Samuel Francis Boys

Samuel Francis Boys

Nicholas C. HandyDepartment of Chemistry, UniVersity of Cambridge, CB2 1EW, U.K.

John A. PopleDepartment of Chemistry, Northwestern UniVersity, EVanston, Illinois 60208-3113

Isaiah ShavittDepartment 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 introductionof Gaussian basis functions in electronic structure calculations. Interestingly, while Boys was fascinatedwith the integrability properties of products of Gaussians on different centers, and used them in otherapplications, including approximation schemes for multicenter integrals over Slater-type orbitals, he did notpursue their use as basis functions in their own right in his own molecular calculations. It was the work ofhis student, Colin M. Reeves, and the latter’s student, Malcolm C. Harrison, which revived interest in Boys’soriginal proposal and started the wide-scale application of Gaussian basis functions in this field. The majorinterest of Boys in the 1950s was the development of the formalism and algorithmic tools needed forconfiguration interaction calculations on atoms and molecules. His earlier work concentrated on atoms andatomic 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 offormulas and the overall organization of the calculations. He devoted much effort to vector coupling, theconstruction of symmetry- and spin-adapted linear combinations of Slater determinants (“co-detors” in histerminology), and the development of practical procedures for “projective reduction”, the reduction of matrixelements 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 projectivereduction formulas, and the formal and numerical parts of the work were integrated on the computer to reducethe need for human intervention and minimize the opportunities for mistakes. On turning his attention tomolecules, he developed and tested a variety of schemes for calculation of the multicenter integrals overSlater-type orbitals and made several pioneering applications of his ideas, notably including the first correctprediction of the geometry of the methylene ground state and a landmark calculation on formaldehyde. Inlater years, concerned about the slow convergence of the configuration interaction expansion, he focused onschemes for the incorporation of interelectronic distances directly into the trial wave function, leading to thedevelopment of the “transcorrelated” method. Other interests included the convergence of nonsymmetricsecular 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 inPudsey, England, in 1911. A compassionate account of his lifeand scientific work was written by Charles A. Coulson afterBoys’s untimely death in 1972 and published as a Royal Societybiographical memoir.1 (Much of the material in this section isbased on that memoir.) The same year, a summary of Boys’swork was given by R. G. Parr.2 Following the example of hisparents, Boys was an avid and eclectic reader from childhoodon. His strong interest in science, and particularly in chemistry,started in his grammar school years in Pudsey, but his wide-ranging interests included archeology, languages, and, particu-larly in his later years, economics.Upon graduation from Pudsey Grammar School, Boys won

a Royal Scholarship and went to Imperial College, London, forhis university studies. At Imperial College he earned a B.Sc.

degree in chemistry with first-class honors in 1932, and he spentthe next three years there taking degree courses in mathematicsand engaging in research. His later interest in molecularelectronic structure was already evident in his first researchtopic, dealing with “the stability of some coordinated cadmiumcompounds with reference to the electronic structure”. He alsostarted research on optical rotatory power, using models basedon classical electromagnetic theory,3 a topic to which he returnedfrom a quantum-mechanical view point in his Ph.D. researchin Cambridge University during 1935-1937.Boys’s move to Trinity College, Cambridge, in 1935 was

made possible by another scholarship, and there he started hisPh.D. work on the quantum theory of optical rotation underProfessor T. M. Lowry. Due to Professor Lowry’s death, Boyscompleted his thesis under the auspices of J. E. Lennard-Jones,the first professor of theoretical chemistry at Cambridge. Themathematical background he earned at Imperial College wasvery helpful to him in this work and strongly colored his laterresearch in electronic structure theory. Cambridge was one ofX Abstract published inAdVance ACS Abstracts,March 15, 1996.

6007J. Phys. Chem.1996,100,6007-6016

0022-3654/96/20100-6007$12.00/0 © 1996 American Chemical Society

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the hotbeds of quantum theory development in the 1920s and1930s, and in the Lennard-Jones group Boys was exposed tothe excitement of the early days of the application of quantumtheory to chemistry. His enthusiasm for quantum chemistrynever waned, and even during the war years, which he spenton munitions research, mostly on rocket propellants, in theBallistics Branch of the Armaments Research Department, hedevoted much free time to electronic structure theory. His notesof that period contain much of the foundations of his later workin 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 themethodology ofab initio electronic structure calculations. Thiswork was continued at Cambridge, where he returned as alecturer in theoretical chemistry in 1948. There he publishedhis landmark series of 12 papers4-15 under the overall title of“Electronic wave functions”, which included the work he haddone in London as well as later work, including research witha 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 hiswork, and his social life revolved around a few close friendsand the Cambridge International Club, of which he waschairman for a number of years. He spent most of his vacationsin Yorkshire, with his mother and sister, and on walking toursin England and on the continent. In the theoretical chemistrydepartment he was the “odd man out”, going his own way,strongly convinced that hisab initio approach was the right oneto pursue, while the rest of the department was engaged in themore fashionable theoretical model building and semiempiricalcalculations. He demanded much of his students, and it wasoften difficult to follow his abstract reasoning style, but whenunderstanding was finally achieved, it was entirely rewarding.

II. Computers at Cambridge

It is difficult to discuss Boys’s research in detail withoutreferring extensively to his utilization of electronic computing.He recognized very early the potential of the electronic computerfor organizing and mechanizing the tedious processes requiredin ab initio electronic structure calculations and even formechanizing much of the mathematical analysis required inderiving formulas for integrals and matrix elements andconstructing and manipulating proper symmetry- and spin-adapted expansion functions in configuration interaction calcula-tions. In 1956 he wrote,16 “In principle, it is possible to predictthe structure and properties of molecules with unlimitedaccuracy merely from Schro¨dinger’s equation and the numbersand properties of the electrons and nuclei defining the molecule.The complicated mathematical formulation, the very intractableintermediate problems and the extent of the numerical operationshave caused this to be both a very confusing and a difficultsubject. It does not appear to have been generally realized thatthe advent of automatic machines has simplified and madepractical such calculations, not only as a means for performingthe arithmetical operations but also for carrying out of much ofthe mathematical analysis of the most formal type.” He saw inthe use of the computer not only an opportunity to carry outlarge-scale calculations which would not have been practicalwithout it but also a means to ensure correctness of thecalculations by avoiding the mistakes which can easily beintroduced in manual calculations, both in the analysis and inthe arithmetic. While the computers of the time were far fromerror free, he demonstrated that correctness could be ensuredby a variety of checks and by extensive repetition.

The first electronic computer in Cambridge was the EDSAC(Electronic Delay-Storage Automatic Calculator).17 Its vacuum-tube-based design was begun at the University MathematicalLaboratory (whose director, Maurice V. Wilkes, had been amember of the Lennard-Jones group before the war1) at the endof 1946, and it carried out its first completely automaticcalculation in May 1949. The machine continued to undergoimprovements and additions in the following years, until itsreplacement 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 develop-ment work by the engineering staff, and almost all scientificapplications work was done at night and on weekends, withthe staff absent, as long as the users could keep the rather ficklemachine operating satisfactorily (which often required adjust-ment of pulse rates and even replacement of vacuum tubes).The machine was not enclosed in cabients but consisted of openracks and was cooled by an exhaust fan in the ceiling. Becauseof the continuing engineering work, there were many temporaryconnections using “alligator clips”.

Boys’s early electronic structure computations,5,11-13 involv-ing atomic configuration interaction wave functions for Be, B,C, F-, Ne, and Na+ (including some excited states), were carriedout by hand using desk calculators. (In one of those papers12

Boys thanks the University of London “for research grantscovering the cost of some professional computational assistanceand for a calculating machine”.) The first work in which theuse of the electronic computer appears14 is No. XI in his“Electronic wave functions” series, with Vector E. Price, onCl, Cl-, S, and S-, for which it “was found possible to adaptseveral sections of the calculation to the automatic calculatingmachine, the EDSAC, which has performed a considerableamount of the computations. In the next paper,15 with R. C.Sahni, concerned with the evaluation of vector couplingcoefficients for atomic configuration interaction calculations,he states that it has been “found possible to develop a methodof calculation which can be performed purely automatically bythe EDSAC, proceeding from the lowest argument valuesindefinitely through all higher values.” By the time molecularelectronic structure calculations were begun in Boys’s groupin the early 1950s (as discussed later in this review), all thecomputational steps and formula derivations were being com-pletely automated.

The EDSAC had no compiler or operating system, andprograms were written in a rudimentary assembler languagewhich performed decimal-to-binary address conversion and hadlimited facilities for relative addressing but no symbolicaddresses or variable names. Thus, the programmer workedvery close to the machine level, manipulating and controllingits registers directly through the machine instructions in theprogram. Considerable ingenuity was often required to fit theprogram into the limited memory of the machine, whichconsisted of 1024 “short” (17 bit) words, each of which couldhold one instruction or numerical value; two short words andan intervening “spacer” bit could be combined to form a 35 bit“long” word, which provided roughly 10 decimal digits ofaccuracy. There was no floating-point arithmetic, and numbershad to be scaled to avoid overflow and to maintain precision.For most of the machine’s existence there was no auxiliarystorage, and intermediate results had to be punched out on papertape and read in again when needed. Much effort was expendedin making each subroutine as short as possible, often at theexpense of operating speed. (It was sometimes stated that “aclever programmer can shorten any program by one storage

6008 J. Phys. Chem., Vol. 100, No. 15, 1996 Handy et al.

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location”, and this process was supposed to be applicablerecursively.) Boys enjoyed programming and wrote manysubroutines and programs himself. He found much satisfactionin designing efficient and compact computational algorithmsand programs. When the new machine, EDSAC II, beganoperations in late 1956, Boys had it mostly to himself for sometime because, unlike most users, he had taken the precautionof writing new programs for it ahead of time.1

III. Research

A. Gaussian Functions (IS).Boys is generally credited withthe introduction of Gaussian basis functions to quantumchemistry calculations.4 Earlier references to the use ofGaussian basis functions to quantum mechanics (other than inthe solution of the harmonic oscillator problem) appeared inRoy McWeeny’s dissertation18 in 1948, applied in momentum-space calculations for H, He, and H2+, and in McWeeny’s workon the nuclear structure problem,19 also in monomentum space.Boys had worked out many of the formulas for using Gaussiansas basis functions in 1942 and earlier,20 but his first publishedwork on Gaussians appeared in his 1950 paper,4 No. I in the“Electronic wave functions” series. Here he specificallyadvocated the use of Gaussians as basis functions in coordinate-space electronic structure calculations for molecules, emphasizedtheir advantage over exponential functions (Slater-type orbitals,STO) in overcoming the difficulties of multicenter integralevaluation, and derived formulas for the necessary integrals.The basis functions he proposed had the form now known as“Cartesian Gaussians”,

whererA ) r - A is the vector from centerA to the pointr,with componentsxA ) x - Ax, etc.,rA ) |rA|, R is a positiveparameter, andl, m, n are nonnegative integers. He includedderivations for the required multicenter integrals forl ) m) n) 0, stating that formulas for other cases can be obtained bydifferentiation with respect to the center coordinates. He pointedout that functions of the form (1), with a single-exponentialparameterR, form a complete system of functions, since theyspan the system of Hermite functions, but that in finite basiscalculations the exponential parameters could be individuallyvaried to produce lower energies.4

The first significant molecular tests of Gaussian basis setswere carried out in Boys’s group by his student Robert K.Nesbet24 on the ground state of methane and at MIT by AlvinMeckler22 on the3Σg

- and1Σg+ states of the oxygen molecule.

Both of these were configuration interaction calculations.Meckler used a minimal Gaussian basis, while Nesbet used abasis almost twice as large as minimal. Nesbet’s calculationprobably was the first trulyab initio calculation of a polyatomicmolecule other than H3 and certainly the first without anyintegral approximations.Interestingly, Boys did not show much enthusiasm for the

use of Gaussian basis sets and concentrated much of his attentionin the 1950s on devising various schemes for the evaluation ofSTO multicenter integrals. Nevertheless, he continued to befascinated by the ease with which various multiple integralsinvolving products of Gaussian functions of interparticlecoordinates could be evaluated and pursued other applicationsof such functions in multidimensional integration problems. Herewe quote from IS:“When I joined this group in 1954, my first project involved

use of least-squares expansions of the Mayerf-function in terms

of Gaussians for the calculation of virial coefficients ofgases.23,24 We also incorporated Gaussians in a flexiblemathematical form we proposed for intermolecular potentials,in the expectation that such a form would facilitate thecalculation of integrals required in the determination of real-gas properties.25 (Boys insisted that my name appear first onthat publication, expecting the proposed potential to be knownas ‘the Shavitt-Boys potential,’ because he thought that ‘TheShavitt 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 multicenterintegrals by an “axial expansion” method, in which each two-center charge distribution (product of two STO’s) was ap-proximated as a linear combination of STO’s centered at severalpoints along the line joining the two centers.26,27 This approachresulted in the expression of a four-center integral as a linearcombination of two-center Coulomb integrals, and while it wasreasonably effective, the accuracy obtained28 would be consid-ered quite inadequate today.In the late 1950s Boys used another approach for the

evaluation of STOmulticenter integrals, based on approximatingeach Slater-type function as a linear combination of Gaussianfunctions on the same center.29-31 Using nine Gaussians foreach expansion, each electron repulsion integral was convertedto a linear combination of 6561 Gaussian integrals, but thenumber of terms in this linear combination was greatly reducedby Schmidt orthogonalization of the Gaussians in each expan-sion, in order of diminishing contribution, followed by truncationof “high-order” products in the multicenter expansion. TheGaussian expansions were not used as a basis set in their ownright but were only employed to evaluate those STO integralswhich could not be computed analytically. The large numberof Gaussians used to fit each STO was needed in order tomaintain high accuracy in the fitting and thus achieve consis-tency between the analytically evaluated integrals and thosecomputed by the Gaussian expansion.IS: “I suggested to Boys in 1959 that the fitting accuracy

requirement could be relaxed substantially, reducing the numberof Gaussian terms needed in each expansion, ifall integralswere evaluated using the Gaussian expansions, so that theseexpansions could be considered the actual basis functions. Boysdid not pursue this approach, but it formed the basis for mylater proposal to use fixed linear combinations of Gaussians asbasis functions.”24

Another fallout from Boys’s Gaussian expansion approachwas a more accurate method for the evaluation of STOmulticenter integrals, the Gaussian transform method.24,32 Shav-itt was led to the idea for this approach by consideration ofBoys’s method of picking the exponential coefficients in theexpansion

Boys used linear least squares to determine the coefficientscibut chose the nonlinear parameterssi by considering theexpansion (2) as a quadrature approximation of the integral

and taking the Gauss-Laguerre quadrature points fors in (3)as the values of thesi in (2). (This approach was also usedlater by Silver.33) The Gaussian transform method is based onusing the integral transform (3), instead of the approximateexpansion (2), as a basis for a procedure for the evaluation of

e-Rr ≈∑i

ci exp(-sir2) (2)

R2xπ∫0∞s-3/2 exp(- R2

4s)e-sr2 ds) e-Rr (3)

ælmn(R;rA) ) xAl yA

mzAn exp(-RrA

2) (1)

Samuel Francis Boys J. Phys. Chem., Vol. 100, No. 15, 19966009

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the STO integrals. A similar transform approach had beenproposed earlier by Kikuchi34 but was not developed furtherby him.Boys returned to Gaussian expansions in electronic wave

function calculations in 1960, when he proposed the use ofproducts of Gaussian basis functions and correlation factors inthe form of Gaussian functions of interelectronic distances.35

He derived the formulas needed for integral evaluation in thisapproach, but did not pursue this idea until later, in the contextof the transcorrelated method. The modern use of Gaussianbasis sets in electronic structure theory can be traced to a largedegree to the impetus provided by the later work of Boys’sstudent, Colin M. Reeves, who with his own students at theUniversity of Leeds, Malcolm C. Harrison and Roger Fletcher,explored the various issues involved in the optimal use of suchbasis sets and demonstrated their effectiveness.36-38 WhenHarrison completed his studies with Reeves, he came to JohnC. Slater’s group at MIT and there, in collaboration with othermembers of that group, including Imre G. Csizmadia, Brian T.Sutcliffe, and Jules W. Moskowitz, started work on what becamethe first widely distributedab initio molecular electronicstructure program, POLYATOM.39,40 This program employedGaussian basis sets, first as individual basis functions and later,41

following the introduction of contracted Gaussian basis sets byClementi and Davis42 in their own program, IBMOL, alsoincluded provisions for contractions. These programs beganthe process of bringingab initio electronic structure calculationsinto the mainstream of chemical research. (A more detailedaccount of these and later developments has been givenelsewhere.43)B. Configuration Interaction (IS). Right from the start,

Boys’s method of choice for electronic structure calculationswas the method of configuration interaction (CI). At that timethis term was mostly applied to the interaction of just a fewterms (Slater determinants) required for the proper descriptionof some electronic states of atoms. Most molecular calculationsbeyond the Hartree-Fock level used the valence bond approachand were mostly semiempirical. Boys saw the CI approach asthe only practical method capable, at least in principle, ofconvergence to the exact solution of the Schro¨dinger equationand proceeded to outline a complete scheme for its implementa-tion.4,5 He called it the “method of variational selection” andintroduced other terms (Table 1), like “detor” for a Slaterdeterminant constructed from orthonormal orbitals and “co-detor” for a symmetry- and spin-adapted linear combination ofdetors, and later referred to the complete procedure as the “poly-detor” method. (It is a pity that his terminology was not takenup by other workers in the field, because his terms are morecompact and precise than those in current usage). He proceededto work out various details of the application of CI,4-6

concentrating initially on application to atoms. In this contextthe problem of vector coupling, i.e., the construction of co-detors adapted to spherical symmetry, looms large, and muchof the “Electronic wave functions” series is devoted to dealing

with this problem and with the reduction of matrix elements ofthe Hamiltonian between vector-coupled co-detors to linearcombinations of one- and two-electron integrals over theorbitals.7-10,15 He called this latter process “projective reduc-tion” and devoted considerable effort to designing efficientalgorithms for its implementation. He also introduced a simpleiterative method for the solution of the matrix eigenvalueproblem.5

The early atomic calculations,5,11-13 which were carried outwithout the benefit of an electronic computer (mostly in London,before Boys’s move to Cambridge, even though all the paperswere submitted from Cambridge), were remarkably successful,considering the rather small expansions used, usually of theorder of a dozen co-detors. They produced good values of therelative energies of several electronic states of the atoms Be,B, and C, using STO basis sets which would now be designatedas (4s4p2d).12,13 Also included was a calculation on the groundstates of F-, Ne, and Na+.11 The first calculation to use acomputer 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 electronaffinities were not well reproduced (not surprisingly, consideringwhat we know now about the difficulties of such calculations).Selection of co-detors on the basis of energy contributions wasalso 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 studentshad already completed molecular research, Robert K. Nesbeton CH421 and Geoffrey B. Cook on BH,16 and had left. Therewas only one other student at the time, Colin M. Reeves, whowas working on H2O.16,26 We were not housed on the premisesof the Theoretical Chemistry Department (where H. ChristopherLonguet-Higgins had just arrived as the new professor oftheoretical chemistry, replacing the recently retired Sir John E.Lennard-Jones), but in the Mathematical Laboratory, where weshared an office with two students of Douglas R. Hartree,Charlotte Froese (now Fischer) and David F. Mayers. Thisarrangement was very convenient, providing easy access toEDSAC and to the expertise of the other computer users. I gotinvolved in electronic structure research only later, whenunforeseen difficulties (eventually surmounted) came up in thevirial coefficients project and Boys, wanting to make sure thatI would have a successful thesis project, suggested calculationson H3.For the co-detor expansion functions in molecular calcula-

tions, Boys adopted a Rumer-like scheme, and with this studentReeves he derived a very efficient projective reduction procedurefor the resulting ‘bonded functions’.26 I used this procedureand Reeves’s programs in my H3 work,16 and continued to usethe same approach in my later work44 until the development ofunitary-group-based methods for ‘direct CI’ calculations in theseventies. In the early work, Boys did not see much point instarting a CI calculation with a self-consistent field (SCF) step.In the early and mid fifties we did not even have an SCF

TABLE 1: Some Terminology Introduced by Boys

term meaning

detor Slater determinant constructed from orthonormal orbitalsco-detor spin- and symmetry-adapted linear combination of detors (i.e., configuration state function)bonded function Rumer-like spin-adapted co-detorvariational selection method,poly-detor method

configuration interaction method (with selection of orbitals and configurations)

poly-detor wave function CI wave functionprojective reduction expression of Hamiltonian matrix elements between co-detors in terms of one- and

two-electron integrals (i.e., determination of “coupling coefficients”)fundamental from first principles,ab initio


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program in the group, and orbitals for CI calculations weresimply chosen as ‘reasonable’ simple linear combinations ofbasis functions. By hindsight, our expectations at that time forthe rate of convergence of the CI method were much toooptimistic. Perhaps this was for the better, because had weknown how many terms were needed for quantitative resultsfor nontrivial problems, we might have been completelydiscouraged and chosen to devote our time to some other pursuit.As it were, perseverance and hard work, plus the developmentof new methods and the astounding advances in the speed andcapacity of electronic computers, brought us to a stage today atwhich theoretical calculations of molecular structures andproperties are a widely accepted major component of chemicalresearch and make major contributions to our understanding ofchemical phenomena.”Two new students came to work with Boys while Shavitt

was in his group, Ivor Jones and John M. Foster. Jones workedinitially on homonuclear diatomics, and particularly C2,45 andlater on HCN.29 Foster worked on H2CO31 and on CH2.46 Mostof their work was completed after the much more efficientEDSAC II had replaced the old computer, allowing morepowerful calculations; the CH2 calculation included up to 128co-detors. At this point an SCF wave function was alreadybeing used to obtain orbitals and to provide the initial term inthe CI expansion.31 Boys reported these calculations and hisgeneral approach47 at the first Boulder conference on theoreticalchemistry in 1959. At that meeting he also presented a neworbital localization scheme48,49which, combined with a methodfor the construction of efficient correlating orbitals,50 could beused to produce a more compact CI expansion. Particularlyinteresting was the CH2 calculation, which was the first topredict a bent structure for the ground (3B1) state, in disagree-ment with the then accepted interpretation of the spectral databy Herzberg. This result was later confirmed by other theoreti-cal treatments and experiments and by a reinterpretation of thespectrum.51

C. Numerical Integration for Molecules (NCH). Anothercontribution was the development of a purely numerical methodfor SCF wave function optimization, which included a particu-larly effective numerical integration scheme for molecularcalculations.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 numericalintegration for molecules. He was driven by the knowledgethat if the expansion set contained the exact wave function, thenit was only necessary to use one quadrature point to evaluatethe matrix elements of the secular equations. This paperdiscusses the advantages of performing calculations usingnumerical integration techniques. He argues that it is simplerto deal with higher angular momentum integrals this way. Hefurther considers that the cost factor is lower if integrals areevaluated numerically.For three-dimensional quadrature he devised the following

scheme for an integral evaluation:

He definedVs(r ) > Vt(r ), all t, whenr was in the region of thesth shell of a given atom:

These ideas parallel the use Voronoi polyhedra of three-dimensional quadrature for DFT today; see for example the oftencited work of Becke.53 With each shell region Boys used aspherical polar grid:

He used equally spaced points inQ space, Legendreθ points,and equally spacedφ points. He used 12 radial points, and thenumber of (θ,φ) points increased asr increased.Boys used the ideas of localized Gaussians to evaluate the

two-electron integrals. The numerical evaluation of two-electronintegrals has not found much favor in recent years because ofthe very fast methods for the evaluation of Gaussian basisfunction integrals. However, Boys rarely mentioned usingGaussian basis functions in his later period.D. The Convergence of Nonsymmetric Secular Equations

(NCH). In another important paper54 Boys examined theconvergence of the disymmetric secular equations:

He was interested in such equations because of his currentinterest in transcorrelated theory and numerical integration. Therelevant equation system can be written

He proved that the computed energyW could be expressed asfollows:

whereµ+ andµ are a measure of the inability of the left andright expansion setsφ and ψ to represent the exact wavefunction, andµQ is a measure of the quadrature error. Theimportant point is that if exact integration is used, then the errorin the energy is the product of the errors in the two expansionsets.Of course, this explains why coupled-cluster theory is so

attractive. In CC theory the equations may be written

Boys’s theory says that the error in the CCSD energy is equalto the product of the CISD wave function error and the CCSDwave function error. The first is already small, and the secondis very small! A paper on this topic has recently been writtenby Kutzelnigg.55

E. The Transcorrelated Method (NCH). As the slowconvergence of the CI expansion became more and moreevident, Boys turned his attention to alternative approaches. Hebecame convinced that the incorporation of interelectroniccoordinates in trial wave function, as in the work of Hylleraas

I )∫F(r ) dr (4)

F(r ) ) ∑s

F(r ) Vs(r )/∑t

Vt(r ) (5)

) ∑s

Fs(r ) (6)

V1(r ) ) j1(r )/rA4 (7)

V2(r ) ) (1- j1(r ))/rA4 (8)

V3(r ) ) 1/rB4 (9)

V4(r ) ) 1/rC4 (10)

r ) aZ4/(Z3 + â) (11)

Z2 ) Q/(1- Q) (12)

∑s

⟨Φr|H - W|Ψs⟩cs ) 0 (13)

∑s

⟨C-1Φr(∑I

δ(r I,r )wI)|H - W|CΦs⟩cs ) 0 (14)

W) Wexact+ (µ+µ + µQµ)(W11 + ...) (15)

⟨Φr|H - W| exp(∑s

Ts)Φ⟩ ) 0 (16)


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on helium and of James and Coolidge on H2, provided the besthope for further progress. This was the motivation for his 1960proposal for Gaussian correlation functions35 (generalized bySinger56) and led later to the development of the transcorrelatedmethod, in which the wave function incorporates an exponentialcorrelation factor depending on interelectronic coordinates.54,57-61

In their initial studies, Boys and Handy considered a wavefunction forπ electrons of benzene with the following form

To evaluate this wave function, the molecular orbitalsφi areexpanded in terms of the atomicπ orbitalsπA, πB, πC, πD, πE,πF. Thus, the one determinant becomes a fixed linear combina-tion of determinants of atomicπ orbitals. The Jastrow factorin front of each of these determinants is then evaluated byassuming that each electron is situated on the C atoms on whichthe atomicπ orbitals are located, and thus∏(1 - µ exp(-γrij)) becomes∏(1 - µ exp(-γRPQ)), for all C atomsP andQon which the atomicπ orbitals are situated in the relevantdeterminant.RPQ is the distance between the C atomsP andQ. Different parametersµ and γ were used, dependent onwhether the associated electrons had parallel or antiparallel spins.In this way the coefficient in front of eachπ determinant isamended by its own Jastrow factor. Thus, a wave function forthe ground state of benzene has been determined which includesdynamic correlation. The energy of this wave function, whichhe wrote asCΦ, was evaluated through the usual variationalexpression⟨CΦ|H|CΦ⟩/⟨CΦ|CΦ⟩, using the Pariser-Parr-Pople values for the one- and two-electron atomicπ integrals.The value of the correlation energy so obtained (-0.75 eV)was very close to the CI value obtained by Kouteckyet al.62

(-0.78 eV) using the same integrals. The method also gavegood agreement for similar calculations on the excited statesof 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 ofthe equationHCΦ ) ECΦ, Boys had written downC-1HCΦ) EΦ, and thus transcorrelated theory was born.” The wavefunction is written

where

The governing equation is obtained by premultiplying theSchrodinger equation byC-1:

The transcorrelated operator may be expressed in terms of one-,two-, and three-electron operators:

although the form of the three-electron operator meant that theintegrals were no worse than two-electron integrals to evaluate.The resultant equations to determine the parameters were the

following:

This last equation may be understood by recognizing that it isequivalent to

which makes the transcorrelated operator as Hermitian aspossible.The first term inCwas fixed to make the wave function obey

the unlike spin correlation cusp condition. This allowednumerical quadrature to be carried out. Results on Ne and Ne+

were given Tables 1-3 of ref 60. The highlights were that thecomplete transcorrelated wave function for Ne was given interms of 26 parameters with an energy of-128.959 hartrees,compared to the exact value of-128.929 hartrees. Theionization energy was calculated to be 0.793 hartree, comparedto the exact value of 0.791 hartree. Schmidt and Moskowitz64

more recently examined the Ne transcorrelated wave functionusing variational Monte Carlo. Using the Ne correlationfunction, but Clementi orbitals, gave 72% of the correlationenergy. Reoptimization of C gave 83%, and adding more termsgave 85%. The principal reason for the difference is that S-Mwere using a variational energy expression, but it is also verylikely that in the 1960s Boys and Handy were using a totallyinadequate set of quadrature points.To perform transcorrelated calculations on molecules, it was

necessary to lean on the quadrature schemes of Boys andRajagopal.52 After the Voronoi separation, spherical polar(r,θ,φ) grid points were developed as follows:

where

with

Such a Diphantine scheme exactly integrates all Fourierfunctions exp[2πi((pqr ( sqθ ( tqφ)] for which p + 2s + 4t< Γ. (See other works of Haselgrove,65 Conroy,66 and Ellis.67)

Ψ ) ∏i>j(1- µ exp(-γrij))A(φ1

Rφ1

âφ2

Rφ2

âφ3

Rφ3

â) (17)

Ψ ) ∏i>jf(r i,r j) A(φ1φ2...φN) (18)

) CΦ (19)

C) ∏i>jexp[∑

l

DlGl(r i,r j) + ∑p

dp(gp(r i) + gp(r j))] (20)

HCΦ ) WCΦ (21)

C-1HCΦ ) WΦ (22)

C-1HC) H - ∑ij

(12∇i2fijfij+∇ifij‚∇ifij

) -1

2∑ijk

∇ifij‚∇ifikfij fik

(23)

energyW: ⟨Φ|C-1HC- W|Φ⟩ ) 0 (24)

orbital coefficientscRi: ⟨ΦiR|C-1HC- W|Φ⟩ ) 0 (25)

correlation coefficientsDl:

⟨∑ij

Gl(r i,r j)Φ|C-1HC- W|Φ⟩ ) 0 (26)

contraction coefficientsdp:

⟨∑i

gp(r i)Φ|C-1HC- CHC-1|Φ⟩ ) 0 (27)

⟨ΦiR|C-1HC|Φ⟩ ) ⟨Φ|C-1HC|Φi

R⟩ (28)

r ) qr/(1- qr) (29)

θ ) π(6qθ5 - 15qθ

4 + 10qθ3) (30)

φ ) 2πqφ (31)

qr ) jl mod 1 (32)

qθ ) jmmod 1 (33)

qφ ) jnmod 1 (34)

(l,m,n) ) ( 1M,DM,EM) (35)


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The valuesM ) 160,D ) 18, andE ) 46 were used. Thetransformations (29)-(31) are justified by the Euler-Maclaurintheorem.68

The correlation factor for LiH involved 39 parameters, andan energy of-8.063 hartrees was obtained, compared to theexact value of-8.070 hartrees, which on the face of it appearsto be quite a success.Why has this apparent success of the transcorrelated method

meant that research in the method has not continued? Here wequote from NCH:“I suppose I was worried by the lack of the variational upper

bound, which over the three years meant that I had obtainedany number of different results dependent upon the expansionset used. The correlation factor which we used treated like spinsand unlike spins in the same manner. The method involvedthree-electron integrals, and we know all the problems we havehad with the two-electron integrals. The correlation expansionfunctionsGk(r i,r j) are not orthogonal, and this makes both theirselection and use very difficult. The number of quadraturepoints which we used for LiH are on today’s experience toosmall by at least a factor 20. Finally the method in those dayswas considered complicated. For all these reasons I doubtwhether the method can compete with the Coupled Clusterapproach.”F. The Basis Set Superposition Error (NCH). One of the

most notable of Boys’s later contributions was the identificationof basis set superposition error (BSSE) and the proposal of thecounterpoise correction method for dealing with it.69 Thisprocedure is widely used today in many molecular calculations,and particularly in the treatment of weak bonds and intermo-lecular interactions, for which even small superposition errorscan be of major consequence.Boys’s term for BSSE wasfunction counterpoise.69 He was

interested in computing

∆ABW(R) ) WAB(R) - WA - WB (36)

Since he was primarily concerned with numerical integrationprocedures, he discussed in detail the “point counterpoise”procedure. He introduced “function counterpoise” as follows:“In this procedure, the calculations of separate energiesWA,

WB are performed with the full set of expansion functions usedin the calculation of the energyWAB of the bimolecular system.It is considered that a deductive justification for this procedurehas not yet been made and this will have to be extensively tested.Results already available appear to justify it .... The energiesof the separate moleculesWA, WB are now slightly dependenton the value ofR between them .... Cost of less than 3 timesthe original cost is well worthwhile if it gives the possibility ofmuch more accurate values.”We all know that over the intervening years there have been

many, many papers purporting deductive justifications andpresenting extensive tests. This is certainly Boys’s most citedpaper of the final period.G. The Calculation of Ro-Vibrational Energy Levels

(NCH). This work70 was completed after Boys’s death, butBoys completely developed the mathematics of the method,which is probably the earliest variational method for the ro-vibrational energy levels of a triatomic molecule.Again he used a numerical approach, introducing quadrature

points which corresponded to distorted positions of the moleculeon the given potential energy surface. In detail

Ri ) G + T(R,â,γ) Ris(Q) (37)

Ris(Q) ) S(Q) Ri

Q (38)

HereRi is the position of theith atom,G is the center of mass,andT(R,â,γ) is the Euler rotation matrix.Ri

Q is an arbitrarydisplaced position of theith atom, andS(Q) is a rotation matrixwhich rotatesRi

Q to give the displaced positionRis(Q), which

corresponds as closely as possible to a pure vibrationaldisplacement. The rotation matrix is

For a triatomic molecule:

Ω was expanded in terms ofQ as follows:

with theωa andωab determined to obey the Eckart conditions:

The wave function is expanded in coordinatesq, which areclosely related to normal coordinates:

The vibration-rotation expansion functions are

whereYJk is a spherical harmonic andHν are Hermite polyno-mials. These expansion functions are then inserted into thescalar equations to obtain vibration-rotation energy levels fora provided potential surfaceV:

Results were presented for the water molecule.

IV. Conclusion

In 1958 Boys wrote,45 “... the future is full of promise, sinceone should expect a reduction ofT [the amount of effort for amolecular calculation] by a factor of the order of 1000 due tothe foreseeable increase in the speed of the machines and theimprovement of the mathematical methods” (translated fromthe French of the published paper). As it is, the increase inspeed of electronic structure calculations since that time hasalready probably been closer to 1 000 000, not to mention thesignificant methodological advances that have taken place. Thesedevelopments have certainly justified Boys’s faith in theab initioapproach, even though his early expectations for the convergenceof 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

(cosΩ -sinΩ 0sinΩ cosΩ 00 0 1) (39)

R1Q ) (Q1, 0, 0) (40)

R2Q ) (Q2,Q3, 0) (41)

R3Q ) -((m1Q1 + m2Q2),m2Q3, 0)/m3 (42)

Ω ) ∑a

ωa(Qa - Qae) + ∑

ab

ωab(Qa - Qae)(Qb - Qb

e) (43)

∑i

miRie×

∂Ris

∂Qa

) 0 (44)

qi ) ∑j

(B-1)ij(Qj - Qje) (45)

YJk(â,γ) Hν1(q1) Hν2(q2) Hν3(q3) exp(-1/2(q1

2 + q22 + q3

2))(46)

∑s

⟨Φr| - ∑i

1

2mi

∇i2 + V- W|Φs⟩cs ) 0 (47)


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was unfortunate that official recognition in the U.K., in the formof a more elevated position at Cambridge University and electionto fellowship in the Royal Society, did not come until about ayear before his death. His resources were therefore limited,and the number of his students was relatively small. For muchof the time he had to overcome great skepticism, and evenantagonism, towardab initio electronic structure theory, bothat Cambridge and in the British theoretical chemistry establish-ment as a whole. It is interesting to speculate how much morehad he been able to accomplish were the situation different andwere it not for his untimely death.

Acknowledgment. We thank R. G. Parr and G. G. Hall forseveral 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 latesummer of 1948, I joined the Cambridge Theoretical Chemistrydepartment, intending to write a thesis on the theory of liquids,a topic on which the Professor, Sir John Lennard-Jones, hadpublished some major papers just before the war. I had hadone or two preliminary conversations with LJ (as we allabbreviated his rather lengthy name), at one of which he hadtold me that there would be two other new members of thedepartment, another research student, Victor Price, and a recentlyappointed 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 firstworking day. “Hello”! I’m Frank Boys” he said, in a friendlymanner. I was very young and shy in those days, so I justreplied “Hello” and returned to the book I was reading as quicklyas possible. He walked through to his new office, which waslocated at the back of the student room, and I heard himunloading his bulky briefcase. Then, after about five minutes,he poked his head out of his door and said to me, “Will youcome in, please?” Somewhat surprised, I went in and sat downin a chair by the side of his desk. He then produced an emptypad of paper and started to tell me, in great detail, everythinghe knew about quantum mechanics.I cannot remember exactly where he started, but it was

probably the Schroedinger equation. However, I do rememberthat my initial reaction was revulsion at his ugly notation. Then,after five or ten minutes, I began to wonder to myself, “Why isthis man telling me all this?” Conversations with Frank Boyswere always one-sided. He did all the talking, and there wasusually plenty of opportunity to think about the background ofwhat he was saying. A bit later, he began making suggestionsabout what I should read; it dawned on me that I was to be hisresearch student! Panic began to set insthis was certainly farfrom my intention. The presentation continued. I recall hearingabout integration of Gaussian functions, detors, codetors,polydetors, and all the other topics that were published severalyears later. All these things were known to him in 1948. ButI was not paying much attention. Frank must have thought thatI was a very unresponsive student, for I was desperatelywondering why LJ had done this to me, had transferred me tothis new person without even letting me know. This seemedimpossible, since LJ was nothing if not a gentleman, and theseactions appeared most uncharacteristic. But I could think ofno other explanation. The presentation went on and on andon, nearly three hours in all, I recall. So, at the end of this firstconversation, I left in a state of total dejection, holding the sheafof Frank’s notes containing all he knew about quantum

mechanics and surveying the wreckage of my planned careerin liquid theory.After a sleepless night, I returned to the student room and

was sitting, gloomily waiting for LJ’s secretary to come andtell 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 teacher-

student relationship ended after only 20 hours. LJ’s reputationwas rehabilitated, and I was able to resume my research plans.My principal regret now is that, in a rush of relief, I discardedFrank’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 Cam-bridge. As a research student (1948-1958), I quickly came torealize how helpful he could be. LJ was an excellent supervisor,but he was rather inaccessible, appointments having to be madetwo weeks in advance. Frank Boys, on the other hand, wasalways there and always willing to talk at length. Whenever Iwas stuck on some point of mathematical technique, I wouldknock on his door and ask for help. He was always willing toput aside what he was doing and give advice on the spot. Oftenhis response covered much more than had been requested forhis interests spread widely, particularly on down-to-earthpractical 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 wasa huge contrast between the scientific styles of these twoindividuals. Christopher’s presentations were always elegantand polished, with clear starting assumptions, a logical argument,and well-rounded conclusions. Frank, on the other hand, tendedto be more scattered, sometimes almost incoherent, with manyconcepts only partially defined and arguments going in manydirections. There were always loose ends. But the loose endsled in interesting directions, I came to learn, and showed howhis mind was always probing toward the future. There wereconflicts between Boys and Longuet-Higgins, sometimes quitepainful to watch. As the older man in the junior position, Frankmust 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, atthe time, was that the role of the theoretical chemist was theapplication of theory to new branches of chemistry as theydeveloped, any necessary approximations being introduced tomake the impact immediate. He, on the other hand, took thelong-term view. Like many others at the time, I was skepticalabout the length of the term.The second conversation took place one evening in early

1958, while Frank and I were walking from the department upSidney street, toward his lodgings in Jesus Lane. By this time,I had become a lecturer in the mathematics department. Frankwas still a lecturer in theoretical chemistry, so we were at thesame career level. I was unhappy earning my living in thisway and was beginning to look for a position in a scientificdepartment; 1958 was a favorable year to do this. The Sovietsatellite 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 yourcareer going?” he asked, “You must be getting many offers”. Itold him that I was, and he asked for details. So he began togive me helpful, fatherly advice about what I should do. Ashe spoke, I wondered at the conversation. I was not smarterthan he, and he was 15 years older. Why was I the one receivingthe offers, and why was I about to overtake him in the academicrat race? I suppose I knew the answer then. I was doing things


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that were fashionable at the timesaromatic electronic excitedstates, free radical spectroscopy, nuclear magnetic resonance,and the like. Frank took no notice of such things; he carriedon with what he thought important, irrespective of publicacclaim. And why was he not envious? He showed no signof it and seemed genuinely pleased to help me. As theconversation 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 whichhe had been invited by Bob Parr. This meeting was the climaxof the group 1/group 2 divide, which separated the quantumchemistry community throughout the 1950s, brilliantly delin-eated by Charles Coulson in his after-dinner speech (publishedin 1960 in theReViews of Modern Physics). Boys’s presentationwas remarkable. In describing his work, he produced a papertape of his whole computer program and unrolled it along thelength of the chemical lecture bench. There, in one roll, wassomething, of which one could ask a chemical question at oneend and it would produce an answer at the other! The conceptof complete program packages, which we all use today, wasquite foreign then, and most of the audience probably thoughtthe demonstration bizarre. But it was prescient; Frank Boyswas the hero of that meeting.Following six years in an administrative position at the

National Physical Laboratory, I moved to Pittsburgh in 1964to devote full attention to electronic structure theory. After threeyears working on zero-differential-overlap methods (CNDO,INDO, etc.), I finally got around, belatedly, toab initio studiesin late 1967. The first version of what became the GAUSSIAN70 program was developed during 1968 by my student WarrenHehre and myself. We reread the old Boys papers anddeveloped a new algorithm that sped up integral evaluation forcontracted Gaussians by a factor of nearly 200, using axisrotation inside inner loops. This was an important developmentwhich finally allowed extensive application, but it was reallythe only new feature. The rest was pure Boyssthe use ofGaussians, the fitting of Slater orbitals by Gaussians, thepackaging of the whole logic into a unified wholesthese allcame from him.And so I come to the third conversation. The Gaussian

material was first presented in a lecture at the Faraday SocietySymposium at University College, London, in December 1968.Frank Boys was in the audience, still only a lecturer atCambridge. As many of you will know, there are times whenthere is somebody in the audience that the speaker wishes wasabsent. This was one such occasion. As I spoke, I kept lookingnervously 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 slightsmile that he often had. I interpreted it as the smile of anevangelist acknowledging a convert.The conversation took place at the reception held later that

evening. I approached Frank and expressed my appreciationfor his past work and things I had learned from him. I shouldhave done this sort of thing years earlier. He asked about someof the technical details, and we discussed them for about 10minutes. And then I asked him, “Frank, do you remember thefirst conversation we had 20 years ago?” “Oh yes! I rememberit well,” he replied. “What do you think would have happenedif it had been for real and I had become your student?”, I askedfurther. He paused for a while and then said, “I don’t know,but I am sure that we both would have had very differentcareers.” I think we shared a moment of mutual regret.I believe that was the last time we spoke before his untimely

death.

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, 144655, 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.Nature1956,

178, 1207.(17) Wilkes, M. V.Automatic Digital Computers; Methuen: London,

1956.(18) McWeeny, R. Dissertation, Oxford University, 1948;Nature1950,

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.Nature1956, 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 WIS-AF-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;QCPE1964,No. 11, 47.

(40) Csizmadia, I. G.; Harrison, M. C.; Moskowitz, J. W.; Sutcliffe, B.T. Theor. Chim. Acta1966, 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;QCPE1971, 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. InMethods of Electronic Structure Theory; Schaefer,

H. F., III, Ed.; Plenum: New York, 1977.(45) Boys, S. F.; Jones, I.; Shavitt, I. InCalcul des Fonctions d’Onde

Moleculaires (Colloques Internationaux du Centre National de la RechercheScientifique); 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. InQuantum Theory of Atoms, Molecules, and Solids;

Lowdin, P.-O., Ed.; Academic Press: New York, 1966.(50) Foster, J. M.; Boys, S. F.ReV. Mod. Phys. 1960, 32, 300.(51) Shavitt, I.Tetrahedron1985, 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. Acta1991, 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.


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(59) Boys, S. F.; Handy, N. C.Proc. R. Soc. London, Ser. A 1969, 310,43.



(62) Koutecky, J.; Cizek, J.; Dubsky, J.; Hlavaty, K.Theor. Chim. Acta1964, 2, 464.

(63) Handy, N. C. Ph.D. Thesis, University of Cambridge, 1967.

(64) Schmidt, K.; Moskowitz, J. W.J. Chem. Phys. 1990, 93, 4172.(65) Haselgrove, C. B.Math. Comput. 1961, 15, 323.(66) Conroy, H.J. Chem. Phys. 1967, 47, 5307.(67) Ellis, D. E.Int. J. Quantum Chem. Symp. 1968, 2, 35.(68) Handy, N. C.; Boys, S. F.Theor. Chim. Acta1973, 31, 195.(69) Boys, S. F.; Bernardi, F.Mol. Phys. 1970, 19, 553.(70) Boys, S. F.; Bucknell, M. G.Mol. Phys. 1974, 28, 759.

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Samuel Francis Boys

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