Douglas Hartree and Early Computations in Quantum Mechanics

Annals Hist Comput (1988) 10:105-l 11 Q American Federation of Information Processing Societies

Douglas Hartree Computations in Mechanics

Quantum

PAUL A. MEDWICK

Douglas Hartree, who was a mathematical physicist at the University of Manchester and the University of Cambridge during the first half of this century, examined the possibilities for numerical solutions of the many-body problem in quantum mechanics which did not permit analytic, closed-form results. In an attempt to surmount the mathematical complexities associated with multielectron atoms, Hat-tree proposed the method of se/f- consistent fields as an approximation scheme which would give numerical forms of atomic wave functions via iterative solution of the Schrlidinger equation. Hartree was quick to recognize the need for automatic computation, both analog and digital, for the practical implementation of the technique. To this end, he investigated the application of analog differential analyzers to the problem before realizing the superiority of digital computation. The question of the first self-consistent field calculations to be run on an electronic digital machine is addressed in an effort to clarif’y misinformation in the existing literature about the use of early computers to perform Hartree computations.

Categories and Subject Descriptors: K.2 [Computing Milieux]: History of Computing-people, software. G. 1.8 [Mathematics of Computing]: Numerical Analysis-partial differential equations. J.2 [Computer Applications]: Physical Sciences and Engineering-physics.

General Terms: Algorithms, Theory, Additional Terms: EDSAC, Hartree, Wave Mechanics, Quantum

Mechanics.

Introduction in the history of the theory that, while indirect observation of such systems through experiment

The introduction of quantum mechanics in the first decades of the 20th century opened up a new realm for thought. and discovery in physics. Pre- viously, physicists could conceptualize and experiment with systems which they could touch, see, or otherwise sense directly. It was discon- certing to many that the new theory required the abandonment of predictions based on direct observation of quantum systems, objects of atomic or subatomic dimensions. It became apparent early

Author’s Address: School of Applied and Engineering Physics, Cornell University, Clark Hall, Ithaca, NY 14853- 2501.

was allowed to some extent, from that time for- ward physicists would have to become more re- liant on pure mathematical analysis to describe and understand the atomic realm.

Many new forms of analytical methods such as Schradinger’s wave mechanics and Heisenberg’s matrix mechanics were developed as part of the overall quantum theory. The one which will be addressed here in relation to Hartree’s work is wave mechanics. This theory gave analytical results describing many quantum systems, most notably hydrogen and other one-electron atoms, in precise mathematical form using known functions. How- ever, due to the complexity of the physics involved,

Annals of the History of Computing, Volume 10, Number 2, 1988 l 105

P. A. Medwick l Early Hartree Calculations

wave mechanics could not, and still cannot, give a complete quantum mechanical description of the structure of multielectron atoms in closed mathematical form. Accurate descriptions of these systems are critical to the understanding of chemis- try, atomic physics, and nuclear physics among other sciences. In the absence of analytical solutions for multielectron atoms, it was recognized that numerical procedures would be necessary to solve the equations of wave mechanics and to describe the structure of these atoms numerically.

This was the realization of Douglas R. Har- tree, a mathematical physicist who had held appointments at the University of Manchester and Cambridge University in Great Britain. Hartree first formulated his thoughts on numerical methods for quantum mechanical calculations begin- ning in 1928 (Eisberg & Resnick 1985, p. 320). He recognized the need for computational power to carry out such analyses and it is this author’s purpose to investigate the early implementation of these techniques on the existing computers of the day. Of particular interest is the question of the first use of an electronic digital machine to do so, since there appears to be a need for clari- fication of the particulars of this event.

The Problem

Wave mechanics is a physical theory which de- scribes quantum systems by a mathematical en- tity known as a wave function, IJJ. The wave function contains all of the relevant physical characteristics of the system such as an atom. The result of any physical observation of the system can, in principle, be calculated probabilistically using the wave function if it is known (either analytically or numerically). As for all physical waves, this function obeys a differential wave equation in three spatial variables (r, 8, $1 and one time variable, t, known as the Schrodinger equation. For a one-electron atom like hydrogen or monovalent ions, the Schriidinger equation can be solved exactly and its results expressed in closed form. This is possible because the equation only involves coordinates for one electron. By sepa- ration of variables, the original partial differential equation can be divided into a set of ordinary differential equations which can be solved. The product of these separated solutions gives +, the solution to the original problem. The reader is advised to consult standard references on quantum mechanics (Schiff 1968) and Hartree’s orig-

inal papers (Hartree 1928a, 1928b) for further background on the subject.

Atoms with several electrons present a diEi- culty in wave mechanics because they electro- magnetically interact with each other as well as the nucleus, Mathematically, this has the effect of making JI a function of each electron’s space coordinates and of increasing the number of partial derivatives in the Schrodinger equation by a factor of three for each additional electron! The result is an equation that is so mathematically complex that it cannot be solved analytically. The lack of a solution for the general many-body problem is well known in physics (March et al. 1967, p. 1). It was Douglas Hartree’s insight to recognize that a numerical solution would be possible, indeed desirable, under the circumstances. Such a numerical solution would involve a large number of calculations; as a result, Hartree soon appreciated the need for computers in mathematical physics and quantum mechanics in particular.

The Solution

Hartree saw that the Schrodinger equation for a multielectron atom could be separated into a set of equations, one for each electron, which could be solved separately. However, this was only possible to do if one neglected the explicit interaction between all of the electrons by making each

Paul A. Medwick recently completed his Bachelor of Science degree in physics from Carnegie Mellon University in Pittsburgh, Pennsylvania. During his undergraduate years, he performed experimental research in high-energy particle physics and

materials science at Carnegie Mellon and has also held research appointments with the Princeton University Plasma Physics Labora tory for four summers. Mr. Medwick’s professional interests include condensed matter and plasma physics and he is current/y pursuing graduate work toward the Ph.D. in these areas at Cornell University.

106 l Annals of the History of Computing, Volume 10, Number 2, 1988

equation dependent only upon the space coordinates of one electron. This seems inconsistent. Above, it was stated that these interactions were too important to ignore and that taking these ef- fects into account explicitly made the Schrtiin- ger equation mathematically unsolvable. Har- tree ventured that each electron could be considered independent of the others if, in each electron’s equation, it was considered to be mov- ing in a net interaction potential due to both the nuclear field and the field of the remaining electron “cloud.” He showed that numerical solutions could be obtained by demanding that the solutions be self-consistent. By this he meant that if one solves the equations with the correct effective potential, the solution obtained must yield the same effective potential if one calculates in reverse. Hence, this came to be known as Har- tree’s Method of the Self-Consistent Field @later 1960, p. 189). Each electron’s equation can be solved by various computational techniques while employing an initial guess for the effective potential. The product of these solutions is the atom’s wave function. Backward calculation using $ permits determination of an effective potential. If not self-consistent with the original guess, one repeats the procedure until self-consistency is achieved. This is an iterative numerical scheme- one which could be implemented using computational devices. Similar algorithms are used widely in other physical problems such as fluid dynamics - a field which was addressed by von Neumann in the 1930s (Goldstine 1972, p. 177). Regarding such calculational techniques, Har- tree remarked in his 1947 inaugural lecture as Plummer Professor of Mathematical Physics at Cambridge University:

Hartree’s first attempt to implement his scheme on a large-scale device was on the state-of-the-art Differential Analyzer first developed by V. Bush at MIT in the early 1930s. He spent a period of time during the summer of 1933 at MIT in order to learn how to apply Bush’s device to calculations for more complex atoms. A 1934 paper de- tailed his work using the MIT machine (Hartree 1934, p. 739). Excited by the prospects for quantum calculations, Hartree returned to the Uni- versity of Manchester and sought to build a prototype differential analyzer to test the ideas which he had gathered from Bush while at MIT. This precursor to the later full-scale device was con- structed by Hartree and Dr. A. Porter (also of Manchester), primarily out of parts from a Mec- cane set (a type of steel construction kit for chil- dren available in Great Britain similar to the “Erector” sets found in the United States). The resulting device not only suggested that a full- scale model would be practical, but itself gave us- able results for calculations not demanding ex- traordinary accuracy. The pleasantly surprised Hartree later remarked in‘ a 1940 paper:

An iterative process such as this, if it converges and not too slowly, is very suitable for mecha- nization, and there seems to be a field for mathematical research here, on the theory of iterative methods in which the result is a whole function, not a single number. (Hartree 1947, p. 35)

Hartree was later to write concerning the failure of means to do quantum mechanical calculations that “in order to have wave functions in forms that are practicable to calculate, to tabulate, and to use so as to evaluate other atomic properties, it is not only desirable but also quite necessary to have recourse to approximations” (Hartree 1957, p. 17).

This [the construction of the Meccano prototype] was originally done more for amusement than with any serious purpose, but the first results were successful beyond my expectations, and suggested that it would be practicable to build such a model to do serious work on problems for which high accuracy was not required in the results. (Hartree 1940, p. 323)

During the early days of the implementation

Thoughts then turned to building a full-scale version out of conventional hardware and the Metropolitan-Vickers Electrical Company, Ltd. was contracted for its construction (Hartree 1940, p. 319). Hartree and his associates at the Uni- versity of Manchester were soon to have their own



of the Hartree method, the computations were carried out using a desk calculator (Herman & Skillman 1963, p. vii). This formidable task was primarily done by Hartree and his father, Wil- liam Hartree, to whom Douglas was to dedicate his 1957 book on atomic computations “in recol- lection of our happy cooperation in work on the calculation of atomic structures.” (Hartree 1957, p. v) One need only consider the large number of operations involved in order to appreciate the need for automatic computers in quantum mechanics. Hartree and his colleagues recognized this point as critical to the practical application of the Har- tree method.

P. A. Medwick l Early Harlree Calculations

mechanical differential analyzer completed in 1935 (Hartree 1949, p. 8). A similar one was also in- stalled at Cambridge University by Maurice Wilkes and his colleagues at the Mathematical Laboratory there around 1939 (Bratt et al. 1939, p, 485). It was during this time before World War II that the Manchester machine was “applied to calculations concerning the structure of atoms.” (Hartree 1949, p. 25). Hartree further noted that during this period his group was examining “an extension of the field of application of the in- strument to partial differential equations” in a generalized sense (Hartree 1949, p. 26). Mean- while, the Cambridge group was also involved with self-consistent field theory:

. . . shortly after it had been built, the machine was set up to solve the wave equation for the hydrogen atom; this involves four integrators and an input table. The energy of the ground state . * . when compared with the theoretical value was found to he correct to within one part in 500. (Bratt et al. 1939, p. 493)

Despite the success achieved with differential analyzers for doing atomic modeling, the analog nature of the device was to make vast numbers of Hartree calculations impractical. This is largely due to the inherent limitation of computational accuracy determined by measurement precision at the device input and in the mechanical procedures involved in the calculation (Hartree 1947, p. 8). Also, the repetitive nature of the method made analysis using analog integrators ineffi- cient because the process called for human op- eration at the input on each iteration (Hartree 1947, p. 9).

The answer to the problems of effective implementation of Hartree calculations was to come with the advent of electronic digital computing. The hallmark of this new technology was the completion of the ENIAC in 1945. During the summer of that year, Hartree visited the Moore School of Electrical Engineering at the Univer- sity of Pennsylvania to become acquainted with electronic digital systems (Goldstine 1972, p. 219). He became very interested in ENIAC and was to continue his work on learning how to use it through 1947. While there, Hartree used ENLK to solve systems of three simultaneous, nonlinear differential equations resulting from the theory of the laminar boundary layer in a compressible fluid (Hartree 1946, p, 505). This is relevant because it laid the groundwork for atomic calcula-

tions on other digital machines since the fluid problems were of a similar mathematical and iterative nature. It seems that none of the atomic calculations were performed on ENIAC, perhaps due to the fact that it was primarily devoted to military computations.

From his education at the Moore School, Har- tree became convinced that digital computation was superior to analog for scientific applications. Hartree realized the importance of electronic digital computers, particularly suited to his uses for the “evaluation of solutions of partial differential equations in three variables . . .” (Hartree 1946, p. 505). The case for digital was that while analog systems can handle continuous variables, their accuracy is limited by measurement. In contrast, digital machines do numerical integration where any amount of accuracy is theoreti- cally possible (Hartree 1947, p. 9). In addition, Hartree favored electronic machines because of their speed for scientific applications. Perhaps the greatest thing to appreciate about the electronic digital realm was the fact that:

. . . the relief from the labour of carrying out the calculations is very marked, and is difficult to appreciate fully before one has experienced it. In the work with which I was concerned for ex- ample, it was necessary to evaluate a number of sets of solutions of differential equations and to combine them in a rather complicated way. In this case almost all that had to be done, as far as the solution of the differential equations was concerned, was to work out the machine set-up for a single interval of the integration. Then all the integrations which would ever he wanted were as good as done; I could leave entirely to the machine the work of doing them and give my whole attention to organizing the way in which the SO- lutions so obtained were to he used. (Hartree 1947, p. 211

Thus, by 1947 or so, Hartree was convinced of the appropriateness of electronic digital computers for scientific calculations, particularly those relating to atomic structure. Such calculations were soon to be performed using the new ele- tronic digital machines being developed at the time.

First Electronic Digital Implementation

There seems to be some question as to which of these new digital machines was the first to run



a self-consistent field model. One might assume Cambridge, recalls that one of the first substan- prematurely that the Manchester Mark I was the tial uses of EDSAC was for atomic and molecular first to do so since Hartree had been affiliated structure calculations performed by Hartree and with the University of Manchestir up until 1947, S.F. Boys shortly after EDSAC first went on line a year before the device became operational (Wilkes 1985, pp. X0-152). It is clear that EDSAC (Lavington 1980, p. 36). Manchester Mark I em- was used extensively for self-consistent field and ployed tubes as switching elements and a com- other scientific calculations. In a 1950 paper, bination of Williams cathode ray tube random Wilkes relates that 't. . . EDSAC has also been ap- access memory with supplementary magnetic plied to problems arising in geophysics, wave me- drum memory (Lavington 1980, p. 37). However, chanics, statistics, electron optics and electrical it does not appear to be likely that the Manches- engineering.” (Wilkes 1950, p. 943). At this time, ter machine can be credited with the first Har- extensive programming had already been done tree calculations because Hartree, the father of on EDSAC bpf3fOrm nUmeriCa intcQatiOnfort,he the technique, had in fact left Manchester by 1947 solution of differential equations (Gill 1951, p. 96). to accept his appointment at Cambridge and be- In his 1957 book summarizing his work on atomic gan working with the Mathematical Laboratory structures to date, Hartree indicates that most of there. There is also an additional reason. In one his data came from EDSAC calculations pro- of their books on quantum physics, Professors grammed by J. M. Watt (Hartree 1957, p. 166). Robert Resnick of Rensselaer Polytechnic Insti- Professor Wilkes has commented that, to the tute and Robert Eisberg of the University of Cal- best of his knowledge, EDSAC was indeed the first ifornia, Santa Barbara, state that “one of the first electronic digital computer to perform self-con- large digital computers was employed to perform sistent field calculations Wilkes, 11 August 1987, Hartree calculations. It used relays as switching private communication). According to Wilkes, elements, instead of the transistors of modern Hartree had “never used a relay machine” to do computers.” (Eisberg & Resnick 1985, p. 322). In such work and that he had already left Manches- a conversation with this author, Professor Res- ter for Cambridge before either Mark I or EDSAC nick could not comment on the validity of this had become operational (Wilkes, 11 August 1987, statement (Resnick, 27 October 1986, private private communication). In fact, Wilkes states that communication). Professor Eisberg related that the bulk of credit for Hartree calculations on ED- he had been told this by “word of mouth” by some SAC should be given to S. F. Boys since Hartree members of the staff of Cambridge University was only peripherially involved, at best, with the during one of his stays there and that the ma- work at this point having moved on to address chine was one of Cambridge’s own (Eisberg, 27 other problems of interest (Wilkes, 11 August October 1986, private communication). It seems 1987, private communication). Boys, a pupil of J. safe to rule out the Manchester devices as the E. Lennard-Jones, was doing work in molecular premier machines for Hartree calculations, and physics using an augmented type of calculation more logical, considering that Hartree had moved called Hartree-Fock theory which accounts for to Cambridge, that one of the Cambridge sys- parities of wave functions (Eisberg & Resnick tans should be selected. This appears to have been 1985, p. 322). The strength of Wilkes’ testimony the case according to Eisberg. However, the sci- leaves little doubt that the EDSAC at Cambridge entific journals and documents that this author was indeed the first electronic digital computer examined cannot attest to the use of an electro- to perform self-consistent field calculations, al- mechanical relay machine at Cambridge or else- though some modifications in the physics in- where to do Hartree calculations. It seems that valved slightly changed the flavor from that of something has been lost in the translation be- the original Hartree tradition. It is hoped that tween the time of the original electronic com- this finding clarifies any misleading or incom- putations and the present, which has confused plete information in the literature on this sub- vacuum tube circuitry with relays. ject.

The first verifiable use of an electronic digital The rest of the 1950s saw the adoption of Har- computer for the implementation of the Hartree tree calculations on other kinds of computers. algorithm was on the Cambridge EDSAC which Notable among these efforts is the work of Piper became operational in 1949 (Lavington 1980, p. and Prener of the General Electric Research Lab- 33). Maurice Wilkes, director of the Mathemati- oratory who did their calculations on an IBM 650 cal Laboratory (now Computer Laboratory) at in 1955 (Piper & Prener 1955, p. 1250). In 1959,


P, A. Medwick l Early tiartree Calculations

a series of Hartree computations was done on the Rayner, had held positions as Superintendent Of

MIT Whirlwind computer developed by J. W. the Electricity Division of the National Physical For-rester (Watson 1959, p. 15). It is also known Laboratory in Teddington and as mayor of that similar work was being done at the Oxford Cambridge. His father, William Hartree, was an _ University Computing Laboratory during the mid- engineering professor at the University of to-late 1950s (Watson 1959, p. 15). A rather ex- Cambridge. It was there that Douglas was to tensive compilation of the results of the 1960-61 enroll as an undergraduate in 7916. His Hartree work by F. Herman and S. Skillman on education was interrupted later that year by a an IBM 7090 at RCA Labs in Princeton, New Jer- three-year tour of military service from 1916 to sey was published in 1963 (Herman & Skillman 1919. By 1919, Hartree had returned to 1963). This technique took little time to become Cambridge to finish his undergraduate work popular throughout the scientific community. The there. Afterward, he obtained a feiiowship to use of the self-consistent field algorithm has in study at St. John’s Coiiege in 7924. Hartree recent years been applied to the calculation of completed his graduate work under R. H. Fowier nuclear structure and continues to be a valuable and was subsequently awarded the Ph.D. in tool of wave mechanics (Eisberg & Resnick 1985, 1926. He stayed on at Cambridge as a research p. 320). fellow until 1929 after spending a year at the

University of Copenhagen from 7928 to 1929. if appears that around this time he became

Conclusion interested in computation as applied to the many- body problem in quantum mechanics, a topic

It is clear that Hartree’s contributions were some which engaged Hat-tree and his associates for of the most important in quantum mechanics and much of his career. in the evolution of scientific computation. It can After accepting an appointment as Beyer be seen how the need for computational power was Professor of Applied Mathematics at the a natural outgrowth of the complexity of the University of Manchester in 1929, Hartree problems to be solved in wave mechanics. This is devoted much of his time in the 1930s to Se/f-

a theme that has recurred several times through- consistent field calculations as implemented on out the early history of computing. In most cases, differential analyzers. in 1932, Hartree received the development of computers was initiated in the prestigious honor of being selected a Feiiow order to solve certain kinds of scientific and en- of the Royal Society of London. He continued t0 gineering problems. While Hartree did not him- hold the applied mathematics chair at self build new devices, his important contribu- Manchester until 1937 when he moved to the tion was the development and implementation of professorship of theoretical physics (also at techniques like the self-consistent field theory to Manchester). successfully solve scientific problems using early The turning point in Hartree’s thoughts on seif- computers. Hartree promoted automatic compu- consistent fields came during the summer of tation because the types of problems that he 7945 when he was introduced to the recently wanted to work on could not be tackled without completed ENIAC and the advantages of digital an increased amount of computational power. His computation. Embracing the digital realm, Hartree work, therefore, helped to catalyze the develop- went on to study numerical solutions of partial ment of more powerful automatic computers like differential equations related to hydrodynamics the EDSAC and its descendents. Indeed, Hartree’s and self-consistent fields. By this time, he had work was unique because it “represented the first accepted an appointment as Piummer Professor concerted attack on extremely complex physical of Mathematical Physics at the University of and engineering phenomena by computational Cambridge in 1946. it was at Cambridge that rather than experimental [or purely analytical] Hartree, along with Maurice Wilkes and his means.” (Goldstine 1972, p. 101). Present-day associates, became involved with the physics and numerical analysis are indebted to development of the EDSAC at the Mathematical Hartree’s pioneering work. Laboratory (later Computer Laboratory) there. He

continued his interest in many-body calculations on the EDSAC and later machines while also

Douglas Rayner Hat-tree was born in Cambridge, addressing other problems such as the England on 27 March 1897. His mother, Eva propagation of radio waves in magneto-ionic


media. Hat-tree continued on at the University until his death in Cambridge on 12 February 1958.

References

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Eisberg, R., and R. Resnick. 1985. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. New York, John Wiley & Sons, Inc., 2nd ed.

Gill, S. 1951. “A Process for the Step-by-Step Integra- tion of Differential Equations in an Automatic Dig- ital Computing Machine.” Proceedings of the Cam- bridge Philosophical Society, 47, pp. 96-108.

Gillispie, C. C. (ed.). 1972. Dictionary of Scientific Bi- ography, Vol. VI, New York, Charles Scribner’s Sons, pp. 147-148.

bridge, England, Cambridge University Press (re- printed in volume VI of the Charles Babbage Insti- tute Reprint Series for the History of Computing, MIT Press and Tomash Publishing, 1984).

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Hartree, D. R. 1957. The Calculation of Atomic Struc- tures. New York, John Wiley & Sons, Inc.

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Hartree, D. R. 1928a. “The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I-Theory and Methods.” Proceedings of the Cambridge Phil- osophical Society, 24, pp. 89-110.

Hartree, D. R. 192813. “The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part II-Some Results and Discussion.” Proceedings of the Cam- bridge Philosophical Society, 24, pp. 111-132.

Hartree, D. R. 1934. “Approximate Wave Functions and Atomic Field for Mercury.” Physical Review, 46, pp. 738-743.

March, N., W. Young, and S. Sampanthar. 1967. The Many-Body Problem in Quantum Mechanics. Cam- bridge, England, Cambridge University Press.

Mehra, J., and H. Rechenberg. 1982. The Historical Development of Quantum Theory, Vol. 4. New York, Springer-Verlag, p. 66.

Piper, W. W., and J. S. Prener. 1955. “Hartree-Fock Wave Functions for Mn’4” Physical Review, 100, p. 1250.

Schiff, L. 1968. Quantum Mechanics. New York, McGraw-Hill Publishing Company, Inc., 3rd ed.

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Hartree, D. R, 1940. “The Bush Differential Analyser and Its Applications.” Nature, 146, pp. 319-323.

Hartree, D. R. 1946. “The ENIAC: An Electronic Com- puting Machine.” Nature, 158, pp. 500-506.

Hartree, D. R. 1947. Calculating Machines. Cam-

Watson, R. E. 1959. Iron Series Hartree-Fock Calcu- lations. Cambridge, MA, MIT Solid State and Mo- lecular Theory Group.

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Wilkes, M. V. 1985. Memoirs of a Computer Pioneer. Cambridge, MA, MIT Press.

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P. A. Medwick l Early Hat-tree Calculations

Douglas Hartree and Early Computations in Quantum Mechanics

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