International Journal of Quantum Chemistry,

© 1999 John Wiley & Sons, Inc.

University of Nottingham,

Nottingham, NG7 2RD United Kingdom

Received 25 November 1998; accepted 16 February 1999

whose love and care have always inspired and supported me.

Most of my primary schooling was in Mountpottinger Public Elementary School, in the east side of Belfast, a school now closed. It had a high reputation for copperplate writing and a very strict discipline. While affected permanently by both of these, my most exciting memory of this period was our introduction to Euclid whose first propositions we had to learn by rote.

From there I went to Regent House School, in Newtownards. There I came under the influence of many good teachers and found myself attracted to mathematics and science, though I was also learning both Latin and Greek. I then had one year in Methodist College, Belfast, to prepare for the entrance scholarships to Queen’s University.

When I entered Queen’s University in 1943, I realized quickly that science was the kind of career I most wanted. The war was still dominating everything, so career planning was in obeyance. Northern Ireland had no conscription, but I joined the Home Guard at a time when an invasion seemed likely. During the undergraduate course, I became increasingly attracted by Prof. P. P. Ewald. He was a good lecturer, not an orator but a stimulating and perceptive innovator. His philosophy of bringing together mathematical techniques and physical situations in order to bring insight into both has deeply influenced me. I also owe much to Prof. K. G. Emeléus, in Physics, and to Prof. H. R. Pitt and Dr. Mostyn, in Mathematics. I graduated, B. Sc., in 1946 with Firsts in both pure mathematics and mathematical physics.

I had begun to read Dirac’s Quantum Mechanics, stimulated by physics lectures on atomic spectra and mathematics courses on Hilbert spaces, but had no proper exposure to quantum mechanics so I remained for another year to complete the degree course in experimental physics, from which I gained another First in 1947. During this year I also started research, with my good friend Herby Deas, on a little investigation suggested by Ewald into the vibrational spectra of coronene. This involved us in learning group representation theory and applying it to the problem. I have remained interested in group theory as a beautiful and useful technique ever since. This interest in coronene was mentioned when I applied to Cambridge to become a research student and was the reason I was assigned Prof. Lennard-Jones as my supervisor. From such small beginnings was I pointed to quantum chemistry.

In 1947, Cambridge was just beginning to recover from the effects of the war. Sir John Lennard-Jones had had a distinguished war career in the Ministry of Supply. I and two other Cambridge students, were his first group of research students on his return. He was anxious to revive his interest in quantum chemistry [173] and move on from the statistical mechanics of liquids, the topic which had dominated his immediate prewar research. He decided to formulate molecular orbital theory rigorously and began by deriving the differential equations for the best orbitals. I found this approach very attractive because the contemporary theories of the subject seemed to me most vague and unsatisfactory in their foundations.

After three years, in 1950, I had the good fortune to be appointed to the post of Assistant in Research in Theoretical Chemistry. This gave me my first experience of university lecturing. Then, in 1953, I succeeded in the competition, open to all young graduates in the university, to be elected to a fellowship in my own college of St. John’s.

About this time, the university was building its first electronic computer, the EDSAC. Frank Boys, who had joined the Theoretical Chemistry Department in 1948, was one of the very first to use it, and his work on molecules would have been impossible without it [175]. I attended the first M.Sc. course on programming and was soon writing my own suite of programs.

On reflection, I can see that the topics on which I started work in Cambridge have remained with me all my life and have surfaced again from time to time as relevant new ideas matured and developed. I still try to introduce new mathematical ideas into the subject. I have continued to use group theory. I have learned much from developments in solid-state physics. Conjugated hydrocarbons haunt me! The biological applications of our work remain a vital, if often distant, goal. Nevertheless, I have a marked tendency to pursue any side issues that present themselves attractively to me. This has led me into many interesting investigations at the cost of a certain incoherence. The many not-otherwise-cited articles in the publication list give proof of this.

My interest in solids led to the offer of a Lecturership in Mathematics in Imperial College, where the solid-state group was flourishing under Prof. H. Jones, E. P. Wolfarth, and others. Because it covers such a wide area, London is a very different university from Cambridge and personal relationships are much harder to develop. Although I enjoyed being part of the solid-state group, I did not feel sufficiently attracted to switch my research entirely to that area.

In 1957, I received an invitation from Per-Olov Löwdin to spend a year in Uppsala, Sweden, and the department gave me permission to accept. On my return to London, I was given the chance to set up my own research group. Initially, Terry Amos joined me and later Don Rees and Arturo Hardisson. We had to take time learning how to exploit the new computer of London University.

The invitation from Per Löwdin to spend a year in Uppsala came at a good time for me. It gave me the opportunity to think over my future plans and to decide how to proceed. It was also a chance to learn fully the ideas and approach of that group. In the group’s determination to be rigorous and innovative, its members were close to my goals, but there were some important differences, such as my continued interest in semi-empirical theories as the best bridge between the then unattainable accurate solutions and the ad hoc regularities often found during the analysis of experimental results.

An important event during this year was the summer school held in Vålådalen, in the north of Sweden. This was the first of these events and set the pattern for many since then. We had many lectures and talks and, during the final week, a number of very distinguished visitors joined us to discuss and debate aspects of the subject [21]. I can remember illuminating sessions with Mulliken and Pauling, as well as with Löwdin, Preuss, Jensen, McWeeny, Matsen, Pauncz, and Shull, and several eminent experimentalists. It was a good illustration of the friendly nature of our colleagues and of the real interaction between ideas that they encouraged. One outcome for me was the letter [22] that Harry Shull and I wrote asking for specific names to be given to the atomic units of energy and length and urging that these should be the hartree and the bohr.

My appointment to the chair of Applied Mathematics in Nottingham, in 1962, opened up a new range of possibilities both for me and for the university. Nottingham had just split off a Theoretical Mechanics Department from its Mathematics Department to take over the teaching of mathematics to engineers, and this department had major research interests in classical mechanics so that the Mathematics Department could allow its applied section to concentrate on quantum theory. This came at a time when university expansion was taking place, so that new staff were being appointed each year and their specialisms could be determined. The Mathematics Department became and remains one whose applied section is predominately concerned with quantum theory.

I brought with me my Imperial College group, and Raphy Levine joined us as a research student. We had the disadvantage that, at that time, Nottingham had no computer so our work became theoretical rather than computational.

In 1970, we held a Quantum Theory Conference in Nottingham. It was designed, in particular, to help young researchers. At its conclusion, it was decided to continue the event, and Coulson and I offered to host the meeting each year in turn. The series continued until recently but, sad to say, has now come to a halt.

It was a great encouragement to me to be elected, in 1974, to the International Academy of Quantum Molecular Science. I also appreciated being elected to serve, from 1977 to 1980, on the Executive Committee of the International Society for Quantum Biology.

Inevitably, I was drawn into university administration and, from 1972 to 1975, I served as Dean of Pure Science. This took considerable time away from research. I represented the university on the Derbyshire Education Committee from 1966 to 1974. I also served on three national committees, on the Computer Consultative Committee of the Department of Education and Science (1972—1976), as Chairman of the Joint Mathematical Council (1979—1981), and on the Mathematics Committee of the Council for National Academic Awards (1980—1983). The validation activity of the latter led to my first visit to Hong Kong in 1982.

From my first term in Nottingham, I have been involved with mathematical education in the schools. The department, then, ran courses for teachers in the "new math," which was being developed and introduced into the school curricula. Professor Halberstam (my pure mathematics colleague) and I thought through our contribution to this work and decided, eventually, that we should concentrate on in-service courses rather than formal vacation courses. For this purpose we lobbied our then vice chancellor, the late Lord Dainton, for support, and he assisted with the negotiations to obtain finance from the Shell Company, in 1967, to enable us to appoint several staff members to help us in this work. The Shell Centre has continued its mathematical education work ever since. The Shell finance has now finished but contracts from other sources, many of them overseas, support research and the dissemination of "best practice" worldwide. The Shell Centre aims to digest the best of the current research on mathematical education and make it available to teachers. It designs teaching material, which is carefully tested in schools, to ensure that it is robust enough to be used by the average teacher. More recently, it is working on ideas of balanced assessment and on the theory of how to disseminate good practice.

In 1982, the universities in England first came under severe external pressure to cut staff. My department was threatened with the loss of several people because the staffing formula said we were over staffed. After much thought, I decided to take early retirement and avoid having to select others for removal. The arrangement allowed me to remain, under reduced teaching load, for one year. I became an Emeritus Professor and as such still retain a few rights in the university.

One visitor to our group, as a post-doc for a year, was Ralph Christoffersen [44]. He, then, took up a professorship in the Chemistry Department of Kansas University at Lawrence. A little later, in 1969, I received an invitation to spend a semester there with his group. I found this a very stimulating experience, partly because of his activity and partly because the department proved to be a friendly, helpful place with an orientation toward both teaching and research. I delivered an advanced course for research students describing and commenting on the principal methods then in use in our field. I had hopes of publishing this, but the result was, inevitably, very similar to the book of McWeeny and Sutcliffe, which appeared about that time, so I abandoned the manuscript. The consequence of my visit was that it was arranged that I would return for a month or two each year, perhaps to teach a course but always to coordinate our research. We kept up this Kansas connection for 11 years despite problems in financing the travel and subsistence. The contact continued until Christoffersen left Kansas University.

During my transition year in 1982, I received several overseas job offers, but the one that interested me most was from Kyoto University. They were setting up a new department devoted to molecular engineering and wanted me to head the quantum molecular science section. This was an exciting opportunity to be in at the start of something new as well as an opportunity to experience a different civilization and culture. I accepted the post and then, almost immediately, experienced the impact of red tape, which seems to be an essential part of any Japanese official activity [171]. I am grateful to my Japanese colleagues for their help through this difficulty.

My time in Kyoto was the most productive time of my life, both as regards producing research and as regards my learning about Japan. Since Kyoto is the former capital, it retains many traces of its past and has been allowed to preserve them. My colleagues helped me to experience the depth of Japanese thinking and feeling. I tried to reciprocate by helping them toward a more idiomatic use of English [170]. I was helped by many good research students whose commitment to the work was often remarkable [165].

I was greatly moved to receive from Kyoto University, in 1989, their first honorary degree of D. Eng.

I have enjoyed two periods of work with Gerd Diercksen in Munich. His programming project is an ambitious one that aims to provide a complete molecular calculation package, OpenMol, which is user-friendly and capable of accepting many add-ons. It uses object-oriented programs and has a expert program embodied to guide the user through its options. We collaborated to write a semitechnical account of it [131].

An immediate consequence of the Lennard-Jones formulation of molecular orbital theory was his realization that the orbitals, when doubly occupied, could be transformed by a unitary transformation without any change of wave function. As John Pople and I worked through the implications of this, we realized that the chemical consequences were very important. For the first time, we had a rigorous definition of a chemical bond as a localized equivalent orbital [3] and could develop its properties. The term "equivalent" combined the ideas that these were unitary transforms of the canonical molecular orbitals and that, when symmetry was present, they could be generated from one another by a permutation. I took special pleasure in introducing [2] the idea of a permutation representation of a group into the subject. "Localized" meant that the total self-energy of the orbitals was a maximum [20]. Because it is independent of basis set, this definition of a bond is still correct. It enables the chemist to know where each electron is and what it is doing. The extra terms, added to the wave function in order to achieve high accuracy, can be described as adding correlation inside each of the localized orbitals and between them. The advantage of localization is that the bonds become more nearly determined by their terminal atoms alone and their energies should then become constant. Since the major error in the theory is its neglect of electron correlation within the localized orbitals, there is reasonable ground for believing that the corrected bond energies. would also be constant.

Some time later, Jane Hylton and I carried out a regression analysis of the Hartree—Fock energies of a long series of molecules [73] and showed that they could be turned into a simple additive set of atomic and bond energies. This is positive evidence in favor of our ideas of bonds in molecules. These bond energies even showed the effect of eLectronegativity. From the results of calculation we could recover concepts of chemical importance.

Unfortunately, in 1950, there was no hope of solving the molecular orbital differential equations so our discussions about the orbital structure of molecules, except in as far as group theory determined them, were rather approximate. I tried to remedy this by projecting the molecular orbital equations into a finite-dimensional function space [4] and so derived the matrix equations from which approximate orbitals could be found systematically. This theory ran into controversy immediately because it contradicted the accepted theory at that time, which assumed linear orbital equations, whereas the proper equations were cubic [121]. I was warned to publish with care and to avoid trouble!

Using ideas from alternant matrix theory, Terry Amos and I later developed [28] the idea of corresponding orbitals, first suggested by Löwdin, for treating unpaired systems, using unrestricted molecular orbital theory, and calculated some examples. Since these wave functions were not eigenfunctions of spin and since eigenfunctions of spin were too complicated for our computer, we had to introduce a compromise wave function in which, by applying an annihilator, the worst of the spin contamination was removed.

My most interesting achievement in Cambridge was my semiempirical theory of ionization potentials. This was a correctly defined theory [4, 7, 8] but, simply because we could not then calculate them accurately, its defining integrals over equivalent orbitals became parameters to be determined from experiment. It assumed that Koopmans’ theorem was exact and, although this is not strictly true, it is a reasonable approximation because its errors are in opposite directions and so tend to cancel. At the time, accurate experimental ionization potentials were becoming available so the theory had a good interaction with experiment and helped to identify the ionic states involved. It was very successful for a certain class of molecule [11, 13] but failed for others.

Later work [56], with Keith Gregson, proved that this failure was due to the neglect of the strong intramolecular electric fields within some ions. When these are included, along with the internal polarization, which is a part of the correlation energy, the agreement with experiment is again impressive. Classical ideas of the electric field and its effects on charge clouds are still sufficient for this purpose. I learned from this the lesson that the electrons in a neutral molecule can be described as localized and are polarized by any electric field they may experience, but that, on ionization, one electron is no longer paired and is subject to quantum delocalization.

I realized in 1970 [58] that the ideal tool for the discussion of ionization potentials was the one-electron Green’s function, but it was not until 1988 that a student of mine, Y. Mizukami, actually did such a calculation [112].

I adapted my semi-empirical theory of saturated molecules to conjugated hydrocarbons [5, 6] by introducing what I called the "standard excited state." In this state all the pi electrons are given the same spin and so require twice as many orbitals. These orbitals can be localized and give one pi function for each conjugating atom. For the ground state, the most stable set of molecular orbitals are then doubly occupied. There is an implicit appeal to the Koopmans result that orbitals are little changed, whether singly or doubly occupied. This procedure enabled me to justify much of what was called Hückel theory in rigorous terms, showing [14] that it approximated closely to the

I defined alternant matrices in general [16], using partitioning, as those with the form

where 0 is a zero matrix and B may be rectangular, and derived the properties of their eigenvectors and eigenvalues. In effect, the matrix B is diagonalized by two unitary matrices.

This result allowed me to give a closed form [16] for the bond orders of alternant hydrocarbons in Hückel theory. In another work [18], I showed how the molecular orbital equations could be solved in a computer and, as a demonstration, listed the bond orders for 25 aromatic hydrocarbons. Since bond orders for hydrocarbons were then the most successful part of molecular orbital theory, because they correlated very well with crystallographic results on molecular dimensions, these were a source of theoretical information for experimentalists. They also gave us important guidance about the relations between molecular shape and molecular properties. The idea of embedding a molecular fragment within a molecule as a means of predicting the degeneracy of certain molecular eigenvalues is one example of these relations. In particular, the calculations on benzpyrene, a potent carcinogen, showed that it had an unusually large bond order in what the Pullmans called its K region. The study also included the results of a self-consistent calculation, which included the extra nonlinear terms given by the theory. The calculation was very similar to those performed under what was later known as PPP (Pariser, Parr, Pople) theory.

The structure and interactions of the tricycloquinazulenes was another topic of biological importance that we considered, later, using this pi-electron formalism [59, 74]. These molecules have some N atoms in their hexagonal rings and include some of the most potent carcinogens known. Our pharmacists were anxious to have information about them and their activity. William Rodwell and I produced approximate electronic structures for these molecules and examined their interactions with other species. The conclusion was reached, on energy grounds, that an intercalation of these into a DNA chain was possible and could account for their cancerous interruption of biological function.

In Kyoto, I started to investigate conjugated hydrocarbons large enough to have a hole surrounded by hexagon rings [110]. This requires the introduction of the idea of the genus of a molecule - the number of its holes. In this connection, I introduced the concept of the bual [154] of a graph, which is a modified form of its graphical dual. By repeatedly taking the bual, a graph can be stripped down, from the outside, in onion fashion. These molecules are now called coronoids and have been well-studied. I also looked [114] at the connection with the superdelocalizability, an index measuring molecular reactivity.

Another investigation begun in Kyoto was into the forms, in Hückel theory, for the energy of infinite periodic molecular systems. My student, S. Arimoto, proved that the total energy tends to a linear function of the number of repeating units and that we could calculate the leading term in this asymptotic expansion using an integral [123]. We then deduced a number of exact asymptotic expansions of this energy for different series of molecules [130]. The same argument applies to the total zero-point energy of periodic molecular systems and, with some changes, to the heats of formation of polymers and solids.

On my return to Nottingham, I took up again my continuing problem about the total pi energy in what is now called the graph theory of conjugated hydrocarbons. My empirical expression for this [89], given years before in 1981, was based on an analysis of many actual calculations and had the form

where n is the number of C atoms, N the number of CC links, and K the algebraic number of Kekulé structures. This can be backed up with some theory [103, 120, 122, 127], using the moments of eigenvalues to define other graphical invariants and expressing the energy in terms of these, but this fails to explain the third term in the equation. Although, working with Ivan Gutman [124, 125, 132] and considering the eigenvalue spectral densities, we have made some progress in understanding this problem and showing the limitations of this expression, it remains a tantalizing result.

Nowadays, this treatment of hydrocarbons is important both as a means of understanding this rich class of molecules and also as a respected branch of graph theory. Its relationship to molecular orbital or PPP theory tends to be ignored.

As an offshoot of my interest in localized equivalent orbitals, I first became interested in the band theory of solids and in diamond in particular [9, 10, 12]. The interaction of bonds in a hydrocarbon and in a solid have much in common. Treating the solid as a set of localized orbitals, which have a strong local interaction but a negligible more distant interaction, leads immediately to an explicit form for the energies of the various bands [19] and the parameters in this, which measure these interactions, can be fitted well using other calculations of the bands.

While in Sweden, I developed [33] a semiclassical version of the theory of excitation waves in a molecular crystal. This showed that, in the crystal, the transition dipoles of different molecules interact in the same long-range way as do the molecular dipoles and must be treated with equal respect.

During a short visit which he paid to the division in Kyoto, I worked with Tom Claxton on the spin density at the nucleus of trapped muonium in a solid using various simple models of the solid [107]. Later, with my student Y. Mizukami, we took up the problem of muonium trapped at a specific site in diamond using a realistic form for the electrostatic potential acting on the muonium and obtained a reasonable agreement with experiment [108].

With the development of computing center facilities in Nottingham our group took on, in 1969, a programming project to produce a package, called OPIT, which used optimized floating spherical Gaussians (FSGO) in a molecular orbital calculation [63]. This used ALGOL and so made extensive use of procedures. In effect, it was an early. form of object-oriented programming. Many people contributed to this project. We had support from Prof. G. Cook at Leeds and Prof. R. A. Sack at Manchester. Among those who spent many hours on it here were Drs. Brailsford, Packer, Ford, Hylton, Schnuelle, and Tait. The purpose was to model molecular calculations by using a small set of spherical Gaussians, well optimized in position and scale. This could not claim high accuracy but it could, and often did, mimic what much more elaborate calculations were doing. As such, it served a didactic purpose [64, 72]. It allowed us to discuss molecular shapes, dipole moments and even some excited states. We had hopes that we could find a reliable correlation between its results and those of more accurate calculations.

Some of our learning was unfortunate. By having exclusive use of the local computer for a brief period, we showed that it had an erratic least significant bit in its arithmetic unit which prevented its results from being exactly reproducible. We also learned that a strict optimization tests many aspects of a program and often highlights minor weaknesses in the numerical analysis. The sad consequence was that many of our results were not fit to publish. This was an important experiment and taught us much, even though, eventually, it became clear that our hypothesis, that we could simply scale up our energies to reproduce the accurate results, although broadly correct, was not sufficiently accurate to be useful.

One of the important and successful investigations, made at this time using OPIT, was into the structure of phenethylamine and amphetamine. Colin Miller, Gary Schnuelle, and I showed [76] that these, when neutral molecules, preferred an extended configuration but that their protonated forms had the side chain folded over the benzene ring. This folding helps to protect the charge during the motion of the molecule through the liquid and so could have important implications for the biological action of these molecules as neurotransmitters. We also used point-charge models of water to investigate the shifts of their spectra in solution [82].

My first study [1], in 1950, used perturbation theory to investigate the accuracy of Koopmans’ theorem and showed that a molecular orbital changed very little whether it was doubly or singly occupied. We have been using forms of perturbation theory ever since. In London, Arturo Hardisson and I performed various computer exercises on the implementation of self-consistent molecular orbital theory including applications to ring currents, chemical shifts, and g factors [32, 36, 37]. These were the first practical applications of coupled sell-consistent pertutbation theory. The results were of immediate interest to the experimentalists, specially L. M. Jackman, with whom we collaborated [30, 34]. In Nottingham, Terry Amos and I again employed coupled self-consistent perturbation theory within the PPP context. The theory of electrical polarizabiity of conjugated hydrocarbons [46, 47], which we developed, is one example.

I found the theory of the transcorrelated wave function, by Boys and Handy in 1969, an attractive challenge to mathematical principles [57]. With Edward Armour, I tried to explain some of its success by investigating nonsymmetric Hamiltonians. This gave some insight but failed to show how the theory could be put on a firmer foundation. I later developed an infinite-order perturbation theory [83]. By writing the perturbed wave function in the product form

If necessary, this can be solved using a variational principle. The perturbing potential V includes a nonlinear feedback term:

Because of the change in the V potential, the equations are nonlinear and must be solved self-consistently. This formalism is powerful enough to deal successfully with some notorious examples where the conventional theory fails. I related this theory to the transcorrelated theory and showed how this form of perturbation theory leads to all the equations and the major ideas of the theory [80]. Cohn Miller and I applied the ideas to the He atom with a simple correlated wave function [85] and showed that these equations did agree substantially with the Boys-Handy ones and led to a good solution.

The form of this perturbed wave function as a product shows up the limitation of this theory. Since e

Ralph Christoffersen, in Kansas, was intent on using Gaussian models to investigate biological-sized molecules and to probe typical biological behavior such as photosynthesis and electron transport. Since these did not usually require great accuracy but did need to show the various possible phenomena, it required a different attitude to molecular calculation [71]. We collaborated in devising methods of including solvation effects and began to develop two approaches to this. One of these was based on a more accurate treatment of the electrostatic field effects inside a cavity large enough to contain a molecule [70, 78]. The other used point-charge models to evaluate the molecular electrostatic potentials acting on a molecule arising from its neighboring molecules [79]. In some examples we combined the two and represented the hydrogen-bonded waters as sets of point charges surrounding a solvated molecule and then set the combination in a large spherical cavity.

Cohn Smith and I, in Kyoto, took a more radical approach to the subject. We considered [115] a model of the molecular surface and found the averages of the molecule’s electrostatic potential and electric field over the surface around each atom. These give measures of the molecule’s interaction with the surrounding solvent molecules. They should also be used in discussions of molecular recognition. With such quantities we showed that the solubility of many molecules could be satisfactorily predicted. In the course of this we applied fractal theory to show that the volume and surface area of a molecule in solution are not fixed but depend on the size of the liquid molecules.

Since I was lecturing in Kyoto on reaction paths, I tried to extend my research [93] in this direction I produced [95] a model reaction surface for hydrogen transfer reactions which assumed three atoms, with internuclear distances x and y, reacting in a straight line. The potential is based on the Morse potential and has the form

where X = e

This form leads to a good correlation with reaction barrier heights and gives a good account of the reaction geometry and the qualitative aspects of these reactions.

Mayumi Okada and I tried [96, 97] to extend this more generally using sums of higher powers of X and Y as potentials. These solve the asymptotic problems of representing reaction surfaces but, because of differencing, are found to be too sensitive to be used freely.

The problem, highlighted by R. C. Woolley, that "all isomers satisfy the same Schrödinger equation", is another major unsolved problem for our subject. During the course of our work on potential surfaces, the advantages of using diabatic reaction surfaces rather then adiabatic ones became apparent [97]. These allow the global symmetry to be broken in defining the species but restored later at the point where the surfaces intersect and the distinct species interconvert. A similar method of breaking symmetry should contribute to the solution of the major problem. I suggested [105] that, when the isomers can be defined as permutations of one another, and the Schrödinger equation is solved with a permutation constraint, the resulting diabatic surfaces would give an approach to this problem.

I first became interested in point charges when, as an undergraduate, I studied the calculation of the capacity in classical electrostatics. I was minded of this when I realized that the electro restatic potential of a Gaussian distribution of charge is asymptotically that of a point charge and, then, that the product of two Gaussians at different points is another Gaussian at a point on the line between them. Thus a wave function, whose electron density p is a quadratic form of spherical Gaussians, can have its p immediately translated into a point-charge model [67]. Since our FSGO calculations using OPIT were of this type, we could investigate the consequences. Andrew Tait and I found [68] that the molecular electrostatic potential was evaluated very well outside the molecule itself using the point charges, though there were a large number of these. The model conserved the total charge and dipole. David Martin and I [87, 88] strengthened this result by proving the theorem that, if the electron density p is a sum of Gaussians, then a good point-charge model is obtained by retaining their charge but shrinking each Gaussian into a point charge. Thus, if M( x, y, z) is a harmonic property (i.e., V2M = 0) to be evaluated, its value is given exactly by the sum over point-charge contributions:

We also extended the theory to’allow nonspherical Gaussians to be included, but the results became less convenient.

When I looked at some of these models of atomic densities in molecules with K. Tsujinaga, we realized [101] that it was important that each of the heavy atoms should have one diffuse Gaussian to represent its outer charge density. Then, when the remaining Gaussians are shrunk to points, this function remains to model the more sensitive outer density. When the molecule is placed in an electric field, this function is moved and so contributes the major part of the molecular polarization [102]. Because of this, the diffuse function also models the dispersion forces between molecules in a practical way. It can also model the surface shape of the molecule and allow for penetration effects [31]. The name, the “current bun” model of an atom in a molecule, in which positive and negative point charges are embedded in a diffuse negative cloud, does homage to the ideas of Lord Kelvin and J. I. Thompson on atomic structure.

A typical wave function, nowadays, is too complicated an object to understand and, often, it is too elaborate even to extract from its home in a computer. Fortunately, however, we know from density functional theory that all the important physical information is contained in the electron density, so this has to be our object for study. The distillation of useful information from the available oversupply is an important problem in this computer age, so we have a chance to study the electron density as an example of this. In more abstract terms, we need to produce a higher level concept from lower level ones, scientific synthesis rather than analysis.

Having surveyed thç alternative definitions on offer [94], I decided that we should consider a new starting point. Cohn Smith and I began [92, 98, 99] by using our FSGO experience as a guide. We approximated the density, p. of a conventional molecular orbital calculation by p~, a linear sum of spherical Gaussians. The positions and scales of the Gaussians could be optimized. We needed a criterion for the fitting, so I suggested [90] that the error self-energy,

should be minimized. This is based on the familiar Dirichlet principle in electrostatics and can be further justified using the theory of splines. Alternatively expressed, it is a least-squares fit of the electric fields produced by the two densities.

Using this criterion, Tsujinaga and I investigated [101] the approximation of a given molecular charge density by spherical Gaussians placed in optimum positions and with optimum charges. We found it helpful to add the constraint of fixing the total charge. Our studies showed that it was important to balance the number of Gaussians associated with each nucleus to obtain the best fits. Where lone pairs were present, a Gaussian at their center represented them and improved the fit. The results tended to converge after a modest number of terms. The resulting charges could, therefore, be defended as physically rather than mathematically based.

Although the Gaussian is well-adapted to locate the charges, within a molecule, it is somewhat artificial. This led Smith and me to introduce [104] the "optimal population analysis." In this, the approximate density is represented as a sum over the squares of the atomic orbitals already in the density matrix. The optimization is limited to the linear coefficients of these squares. This gives a procedure only a little more complicated than the Mulliken populations but much more clearly based. We showed its realistic advantages by evaluating the different contributions to the total molecular energy using this density. The result was a considerable improvement over the corresponding Mulliken estimates. We found, however, that the results were not independent of hybridization and that localized atomic hybrids gave an even better representation of the density and of the energies. Atomic hybrids allow us to give a more detailed description of the location of the charges but force us to allow charges not to be on nuclei. Chemically speaking, atoms are not spherical and can have dipoles as well as charges. By using very localized functions, we can turn most of the energy contributions into classical electrostatic energies. One consequence is that the exchange and correlation energies then become very small. When the theory can describe where the electrons are localized and show that almost all the molecular potential energy is then electrostatic [118], an understandable account of a molecule becomes possible.

The goal of an agreed definition of atomic charge, which is independent of the basis set being used and which facilitates a physically meaningful analysis of the forces within a molecule, has yet to be reached. Nevertheless, its shape is slowly becoming more clear. Optimization will lead to more stable estimates and a better understanding of localization will lead to energy contributions that explain the structure and properties of the molecule.

Recently, I have begun to work again with Don Rees. We started by thinking about atomic hybrids [133—139]. The operators x, y, and z clearly commute but, when we take their matrix representatives in a finite-dimensional function space, these, in general, do not commute. One modification which often produces commutation involves a scaling of some of the basis functions [136]. In other examples [134], we must add extra operators to compensate for the finite dimensions. The atomic hybrids are the simultaneous eigerifunctions of these commuting matrices. These hybrids are the most localized functions according to the definition given by Boys. Their use in describing the structure of a molecule and evaluating its energy contributions will be a contribution to our understanding of molecules.

Our work, which is now progressing actively [141—143], can also be described as the generalization of Gaussian quadrature to higher dimensions since the Gaussian grid points are at the centers of the hybrids and their weights can be derived from the value of the hybrid at its center. It is an important property of these hybrids that each has nodes at all the other grid points. The systematic use of Gaussian quadrature in molecular calculations, which we hope to encourage, would give a greatly increased flexibility in the choice of basis functions. It would also eliminate many of the integral complexities from programs.

In its quadrature form, our theory can be generalized to solid-state calculations by using cosines as the basis functions rather than polynomials [140]. The resulting grid points are the "special points" used extensively in solid-state calculations of integrals over reciprocal space.

During my stay in Sweden, I showed how to make the radial variable a function of another auxiliary variable and so produced a two-parameter orbital function for the helium isoelectronic series which reproduced the Hartree—Fock results [23].

I was specially pleased to prove [25] that, for such isoelectronic ions, the electron-nuclear potential energy can be defined as Z a W/dZ, where W(Z) is the energy as a function of nuclear charge Z. Chemistry may be confined to integral values of Z but mathematics is not. Don Rees and I later used this expression [29] in our calculation of magnetic shielding constants for atoms.

While he was with us, Raphy Levine used operators extensively in his work on time-dependent properties, and we learned from his example [45]. In particular, the use of Green’s operators and functions was a natural development of working in the town where Green had lived [176]. I proposed a variational principle for the energy [50, 55] using a Green’s function and proved that it always gave better energies than the Ritz—Rayleigh ones. As Don Rees, John Hyslop, and I discovered [54, 60], its use for problems with more than one electron proved difficult.

One of my colleagues in Nottingham was Kathleen Collard, and we become interested in the application of differential topology to analyze the shapes of the electron densities of molecules. We suggested [84] the use of orthogonal trajectories to define the bonds between nuclei. We could also relate the numbers of turning points of different kinds in the density by using the Poincaré-Hopf theorem. We looked carefully at possible catastrophe points. I later [100] extended these ideas to the molecular electrostatic potential.

H. Nobutoki and I tried [113] to understand the pattern of vibrational energy loss associated with the dephasing that occurs at the start of an electronic transition. We calculated many examples of the time evolution of the water molecule from various initial vibrational states and tried to find the common pattern. With a little help from information theory, we could rationalize some of the observations but a good theory eluded us.

N. Saito and I [117] looked at the effect on a molecule of embedding it in an intense laser field. This became an interesting exercise in radiation theory but seemed to be of small chemical impor- tance since the only significant effects were for hydrogen atoms.

I have tried to contribute to the work of the Shell Centre through some research as well as in various other ways. One research piece which influenced me considerably [161] was an investigation into the motivation of mathematics students. This showed conclusively that their university experience was killing their strong initial mathematical motivation quite rapidly. I then tried to reverse this by redesigning courses in our degree program [149, 162] so that they concentrated on tackling realistic practical problems by model-building, but the initial improvements were not sustained and did not convince my colleagues.

I also support the aims of the Shell Centre by producing small examples of original applications [177—206], principally for the Bulletin of the Institute of Mathematics and its Applications, now called Mathematics Today. These follow a common pattern which I have developed. Each must arise from some everyday observation or reflection. Each must have one basic mathematical point to make. The length should not exceed one page in print and is often about half a page. I see these almost as a new art form!

To look back is to look forward. It is rarely possible to look at scientific work and know that the last word on it has been spoken. When I look at all the topics on which I have worked, I see so much that still needs to be done both to extend the theory and to produce examples which publicize it and prove its worth. While I have every intention to continue working on many of these, I can offer no guarantee that I will not be deflected into some entirely different direction!

While in London I started to write a book on matrices and tensors. The book was intended for chemists beginning our subject. It combined two aims. The first was to explain, as simply as possible, how practical calculations on matrices can be performed by hand or on a computer. The second was an attempt to explain how tensor ideas provide justification for the "vector model" used in angular momentum theory. The book, which is a volume in the International Encyclopedia of Physical Chemistry and Chemical Physics, did not try to cover all aspects of the topics. it is still used but no longer sold!

This is a graduate textbook. I had become dissatisfied with all the current texts on group theory. They seemed to me overelaborate and not compatible with the ideas of quantum theory. Starting from the brief hints in Dirac's book, I started to develop the subject using class operators. This involved me in formulating and proving many theorems for the first time. The result, I believe, is a class algebra approach still not found elsewhere. I enlarged this discussion of finite groups by bringSing in sections of my Cambridge thesis on the groups of polymers. This is another topic rarely mentioned. My account of spherical symmetry is less original but aims to bring together several different aspects of the subject. The book has had a reasonable life and was translated into Hungarian.

The Division of Molecular Engineering was set up with the intention of furthering the development of this subject into a new discipline. For this purpose we designed new courses which brought together the many varied aspects of the subject. Thus, experimental work on ceramics, the design of miniature batteries, the structure of high-temperature superconductors were all combined with related theoretical courses. As a contribution to this defining of a new subject, I wrote a book, based on my lectures on molecular solid-state physics which tried to introduce, to chemists, enough solid-state theory to understand the variety of interesting new materials. The book has been a failure, perhaps because the subject is, as yet, very unfamiliar.

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Copyright © June 18, 2003 by U. Anders, Ph.D.

e-mail Udo Anders : udo39@t-online.de

Copyright © June 18, 2003 by U. Anders, Ph.D.

e-mail Udo Anders : udo39@t-online.de

Last updated : June 19, 2003 - 18:52 CET