. in-out excitations when including the angular excita- Bender and Davidson25 introduced in 1966 the tions. In view of his pioneering work on the helium iterative natural orbital (INO) method, in which the atom, Hylleraas can be called the father of the CI the- advantage of using NSOs for CI calculations was ex- ory (as he may also be called the father of the R12 ploited without having to calculate first the CI density methods in view of his second paper on the helium matrix. They started from approximate NSOs calcu- ground state, in which he explicitly introduced the 20 lated for a less-than-complete initial set of configura- interelectronic distance r12 into the wavefunction ). tions. The analysis of the occupation numbers of these Despite the fact that CI dates back to the very NSOs made it possible to eliminate configurations beginning of quantum chemistry, the lack of compu- which turned out to be unimportant. New tational power and the stagnancy in many parts of rations were then added, and the procedure repeated science during World War II delayed developments for an improved CI calculation. When successive iter- until the 1950s. In 1950, S. F. Boys published a, by ations (mostly four or five) did not yield any change today's standards, relatively small CI study of the in the CI expansion, the INO-CI result was obtained. ground state of the beryllium atom,21 which led to an In this way, these authors could obtain 89% of the energy clearly below the HF limit (i.e., the HF energy correlation energy of LiH by using in the end just 45 obtained with an infinitely large basis set) and which configurations, which included only 35 of the origi- in terms of computational cost, was less costly nal 50 configurations in the initial CI guess. The same numerical HF limit calculation. Boys concluded in his authors26 could describe diatomic hydrides of the first paper that CI is the only feasible method for calcu- row with more than 3000 configurations, recovering lating, analyzing, and predicting the electronic struc- about 75% of the correlation energy, which had been ture of atoms and molecules. In the 1950s, Boys and unthinkable before that time. his students at Cambridge (among them I. Shavitt) An alternative approach for exploiting the ad- pushed forward the algorithms and computational vantages of NSO was suggested in 1966 by Edmin- techniques to carry out CI calculations. ston and Krauss27 in the form of the pseudonatural In 1952, Taylor and Parr22 published CI calcu- orbital CI (PNO-CI; later also called pair-natural or- lations on the helium atom with four configurations bital CI). Shull and Lowdin24 had shown that for a that accounted for 88% of the correlation energy and two-electron system such as He the use of NSOs re- underlined the importance of angular electron corre- duces the length of the CI expansion from p(p + 1) lation (excitations from 1s to np AOs) thus correcting configurations to just p. Edminston and Krauss ex- Hylleraas' original CI description of the He atom. ploited this by describing in a many-electron system two electrons located in the same region of space and ADVANTAGES OF NATURAL SPIN experiencing the mean field of all other N - 2 elec- ORBITALS trons by a set of approximate NSOs obtained in a small CI calculation. Then, the actual CI calculation In 1955, Lowdin23 introduced the natural spin or- was carried out with the PNOs of all electron pairs. bitals (NSOs), which helped to analyze and under- Edminston and Krauss investigated in this way just 27 stand the wavefunction of a CI calculation. NSOs some three-electron systems (Hei, H 3) ; however,

Volume 3, September/October 2013 © 2013 John Wiley & Sons, Ltd. 485 Advanced Review wires.wiley.com/wcms their ideas could be easily generalized if one was pre- as POLYATOM,34 IBMOL,35 MOLE,36 and Pople's pared to accept that the external orbitals expressed in Gaussian70,37 where the latter stood at the beginning terms of PNOs are no longer orthogonal for differ- of a development that changed the world of quan- ent pairs and therefore an overlap matrix had to be tum chemistry to what it is today: no more tedious calculated. and extremely lengthy hand computations on tiny It soon turned out that the additional calcula- molecules, but extremely intensive high-performance tions caused by a PNO-CI are less time consuming computing including trillions of calculational steps than was originally believed. In an N-electron sys- on large molecules such as steroids, proteins, or man- tem, there are N(N- 1)/2 pairs, which can be singlet made polymers. or triplet coupled. Each of these pairs is described by Important steps in this direction were again a set of PNOs where the PNOs are properly located made by Nesbet as documented in two publications in space. Accordingly, they are superior to HF vir- during the 1960s. 38·39 In the first, he showed that a tual orbitals as external orbitals in a CI calculation, monstrous O(M8) calculational bottleneck inherent which gives a PNO-CI method a significant advan- to CI and all post-HF calculations could be avoided tage. Meyer developed the first variational PNO-CI by a much less costly computational procedure. The method, which was used for a number of accurate bottleneck results from the enormous number of two- CI calculations of the properties of various first row electron integrals (approximately M4/8) that have to hydrides. 28·29 be calculated for a HF description of an electronic The PNO-CI approach made an important con- system with M basis functions. The solution of the tribution to the development of more efficient CI HF-SCF problem requires the calculation of electron methods. However, soon it became clear that the use interaction integrals expressed in terms of basis func- of NSOs only pays out for relatively small CI ex- tions whereas post-HF methods need the very pansions whereas it is less useful when one considers same electron interaction integrals expressed in terms millions of configurations. NSOs are used nowadays of spin orbitals. This implies a transformation accord- more indirectly when setting up a basis set for a CI ing to Eq. (9): calculation. Almlof and Taylor30·31 performed CISD calculations on atoms, determined the corresponding (ijlkl) = L L L L.>ii-'CjvCkACiu (!tvi.A.a) (9) atomic natural orbitals (ANOs), and used their form 1-' A u to set up generally contracted Gaussian basis sets. The corresponding ANO basis sets facilitate the cal- which corresponds to an O(M8) calculational load. culation of the correlation energy in a CI expansion Nesbet38 made a first step to reduce this load by because they improve the convergence to the basis set carrying out the transformation in two steps, which limit. required an intermediate array for storing the semi- transformed integrals. Later, Nesbet's idea was per- fected by Bender40 who showed that by carrying out the transformation in four steps the calculational BOTTLENECKS OF THE CI 8 5 CALCULATION costs are reduced from O(M ) to O(M ) thus circum- venting one of the bottlenecks of a CI calculation. A prerequisite for any CI method is the efficient cal- The other bottleneck of CI calculations resulted culation of the matrix elements HPQ of Eq. (6). First from the diagonalization of the large matrix H of steps in this direction were already made by Nesbet32 Eq. (5) where often only the lowest eigenvalue is re- (a student of Boys) in 1955 when he showed how the quired to obtain the GS energy of an electronic system. calculation of the matrix elements could be simpli- Nesbet39 was the first to realize that this simplifies fied by focusing just on the few spin orbitals differing the diagonalization procedure considerably, and he in the reference and excited determinant. A big step worked out an algorithm to obtain just that lowest forward in the calculation of a CI expansion and the eigenvalue and the associated eigenvector rather than corresponding CI energy was made when the Boys all eigenvalues of a large CI matrix. Shavitt and co- group in Cambridge, UK, programmed each part of workers41 generalized the Nesbet method to also cal- a CI calculation for the EDSAC (Electronic Delay culate higher lying eigenvalues. In 1975, Davidson42 Storage Automatic Calculator) electronic computer published the most general solution to the problem by and presented in 1956 results for small molecules proposing an algorithm that was based on the Lancoz with more than just two electrons. 33 The program- tridiagonalization scheme, 43 combined with features ming work of the Boys group triggered the develop- of previous solutions to the problem. The Davidson ment of quantum chemical program packages such method (which is still used today) converged much

486 © 2013 John Wiley & Sons, Ltd. Volume 3, September/October 2013 WIREs Computational Molecular Science Configuration interaction and quadratic configuration interaction faster, had moderate storage requirements, worked ment, and inverse are defined) can be used to construct for nearly degenerate eigenvalues, and could also be an orthonormal basis and to evaluate the matrix ele- applied to higher eigenvalues without calculating all ments of the generators of this basis. The importance lower eigenvalues.42 of this discovery was recognized for the nuclear and the many-electron problem by Moshinsky47 and discussed by various authors as for example Matsen and 48 13 49 DIRECT CI AND GUGA CI Pauncz. In 1974, it was Paldus • who achieved a considerable simplification of UGA in connection A revolution in CI was the introduction of the di- with its application to CI (and perturbation theory) rect CI method by Roos in 1972.44 When using con- calculations of many-electron systems. He demon- ventional CI methodology, one constructed first each strated that the calculation of the Hamiltonian matrix matrix element HpQ from one- and two-electron inte- elements of a CI expansion could be carried out with grals each multiplied by coupling coefficients, which the help of matrix representatives of the generators reflect the occupation and spin-coupling of the spin 13 50 EPQ of the Gel'fand-Tsetlin formulas. • In this ap- orbitals in the interacting CSFs lp) and IQ). proach, individual CSFs were represented by a Paldus M M tableaux. HpQ = L L The significance of Paldus' work on UGA was not directly recognized by the CI community, and M M M M therefore it needed a more transparent approach to + (10) make UGA attractive as a powerful tool for gener- k ating the huge number of matrix elements in the CI calculation of nontrivial molecules. In 1977, Shavitt51 The matrix elements were derived by hand and in- developed a compact digital representation of excited dividually coded into a computer program. In this state configurations in terms of Gel'fand states, which way, an expansion of just 10,000 configurations led 51 53 (DRT). - 7 he called the distinct row table In ad- to a total of 5 x 10 matrix elements, which required dition, Shavitt introduced a graphical complement still millions of HPQ values if just 10% of them was of UGA (GUGA). With the help of GUGA and the nonzero. These had to be stored on external storage Shavitt graphs as they are called today, it was pos- devices before they were read back to solve the CI 44 sible to represent the DRT in a pictorial way thus eigenvalue problem. Roos avoided the tedious con- yielding a detailed insight into the structure of the CI struction of elements HPQ and their storing by directly Hamiltonian matrix. working with the one- and two-electron integrals Brooks and Schaefer54 soon presented the first M M implementation of the GUGA-CI method, which al- CTp = ready included the possibility for multireference CI Q (MR-CI) calculations. When discussing the possibil- M M M M ity of GUGA-CI, the authors pointed out that even + LLLLLBijklPQ(ijlkl)cQ (11) with a minicomputer CI expansions with 25,000 con- Q k figurations would soon become accessible. In a second paper, 1 year later, Brooks and co-workers55 which leads to replaced the previously preferred integral-driven ap- cr p = He p = E pc p (12) proach (i.e., for each molecular integral all productive loops of a Shavitt graph are constructed and all upper Accordingly, the major task of the direct CI calcu- and lower walks (configurations) are determined; the lation (direct because the eigenvalues and eigenvec- next integral on the integral list is processed in the tors are calculated directly from the molecular inte- same way) by a loop-driven algorithm, in which that grals) is to determine the coupling coefficients loop, which can be generated in the easiest way from and BijklPQ. This problem could be facilitated by us- the previous one, is executed first. The generation of ing the graphical unitary group approach (GUGA) closely related groups of loops and the corresponding or, alternatively, perturbation theory. 1 blocks of integrals needed for the execution of these GUGA is based on the unitary group approach loops led to significant time savings. (UGA), which has its roots in nuclear physics. Already By connecting Shavitt's GUGA-CI and Roos' di- 45 46 in 1950, Gel'fand and Tsetlin • had demonstrated rect CI, the coupling coefficients of the later method how the unitary group U(M) (set of all M-dimensional could be effectively evaluated and used for the calcula- unitary matrices, for which multiplication, unity ele- tion of the vector cr p needed for determination of the

Volume 3, September/October 2013 © 2013 John Wiley & Sons, Ltd. 487 Advanced Review wires.wiley.com/wcms corresponding eigenvalue and eigenvector. In two im- and portant publications, Siegbahn56·57 showed that the (Pl - """ IJ c(p-1) coupling coefficients involving external orbitals adopt (J p - L,.- LIPQ Q (19) a simple structure so that all coupling coefficients can Q be expressed in terms of equations that involve the In this way, the coefficients of the CI expansion are 58 relatively small internal space. calculated iteratively with the help of Eq. (15), i.e. Eq. (15) connects the direct CI method (the u vector, Eq. (19), appears on the right side of Eq. (15) FACILITATING THE SOLUTION OF and is expressed via Eq. (11)) with the diagonaliza- THE CI PROBLEM WITH THE HELP OF tion methods for large matrices as required by the CI PERTURBATION THEORY eigenvalue problem. The direct CI method can be further improved by using perturbation theory. Already at an early stage of CALCULATION OF MOLECULAR the CI development, perturbation theory was used for a prescreening of individual CSFs and their selection PROPERTIES BY CI according to their weight in the CI expansion or their A quantum chemical method will be considered to contribution to the energy. This can be done in a be ready for routine use if first- and second-order re- preliminary calculation by using first-order Rayleigh- sponse properties can be routinely calculated. This 1 Schrodinger perturbation theory according to is greatly aided by the derivation and programming of analytical first and second energy derivatives. In c p = --.,------,--- (13) 1980, two back-to-back papers reported the analyt- (oiHIo)- (piHIp) ical derivation of the CI gradient. Brooks and co- workers55 based their derivation on the loop-driven or for the energy contribution GUGA CI, which provided a calculationally feasible way for obtaining the two-particle density ma- (14) trix and expressing the CI-gradient in terms of the latter, the one-particle density matrix, the derivatives If one or both parameters are below a preset thresh- of the molecular integrals, and the Lagrangian ma- old, the corresponding CSF is eliminated. More so- trix needed in connection with the changes in the HF phisticated ways of preselecting the CSFs of a CI orbitals. expansion have been described in various review The approach of Krishnan and co-workers60 fo- articles. 1·4 cused on the CISD gradient, for which an explicit Perturbation theory59 can also be used for the analytical expression in terms of integral derivatives, calculation of the coefficients of the CI expansion (for the first-order changes of the HF orbitals, and the CI convergence problems in the perturbation series, see expansion coefficients were derived. The analytical Ref 15): calculation of the CI gradient reduced the computational costs of the previously numerical calculations p-1 c(p)- (Eo - E0)-1 """E(p-r)c(p-r) significantly, so that routine calculations of molecu- p - p 0 L..- p [ lar geometries, dipole moments, and other first order r=O response properties became possible at the CISD level of theory. 11 11 0 - """HL,.- PQ c(P-Q + c(P-p E p ] (15) The Schaefer group was actively involved in the Q development of first and second derivatives for the CI and MR-CI energies and summarized their expe- where rience in 1994 in a monograph61 that contains all (16) formulas for determining the analytic energy gradient and Hessian matrix. These are needed in connection with the calculation of the molecular forces (to carry

0 A out geometry optimizations) or for the calculation of (17) E0 = (oiHoio) molecular vibrational frequencies, which in turn are needed for the characterization of stationary points of the potential energy surface, the determination 0 A Ep = (pll-loio) (18) of the zero-point energy and other thermochemical

488 © 2013 John Wiley & Sons, Ltd. Volume 3, September/October 2013 WIREs Computational Molecular Science Configuration interaction and quadratic configuration interaction corrections or for the analysis of vibrational spec- size-consistency errors become rapidly smaller with tra. Another summary on this topic was given by increasing length of the CI expansion. Only FCI ful- Shepard.62 fills the requirements of size extensivity and size consistency. The deficiencies of truncated CI have been THE SIZE-EXTENSIVITY PROBLEM traced back to product terms in the CI projection equations where both parts of a product scale with 2 The size extensivity of a given quantum chemical N and accordingly the product with N . These terms method guarantees that the energy calculated for an correspond to unlinked diagrams in the diagrammatic electronic system with this method scales linearly with representation of the theory. The appearance of un- the number N of electrons. In thermodynamics, a linked diagrams is in itself not problematic as long property is called extensive if it is additive for indepen- as the unlinked diagrams are canceled out by other dent, noninteracting subsystems whereas an intensive unlinked diagrams in the projection equations. This property does not depend on the size of the system is the case for truncated CC methods, which fulfill the considered (e.g., the temperature is the same for a linked diagram theorem, 65 whereas the truncated CI system and all its subsystems). Since the energy of an methods fail to do so. electronic system is an extensive property, quantum There have been numerous methods to rem- chemical methods must describe it in this way. edy the size-extensivity problem of truncated CI. Best 63 64 The term size consistency • is used in the case known is the Langhoff-Davidson correction, 66 that is of chemical reactions, for example, when the sub- based on the energy of a CID calculation. systems are separated to infinitely large distance in the course of a bond-breaking reaction (dissociation) t:,E(LD) = [E(CJD)- E(HF)](1- c6) leading to two fragments of a molecule. In so far, = t:,E(CID, corr)(1- c6) (21) the property of size consistency can be considered as a special case of size extensivity. However, size con- where t:,E(LD) gives the size-extensivity correction, sistency is often also considered to imply the correct E(CID) - E(HF) is the CID correlation energy description of the dissociation products. On this ba- t:,E( CID, carr), and co is the coefficient of the HF sis, restricted HF (RHF) is size extensive, however not reference in the CI expansion. It is common to ap- size consistent because it fails to describe homolytic ply the same formula also for CISD calculations or to dissociation (yielding two radicals for single bond dis- use improvements of it in form of the renormalized sociation) correctly whereas unrestricted HF (UHF) is Davidson (RD) correction67 both size extensive and size consistent. 2 63 1- c0 It was Pople who pointed out that CID and t:,E(RD) = t:,E(CID, corr)--- (22) 2 CISD are not size consistent (and also not size exten- co sive). For a reaction AB --+ A + B, described with the Davidson-Silver correction,67 the Pople the CID method, the products are calculated sepa- correction, 64 or a more elaborate form by Duch rately so that the sum of product energies effectively and Diercksen. 68 Using fourth-order many-body contains disconnected Q excitations and is much bet- perturbation theory (MBPT), Barlett and Shavitt69 ter described than the reactant AB, for which just showed that the Langhoff-Davidson correction66 D excitations are used. In this way, an endothermic accounts for the correlation effects of unlinked (exothermic) reaction energy is underestimated (over- quadruple excitations, which is a major part of the estimated). The property of size consistency can be size-extensivity error of CID and CISD. checked in the following way: Another way of correcting for the size- If two noninteracting systems A and B are cal- consistency error of truncated CI has been worked out culated as a supersystem {A · · · B, 10} (e.g., with a by Ahlrichs and co-workers70 who related the CISD distance of 10 A between A and B), then the sum of approach to the coupled electron pair approximation the energies E(A) + E(B) must fulfill the following CEPA -1 and developed the coupled pair functional condition: (CPF) approach. CPF is based on a correct descrip- E(A- · · B, 10) = E(A) + E(B) (20) tion of separated electron pairs and uses invariance requirements with regard to unitary transformations within numerical accuracy. This condition is fulfilled of equivalent orbitals of identical subsystems. It in- by any truncated coupled cluster (CC) method; how- troduces products of S and/or D excited terms, which ever, truncated CI methods do not fulfill the con- cancel out some of the higher unlinked excitation dition due to the lack of size consistency although terms of a truncated CI expansion.

Size-extensivity and size-consistency corrections algorithm. The dimension of the CSF space was can be useful. They are best documented forD excited 1.4 x 109 and involved more than a trillion dis- configurations, whereas there is a lack of correction tinct Slater determinants. Hence, teraflops comput- possibilities for S and T excited configurations (note ing makes these computation possible and provides that when correcting CISD energies, one assumes that a possibility for these highly correlated configuration the S excitations make only a minor contribution interaction calculations. 75 to size-extensivity error, which is not always justi- CI methods have retained their importance for fied). Also, there is no generally applicable way of quantum chemistry in form of FCI and MR-CI ap- correcting size-extensivity errors for other properties proaches. FCI is an approach that guarantees that than the energy. Size-extensivity problems of the trun- all errors are removed from the N-electron space so cated CI can be partly cured when choosing a MR-CI that the accuracy of the one-electron basis set can method. be assessed. This was first emphasized already in Duch and Diercksen68 have pointed out that 1981 by Saxe et al?6 who solved the first 1 million- quantum mechanics is a holistic theory and accord- dimensional FCI eigenvalue problem. In the 1980s, ingly does not provide a well-defined way of describ- new methods for solving the FCI problem were de- ing subsystems. In view of this fact, they consider veloped where especially the work by Knowles and size extensivity as not the most important property Handy,77 Siegbahn,78 and Olsen and co-workers79 of a quantum chemical method. If the lack of size has to be mentioned. Bauschlicher and Taylor80 car- consistency is approximately corrected, the method is ried out FCI benchmark calculations having matrix considered still valuable as long as it can describe the dimensions up to 28 millions. Knowles and Handy81 dissociation channels of a chemical system correctly. reported FCI calculations with 210 million Slater determinants. An impressive breakthrough was ac- complished when Olsen et al. 82 presented the FCI MODERN USE OF CI METHODS description of the Mg atom, which for 10 active electrons and 30 active orbitals required the solu- CISD calculations improved by the Langhoff- tion of a 1 billion-dimensional eigenvalue problem Davidson or other similar, approximate corrections (1,016,018,176 Slater determinants). have been used in the 1970s and 1980s, however In the past decade, FCI benchmark calculations were soon replaced by the more economic (nonvari- with more than 10 billion determinants have been ational) MBPT and more accurate CC methods. Yet, carried out, for example, for the N2 molecule83 , the this did not stop the development of new CI meth- CN anion,84 or different states of the diatomics BN ods. Harrison and Handy71 found on the basis of FCI and AlN.85 One of the largest FCI calculation done calculations that CISD recovers 95-96% of the total so far seems to be the investigation of c2 based on correlation energy determined with a specific basis set. 65 billion Slater determinants. 86 The addition ofT excitations (CISDT) approximately Most of the FCI methods developed in the past accounts for an additional 1% of the total correla- three decades are based on the use of Slater deter- tion energy, whereas the addition of all quadruples minants rather than CSFs because the coupling co- (CISDTQ) recovers more than 99%. efficients for Slater determinants can only be 0 or Various attempts were made to include T and ±1. There is however still the need for using CFSs Q excitations in an approximate or indirect way. 72 because of the troubling input-output bottleneck of Sherill and Schaefer73 were the first to solve the the FCI eigenvalue problem. Tens to hundreds of ter- O(M10 ) problem of CISDTQ in an efficient way by abytes of disk space are needed to store the expen- choosing the natural spin orbitals of a CISD calcula- sive parts for the iterative calculation of the direct CI tion and using them to identify those classes ofT- and vector cr. This number can be significantly reduced Q-substituted configurations, which deemed most im- when using CSFs because the FCI matrix contains portant in view of the natural orbital populations. in this case a lower number of nonzero elements. Their results for a cc-p VTZ basis set description of Further progress can be expected by exploiting the water show impressively how the number of Slater sparse nature of the FCI expansion vector (sparse determinants increases from CISD to CISDT and FCI) with the aim of avoiding extensive disk read- CISDTQ: 15,939; 938,679; 28,085,271.73 In 2006, ing/writing operations. The basic idea of these ap- Bunge and Carbo-Dorca74 carried out a CISDTQ proaches is to avoid the treatment of unimportant de- calculation of the ground state of the Ne atom us- terminants in the expansion vector c when performing ing a 12s12plld10f10g9h8i7k615m4n3o3q3r (!) ba- the linear transformation cr =He (see, e.g., Rolik and sis set and applying a selected divide-and-conquer co-workers87 ).

The huge FCI calculations done in recent years the electron correlation energy is normally less than are of course a result of the adaptation of the FCI 1%. algorithm to parallel computer architectures and the In view of the exponential form of the CC wave extensive use of powerful supercomputers. Further function, there are also a large number of nonlinear 16 17 88 progress can be expected with the advent of petaflops cluster terms • • computing. l\llcc) = exp(T)Io) = (1 + T + T2j2 + T3j3! + .. ·)Io) (29) COMPARISON OF THE CONFIGURATION INTERACTION which lead to disconnected, but still linked contributions to the exact wavefunction involving products of AND COUPLED CLUSTER METHODS the cluster operators such as T1 t, T1 T2, 1212. Con- CI and CC theory are closely related to each other sidering that S excitations lead to an improvement of in so far as the FCI and full-coupled cluster (FCC) orbitals under the impact of dynamic electron cor- approach both lead to the exact wavefunction and relation, these contributions correspond to indepen- the exact energy for an N-electron system described dent orbital rotations, an orbital rotation taking place with a given basis set of finite size (i.e., the term exact at the same time with pair correlation, independent refers to the one-particle basis used): pair-pair correlations, etc. The CI excitation operators Cp contain effects of l\11) = CIo) = exp(T)Io) (23) the connected cluster operator Tp plus all nontrivial where C and Tare the CI and CC excitation opera- partitions of Tp in products of disconnected cluster tors, respectively, defined in the following way: operators: n (30) C= 1+ LCP (24) p

(31) (25) with

(26) (32) i,a

A A 1A2 AA 1A2A 1A4 'tio) = (27) C4 = 14 + T2 + Tt T3 + T2 + ! T1 (33) 2 2 71 4 Hence, a pattern of the different electron correlation effects emerges out of the cluster decomposition of the tio) = (28) electronic wavefunction. It is easy to foresee which correlation effects are more important than others. All 't contributions are connected and lead to the rep- For example, pair-pair correlation effects represented resentation of n-electron correlation effects, i.e., the by 1/2 Tf are more important than the connected correlation of n electrons at the same time. Already four-electron correlation effects associated with T4. for three electrons, there is a small probability of a The advantages and disadvantages of CI and simultaneous correlated movement and accordingly CC theory become obvious when truncating the CI or the connected three-electron correlation energies are CC expansion. In view of the discussion given above, small. However in view of the fact that there is a large these can be summarized as follows: number of such connected three-electron correlations, the total of all T3 correlation contributions can rep- 1. Truncated CI is variational and yields an up- resent an important part of the molecular correla- per bound to the true energy. Truncated CC is tion energy, especially if molecules with electron-rich not variational, which could be considered as or strongly electronegative atoms, multiple bonds, or a disadvantage. However, practice has shown electron delocalization are investigated. There is an that the variational property of a wavefunc- even lower probability that connected four-electron tion is less important than the size extensivity correlations take place. Their total contribution to of calculated energies and other properties.

2. Truncated CI describes dynamic (short-range) THE QUADRATIC CONFIGURATION electron correlation and for truncation at the INTERACTION APPROACH: T and Q level still a significant amount of non- MOTIVATION AND CRITICISM dynamic (long-range) electron correlation, for example, caused by a (quasi)degeneracy of J. Pople was not only a brilliant quantum chemist, electronic states. Truncated CC methods ac- but also he was perfectly capable of selling new meth- count for dynamic and also some nondynamic ods of the Pople-group to the community of quantum electron correlation effects, however, fail in chemists in an effective way. He had two successful the case of degeneracy of the reference Io) soon available via the Gaussian program package). In the first paper from 1987, the relation of QCISD (34) to CCSD theory was discussed and its corrections for T substitutions via a perturbative approach was out- lined, thus leading to QCISD(T), the QCI analogue of which indicates that for both reactant and CCSD(T). 92 Both QCI procedures were tested by ap- products the disconnected pair-pair (de- plication to some small electronic systems for which scribed by Tf ), pair-pair-pair correlation ef- FCI results were available. fects (described by T{), etc. are all included In the following years, Pople and/or his co- thus fulfilling the condition given by Eq. (20) workers rapidly published several QCI applications in the way that the same correlation effects for calculationally challenging electronic systems such are assessed for reactant and products. This as ozone,93 the excitation and ionization energies of 94 95 96 property of truncated CC is most important transition metal atoms, • fluorine peroxide, and for the studies of molecules and chemical re- the bond dissociation energies (BDEs) of a variety of actions and makes therefore CC theory su- small molecules.97 These studies illustrated the use- perior to CI theory. CISD energies, which fulness of the QCI approach. Especially, with the are corrected with the help of the Langhoff- study of BDEs a long-term project of the Pople group Davidson correction66 or other correction was started, which closely connected QCI and the Gn schemes, are denoted as CISD+Q energies, (Gaussian) theory. The Gn studies led to a series of E(CISD + Q). subsequent publications on thermochemical data, in which QCI played an essential role and thus guaran- t:,E(CISD+ Q) = E(CISD) + (1- c6)t:,E(CJSD) teed the visibility and use of QCI. For three genera- tions of Gn methods, QCISD(T) was the basic tool for (35) determining high-order correlation effects. Finally, in 2007, CCSD(T) 92 replaced QCISD(T) as a method of Since these corrections do not leads to pre- choice for high-order correlation corrections in G4. 98 cise size extensivity, Pople and co-workers90 Additional proof for the usefulness of QCI was developed a more systematic improvement of provided by the Cremer group. Gauss and Cremer truncated CI methods, which will be discussed developed in the years 1988 and 1989 the analyti- 99 100 101 in the following. cal energy gradients of QCISD and QCISD(T) •

492 © 2013 John Wiley & Sons, Ltd. Volume 3, September/October 2013 WIREs Computational Molecular Science Configuration interaction and quadratic configuration interaction where these developments were based on their previ- thanked not less than 19 well-known colleagues for ous work on the gradients for third- and fourth-order valuable comments and suggestions for their Reply. Moller-Plesset (MP3=MBPT3 and MP4=MBPT4) This indicated that more criticisms on QCI were perturbation theory.102·103 (Note that we use in this about to be published. review the terms MP and MBPT synonymously.) Scuseria and Schaefer112 published in 1989 an Cremer and co-workers demonstrated the useful- article, in which they demonstrated that the compu- ness of QCI when calculating QCI response prop- tational costs for both CCSD and QCISD are pro- erties (electron density distributions, atomic charges, portional to O(M6) and that the terms to be deleted dipole moments, quadrupole moments) for 20 small in CCSD to obtain QCISD scale as O(M5). If suit- molecules104 or the electric field gradients and nu- able intermediate arrays are formulated, the algebraic clear quadrupole coupling constants of isonitriles. 105 equations of both approaches lead to similar calcula- These authors emphasized that QCI could be used for tionalloads. In a 1989 feature article on CC theory, getting suitable reference values when testing the ac- Bartlett88 mentioned QCI shortly and stated that QCI curacy and reliability of MPn results for low values is an approximate CC method and that the choice of order n. of the term quadratic CI is unfortunate because QCI At various conferences in the late 1980s, criti- contrary to CI is not based on an eigenvalue equation, cism was raised with regard to the usefulness of the nor is it variational, nor does it include all quadratic QCI approach of Pople and co-workers. These con- terms at the QCISD level. In an article from 1995, cerns were first summarized in a 1989 publication of Bartlett and co-workers 113 reported a serious fail- Paldus and co-workers106 and triggered a scientific ure of QCISD and QCISD(T) as compared to CCSD dispute in form of several back-and-forth Comments and CCSD(T), respectively, when calculating spectro- 107 109 and Replies. - Paldus and co-workers pointed scopic and electric properties of BeO in its GS. The out that QCISD, when analyzing its cluster contri- authors traced back the failure of QCI to the S exci- butions in terms of low-order perturbation theory, is tation cluster operator Ii, as several terms involving very close to CPMET( C) (coupled pair many-electron- 't are neglected in the QCISD approach contrary to theory) 110 or CCSD-1 111 and therefore should be con- CCSD. sidered as a CC method. Pople and co-workers 107 He and Cremer114·115 compared CC and QCI confirmed this but replied that the second of their methods in terms of sixth, seventh, eighth, and two conditions (correctness for a two-electron sys- infinite-order MP perturbation theory. They deter- tem) was not fulfilled by either CPMET(C) or CCSD- mined how many of the S, D, T, Q, P (pentuple), 1 and that the latter methods differed from QCISD H (hextuple) excitations are accounted for by a given by an additional 1/2 term. QCI or CC method. Figures 1-3 summarize their re- A more serious concern raised by Paldus and sults in terms of sixth order MP (MP6) theory.11 6·117 co-workers106 was the fact that QCISDT, when con- The analysis revealed that CCSD and QCISD are structed in the same way as QCISD, was no longer equivalent in the truncated space of S, D, Q, H, · · · X size extensive (size consistent). Again, Pople and co- excitations, where X is any even excitation generated workers107 agreed with regard to this point, how- by the cluster operator Since these excitations ever pointing out that they never had claimed that describe orbital relaxation and electron pair correla- it would be. Their development90 focused on QCISD tion effects, which are the most important correla- and QCISD(T), and both methods were size extensive. tion effects for relatively small closed-shell molecules There was additional controversy with regard to with just single bonds, it became understandable that the reliability of QCISD as compared to CCSD and QCISD and CCSD perform equally well in these the use of the term quadratic CI for a method that cases. was essentially a CC method. Pople and co-workers However for larger molecules and for molecules argued that, because of the omission of certain cluster with distinct T effects, CCSD outperforms QCISD terms in the CCSD equations, QCISD was not neces- because of the following three reasons: (i) QCISD sarily the less reliable method. They showed also that rapidly falls back behind CCSD at higher values of the QCISD equations could be derived from the FCI n with regard to the total number of energy contri- equations thus stressing that the name of their method butions covered (Figures 1 and 2). (ii) Part of the T, was a reasonable descriptive of the derivation of QCI P (pentuple), ... Y contributions (Y is any odd order from FCI.107 excitation) generated by the cluster operators T1 72 are 108 109 Another round of Replies • did not lead to delayed at the QCI level by one order of perturbation any substantial change in the arguments. It is note- theory since they have to be introduced by S excita- worthy in this connection that Paldus and co-workers tion coupling. (iii) Another part of the T, P, ... Y

1000 Number of TTA, ATT, and TAT Comparison of MPn coupling terms at sixth order 900 QCI with CC theory • Completely contained 800 • Partly contained

"' e... 700 .....0,> = ..... 600

... c c I=" 1- 1: ... ...... FIGURE 3 Number of MPn (n = 6) energy contributions of the 0 I ... TTA. TAT, and TTT (A= S, D, Q, P) coupling type as accounted for by 0,> 300 ..c QCI and CC methods. The first entry (black) gives the total number, e whereas the second entry (red) gives the number of terms only partly = 200 z contained. Note that MP6 is the reference defining the total number of TT coupling contributions (11 ). 100 contributions accounted for by CCSD is not contained in QCISD at all.114,115 5 6 7 8 9 Order n of perturbation theory A noniterative improvement of QCISD by T excitations is more important for QCISD than for FIGURE 1 I Number of energy contributions pnl(ABC . . ·) CCSD. However, QCISD(T) exaggerates T effects accounted for by QCI, CC, and MPn (n = 4, 5, 6, 7, 8, ... ) methods, since it does not contain any of the TT coupling terms where A B, C. ... denote 5, D, T. Q, ... contributions, respectively. (see Figure 3 ), which normally reduce the influence of connected three-electron correlation effects. As for the total number of energy contributions, QCISD(T) falls back behind CCSD at higher orders of perturba- Number of T contributions at sixth order tion theory (Figure 1). CCSD(T) is clearly the better method because it contains, at least partly, two of the TT coupling effects at sixth order. QCISD(T) easily tends to an exaggeration ofT effects. The difference between QCI and CC is considerably decreased at the CCSD(TQ) and QCISD(TQ) level of theory if one considers in particular MP6 correlation effects (Figure 2). However, if one considers the total number of energy contributions cov-

c c 1- 1: ered for larger n, then QCISD(TQ) is even inferior to

494 © 2013 John Wiley & Sons, Ltd. Volume 3, September/October 2013 WIREs Computational Molecular Science Configuration interaction and quadratic configuration interaction levels. 107 However in 1990, he and his group did (oiH721o) = Ecorr (43) and QCISDTQ are identical with the corresponding CC methods CCD and CCSDTQ, respectively, - A A aECISD (44) whereas size-extensive QCISDT is neither identical (fiH(Tt + 12)1o) = Ci corr with CCSDT nor with a QCISDT method derived according to Pople's concept of converting CI into size- (ab1H(1 + 1; + T)I) = cabECISD (45) extensive QCI.118 Their derivation of size-extensive i j 1 2 0 11 corr CI (ECI) methods provides an insight into the ba- The unlinked diagrams result now from cf and sic equations of the various methods and is shortly cabECISD z1 corr in Eqs. (44) and (45). They can be can- sketched here. celed by adding (f IHT1 12 Ifl1HTi/2)1o) on the left of Eq. (45). In this way, C and T). Instead, a distinction is made by denoting the ECISD (size-extensive CISD) projection equations CI amplitudes by cf/k.::· and CC (QCI, ECI) amplitudes are obtained. by af/k.::·. The operators b+ and b are creation and - A ECISD annihilation operators. (oiH721o) = Ecorr (46) The CID projection equations can be written as

- A CJD - A A A A a ECI SD (oiHT21o) = Ecorr (36) (fiH(Tt + 12 + Tt12)1o) = ai Ecorr (47)

ab - (1 ) abECID (37) (ij IH + L2 ff1H(1 + t1 + t oiHT2Io) = Ecorr (49) ECIDcorr corresponds to the CID correlation energy EciD = E(CID)- E(HF). (40) corr - A A A a CISDT (fiH(Ii + T2 + 73)1o) = Ci Ecorr (50)

In Eq • (37), cabz1 ECIDcorr corresponds to a discon-

b - A A A a b CIS DT nected term that leads to unlinked diagrams. Hence, (fj IH(1 + 7i + T2 + 73)1o) = Cij Ecorr (51) the CID method is not size extensive. The simplest matrix element that contains the same unlinked dia-

b - A A A abc CISDT (52) grams is (fl1HTi/21o). By inserting this term on (fjtiH(Tt + 12 + T3)Io) = Cijk Ecorr · the left side of Eq. (37), size-extensive CID (denoted From Eqs. (50) and (51), the ECISDT projection equa- as ECID) is obtained: tions for S and D excitations are obtained in the (41) same way as the corresponding ECISD equations.

The T projection equation contains two disconnected The CISDTQ projection equations are given by terms: (fftl Hii I 0 IHT2 I 0 ) = ECISDTQcarr (60) of Eq. (52). However by canceling cf/'k one gets the new disconnected term (fft I't ( HT3 )c I o) = - A A A CISDTQ s b A - A (61) Ls (fjk 1121s) (s I( HT3)cl fiH(Tt + T2 + T3)1o) = cf Ecarr denotes a restriction to connected terms. This requires the inclusion of further terms for which the single ex- ab - A A A A _ ab CI SDTQ citations multiplied by T2 and the double excitations (ii IH(1 + Tt + T2 + T3 + nJIo)- cii Ecarr multipliedA A by 1i Aare A2 used A2thus leadingA A to the extra (62) terms Tt T2, 1/2 Tt T2 , 1/2 T1, and Tt T3. In this way, all disconnected terms are cancelled in Eq. (52) and 8 abc - A A A A _ abc CI SDTQ size extensivity is enforced.11 This leads to (iik IH(Tt + T2 + T3 + nJIo)- ciik Ecarr

EECISDT ( 0 IHT2 I 0 ) = carr (53) (63)

abcdlfl(T + T + j; )I ) = cabcdECISDTQ (f1H(T1 + t2 + t + t1 T2lio) ( 11kl 2 3 4 0 11kl carr · = aa EECISDT (64) z corr (54) Extending these equations by terms, which cancel the unlinked diagrams, the ECISDTQ projection equa- b - A A A 1 A2 tions are obtained, which turn out to be identical with (fi IH(1 + T1 + 12 + T3 + 212 )Io) the CCSDTQ equations: = aabEECISDT (55) zJ carr ECCSDTQ =( IH(T + y;2)1 ) (65) carr 0 2 2 1 0

b - A A A 1 A2 (fitiH(Tt + T2 + T3 + 2 T1

A A A A 1 A A2 A A 1 A2 = aa ECCSDTQ +T1T2 + T1T3 + 2TtT2 + 12T3 + 212 )Io) z corr (66)

= (aabc + (abclj; j: ))EECISDT 11k 11k 1 2 carr (56) or, alternatively, to (fiH(Ii + T2 + t + T1 'tlio)c = 0 (57)

= (aab + (abl t2I

b - A A A A 1 A2 A A A A b -A A AA AA AA (fit1H(T1 + 12 + T3 + n + 'ft + T1 T2 + T1 T3 (fit1H(72 + T3 + Tt 12 + T1 T3 + T2 T3 2 AA 1A2 AA 1A3 1A2A 1A2 1AA2 + T1 n + 12 + 12 T3 + ! 'ft + T1 12 +212 +2T172lio)c=0 (59) 2 3 2 1Ar 1Ar 1AA 2 AAA 1A 3 + T1 T3 + 'ft n + TtT2 + TtT2T3 + ! T i.e., when constraining to connected terms. 2 2 2 3 2 Clearly, ECISDT (size-extensive CISDT) is not 1A4 1A3A 1A3A 1A2A2 identical to the QCISDT method constructed in the + 4! T1 + 3! T1 T2 + 3! T1 T3 + 4 T1 12 same way as QCISD. 90 Since ECISDT is also not identical to CCSDT, it is intermediate between QCISDT 1 A5 1 A6 and CCSDT, but closer to the latter than the former + 5! T1 + 6! T1 )Io) method. ECISDT and CCSDT are both size-extensive 118 = (aabc + (abclj; j: + _.!._ t3I

bd-A A A 1A2 AA AA (ctJfitJ IH(T2 + T3 + + 'ft + T1 n + T1 T3 n 2 (73) AA 1A2 AA AA 1A2 +7in+ 2r2 1A3 1Ar 1Ar 1Ar 1AA2 min[p+2,n] + ! 'ft + r1 n + r1 T3 + r1 + T1T - ""' A CI 3 2 2 2 n 2 2 (clJpiH( L,.- 7;)1¢o) = CpEeorr (n?:,p?:,3) AAA 1A3 11:4 1A3A 1A3A i=p-2 + 7i n T3 + 3! 72 + 4! 1 + 3! T1 n + 3! 'ft T3 (74) 1A2A2 1A5 1A4A 1A6 where s, d, and p are S, D, and general excitation +-'ft 72 +-'ft +-'ft n +-'ft )lctJo) 4 5! 4! 6! indices. abed abed 1 A2 A A 1 A2 A As shown above, it suffices to add Ii 72 and = (aiikl + (¢iikl 1272 + T1T3 + 2T1 n Ti/2 to the Sand D projection Eqs. (72) and (73), respectively, to eliminate all disconnected terms from +_.!._ 1;41¢ ))ECCSDTQ 4! 1 0 eorr (69) these equations. For any excitation index p higher than d, just the disconnected terms (¢pi HTp-2lclJo) This derivation underlines two important facts: and appear in the corresponding projection equations. Introduction of - HTp-2 and parts of the • A size-extensive truncated CI (ECI) method, term H't Tp, namely (H't Tp)c and Tp(H't)c, on which contains P-fold substitutions (P = S, D, T, Q, ... ) cannot be achieved by just con- the left side of Eq. (74) leads to a cancellation of all disconnected terms and to the QCic equations in their sidering quadratic correction terms in the two 118 highest projection equations. general form :

QCI - A • Size-extensive ECI methods do not form a Eeorr e = (ctJoiHnlctJo) (75) hierarchy of independent CC methods, and, therefore, the concept of improving truncated CI to size-extensive CI is not a generally useful concept. The same holds for the quadratic CI approach in the sense as it was originally derived. 90 _ A A A A 1 - A2 (¢d1H(1 + T1 + n + T3 + nJ + 2(HT2 )clctJo) = 0 118 Cremer and He used the ECI projection equa- (77) tions to develop a hierarchy of size-extensive CI methods for a reference wavefunction constructed from HF orbitals. Since these methods are also based on min[p+2,n] the addition of quadratic terms to the CI projection (ctJpiH( L Ii) + (H'tTp)clctJo) equations where however the addition is done in such i=p-1 a way that all disconnected terms are canceled and = 0 (n ?:, p ?:, 3) (78) just connected terms remain in the new QCI projection equations, the term QCic (c: connected) was These projection equations lead to a hierarchy of coined. 118 Formally, the QCic equations can be de- QCic methods that are all size extensive. rived directly from the corresponding CI equations. On the basis of this approach, Cremer and He121 For this purpose, the projection equations of a trun- developed size-extensive QCISDTc: cated CI method that includes up top-fold excitations QCISDT- - A Ecorr - (¢o1HT21¢o) (79) are written as follows: (70) (80)

(ctJpiH(1 + Ii + t + · · · + YnllctJo) b - A A A 1 A2 = CpE;;_r (p = 1, 2, ... , n) (71) (ctJfi IH(1 + 7i + T2 + T3 + 2 T2 )lctJo)c = 0 (81) or, alternatively, as - A A A CI (¢s1H(7i + T2 + T3)1¢o) = CsEeorr (72) (82)

The authors demonstrated that QCISDTc leads to mathematical problems of carrying out for example results comparable to those obtained with CCSDT, a FCI calculation for just a 10-electron atom with a which has to do with the fact that up to fifth-order larger basis set, they probably would have stopped perturbation theory just the TQ coupling term of their efforts, which would have been a tremendous MPS is missing. 121 Although QCISDTc and CCSDT loss to quantum chemistry as a whole. Many of the 8 both scale with O(M ) (actually O(N3(M - N)5)), heroic attempts to improve the performance of CI the former method offers time saving because of its triggered activities aimed at the development of alter- faster convergence in the QCic iterations. He and co- native methods, deepened the understanding of the workers122 showed in a follow-up publication that quantum chemical methodology, and provided an in- QCISDTc is clearly superior to QCISD(T), and they sight into calculational outcomes. traced performance advantages back to a better de- Today, the continuous value of CI theory results scription of TT coupling terms contained in the for- from two basic facts: (i) Single reference CI discussed mer, however not in the latter method. in this article can be easily extended to MR-CI (at In 2000, He and co-workers123 extended the least conceptually), which is the method of choice in size-extensive QCic methods by developing QCIS- many cases. This is more difficult for multireference DTQc: CC theory. 89 (ii) There is a continuous interest in FCI calculations because of the need of reliable reference EQCISDT =( IHT I ) (83) corr 0 2 0 values. Also new developments such as the connection of FCI with quantum Monte Carlo124 calculations have opened new avenues for FCI applications. (flfl(Ti + T2 + t + T1 Ilio)c = 0 (84) However, one should not forget that a system with 12 electrons in 20 orbitals or 10 electrons in 30 orbitals

b - A A A A 1 A2 already leads to a billion Slater determinants so that (fi IH(1 + Ti + 12 + T3 + 14 + 2 72 )Io)c = 0 FCI reference calculations are rather limited. (85) Truncated CI is used today mostly to demon- strate the shortcomings of CI. There is however an interest in using truncated CI in connection with relativistic theory, for example, when calculating spin orbit coupling effects. 125 The situation is somewhat different for highly correlated CI methods such as bd-A A AA (fiki IH(T3 + 14 + T2 14)1o)c = 0 (87) CISDTQ, which leads to energies close to FCI energies and therefore is an interesting reference method. These authors also developed QCISDTQc(6) as a In view of the costs of a CISDTQ calculation, the use size-extensive Q excitation method that is exact at of this and similar CI methods is also rather limited. sixth-order perturbation theory, leads to results which The QCI approach is certainly closer to CC are better than those from CCSD(T), QCISDTc, or rather than CI theory. This may be disputable for the CCSDT calculations and close to CCSDTQ and FCI original QCISD method of Pople and co-workers.90 results. QCISDTQc( 6) is the cheapest of all Q ex- However, this statement is definitely true if this ap- citation containing methods accounting for infinite proach is converted into a general, size-extensive order correlation effects introduced by the 14 clus- 123 (size-consistent) QCic method as done by Cremer and ter operator. He and co-workers also showed the 118 co-workers. •121-123 The original QCISD approach importance of connected four-electron correlation ef- does not lead to a set of independent methods because fects, which especially in the case of multireference converting truncated CI into Pople-type QCI meth- electronic systems containing electronegative atoms ods only leads to QCISD, whereas QCID = CCD90 are needed to closely approximate FCI reference en- and QCISDT, QCISDTQ, etc. are not size extensive ergies. (however, ECISDT, ECISDTQ, QCISDTc, and QCIS- DTQc are size extensive).11 8 CONCLUSIONS AND OUTLOOK With CCSD(T) becoming the gold standard for quantum chemical calculations, 126 the use of The research on CI theory was for many decades one QCISD(T) has dramatically decreased. QCISD(T) of the major driving forces for the development of does not lead to significant cost advantages, has de- accurate and routinely applicable quantum chemi- ficiencies with regard TT coupling effects, and fails cal methods. If the early pioneers of the CI theory if systems with multireference character are investi- had known about all the obstacles and the enormous gated. This is also documented by the fact that the

G4-theory of Curtiss98 is based on CCSD(T) rather the approximate CC methods offered in the Gaus- than QCISD(T). sian program is QCI. Therefore, it is not surprising Clearly, the scientific dispute between Pople and that even in our time QCI is discussed and reviewed. his group on the one side and a large number of well- In 2006, Bartlett and MusiaP27 reconsidered the pos- known colleagues headed by Paldus on the other side sibility of deriving truncated, but size-extensive CI has been fruitful, especially with regard to realizing methods. Paldus126 wrote in 2010 a retrospection more clearly the problems of truncated CI methods, on QCI and the scientific dispute he had with Pople the importance of size extensivity, and the advantages where he points out in a humorous way that the oppo- of CC theory. Some of the scientific dispute on the nents of the dispute never lost their mutual respect and usefulness of the QCI approach has reached into our appreciation. Clearly, QCI is worth to be reviewed in days because the Gaussian series of programs started the year 2012, but the current summary may be one by Pople37 is still heavily used today, and one of of the last reviews on the topic.

ACKNOWLEDGMENTS This work was financially suppor ted by the National Science Foundation, Grant CHE 1152357. We thank SMU for providing computational resources. I thank A. Humason for useful comments.

REFERENCES

1. Shavitt I. The method of configuration interaction, in photoionization cross sections: the first-row hydrides. methods of electronic structure theory. In: Schaefer J Chern Phys 2004, 121:4143. HF, ed. Modern Theoretical Chemistry. New York: 9. Brillouin L. La methode du champ self-consistent. Ac- Plenum Press; 1977, 189-275. tual Scie Ind 1933, 71:46. 2. Roos B, Siegbahn P. The direct configuration inter- 10. Slater JC. The theory of complex spectra. Phys Rev action method from molecular integrals. In: Schaefer 1929,34:1293-1322. HF ed. Modern Theoretical Chemistry. New York: 11. Slater JC. Molecular energy levels and valence bonds. Plenum Press; 1977, Vol. 3, Chap. 7, pp. 277- Phys Rev 1931, 38:1109-1144. 318. 12. Weyl H. Gruppentheorie und Quantenmechanik. 3. Meyer W. Configuration expansion by means of Leipzig, Germany: Hirzel; English translation: The pseudo natural orbitals, in methods of electronic struc- Theory of Groups and Quantum Mechanics. 1st ed. ture theory. In: Schaefer HF, ed. Modern Theoretical London: Methuen; 1928. Chemistry. New York: Plenum Press; 1977,413--455. 13. Paldus ]. Group theoretical approach to the config- 4. Carsky P. Configuration interaction. In: Schleyer P, uration interaction and perturbation theory calcula- Allinger NL, Clark T, Gasteiger J, Kollman PA, Schae- tions for atomic and molecular systems. J Chern Phys fer HF, Schreiner PR. eds. Encyclopedia of Compu- 1974, 61:5321-5330. tational Chemistry. Chichester, UK: John Wiley and 14. Cremer D. Moller-Plesset perturbation theory. In: Sons; 1998, Vol. 1, 485--497. Schleyer P, Allinger NL, Clark T, Gasteiger J, Koll- 5. Sherrill CD, Schaefer HF III. The configuration in- man PA, Schaefer HF, Schreiner PR, eds. Encyclo- teraction method: advances in highly correlated ap- pedia of Computational Chemistry. Chichester, UK: proaches. Adv Quant Chern 1999, 34:143-269. John Wiley & Sons; 1998, Vol. 3, 1706-1735. 6. Szabo A, Ostlund NS. Modern Quantum Chemistry: 15. Cremer D. Moller-Plesset perturbation theory, from Introduction to Advanced Electronic Structure The- small molecule methods to methods for thousand of ory. New York: McGraw-Hill, 1989. atoms. Wiley Interdiscip Reviews: Comput Mol Sci 7. Lowdin P-0. Quantum theory of many-particle sys- 2011, 1:509-530. tems. III. extension of the hartree-fock scheme to in- 16. Crawford TD, Schaefer HF. An introduction to cou- clude degenerate systems and correlation effects. Phys pled cluster theory for computational chemists. In: Rev 1955, 97:1509-1520. Lipkowics KB Boyd DB, eds. Reviews in Computa- 8. Seabra GM, Kaplan IG, Zakrzewski VG, Ortiz JV. tional Chemistry. New York: John Wiley & Sons; Electron propagator theory calculations of molecular 2000, Vol. 14, 33-136.

17. Bartlett RJ, ed. Recent Advances in Computational tions on organic molecules with gaussian-type func- Chemistry- Volume 3: Recent Advances in Coupled tions. Introductory review and mathematical formal- Cluster Methods. Singapore: World Scientific; 1997. ism. Theor Chim Acta 1966, 6:191-216. 18. Parr RG, Yang W. International Series of Mono- 35. Davies DR, Clementi E. IBMOL: Computation of graphs on Chemistry -16. Density Functional Theory Wave Functions of General Geometry. San Jose, CA: of Atoms and Molecules. New York: Oxford Univer- Special IBM-Research Report, 1965. sity Press; 1989. 36. Pendergast P, Hayes EF, Liedtke RC, Schwartz ME, 19. Hylleraas EA. Uber den grundzustand des heliu- Rothenberg S, Kollman PA. MOLE: a system for matoms. Physik Z 1928, 48:469-494. quantum chemistry. II. recent developments. Int J 20. Hylleraas EA. Neue berechnung der energie des heli- Quant Chern 1976, 10:77-83. ums in grundzustande, sowie des tiefsten terms von 37. Hehre WJ, Lathan WA, Ditchfield R, Newton MD, orthohelium. Physik Z 1929, 54:347-366. Pople JA. Gaussian 70, quantum chemistry program 21. Boys SF. Electronic wave functions. II. A calculation exchange, program no. 237, 1970. for the ground state of the beryllium atom. Proc Roy 38. Nesbet RK. Computer programs for electronic wave- Soc A (London) 1950, 201, 125-137. function calculations. Rev Mod Phys 1963, 35:552- 22. Taylor GR, Parr RG. Superposition of configurations: 557. the helium atom. Proc USA Natl Acad Sci 1952, 39. Nesbet RK. Algorithm for diagonalization of large 38:154-160. matrices. J Chern Phys 1965, 43:311-312. 23. Lowdin PO. Quantum theory of many-particle sys- 40. Bender CF. A bottleneck in molecular quantum me- tems. 1. Physical interpretations by means of den- chanical calculations. Comput Phys 1972, 9:54 7- sity matrices, natural spin-orbitals, and convergence 554. problems in the method of configuration interaction. 41. Shavitt I, Bender CF, Pipano A, Hosteny RP. The it- Phys Rev 1955, 97:1474-1489. erative calculation of several of the lowest or high- 24. Shull H, Lowdin PO. Natural spin orbitals for helium. est eigenvalues and corresponding eigenvectors of J Chern Phys 1955,23:1565-1573. very large symmetric matrices. J Comput Phys 1973, 25. Bender CF, Davidson ER. A natural orbital based en- 11:90-108. ergy calculation for helium hydride and lithium hy- 42. Davidson ER. The iterative calculation of a few of the dride. J Phys Chern 1966, 70:2675-2685. lowest eigenvalues and corresponding eigenvectors of 26. Bender CF, Davidson ER. Studies in configuration large real-symmetric matrices. J Comput Phys 1975, interaction: the first-row diatomic hydrides. Phys Rev 17:87-94. 1969, 183:23-30. 43. Lancoz C. An iteration method for the solution of the 27. Edmiston C, Krauss M. Pseudonatural orbitals as a eigenvalue problem of lineal differential and integral basis for the superposition of configurations. I. He2. operators. J Res Nat Bur Stand 1950, 45:255-282. J Chern Phys 1966,45:1833-1839. 44. Roos B. A new method for large-scale CI calculations. 28. Meyer W. Ionization energies of water from pno-ci Chern Phys Lett 1972, 15:153-159. calculations. Int J Quant Chern Symp 1990, 5:341- 45. Gel'fand IM, Tsetlin ML. Finite-dimensional rep- 348. resentations of the group of unimodular matrices. 29. Meyer W. Pno-ci studies of electron correlation Doklady Akad Nauk SSSR (N S) 1950, 71:825- effects. I. Configuration expansion by means of 828. nonorthogonal orbitals and application to the ground 46. Gel'fand IM, Tsetlin ML. Finite-dimensional repre- state and ionized states of methane. J Chern Phys sentations of groups of orthogonal matrices. Doklady 1973, 58:1017. Akad Nauk SSSR (N S) 1950, 71:1017-1020. 30. Almlof ], Taylor PR. General contraction of gaussian 47. Moshinsky M. Group Theory and the Many-Body basis sets. I. Atomic natural orbitals for first and sec- Problem. New York: Gordon and Breach; 1968. ond row atoms. J Chern Phys 1987, 86:4070-4077. 48. Matsen FA, Pauncz R. The Unitary Group in Quan- 31. Almlof ], Taylor PR. Atomic natural orbital (ANO) tum Chemistry. Amsterdam, the Netherlands: Else- basis sets for quantum chemical calculations. Adv vier; 1986. Quant Chern 1991, 22:301-373. 49. Paldus ]. Many-electron correlation problem: a group 32. Nesbet RK. Configuration interaction in orbital theo- theoretical approach. In: Eyring H, Henderson D, eds. ries. Proc Roy Soc A (London) 1955, 230, 312-318. Theoretical Chemistry: Advances and Perspectives. 33. Boys SF, Cook GB, Reeves CM, Shavitt I. Automatic New York: Academic Press; 1976, Vol. 2, 131-290. fundamental calculations of molecular structure. Na- 50. Pauncz R. Spin Eigenfunctions, Construction and ture (London) 1956, 178:1207-1209. Use. New York: Plenum Press; 1979. 34. Csizmadia IG, Harrison MC, Moskowitz JW, Sut- 51. Shavitt I. Graph theoretical concepts for the uni- cliffe BT. Nonempirical LCAO-MO-SCF-CI calcula- tary group approach to the many-electron correlation

problem. Int] Quantum Chern Symp 1977, 11:131- 65. Lindgren I, Morrison]. Atomic Many-Body Theory. 148. 2nd ed. Berlin: Springer; 1986. 52. Shavitt I. Matrix element evaluation in the unitary 66. Langhoff SR, Davidson ER. Configuration interaction group approach to the electron correlation problem. calculations on the nitrogen molecule. Int J Quant Int J Quantum Chern Symp 1978, 12:5-32. Chern 1974, 8:61-72. 53. Shavitt I. The graphical unitary group approach and 67. Davidson ER, Silver DW. Size consistency in dilute its application to direct configuration interaction cal- helium gas electronic-structure. Chern Phys Lett 52, culations. In: Hinze J, ed. The Unitary Group for 1977:403-406. the Evaluation of Electronic Energy Matrix Ele- 68. Duch W, Diercksen GHF. Size-extensivity corrections ments (Lecture Notes in Chemistry, Vol. 22), Berlin: in configuration interaction methods. J Chern Phys Springer-Verlag; 1981, 51-99. 1994, 101:3018-3030. 54. Brooks BR, Schaefer HF. The graphical unitary group 69. Bartlett RJ, Shavitt I. Determination of size- approach to the electron correlation problem. Meth- consistency error in single and double excitation con- ods and preliminary applications. J Chern Phys 1979, figuration interaction-model. Int J Quant Chern 1977, 70:5092-5106. 12:165-173. 55. Brooks BR, Laidig WD, Saxe P, Goddard JD, Yam- 70. Ahlrichs R, Scharf P, Ehrhardt C. The coupled pair aguchi Y, Schaefer HF. Analytic gradients from cor- functional (CPF). A size consistent modification of the related wave functions via the two-particle density CI(SD) based on an energy functional. J Chern Phys matrix and the unitary group approach. J Chern Phys 1985, 82:890-898. 1980, 72:4652-4653. 71. Harrison RJ, Handy NC. Full CI results for Be2 56. Siegbahn PEM. Generalizations of the direct CI and (H2h in large basis-sets. Chern Phys Lett 1983, method based on the graphical unitary group ap- 98:97-101. proach. I. Single replacements from a complete CI 72. Grev RS, Schaefer HF. Natural orbitals from single root function of any spin, first order wave functions. and double excitation configuration interaction wave J Chern Phys 1979, 70:5391-5393. functions: their use in second order configuration in- 57. Siegbahn PEM. Generalizations of the direct CI teraction and wave functions incorporating limited method based on the graphical unitary group ap- triple and quadruple excitations. J Chern Phys 1992, proach. II. single and double replacements from any 96:6850-6856. set of reference configurations. J Chern Phys 1980, 73. Sherrill CD, Schaefer HF III. Compact varia- 72:1647-1656. tional wave functions incorporating limited triple 58. Siegbahn PEM. The direct CI method. In: Dierk- and quadruple substitutions. J Chern Phys 1996, sen GHF, WilsonS, eds. Methods in Computational 100:6069-6075. Molecular Physics. Dordrecht, the Netherlands: D. 74. Bunge CF, Carb6-Dorca R. Select-divide-and- Reidel; 1983, 189-207. conquer method for large-scale configuration inter- 59. Landau L, Lifschitz EM. Quantum Mechanics: Non- action.] Chern Phys 2006, 125:014108-014114. relativistic Theory, Volume III. 3rd ed. Amsterdam, 75. Bunge CF. Selected configuration interaction with the Netherlands: Butterworth-Heineman; 2003. truncation energy error and application to the Ne 60. Krishnan R, Schlegel HB, Pople JA. Derivative stud- atom.] Chern Phys 2006, 125:014107-014110. ies in configuration-interaction theory. J Chern Phys 76. Saxe P, Schaefer III HF, Handy NC. Exact solution 1980, 72:7654-7655. (within a double-zeta basis set) of the schrodinger 61. Yamaguchi Y, Osamura Y, Goddard JD, Schaefer electronic equation for water. Chern Phys Lett 1981, HF. A New Dimension to Quantum Chemistry: Ana- 79:202-204. lytic Derivative Methods in Ab Initio Molecular Elec- 77. Knowles PJ, Handy NC. A new determinant-based tronic Structure Theory. New York: Oxford Univer- full configuration-interaction method. Chern Phys sity Press; 1994. Lett 1984, 111:315-321. 62. Shepard R. The analytic gradient method for config- 78. Siegbahn PEM. A new direct CI method for large CI uration interaction wave functions. In: Yarkony DR, expansions in a small orbital space. Chern Phys Lett ed. Modern Electronic Structure Theory, Part 1. Sin- 1984,109:417-423. gapore: World Scientific; 1995, 345--458. 79. Olsen J, Roos BO, Jorgensen P, Jensen HJA. 63. Pople JA. Theoretical models for chemistry. In: Smith Determinant based configuration interaction algo- DW, McRae WB, eds. Energy, Structure and Reactiv- rithms for complete and restricted configuration ity. New York: John Wiley & Sons; 1973, 51-61. interaction spaces. J Chern Phys 1988, 89:2185- 64. Pople JA, Seeger R, Krishnan R. Variational con- 2192. figuration interaction methods and comparison with 80. Bauschlicher CW Jr, Taylor PR. Benchmark full perturbation-theory. Int J Quant Chern 1977, configuration-interaction calculations on H2 0, F, and 12:149-163. p-. J Chern Phys 1986, 85:2779-2783.

81. Knowles PJ, Handy NC. Unlimited full configuration- 95. Raghavachari K Trucks GW. Highly correlated sys- interaction calculations.] Chern Phys 1989,91:2396- tems. Ionization energies of first row transition metals 2398. Sc-Zn. J Chern Phys 1989, 91:2457-2460. 82. Olsen], Jorgensen P, Simons]. Passing the one-billion 96. Raghavachari K Trucks GW. Highly correlated sys- limit in full configuration-interacfion (FCI) calcula- tems. Structure and harmonic force field of F202. tions. Chern Phys Lett 1990, 169:463--472. Chern Phys Lett 1989, 162:511-516. 83. Rossi E, Bendazzoli G, Evangelisti S, Maynau D. 97. Pople JA, Head-Gordon M, Fox DJ, Raghavachari K, Passing the one-billion limit in full configuration- Curtiss LA. Gaussian1 theory: A general procedure interacfion (FCI) calculations. Chern Phys Lett 1999, for prediction of molecular energies. J Chern Phys 310:530-536. 1989, 90:5622-5629. 84. Thogersen L Olsen J. A coupled cluster and full con- 98. Curtiss LA. Gaussian-4 theory. J Chern Phys 2007, figuration interaction study of CN and eN-. Chern 126:084108. Phys Lett 2004, 393:36-43. 99. Gauss], Cremer D. Analytical evaluation of energy 85. Gan Z, Grant DJ, Harrison RJ, Dixon DA. The lowest gradients in quadratic interaction theory. Chern Phys energy states of the group-IIIA-group-VA heteronu- Lett 1988, 150:280-286. clear diatomics: BN, BP, AlN, and AlP from full 100. Gauss ], Cremer D. Analytical differentiation of configuration interaction calculations. J Chern Phys the energy contribution due to triple excitations 2006, 125:124311. in quadratic configuration interaction theory. Chern 86. Gan Z, Harrison RJ. Calibrating quantum chemistry: Phys Lett 1989, 163:549-554. a multi-teraflop, parallel-vector, full-configuration in- 101. Gauss ], Cremer D. Analytical energy gradients in teraction program for the cray-Xl. In: Proceedings of Moller-Plesset perturbation and quadratic configura- the 2005 ACM/IEEE conference on Supercomputing. tion interaction methods: theory and application. Adv Washington, DC, USA: IEEE Computer Society; 2005 Quant Chern 1992,23:205-299. 22. 102. Gauss], Cremer D. Implementation of analytical en- 87. Rolik Z, Szabados A, Surjan PR. A sparse ma- ergy gradients at third and fourth Order Moller- trix based full-configuration interaction algorithm. J Plesset perturbation theory. Chern Phys Lett 1987, Chern Phys 2008, 128:144101. 138:131-140. 88. Bartlett RJ. Coupled-cluster approach to molecu- 103. Gauss], Cremer D. Analytical differentiation of the lar structure and spectra: a step toward predictive energy contribution due to triple excitations in fourth- quantum chemistry. J Phys Chern 1989, 93:1697- order Moller-Plesset perturbation theory. Chern Phys 1708. Lett 1989, 153:303-308. 89. Paldus ], Pittner J, Carsky P. New developments in 104. Kraka E, Gauss], Cremer D. Determination and use many body perturbation theory and coupled clus- of response-densities. J Mol Struct (Theochem) 1991, ter theory. In: Carsky P, Paldus ], Pittner J, eds. 234:95-126. Recent Progress in Coupled-Cluster Methods: The- 105. Cremer D, Kruger M. Electric field gradients and ory and Applications. Berlin: Springer; 2010, 455- nuclear quadrupole coupling constants of isonitriles 489. obtained from Moller-Plesset and quadratic config- 90. Pople JA, Head-Gordon M, Raghavachari K. uration interaction calculations. J Phys Chern 1992, Quadratic configuration interaction. A general tech- 96:3239-3245. nique for determining electron correlation energies. J 106. Paldus ], Cizek ], Jeziorski B. Coupled cluster ap- Chern Phys 1987, 87:5968-5975. proach or quadratic configuration interaction? J 91. Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Chern Phys 1989, 90:4356-4362. Robb MA, Cheeseman JR, Scalmani G, Barone V, 107. Pople JA, Head-Gordon M, Raghavachari K. Mennucci B, Petersson GA., et a!., Gaussian 09 Revi- Quadratic configuration interaction: Reply to com- sion A.l. Wallingford CT: Gaussian Inc.; 2009. ment by Paldus, Cizek, and Jeziorski. ] Chern Phys 92. Raghavachari K, Trucks GW, Pople JA, Head- 1989, 90:4635-4636. Gordon M. Fifth-order perturbation comparison of 108. Paldus ], Cizek ], Jeziorski B. Coupled cluster electron correlation theories. Chern Phys Lett 1989, approach or quadratic-configuration interaction?: 157:4 79-483. Reply to comment by Pople, Head-Gordon, and 93. Raghavachari K, Trucks GW, Pople JA, Replogle E. Raghavachari.J Chern Phys 1990, 93:1485-1486. Highly correlated systems: Structure, binding energy 109. Raghavachari K, Head-Gordon M, Pople JA. Re- and harmonic vibrational frequencies of ozone. Chern ply to comment on: Coupled cluster approach or Phys Lett 1989, 158:207-212. quadratic configuration interaction?. J Chern Phys 94. Raghavachari K Trucks GW. Highly correlated sys- 1990,93:1486-1487. tems. Excitation energies of first row transition metals 110. Paldus ], Cizek ], Shavitt I. Correlation problems Sc-CuJ Chern Phys 1989, 91:1062-1065. in atomic and molecular systems. IV. extended

coupled-pair many-electron theory and Its applica- in Quantum Chemistry, Volume III. Dordrecht, the tion to the BH3 molecule. A PhysRev 1972, 5:50-67. Netherlands: Kluwer; 1997,239-318. 111. Purvis GD, Bartlett RJ. The reduced linear equation 120. Raghavachari K, Pople JA, Replogle E, Head-Gordon method in coupled cluster theory.] Chern Phys 1981, M. Fifth order Moller-Plesset perturbation theory: 75:1284-1292. comparison of existing correlation methods and im- 112. Scuseria GE, Schaefer HF. Is coupled cluster singles plementation of new methods correct to fifth order. J and doubles (CCSD) more computationally intensive Phys Chern 1990, 94:5579-5586. than quadratic configuration interaction (QCISD)? J 121. Cremer D, He Z. Size-extensive QCISDT - imple- Chern Phys 1989, 90:3700-3703. mentation and application. Chern Phys Lett 1994, 222:40-45. 113. Watts JD, Urban M, Bartlett RJ. Accurate electri- cal and spectroscopic properties of X1 I;+ BeO from 122. He Z, Kraka E, Cremer D. Application of quadratic coupled-cluster methods. Theor Chim Acta 1995, CI with singles, doubles, and triples (QCISDT): an 90:341-355. attractive alternative to CCSDT. Int J Quant Chern 1996, 57:157-172. 114. He Z, Cremer D. Analysis of coupled cluster and quadratic configuration interaction theory in terms 123. He Y, He Z, Cremer D. Size-extensive quadratic of sixth-order perturbation theory. Int J Quant Chern CI methods including quadruple excitations: 1991, 25:43-70. QCISDTQ and QCISDTQ(6) - On the importance of four-electron correlation effects. Chern Phys Lett 115. He Cremer D. Analysis of coupled cluster meth- Z, 2000, 317:535-544. ods. II. what is the best way to account for triple ex- 124. Booth GH, Alavi A. Approaching chemical accuracy citations in coupled cluster theory? Theor Chim Acta using full configuration-interaction quantum Monte 1993, 85:305-323. Carlo: A study of ionization potentials. J Chern Phys 116. He Z, Cremer D. Sixth-order many-body perturba- 2010,132:1741041-1741047. tion theory I. basic theory and derivation of the energy 125. Alekseyev AB, Liebermann HP, Buenker RJ. Spin- formula. Int J Quant Chern 1996, 59:15-29. orbit multireference configuration interaction method 117. He Z, Cremer D. Sixth-order many-body perturba- and applications to systems containing heavy atoms. tion theory II. implementation and application. Int J In: Hirao K, Ishikawa M, eds. Relativistic Molecu- Quant Chern 1996, 59:31-55. lar Calculations. Singapore: World Scientific; 2003, 118. Cremer D, He Z. Analysis of coupled cluster methods. 65-105. IV. size-extensive quadratic CI methods - quadratic 126. Paldus ]. QCI and related CC approaches: a retro- CI with triple and quadruple excitations. Theor Chim spection. Mol Phys 2010, 108:2941-2950. Acta 1994, 88:47-67. 127. Bartlett RJ, Musial M. Addition by subtraction in 119. Cremer D, He Z. New developments in many body coupled-cluster theory: A reconsideration of the CC perturbation theory and coupled cluster theory. In: and CI interface and the nCC hierarchy. J Chern Phys Calais J-L, Kryachko E, eds. Conceptual Perspectives 2006, 125:204105.