J. Phys. Chem. 1996, 100, 6007-6016
6007
Samuel Francis Boys Nicholas C. Handy Department of Chemistry, UniVersity of Cambridge, CB2 1EW, U.K.
John A. Pople Department of Chemistry, Northwestern UniVersity, EVanston, Illinois 60208-3113
Isaiah Shavitt Department of Chemistry, The Ohio State UniVersity, Columbus, Ohio 43210 ReceiVed: September 28, 1995X
The development most often associated with the name of S. F. Boys in quantum chemistry is the introduction of Gaussian basis functions in electronic structure calculations. Interestingly, while Boys was fascinated with the integrability properties of products of Gaussians on different centers, and used them in other applications, including approximation schemes for multicenter integrals over Slater-type orbitals, he did not pursue their use as basis functions in their own right in his own molecular calculations. It was the work of his student, Colin M. Reeves, and the latter’s student, Malcolm C. Harrison, which revived interest in Boys’s original proposal and started the wide-scale application of Gaussian basis functions in this field. The major interest of Boys in the 1950s was the development of the formalism and algorithmic tools needed for configuration interaction calculations on atoms and molecules. His earlier work concentrated on atoms and atomic ions and initially employed desk calculators, but he soon realized the potential of the electronic computer, not only for the arithmetic calculations but also for the formal manipulations required in the derivation of formulas and the overall organization of the calculations. He devoted much effort to vector coupling, the construction of symmetry- and spin-adapted linear combinations of Slater determinants (“co-detors” in his terminology), and the development of practical procedures for “projective reduction”, the reduction of matrix elements of the Hamiltonian between co-detors to linear combinations of one- and two-electron integrals. The computer was used to develop the formulas for the various basis set integrals and for the projective reduction formulas, and the formal and numerical parts of the work were integrated on the computer to reduce the need for human intervention and minimize the opportunities for mistakes. On turning his attention to molecules, he developed and tested a variety of schemes for calculation of the multicenter integrals over Slater-type orbitals and made several pioneering applications of his ideas, notably including the first correct prediction of the geometry of the methylene ground state and a landmark calculation on formaldehyde. In later years, concerned about the slow convergence of the configuration interaction expansion, he focused on schemes for the incorporation of interelectronic distances directly into the trial wave function, leading to the development of the “transcorrelated” method. Other interests included the convergence of nonsymmetric secular equations, studies of basis set superposition error, and the calculation of ro-vibrational energy levels.
I. Early History Samuel Francis Boys (Frank to his friends) was born in Pudsey, England, in 1911. A compassionate account of his life and scientific work was written by Charles A. Coulson after Boys’s untimely death in 1972 and published as a Royal Society biographical memoir.1 (Much of the material in this section is based on that memoir.) The same year, a summary of Boys’s work was given by R. G. Parr.2 Following the example of his parents, Boys was an avid and eclectic reader from childhood on. His strong interest in science, and particularly in chemistry, started in his grammar school years in Pudsey, but his wideranging interests included archeology, languages, and, particularly in his later years, economics. Upon graduation from Pudsey Grammar School, Boys won a Royal Scholarship and went to Imperial College, London, for his university studies. At Imperial College he earned a B.Sc. X
Abstract published in AdVance ACS Abstracts, March 15, 1996.
0022-3654/96/20100-6007$12.00/0
degree in chemistry with first-class honors in 1932, and he spent the next three years there taking degree courses in mathematics and engaging in research. His later interest in molecular electronic structure was already evident in his first research topic, dealing with “the stability of some coordinated cadmium compounds with reference to the electronic structure”. He also started research on optical rotatory power, using models based on classical electromagnetic theory,3 a topic to which he returned from a quantum-mechanical view point in his Ph.D. research in Cambridge University during 1935-1937. Boys’s move to Trinity College, Cambridge, in 1935 was made possible by another scholarship, and there he started his Ph.D. work on the quantum theory of optical rotation under Professor T. M. Lowry. Due to Professor Lowry’s death, Boys completed his thesis under the auspices of J. E. Lennard-Jones, the first professor of theoretical chemistry at Cambridge. The mathematical background he earned at Imperial College was very helpful to him in this work and strongly colored his later research in electronic structure theory. Cambridge was one of © 1996 American Chemical Society
6008 J. Phys. Chem., Vol. 100, No. 15, 1996 the hotbeds of quantum theory development in the 1920s and 1930s, and in the Lennard-Jones group Boys was exposed to the excitement of the early days of the application of quantum theory to chemistry. His enthusiasm for quantum chemistry never waned, and even during the war years, which he spent on munitions research, mostly on rocket propellants, in the Ballistics Branch of the Armaments Research Department, he devoted much free time to electronic structure theory. His notes of that period contain much of the foundations of his later work in London and Cambridge in this field. After the war Boys obtained an I.C.I. fellowship at Imperial College, where he devoted full time to the development of the methodology of ab initio electronic structure calculations. This work was continued at Cambridge, where he returned as a lecturer in theoretical chemistry in 1948. There he published his landmark series of 12 papers4-15 under the overall title of “Electronic wave functions”, which included the work he had done in London as well as later work, including research with a number of graduate students, at Cambridge. At Cambridge, where Boys spent the rest of his short life, he was not a college man. He devoted most of his time to his work, and his social life revolved around a few close friends and the Cambridge International Club, of which he was chairman for a number of years. He spent most of his vacations in Yorkshire, with his mother and sister, and on walking tours in England and on the continent. In the theoretical chemistry department he was the “odd man out”, going his own way, strongly convinced that his ab initio approach was the right one to pursue, while the rest of the department was engaged in the more fashionable theoretical model building and semiempirical calculations. He demanded much of his students, and it was often difficult to follow his abstract reasoning style, but when understanding was finally achieved, it was entirely rewarding. II. Computers at Cambridge It is difficult to discuss Boys’s research in detail without referring extensively to his utilization of electronic computing. He recognized very early the potential of the electronic computer for organizing and mechanizing the tedious processes required in ab initio electronic structure calculations and even for mechanizing much of the mathematical analysis required in deriving formulas for integrals and matrix elements and constructing and manipulating proper symmetry- and spinadapted expansion functions in configuration interaction calculations. In 1956 he wrote,16 “In principle, it is possible to predict the structure and properties of molecules with unlimited accuracy merely from Schro¨dinger’s equation and the numbers and properties of the electrons and nuclei defining the molecule. The complicated mathematical formulation, the very intractable intermediate problems and the extent of the numerical operations have caused this to be both a very confusing and a difficult subject. It does not appear to have been generally realized that the advent of automatic machines has simplified and made practical such calculations, not only as a means for performing the arithmetical operations but also for carrying out of much of the mathematical analysis of the most formal type.” He saw in the use of the computer not only an opportunity to carry out large-scale calculations which would not have been practical without it but also a means to ensure correctness of the calculations by avoiding the mistakes which can easily be introduced in manual calculations, both in the analysis and in the arithmetic. While the computers of the time were far from error free, he demonstrated that correctness could be ensured by a variety of checks and by extensive repetition.
Handy et al. The first electronic computer in Cambridge was the EDSAC (Electronic Delay-Storage Automatic Calculator).17 Its vacuumtube-based design was begun at the University Mathematical Laboratory (whose director, Maurice V. Wilkes, had been a member of the Lennard-Jones group before the war1) at the end of 1946, and it carried out its first completely automatic calculation in May 1949. The machine continued to undergo improvements and additions in the following years, until its replacement by a much-improved transistor-based machine, EDSAC II, in late 1956. During the first EDSAC’s existence, most of the day shift was devoted to maintenance and development work by the engineering staff, and almost all scientific applications work was done at night and on weekends, with the staff absent, as long as the users could keep the rather fickle machine operating satisfactorily (which often required adjustment of pulse rates and even replacement of vacuum tubes). The machine was not enclosed in cabients but consisted of open racks and was cooled by an exhaust fan in the ceiling. Because of the continuing engineering work, there were many temporary connections using “alligator clips”. Boys’s early electronic structure computations,5,11-13 involving atomic configuration interaction wave functions for Be, B, C, F-, Ne, and Na+ (including some excited states), were carried out by hand using desk calculators. (In one of those papers12 Boys thanks the University of London “for research grants covering the cost of some professional computational assistance and for a calculating machine”.) The first work in which the use of the electronic computer appears14 is No. XI in his “Electronic wave functions” series, with Vector E. Price, on Cl, Cl-, S, and S-, for which it “was found possible to adapt several sections of the calculation to the automatic calculating machine, the EDSAC, which has performed a considerable amount of the computations. In the next paper,15 with R. C. Sahni, concerned with the evaluation of vector coupling coefficients for atomic configuration interaction calculations, he states that it has been “found possible to develop a method of calculation which can be performed purely automatically by the EDSAC, proceeding from the lowest argument values indefinitely through all higher values.” By the time molecular electronic structure calculations were begun in Boys’s group in the early 1950s (as discussed later in this review), all the computational steps and formula derivations were being completely automated. The EDSAC had no compiler or operating system, and programs were written in a rudimentary assembler language which performed decimal-to-binary address conversion and had limited facilities for relative addressing but no symbolic addresses or variable names. Thus, the programmer worked very close to the machine level, manipulating and controlling its registers directly through the machine instructions in the program. Considerable ingenuity was often required to fit the program into the limited memory of the machine, which consisted of 1024 “short” (17 bit) words, each of which could hold one instruction or numerical value; two short words and an intervening “spacer” bit could be combined to form a 35 bit “long” word, which provided roughly 10 decimal digits of accuracy. There was no floating-point arithmetic, and numbers had to be scaled to avoid overflow and to maintain precision. For most of the machine’s existence there was no auxiliary storage, and intermediate results had to be punched out on paper tape and read in again when needed. Much effort was expended in making each subroutine as short as possible, often at the expense of operating speed. (It was sometimes stated that “a clever programmer can shorten any program by one storage
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location”, and this process was supposed to be applicable recursively.) Boys enjoyed programming and wrote many subroutines and programs himself. He found much satisfaction in designing efficient and compact computational algorithms and programs. When the new machine, EDSAC II, began operations in late 1956, Boys had it mostly to himself for some time because, unlike most users, he had taken the precaution of writing new programs for it ahead of time.1 III. Research A. Gaussian Functions (IS). Boys is generally credited with the introduction of Gaussian basis functions to quantum chemistry calculations.4 Earlier references to the use of Gaussian basis functions to quantum mechanics (other than in the solution of the harmonic oscillator problem) appeared in Roy McWeeny’s dissertation18 in 1948, applied in momentumspace calculations for H, He, and H2+, and in McWeeny’s work on the nuclear structure problem,19 also in monomentum space. Boys had worked out many of the formulas for using Gaussians as basis functions in 1942 and earlier,20 but his first published work on Gaussians appeared in his 1950 paper,4 No. I in the “Electronic wave functions” series. Here he specifically advocated the use of Gaussians as basis functions in coordinatespace electronic structure calculations for molecules, emphasized their advantage over exponential functions (Slater-type orbitals, STO) in overcoming the difficulties of multicenter integral evaluation, and derived formulas for the necessary integrals. The basis functions he proposed had the form now known as “Cartesian Gaussians”,
φlmn(R;rA) ) xAlyAmzAn exp(-RrA2)
(1)
where rA ) r - A is the vector from center A to the point r, with components xA ) x - Ax, etc., rA ) |rA|, R is a positive parameter, and l, m, n are nonnegative integers. He included derivations for the required multicenter integrals for l ) m ) n ) 0, stating that formulas for other cases can be obtained by differentiation with respect to the center coordinates. He pointed out that functions of the form (1), with a single-exponential parameter R, form a complete system of functions, since they span the system of Hermite functions, but that in finite basis calculations the exponential parameters could be individually varied to produce lower energies.4 The first significant molecular tests of Gaussian basis sets were carried out in Boys’s group by his student Robert K. Nesbet24 on the ground state of methane and at MIT by Alvin 1 + Meckler22 on the 3Σg and Σg states of the oxygen molecule. Both of these were configuration interaction calculations. Meckler used a minimal Gaussian basis, while Nesbet used a basis almost twice as large as minimal. Nesbet’s calculation probably was the first truly ab initio calculation of a polyatomic molecule other than H3 and certainly the first without any integral approximations. Interestingly, Boys did not show much enthusiasm for the use of Gaussian basis sets and concentrated much of his attention in the 1950s on devising various schemes for the evaluation of STO multicenter integrals. Nevertheless, he continued to be fascinated by the ease with which various multiple integrals involving products of Gaussian functions of interparticle coordinates could be evaluated and pursued other applications of such functions in multidimensional integration problems. Here we quote from IS: “When I joined this group in 1954, my first project involved use of least-squares expansions of the Mayer f-function in terms
of Gaussians for the calculation of virial coefficients of gases.23,24 We also incorporated Gaussians in a flexible mathematical form we proposed for intermolecular potentials, in the expectation that such a form would facilitate the calculation of integrals required in the determination of realgas properties.25 (Boys insisted that my name appear first on that publication, expecting the proposed potential to be known as ‘the Shavitt-Boys potential,’ because he thought that ‘The Shavitt Boys’ sounded like the name of a musical band.)” The molecular calculations carried out in Boys’s group at the time16 employed STO basis sets, evaluating the multicenter integrals by an “axial expansion” method, in which each twocenter charge distribution (product of two STO’s) was approximated as a linear combination of STO’s centered at several points along the line joining the two centers.26,27 This approach resulted in the expression of a four-center integral as a linear combination of two-center Coulomb integrals, and while it was reasonably effective, the accuracy obtained28 would be considered quite inadequate today. In the late 1950s Boys used another approach for the evaluation of STO multicenter integrals, based on approximating each Slater-type function as a linear combination of Gaussian functions on the same center.29-31 Using nine Gaussians for each expansion, each electron repulsion integral was converted to a linear combination of 6561 Gaussian integrals, but the number of terms in this linear combination was greatly reduced by Schmidt orthogonalization of the Gaussians in each expansion, in order of diminishing contribution, followed by truncation of “high-order” products in the multicenter expansion. The Gaussian expansions were not used as a basis set in their own right but were only employed to evaluate those STO integrals which could not be computed analytically. The large number of Gaussians used to fit each STO was needed in order to maintain high accuracy in the fitting and thus achieve consistency between the analytically evaluated integrals and those computed by the Gaussian expansion. IS: “I suggested to Boys in 1959 that the fitting accuracy requirement could be relaxed substantially, reducing the number of Gaussian terms needed in each expansion, if all integrals were evaluated using the Gaussian expansions, so that these expansions could be considered the actual basis functions. Boys did not pursue this approach, but it formed the basis for my later proposal to use fixed linear combinations of Gaussians as basis functions.”24 Another fallout from Boys’s Gaussian expansion approach was a more accurate method for the evaluation of STO multicenter integrals, the Gaussian transform method.24,32 Shavitt was led to the idea for this approach by consideration of Boys’s method of picking the exponential coefficients in the expansion
e-Rr ≈ ∑ci exp(-sir2)
(2)
i
Boys used linear least squares to determine the coefficients ci but chose the nonlinear parameters si by considering the expansion (2) as a quadrature approximation of the integral
( )
R R2 -sr2 ∞ -3/2 exp e ds ) e-Rr ∫ 0s 4s 2xπ
(3)
and taking the Gauss-Laguerre quadrature points for s in (3) as the values of the si in (2). (This approach was also used later by Silver.33) The Gaussian transform method is based on using the integral transform (3), instead of the approximate expansion (2), as a basis for a procedure for the evaluation of
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TABLE 1: Some Terminology Introduced by Boys term
meaning
detor co-detor bonded function variational selection method, poly-detor method poly-detor wave function projective reduction
Slater determinant constructed from orthonormal orbitals spin- and symmetry-adapted linear combination of detors (i.e., configuration state function) Rumer-like spin-adapted co-detor configuration interaction method (with selection of orbitals and configurations)
fundamental
CI wave function expression of Hamiltonian matrix elements between co-detors in terms of one- and two-electron integrals (i.e., determination of “coupling coefficients”) from first principles, ab initio
the STO integrals. A similar transform approach had been proposed earlier by Kikuchi34 but was not developed further by him. Boys returned to Gaussian expansions in electronic wave function calculations in 1960, when he proposed the use of products of Gaussian basis functions and correlation factors in the form of Gaussian functions of interelectronic distances.35 He derived the formulas needed for integral evaluation in this approach, but did not pursue this idea until later, in the context of the transcorrelated method. The modern use of Gaussian basis sets in electronic structure theory can be traced to a large degree to the impetus provided by the later work of Boys’s student, Colin M. Reeves, who with his own students at the University of Leeds, Malcolm C. Harrison and Roger Fletcher, explored the various issues involved in the optimal use of such basis sets and demonstrated their effectiveness.36-38 When Harrison completed his studies with Reeves, he came to John C. Slater’s group at MIT and there, in collaboration with other members of that group, including Imre G. Csizmadia, Brian T. Sutcliffe, and Jules W. Moskowitz, started work on what became the first widely distributed ab initio molecular electronic structure program, POLYATOM.39,40 This program employed Gaussian basis sets, first as individual basis functions and later,41 following the introduction of contracted Gaussian basis sets by Clementi and Davis42 in their own program, IBMOL, also included provisions for contractions. These programs began the process of bringing ab initio electronic structure calculations into the mainstream of chemical research. (A more detailed account of these and later developments has been given elsewhere.43) B. Configuration Interaction (IS). Right from the start, Boys’s method of choice for electronic structure calculations was the method of configuration interaction (CI). At that time this term was mostly applied to the interaction of just a few terms (Slater determinants) required for the proper description of some electronic states of atoms. Most molecular calculations beyond the Hartree-Fock level used the valence bond approach and were mostly semiempirical. Boys saw the CI approach as the only practical method capable, at least in principle, of convergence to the exact solution of the Schro¨dinger equation and proceeded to outline a complete scheme for its implementation.4,5 He called it the “method of variational selection” and introduced other terms (Table 1), like “detor” for a Slater determinant constructed from orthonormal orbitals and “codetor” for a symmetry- and spin-adapted linear combination of detors, and later referred to the complete procedure as the “polydetor” method. (It is a pity that his terminology was not taken up by other workers in the field, because his terms are more compact and precise than those in current usage). He proceeded to work out various details of the application of CI,4-6 concentrating initially on application to atoms. In this context the problem of vector coupling, i.e., the construction of codetors adapted to spherical symmetry, looms large, and much of the “Electronic wave functions” series is devoted to dealing
with this problem and with the reduction of matrix elements of the Hamiltonian between vector-coupled co-detors to linear combinations of one- and two-electron integrals over the orbitals.7-10,15 He called this latter process “projective reduction” and devoted considerable effort to designing efficient algorithms for its implementation. He also introduced a simple iterative method for the solution of the matrix eigenvalue problem.5 The early atomic calculations,5,11-13 which were carried out without the benefit of an electronic computer (mostly in London, before Boys’s move to Cambridge, even though all the papers were submitted from Cambridge), were remarkably successful, considering the rather small expansions used, usually of the order of a dozen co-detors. They produced good values of the relative energies of several electronic states of the atoms Be, B, and C, using STO basis sets which would now be designated as (4s4p2d).12,13 Also included was a calculation on the ground states of F-, Ne, and Na+.11 The first calculation to use a computer was for S, S-, Cl, and Cl-.14 It employed a (6s5p2d) basis and up to 32 co-detors and included relativistic corrections. Again, good energy splittings were obtained, but electron affinities were not well reproduced (not surprisingly, considering what we know now about the difficulties of such calculations). Selection of co-detors on the basis of energy contributions was also introduced in the course of these atomic calculations.13 IS: “When I came to work with Boys in the autumn of 1954, his attention had already turned to molecules. Two students had already completed molecular research, Robert K. Nesbet on CH421 and Geoffrey B. Cook on BH,16 and had left. There was only one other student at the time, Colin M. Reeves, who was working on H2O.16,26 We were not housed on the premises of the Theoretical Chemistry Department (where H. Christopher Longuet-Higgins had just arrived as the new professor of theoretical chemistry, replacing the recently retired Sir John E. Lennard-Jones), but in the Mathematical Laboratory, where we shared an office with two students of Douglas R. Hartree, Charlotte Froese (now Fischer) and David F. Mayers. This arrangement was very convenient, providing easy access to EDSAC and to the expertise of the other computer users. I got involved in electronic structure research only later, when unforeseen difficulties (eventually surmounted) came up in the virial coefficients project and Boys, wanting to make sure that I would have a successful thesis project, suggested calculations on H3. For the co-detor expansion functions in molecular calculations, Boys adopted a Rumer-like scheme, and with this student Reeves he derived a very efficient projective reduction procedure for the resulting ‘bonded functions’.26 I used this procedure and Reeves’s programs in my H3 work,16 and continued to use the same approach in my later work44 until the development of unitary-group-based methods for ‘direct CI’ calculations in the seventies. In the early work, Boys did not see much point in starting a CI calculation with a self-consistent field (SCF) step. In the early and mid fifties we did not even have an SCF
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program in the group, and orbitals for CI calculations were simply chosen as ‘reasonable’ simple linear combinations of basis functions. By hindsight, our expectations at that time for the rate of convergence of the CI method were much too optimistic. Perhaps this was for the better, because had we known how many terms were needed for quantitative results for nontrivial problems, we might have been completely discouraged and chosen to devote our time to some other pursuit. As it were, perseverance and hard work, plus the development of new methods and the astounding advances in the speed and capacity of electronic computers, brought us to a stage today at which theoretical calculations of molecular structures and properties are a widely accepted major component of chemical research and make major contributions to our understanding of chemical phenomena.” Two new students came to work with Boys while Shavitt was in his group, Ivor Jones and John M. Foster. Jones worked initially on homonuclear diatomics, and particularly C2,45 and later on HCN.29 Foster worked on H2CO31 and on CH2.46 Most of their work was completed after the much more efficient EDSAC II had replaced the old computer, allowing more powerful calculations; the CH2 calculation included up to 128 co-detors. At this point an SCF wave function was already being used to obtain orbitals and to provide the initial term in the CI expansion.31 Boys reported these calculations and his general approach47 at the first Boulder conference on theoretical chemistry in 1959. At that meeting he also presented a new orbital localization scheme48,49 which, combined with a method for the construction of efficient correlating orbitals,50 could be used to produce a more compact CI expansion. Particularly interesting was the CH2 calculation, which was the first to predict a bent structure for the ground (3B1) state, in disagreement with the then accepted interpretation of the spectral data by Herzberg. This result was later confirmed by other theoretical treatments and experiments and by a reinterpretation of the spectrum.51 C. Numerical Integration for Molecules (NCH). Another contribution was the development of a purely numerical method for SCF wave function optimization, which included a particularly effective numerical integration scheme for molecular calculations.52 In the early part of this paper Boys introduces “chemical accuracy”, which he defines as “1 kcal/mol”. Throughout this period Boys was fascinated with numerical integration for molecules. He was driven by the knowledge that if the expansion set contained the exact wave function, then it was only necessary to use one quadrature point to evaluate the matrix elements of the secular equations. This paper discusses the advantages of performing calculations using numerical integration techniques. He argues that it is simpler to deal with higher angular momentum integrals this way. He further considers that the cost factor is lower if integrals are evaluated numerically. For three-dimensional quadrature he devised the following scheme for an integral evaluation:
(4)
F(r) ) ∑F(r) Vs(r)/∑Vt(r) ) ∑Fs(r)
(7)
V2(r) ) (1 - j1(r))/rA4
(8)
V3(r) ) 1/rB4
(9)
V4(r) ) 1/rC4
(10)
These ideas parallel the use Voronoi polyhedra of threedimensional quadrature for DFT today; see for example the often cited work of Becke.53 With each shell region Boys used a spherical polar grid:
r ) aZ4/(Z3 + β)
(11)
Z2 ) Q/(1 - Q)
(12)
He used equally spaced points in Q space, Legendre θ points, and equally spaced φ points. He used 12 radial points, and the number of (θ,φ) points increased as r increased. Boys used the ideas of localized Gaussians to evaluate the two-electron integrals. The numerical evaluation of two-electron integrals has not found much favor in recent years because of the very fast methods for the evaluation of Gaussian basis function integrals. However, Boys rarely mentioned using Gaussian basis functions in his later period. D. The Convergence of Nonsymmetric Secular Equations (NCH). In another important paper54 Boys examined the convergence of the disymmetric secular equations:
∑s 〈Φr|H - W|Ψs〉cs ) 0
(13)
He was interested in such equations because of his current interest in transcorrelated theory and numerical integration. The relevant equation system can be written
∑s 〈C-1Φr(∑I δ(rI,r)wI)|H - W|CΦs〉cs ) 0
(14)
He proved that the computed energy W could be expressed as follows:
W ) Wexact + (µ+µ + µQµ)(W11 + ...)
(15)
where µ+ and µ are a measure of the inability of the left and right expansion sets φ and ψ to represent the exact wave function, and µQ is a measure of the quadrature error. The important point is that if exact integration is used, then the error in the energy is the product of the errors in the two expansion sets. Of course, this explains why coupled-cluster theory is so attractive. In CC theory the equations may be written
〈Φr|H - W| exp(∑Ts)Φ〉 ) 0
(16)
s
I ) ∫F(r) dr s
V1(r) ) j1(r)/rA4
(5)
t
(6)
s
He defined Vs(r) > Vt(r), all t, when r was in the region of the sth shell of a given atom:
Boys’s theory says that the error in the CCSD energy is equal to the product of the CISD wave function error and the CCSD wave function error. The first is already small, and the second is very small! A paper on this topic has recently been written by Kutzelnigg.55 E. The Transcorrelated Method (NCH). As the slow convergence of the CI expansion became more and more evident, Boys turned his attention to alternative approaches. He became convinced that the incorporation of interelectronic coordinates in trial wave function, as in the work of Hylleraas
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on helium and of James and Coolidge on H2, provided the best hope for further progress. This was the motivation for his 1960 proposal for Gaussian correlation functions35 (generalized by Singer56) and led later to the development of the transcorrelated method, in which the wave function incorporates an exponential correlation factor depending on interelectronic coordinates.54,57-61 In their initial studies, Boys and Handy considered a wave function for π electrons of benzene with the following form
Ψ ) ∏(1 - µ i>j
exp(-γrij))A(φR1 φβ1 φR2 φβ2 φR3 φβ3 )
(18)
i>j
) CΦ
〈Φ|C-1HC - W|Φ〉 ) 0
energy W:
〈ΦiR|C-1HC - W|Φ〉 ) 0
orbital coefficients cRi:
(24) (25)
correlation coefficients Dl:
〈∑Gl(ri,rj)Φ|C-1HC - W|Φ〉 ) 0 (26)
(17)
To evaluate this wave function, the molecular orbitals φi are expanded in terms of the atomic π orbitals πA, πB, πC, πD, πE, πF. Thus, the one determinant becomes a fixed linear combination of determinants of atomic π orbitals. The Jastrow factor in front of each of these determinants is then evaluated by assuming that each electron is situated on the C atoms on which the atomic π orbitals are located, and thus ∏(1 - µ exp(γrij)) becomes ∏(1 - µ exp(-γRPQ)), for all C atoms P and Q on which the atomic π orbitals are situated in the relevant determinant. RPQ is the distance between the C atoms P and Q. Different parameters µ and γ were used, dependent on whether the associated electrons had parallel or antiparallel spins. In this way the coefficient in front of each π determinant is amended by its own Jastrow factor. Thus, a wave function for the ground state of benzene has been determined which includes dynamic correlation. The energy of this wave function, which he wrote as CΦ, was evaluated through the usual variational expression 〈CΦ|H|CΦ〉/〈CΦ|CΦ〉, using the Pariser-ParrPople values for the one- and two-electron atomic π integrals. The value of the correlation energy so obtained (-0.75 eV) was very close to the CI value obtained by Koutecky et al.62 (-0.78 eV) using the same integrals. The method also gave good agreement for similar calculations on the excited states of benzene.63 With this work in place, a crucial advance took place when, in the words of NCH, “One morning I walked in and instead of the equation HCΦ ) ECΦ, Boys had written down C-1HCΦ ) EΦ, and thus transcorrelated theory was born.” The wave function is written
Ψ ) ∏f(ri,rj) A(φ1φ2...φN)
following:
ij
contraction coefficients dp:
〈∑gp(ri)Φ|C-1HC - CHC-1|Φ〉 ) 0 (27) i
This last equation may be understood by recognizing that it is equivalent to
〈ΦiR|C-1HC|Φ〉 ) 〈Φ|C-1HC|ΦiR〉
(28)
which makes the transcorrelated operator as Hermitian as possible. The first term in C was fixed to make the wave function obey the unlike spin correlation cusp condition. This allowed numerical quadrature to be carried out. Results on Ne and Ne+ were given Tables 1-3 of ref 60. The highlights were that the complete transcorrelated wave function for Ne was given in terms of 26 parameters with an energy of -128.959 hartrees, compared to the exact value of -128.929 hartrees. The ionization energy was calculated to be 0.793 hartree, compared to the exact value of 0.791 hartree. Schmidt and Moskowitz64 more recently examined the Ne transcorrelated wave function using variational Monte Carlo. Using the Ne correlation function, but Clementi orbitals, gave 72% of the correlation energy. Reoptimization of C gave 83%, and adding more terms gave 85%. The principal reason for the difference is that S-M were using a variational energy expression, but it is also very likely that in the 1960s Boys and Handy were using a totally inadequate set of quadrature points. To perform transcorrelated calculations on molecules, it was necessary to lean on the quadrature schemes of Boys and Rajagopal.52 After the Voronoi separation, spherical polar (r,θ,φ) grid points were developed as follows:
(19)
r ) qr/(1 - qr)
(29)
θ ) π(6qθ5 - 15qθ4 + 10qθ3)
(30)
φ ) 2πqφ
(31)
qr ) jl mod 1
(32)
qθ ) jm mod 1
(33)
qφ ) jn mod 1
(34)
where
C ) ∏exp[∑DlGl(ri,rj) + ∑dp(gp(ri) + gp(rj))] i>j
l
(20)
p
The governing equation is obtained by premultiplying the Schro¨dinger equation by C-1:
HCΦ ) WCΦ
(21)
C-1HCΦ ) WΦ
(22)
where
The transcorrelated operator may be expressed in terms of one-, two-, and three-electron operators:
C-1HC ) H - ∑ ij
(
2 1 ∇i fij
2 fij
+
)
∇ifij‚∇i fij
-
1
∑
2 ijk
with
∇ifij‚∇ifik fijfik
(23)
although the form of the three-electron operator meant that the integrals were no worse than two-electron integrals to evaluate. The resultant equations to determine the parameters were the
(l,m,n) )
(M1 , MD, ME )
(35)
Such a Diphantine scheme exactly integrates all Fourier functions exp[2πi((pqr ( sqθ ( tqφ)] for which p + 2s + 4t < Γ. (See other works of Haselgrove,65 Conroy,66 and Ellis.67)
Samuel Francis Boys
J. Phys. Chem., Vol. 100, No. 15, 1996 6013
The values M ) 160, D ) 18, and E ) 46 were used. The transformations (29)-(31) are justified by the Euler-Maclaurin theorem.68 The correlation factor for LiH involved 39 parameters, and an energy of -8.063 hartrees was obtained, compared to the exact value of -8.070 hartrees, which on the face of it appears to be quite a success. Why has this apparent success of the transcorrelated method meant that research in the method has not continued? Here we quote from NCH: “I suppose I was worried by the lack of the variational upper bound, which over the three years meant that I had obtained any number of different results dependent upon the expansion set used. The correlation factor which we used treated like spins and unlike spins in the same manner. The method involved three-electron integrals, and we know all the problems we have had with the two-electron integrals. The correlation expansion functions Gk(ri,rj) are not orthogonal, and this makes both their selection and use very difficult. The number of quadrature points which we used for LiH are on today’s experience too small by at least a factor 20. Finally the method in those days was considered complicated. For all these reasons I doubt whether the method can compete with the Coupled Cluster approach.” F. The Basis Set Superposition Error (NCH). One of the most notable of Boys’s later contributions was the identification of basis set superposition error (BSSE) and the proposal of the counterpoise correction method for dealing with it.69 This procedure is widely used today in many molecular calculations, and particularly in the treatment of weak bonds and intermolecular interactions, for which even small superposition errors can be of major consequence. Boys’s term for BSSE was function counterpoise.69 He was interested in computing
∆ABW(R) ) WAB(R) - WA - WB
(36)
Since he was primarily concerned with numerical integration procedures, he discussed in detail the “point counterpoise” procedure. He introduced “function counterpoise” as follows: “In this procedure, the calculations of separate energies WA, WB are performed with the full set of expansion functions used in the calculation of the energy WAB of the bimolecular system. It is considered that a deductive justification for this procedure has not yet been made and this will have to be extensively tested. Results already available appear to justify it .... The energies of the separate molecules WA, WB are now slightly dependent on the value of R between them .... Cost of less than 3 times the original cost is well worthwhile if it gives the possibility of much more accurate values.” We all know that over the intervening years there have been many, many papers purporting deductive justifications and presenting extensive tests. This is certainly Boys’s most cited paper of the final period. G. The Calculation of Ro-Vibrational Energy Levels (NCH). This work70 was completed after Boys’s death, but Boys completely developed the mathematics of the method, which is probably the earliest variational method for the rovibrational energy levels of a triatomic molecule. Again he used a numerical approach, introducing quadrature points which corresponded to distorted positions of the molecule on the given potential energy surface. In detail
Ri ) G + T(R,β,γ) Ris(Q)
(37)
Ris(Q) ) S(Q) RiQ
(38)
Here Ri is the position of the ith atom, G is the center of mass, and T(R,β,γ) is the Euler rotation matrix. RiQ is an arbitrary displaced position of the ith atom, and S(Q) is a rotation matrix which rotates RiQ to give the displaced position Ris(Q), which corresponds as closely as possible to a pure vibrational displacement. The rotation matrix is
(
-sin Ω cos Ω 0
cos Ω sin Ω 0 For a triatomic molecule:
0 0 1
)
(39)
R1Q ) (Q1, 0, 0)
(40)
R2Q ) (Q2, Q3, 0)
(41)
R3Q ) -((m1Q1 + m2Q2), m2Q3, 0)/m3
(42)
Ω was expanded in terms of Q as follows:
Ω ) ∑ωa(Qa - Qae) + ∑ωab(Qa - Qae)(Qb - Qbe) (43) a
ab
with the ωa and ωab determined to obey the Eckart conditions:
∑i miRi
e
×
∂Ris ∂Qa
)0
(44)
The wave function is expanded in coordinates q, which are closely related to normal coordinates:
qi ) ∑(B-1)ij(Qj - Qje)
(45)
j
The vibration-rotation expansion functions are
YJk(β,γ) Hν1(q1) Hν2(q2) Hν3(q3) exp(-1/2(q12 + q22 + q32)) (46) where YJk is a spherical harmonic and Hν are Hermite polynomials. These expansion functions are then inserted into the scalar equations to obtain vibration-rotation energy levels for a provided potential surface V:
〈
〉
∑s Φr| - ∑i 2m ∇i2 + V - W|Φs cs ) 0 1
i
(47)
Results were presented for the water molecule. IV. Conclusion In 1958 Boys wrote,45 “... the future is full of promise, since one should expect a reduction of T [the amount of effort for a molecular calculation] by a factor of the order of 1000 due to the foreseeable increase in the speed of the machines and the improvement of the mathematical methods” (translated from the French of the published paper). As it is, the increase in speed of electronic structure calculations since that time has already probably been closer to 1 000 000, not to mention the significant methodological advances that have taken place. These developments have certainly justified Boys’s faith in the ab initio approach, even though his early expectations for the convergence of the CI expansion were overoptimistic. It is clear that Frank Boys has had a remarkable impact on the progress of electronic structure theory and methodology. It
6014 J. Phys. Chem., Vol. 100, No. 15, 1996 was unfortunate that official recognition in the U.K., in the form of a more elevated position at Cambridge University and election to fellowship in the Royal Society, did not come until about a year before his death. His resources were therefore limited, and the number of his students was relatively small. For much of the time he had to overcome great skepticism, and even antagonism, toward ab initio electronic structure theory, both at Cambridge and in the British theoretical chemistry establishment as a whole. It is interesting to speculate how much more had he been able to accomplish were the situation different and were it not for his untimely death. Acknowledgment. We thank R. G. Parr and G. G. Hall for several valuable contributions to this paper. Appendix. Three Conversations with Frank Boys (1948-1958-1968) by John Pople My first meeting with Frank was quite dramatic. In the late summer of 1948, I joined the Cambridge Theoretical Chemistry department, intending to write a thesis on the theory of liquids, a topic on which the Professor, Sir John Lennard-Jones, had published some major papers just before the war. I had had one or two preliminary conversations with LJ (as we all abbreviated his rather lengthy name), at one of which he had told me that there would be two other new members of the department, another research student, Victor Price, and a recently appointed University Lecturer, Dr. S. F. Boys. One afternoon in the middle of September, I was sitting alone in the student room, when Frank walked in to start his first working day. “Hello”! I’m Frank Boys” he said, in a friendly manner. I was very young and shy in those days, so I just replied “Hello” and returned to the book I was reading as quickly as possible. He walked through to his new office, which was located at the back of the student room, and I heard him unloading his bulky briefcase. Then, after about five minutes, he poked his head out of his door and said to me, “Will you come in, please?” Somewhat surprised, I went in and sat down in a chair by the side of his desk. He then produced an empty pad of paper and started to tell me, in great detail, everything he knew about quantum mechanics. I cannot remember exactly where he started, but it was probably the Schroedinger equation. However, I do remember that my initial reaction was revulsion at his ugly notation. Then, after five or ten minutes, I began to wonder to myself, “Why is this man telling me all this?” Conversations with Frank Boys were always one-sided. He did all the talking, and there was usually plenty of opportunity to think about the background of what he was saying. A bit later, he began making suggestions about what I should read; it dawned on me that I was to be his research student! Panic began to set insthis was certainly far from my intention. The presentation continued. I recall hearing about integration of Gaussian functions, detors, codetors, polydetors, and all the other topics that were published several years later. All these things were known to him in 1948. But I was not paying much attention. Frank must have thought that I was a very unresponsive student, for I was desperately wondering why LJ had done this to me, had transferred me to this new person without even letting me know. This seemed impossible, since LJ was nothing if not a gentleman, and these actions appeared most uncharacteristic. But I could think of no other explanation. The presentation went on and on and on, nearly three hours in all, I recall. So, at the end of this first conversation, I left in a state of total dejection, holding the sheaf of Frank’s notes containing all he knew about quantum
Handy et al. mechanics and surveying the wreckage of my planned career in liquid theory. After a sleepless night, I returned to the student room and was sitting, gloomily waiting for LJ’s secretary to come and tell me of the new arrangements. Then in walked Frank Boys. “Hello!” he said brightly. “I think there was a mixup yesterday. You’re not Victor Price, are you?” And so the crisis was resolved, and the Boys-Pople teacherstudent relationship ended after only 20 hours. LJ’s reputation was rehabilitated, and I was able to resume my research plans. My principal regret now is that, in a rush of relief, I discarded Frank’s sheaf of notes, outlining all he knew in 1948. For the next ten years (1948-1958), Frank and I spent most of our time in the theoretical chemistry department at Cambridge. As a research student (1948-1958), I quickly came to realize how helpful he could be. LJ was an excellent supervisor, but he was rather inaccessible, appointments having to be made two weeks in advance. Frank Boys, on the other hand, was always there and always willing to talk at length. Whenever I was stuck on some point of mathematical technique, I would knock on his door and ask for help. He was always willing to put aside what he was doing and give advice on the spot. Often his response covered much more than had been requested for his interests spread widely, particularly on down-to-earth practical computational techniques. I listened and learned a lot. In 1954, LJ was succeeded as professor by Christopher Longuet-Higgins, Frank Boys remaining as lecturer. There was a huge contrast between the scientific styles of these two individuals. Christopher’s presentations were always elegant and polished, with clear starting assumptions, a logical argument, and well-rounded conclusions. Frank, on the other hand, tended to be more scattered, sometimes almost incoherent, with many concepts only partially defined and arguments going in many directions. There were always loose ends. But the loose ends led in interesting directions, I came to learn, and showed how his mind was always probing toward the future. There were conflicts between Boys and Longuet-Higgins, sometimes quite painful to watch. As the older man in the junior position, Frank must have found the going hard, but he persisted. We did not collaborate on any scientific projects, although I followed and came to admire what he was doing. My view, at the time, was that the role of the theoretical chemist was the application of theory to new branches of chemistry as they developed, any necessary approximations being introduced to make the impact immediate. He, on the other hand, took the long-term view. Like many others at the time, I was skeptical about the length of the term. The second conversation took place one evening in early 1958, while Frank and I were walking from the department up Sidney street, toward his lodgings in Jesus Lane. By this time, I had become a lecturer in the mathematics department. Frank was still a lecturer in theoretical chemistry, so we were at the same career level. I was unhappy earning my living in this way and was beginning to look for a position in a scientific department; 1958 was a favorable year to do this. The Soviet satellite Sputnik had been launched a few months previously, and Western governments were scrambling to boost science. As we walked, Frank brought up the subject. “How is your career going?” he asked, “You must be getting many offers”. I told him that I was, and he asked for details. So he began to give me helpful, fatherly advice about what I should do. As he spoke, I wondered at the conversation. I was not smarter than he, and he was 15 years older. Why was I the one receiving the offers, and why was I about to overtake him in the academic rat race? I suppose I knew the answer then. I was doing things
Samuel Francis Boys that were fashionable at the timesaromatic electronic excited states, free radical spectroscopy, nuclear magnetic resonance, and the like. Frank took no notice of such things; he carried on with what he thought important, irrespective of public acclaim. And why was he not envious? He showed no sign of it and seemed genuinely pleased to help me. As the conversation ended, I thanked him and admired him a lot. I left Cambridge later that year and saw less of Frank from then on. We did meet at the 1959 Boulder conference, to which he had been invited by Bob Parr. This meeting was the climax of the group 1/group 2 divide, which separated the quantum chemistry community throughout the 1950s, brilliantly delineated by Charles Coulson in his after-dinner speech (published in 1960 in the ReViews of Modern Physics). Boys’s presentation was remarkable. In describing his work, he produced a paper tape of his whole computer program and unrolled it along the length of the chemical lecture bench. There, in one roll, was something, of which one could ask a chemical question at one end and it would produce an answer at the other! The concept of complete program packages, which we all use today, was quite foreign then, and most of the audience probably thought the demonstration bizarre. But it was prescient; Frank Boys was the hero of that meeting. Following six years in an administrative position at the National Physical Laboratory, I moved to Pittsburgh in 1964 to devote full attention to electronic structure theory. After three years working on zero-differential-overlap methods (CNDO, INDO, etc.), I finally got around, belatedly, to ab initio studies in late 1967. The first version of what became the GAUSSIAN 70 program was developed during 1968 by my student Warren Hehre and myself. We reread the old Boys papers and developed a new algorithm that sped up integral evaluation for contracted Gaussians by a factor of nearly 200, using axis rotation inside inner loops. This was an important development which finally allowed extensive application, but it was really the only new feature. The rest was pure Boyssthe use of Gaussians, the fitting of Slater orbitals by Gaussians, the packaging of the whole logic into a unified wholesthese all came from him. And so I come to the third conversation. The Gaussian material was first presented in a lecture at the Faraday Society Symposium at University College, London, in December 1968. Frank Boys was in the audience, still only a lecturer at Cambridge. As many of you will know, there are times when there is somebody in the audience that the speaker wishes was absent. This was one such occasion. As I spoke, I kept looking nervously at Frank, sitting in the second row to my left, wondering what he thought of my trespass into his territory. He was certainly paying close attention. There was a slight smile that he often had. I interpreted it as the smile of an evangelist acknowledging a convert. The conversation took place at the reception held later that evening. I approached Frank and expressed my appreciation for his past work and things I had learned from him. I should have done this sort of thing years earlier. He asked about some of the technical details, and we discussed them for about 10 minutes. And then I asked him, “Frank, do you remember the first conversation we had 20 years ago?” “Oh yes! I remember it well,” he replied. “What do you think would have happened if it had been for real and I had become your student?”, I asked further. He paused for a while and then said, “I don’t know, but I am sure that we both would have had very different careers.” I think we shared a moment of mutual regret. I believe that was the last time we spoke before his untimely death.
J. Phys. Chem., Vol. 100, No. 15, 1996 6015 References and Notes (1) Coulson, C. A. Biog. Mem. Fellows R. Soc. 1973, 19, 95. (2) Parr, R. G. Int. J. Quantum Chem. Symp. 1973, 7, 123. (3) Boys, S. F. Proc. R. Soc. London, Ser. A 1934, 144 655, 675. (4) Boys, S. F. Proc. R. Soc. London, Ser. A 1950, 200, 532. (5) Boys, S. F. Proc. R. Soc. London, Ser. A 1950, 201, 125. (6) Boys, S. F. Proc. R. Soc. London, Ser. A 1951, 206, 489. (7) Boys, S. F. Proc. R. Soc. London, Ser. A 1951, 207, 181. (8) Boys, S. F. Proc. R. Soc. London, Ser. A 1951, 207, 197. (9) Boys, S. F. Philos. Trans. R. Soc. A 1952, 245, 95. (10) Bernal, M. J. M.; Boys, S. F. Philos. Trans. R. Soc. London, Ser. A 1952, 245, 116. (11) Bernal, M. J. M.; Boys, S. F. Philos. Trans. R. Soc. London, Ser. A 1952, 245, 139. (12) Boys, S. F. Proc. R. Soc. London, Ser. A 1953, 217, 136. (13) Boys, S. F. Proc. R. Soc. London, Ser. A 1953, 217, 235. (14) Boys, S. F.; Price, V. E. Philos. Trans. R. Soc. London, Ser. A 1954, 246, 451. (15) Boys, S. F.; Sahni, R. C. Philos. Trans. R. Soc. London, Ser. A 1954, 246, 463. (16) Boys, S. F.; Cook, G. B.; Reeves, C. M.; Shavitt, I. Nature 1956, 178, 1207. (17) Wilkes, M. V. Automatic Digital Computers; Methuen: London, 1956. (18) McWeeny, R. Dissertation, Oxford University, 1948; Nature 1950, 166, 21. (19) McWeeny, R. Proc. Cambridge Philos. Soc. 1949, 45, 315. (20) Parr, R. G. Private communication. (21) Nesbet, R. K. Dissertation, Cambridge University, 1954; J. Chem. Phys. 1960, 32, 1114. (22) Meckler, A. J. Chem. Phys. 1953, 21, 1750. (23) Boys, S. F.; Shavitt, I. Proc. R. Soc. London, Ser. A 1960, 254, 487, 499. (24) Shavitt, I. Methods Comput. Phys. 1963, 2, 1. (25) Shavitt, I.; Boys, S. F. Nature 1956, 178, 1340. (26) Reeves, C. M. Dissertation, Cambridge University, 1957. (27) Shavitt, I. Dissertation, Cambridge University, 1957. (28) Boys, S. F.; Shavitt, I. A Fundamental Calculation of the Energy Surface for the System of Three Hydrogen Atoms, Technical Report WISAF-13; University of Wisconsin Naval Research Laboratory: Madison, WI, 1959. (29) Jones, I.; Boys, S. F. Unpublished work, 1958. (30) Foster, J. M. Dissertation, Cambridge University, 1961. (31) Foster, J. M.; Boys, S. F. ReV. Mod. Phys. 1960, 32, 303, 305. (32) Shavitt, I.; Karplus, M. J. Chem. Phys. 1962, 36, 550; 1965, 43, 398. (33) Silver, D. M. Mol. Phys. 1971, 22, 1069. (34) Kikuchi, R. J. Chem. Phys. 1954, 22, 148. (35) Boys, S. F. Proc. R. Soc. London, Ser. A 1960, 258, 402. (36) Reeves, C. M. J. Chem. Phys. 1963, 39, 1. (37) Reeves, C. M.; Harrison, M. C. J. Chem. Phys. 1963, 39, 11. (38) Reeves, C. M.; Fletcher, R. J. Chem. Phys. 1965, 42, 4073. (39) Csizmadia, I. G.; Harrison, M. C.; Moskowitz, J. W.; Seung, S.; Sutcliffe, B. T.; Barnett, M. P. POLYATOM, Program 47; QCPE 1964, No. 11, 47. (40) Csizmadia, I. G.; Harrison, M. C.; Moskowitz, J. W.; Sutcliffe, B. T. Theor. Chim. Acta 1966, 6, 191; 1967, 7, 156. (41) Neumann, D. B.; Basch, H.; Kornegay, R. L.; Snyder, L. C.; Moskowitz, J. W.; Hornback, C.; Leibman, S. P. POLYATOM, Version 2, Program 199; QCPE 1971, No. 11, 199. (42) Clementi, E.; Davis, D. R. J. Chem. Phys. 1966, 45, 2593; J. Comput. Phys. 1966, 1, 223. (43) Shavitt, I. Isr. J. Chem. 1993, 33, 357. (44) Shavitt, I. In Methods of Electronic Structure Theory; Schaefer, H. F., III, Ed.; Plenum: New York, 1977. (45) Boys, S. F.; Jones, I.; Shavitt, I. In Calcul des Fonctions d’Onde Mole´ culaires (Colloques Internationaux du Centre National de la Recherche Scientifique); Daudel, R., Ed.; C.N.R.S.: Paris, 1958. (46) Foster, J. M.; Boys, S. F. ReV. Mod. Phys. 1960, 32, 305. (47) Cook, G. B.; Boys, S. F. ReV. Mod. Phys. 1960, 32, 285. (48) Boys, S. F. ReV. Mod. Phys. 1960, 32, 296. (49) Boys, S. F. In Quantum Theory of Atoms, Molecules, and Solids; Lo¨wdin, P.-O., Ed.; Academic Press: New York, 1966. (50) Foster, J. M.; Boys, S. F. ReV. Mod. Phys. 1960, 32, 300. (51) Shavitt, I. Tetrahedron 1985, 41, 1531. (52) Boys, S. F.; Rajagopal, P. AdV. Quantum Chem. 1965, 2, 1. (53) Becke, A. D. J. Chem. Phys. 1988, 88, 2547. (54) Boys, S. F. Proc. R. Soc. London Ser. A 1969, 309, 195. (55) Kutzelnigg, W. Theor. Chim. Acta 1991, 80, 349. (56) Singer, K. Proc. R. Soc. London, Ser. A. 1960, 258, 412. Longstaff, J. V. L.; Singer, K. Proc. R. Soc. London, Ser. A 1960, 258, 421. (57) Boys, S. F. Symp. Faraday Soc. 1968, 2, 95. (58) Boys, S. F.; Handy, N. C. Proc. R. Soc. London, Ser. A 1969, 309, 209.
6016 J. Phys. Chem., Vol. 100, No. 15, 1996 (59) Boys, S. F.; Handy, N. C. Proc. R. Soc. London, Ser. A 1969, 310, 43. (60) Boys, S. F.; Handy, N. C. Proc. R. Soc. London, Ser. A 1969, 310, 63. (61) Boys, S. F.; Handy, N. C. Proc. R. Soc. London, Ser. A 1969, 311, 309. (62) Koutecky, J.; Cizek, J.; Dubsky, J.; Hlavaty, K. Theor. Chim. Acta 1964, 2, 464. (63) Handy, N. C. Ph.D. Thesis, University of Cambridge, 1967.
Handy et al. (64) (65) (66) (67) (68) (69) (70)
Schmidt, K.; Moskowitz, J. W. J. Chem. Phys. 1990, 93, 4172. Haselgrove, C. B. Math. Comput. 1961, 15, 323. Conroy, H. J. Chem. Phys. 1967, 47, 5307. Ellis, D. E. Int. J. Quantum Chem. Symp. 1968, 2, 35. Handy, N. C.; Boys, S. F. Theor. Chim. Acta 1973, 31, 195. Boys, S. F.; Bernardi, F. Mol. Phys. 1970, 19, 553. Boys, S. F.; Bucknell, M. G. Mol. Phys. 1974, 28, 759.
JP963465D
