Computational quantum chemistry - then and now

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George G. Hall
Shell Centre, School of Education,
University of Nottingham,
Nottingham, NG7 2RD United Kingdom
(Received 29 June 1990)


Abstract:

This paper indulges in the luxury of reflecting on the past, present and future
of computational quantum chemistry from a personal point of view. It offers advice and comment
but makes no claim to infallibility.


THE PAST

I arrived in Cambridge as a research student of Sir John Lennard-Jones in 1947. He had just returned to university life from his wartime activity in the Ministry of Supply and was anxious to make a new beginning in academic research. He decided to return to the subject of the electronic structure of molecules which he had done so much to initiate in earlier years. Within a year he had written the first [1] in his long series of Royal Society papers on the molecular orbital theory of chemical valency. In this, he derived the differential equations for the best molecular orbitals so that their properties could be investigated rigorously. At the time it seemed strange that this had not been done many years before since the atomic equations were well-known. In 1938 Coulson had calculated [2] the molecular orbital for H2 using a James-Coolidge type of expansion and had performed several calculations, using various integral approximations, on species such as CH4 and H20. These calculations were based on the optimization of the total energy and so did not use any orbital equations. Since previous discussions of the form of the orbitals were almost all based on physical argument rather than mathematical derivation there were several controversies in the literature which had not been settled. One of these concerned the localization of the orbitals. Some authors argued strongly for delocalized orbitals while others looked for localized ones which could be a realization of chemical bonds. In his second paper [3] Lennard-Jones settled this issue by showing that the orbitals could be subjected to a unitary transformation without changing the total wavefunction or its energy and that this permitted both kinds of orbital to be used.

My first paper, which was the third in the series [4], suggested that the molecular orbitals, in the strict sense (Roothaan later called them canonical molecular orbitals), should be defined as the eigenfunctions of the Fock Hamiltonian. They are usually delocalized. The localized orbitals obtained from these by a unitary transformation into spatially equivalent functions were called equivalent orbitals. We contrasted the properties of the two kinds of orbital by showing that ionization had to be described using the delocalized molecular orbitals whereas properties of the ground state alone could have alternative descriptions. In the fourth paper [5], Lennard-Jones and Pople showed that the localized orbitals had a much smaller exchange energy than the molecular orbitals and so could be used to discuss molecular structure electrostatically. As I would now phrase it, there are two equivalent representations of the set of orbitals. The molecular orbitals give a representation in which the electrons move independently of one another whereas the equivalent orbitals give a representation (defined by maximizing the sum of their self-energies) in which the electrons act like charge clouds in classical electrostatics.

Since the most successful part of MO theory, at that time, was the application of Hückel theory to conjugated molecules, I tried to deduce the LCAO equations from the MO equations. It was easy enough to apply the LCAO approximation to the solution of the MO equations but the results were not what we expected. The equations for the coefficients were not linear, as Hückel theory had assumed, but cubic and the effective potential took a very different form when the solutions moved from the occupied orbitals to the virtual orbitals. These results seemed to undermine the basis of most of the previously successful applications. I had to find a new basis. At about that time an experimental paper by Honig [6] appeared reporting ionization potentials for alkanes so these became a natural target. I then set up my equivalent orbital theory of ionization potentials as a semi-empirical theory and showed that very good agreement with experiment could be obtained. Since this relied on a unitary transformation, the equations were strictly linear and so relatively simple to manipulate and understand.
  When I came to publish my results, in 1950, I found Lennard-Jones enthusiastic about the final results but dubious about publishing the derivation of the LCAO equations. He argued that Coulson must already have published them and Dr. S.F. Boys made the same comment. I could not proceed [7] until I had done a full literature search and could assure them both that Coulson had not done so. Coulson had objected to a previous paper in the series so both were anxious not to cause him any offence. We first became aware of the parallel work of Roothaan when a preprint of his thesis, which became his paper published in Reviews of Modern Physics, arrived in the department. I remember receiving it the same day as I corrected the proofs of my paper. In fact, it appeared in print [8] a year later than mine.

{Note by this website: Therefore it should at least be the Hall-Roothaan-Method, as it was at some time also generally being called in the literature. Unfortunately, the work Hall had done - as he states it above - became a bit forgotten.}         
In later papers [9] I showed how the Hückel theory could be rescued, although extra energy terms were required. I used what I called the standard excited state (and now call the ferromagnetic state) of the conjugated molecule. Since this state is highly excited, it has seemed irrelevant to other workers but I still feel it offers the best hope of building a rigorous semi-empirical theory which can be checked by ab initio calculations. The equations which it produces are the same as those used in PPP theory.

The possibility of using the LCAO equations numerically seemed unrealistic at the time. We had electric calculators but the Cambridge electronic computer was still being built. The integral evaluation problems were known to be formidable. Boys had just produced his Gaussian paper [10] which he regarded as an existence theorem rather than a practical method. The intellectual pressure, especially from experimental chemists, was for explanations of spectra and predictions of the structure of large complicated species. These could only be satisfied through improved semi-empirical methods. Within a few years, the EDSAC became available and I managed, in 1957, to produce a program to perform both Hückel and PPP calculations for hydrocarbons larger than had previously been possible [11]. By this time, Boys had acquired a research team and they were writing the suite of programs necessary to perform full ab initio molecular calculations. Their efforts were strongly opposed by many, including the experimental chemists, and I was dissuaded from following this aspect of the subject.


THE PRESENT


Forty years on gives me a good vantage point from which to look back. It is now clear that the electronic computer has succeeded in changing the face of quantum chemistry. Its power and speed have been harnessed to produce results on molecules which are more accurate than we dared to hope for. In some areas it can now provide results with an accuracy approaching that of experiment and with a freedom to consider rare or impossible species and configurations which experiment cannot match.
  I will draw one personal lesson. I am sure that nobody pioneering a radical new method should be deflected by the criticism of those who do not understand the subject, however exalted their positions and their influence.         
The major lesson has to be that the foundations of a subject are vital to its long-term success or failure. In the short term, some theories may appear attractive because of their simplicity and their immediate success but if they cannot be rigorously defended, their eventual fate is sealed. Computational quantum chemistry is firmly based on the study of practical solutions of the Schr鰀inger equation for a molecule. It can operate at different levels of accuracy according to the size of the molecule and the property of interest. Any advance in the relevant numerical analysis may change its procedures just as any major advance in the design of computers will affect its scope and accuracy.

It is interesting to observe the way in which the subject becomes transmuted in the course of time. The original emphasis on using atomic orbitals, e.g. in LCAO, came from the hope that the wealth of atomic information could be used to give the missing molecular information without the need for integration. Their continued use is due to the thought that they are localized basis sets which can be expected to persist from molecule to molecule and from state to state. Nobody expects now that atomic spectra will play any part in the calculations. Similarly, the original reason for Gaussians was to prove that the expansion method, using many interacting configurations, could converge to provide a solution, in principle, to the molecular Schr鰀inger equation. It was not thought to be practical since the Gaussian has no nuclear cusp and all our experience of power series expansions said that convergence is determined by the presence of singularities and the ability of the expansion functions to reproduce those singularities. We now realise that the advantage of having explicit integral formulae outweighs this objection. Convergence is still slow but the speed of the computer has made this limitation less and less relevant. Even the old argument over the relative merits of MO and VB method is out-dated. Modern methods go beyond both in their accuracy. On the other hand, the present emphasis on numerical and computing methods has weakened the connection which these theories made with physical insight. The scope for a physical interpretation of the calculations has greatly decreased just at the time when the volume of numerical information available from the computer threatens to drown us. The need for a stronger interaction between calculation and understanding is still great!


THE FUTURE


It is always a challenge to speculate on the future of one’s subject in print. The result may be more significant as self-revelation than as prediction!

We are entering a period when the packages for molecular calculation come close to giving a chemist a means of approaching his subject from a purely theoretical point of view. He can explore compounds, e.g. containing Li or B, which it would be difficult or dangerous to use experimentally. He can ask questions about the molecule which experiment could never answer. Questions such as the comparison between the relativistic and non-relativistic results, the changing charge on a particular atom at each point of a reaction path or the relative importance of σ and π electrons for the intensity of a vibrational transition. In effect, the experimentalist now has a new instrument at his command. As with all other instruments, it has advantages and disadvantages and so has to be evaluated in relation to the research objectives. This is close to the ideal situation envisaged by Boys 40 years ago!

There is always a trade-off in a calculation between the size of the molecule and the required accuracy. For small molecules we can now approach spectroscopic accuracy but for biological molecules we have to be content with cruder treatments. The computational difficulties increase as a high power of the number of electrons and there will always be a barrier to what can be calculated even if it is slowly pushed back. As the calculations become more sophisticated and longer, the inevitable rounding-off errors must increase and the significance of the numerical answers will decrease. This seems to me to be the continuing case for the development of \"semi\" methods. The semi-empirical methods have helped the subject in many ways. The development of semi-theoretical methods has hardly begun. These would use rigorous theoretical equations along with accurate calculations on small systems to extrapolate to large systems. We have good reason to believe that such methods would be successful.

It is hard to believe that a radically new method will suddenly appear now. So many avenues have been explored that none seems to remain undisturbed. But this does not mean that one that has been neglected may not become attractive again. Most of our computational work rests on the narrow base of the optimization of the energy derived from a many-particle wavefunction. The lesson from the rise of density-functional methods is that this wavefunction contains much information which we do not need. If practical density-functional methods are developed, then we can enlarge greatly the scope of what is easily calculated.

I would appeal also for the need to develop new theoretical insights. The molecular computer is now a feasible objective but it forces us to look at molecules from a different perspective, as dynamic elements in an electronic system. We approach, in quantum biochemistry and pharmacology, the edge of the biological sphere. In this we know that many new phenomena are possible. The living cell is an energy-information transducer, a self-replicating system and exercises a degree of control over its functions. The chemical species involved are slowly becoming evident but the nature of their activity is still obscure. Major discoveries will be made in this area. Our subject can be of immense value in this search.

Finally, I would like to add that part of my interest in quantum chemistry has always been that it reverses so much of the usual scientific method. Quantum chemistry is deductive, not inductive. We know how electrons behave and we have to deduce how molecules behave. We are building up larger and more complex systems rather than breaking them down to simpler ones. We need new mathematical methods for this task, knowing that most existing methods were designed for the analytical aims of other sciences. We have stimulated some advances, not least through our constant ability to saturate any computing facilities that are developed! But much more remains to be achieved.


REFERENCES


1  Sir John Lennard-Jones,
Proc. Roy. Soc. [London], A 198, 1, (1949).

2  C. A. Coulson,
Proc. Cambridge Philos. Soc., 34, 204, (1938).

3  Sir John Lennard-Jones,
Proc. Roy. Soc. [London], A 198, 14, (1949).

4  G. G. Hall and Sir John Lennard-Jones,
Proc. Roy. Soc. [London], A 202,155. (1950).

5  Sir John Lennard-Jones and J. A. Pople,
Proc. Roy. Soc. [London], A 202, 166, (1950).

6  R. E. Honig,
J. Chem. Phys., 16, 105, (1948).

7  G. G. Hall,
The molecular orbital theory of chemical valency.
VIII - A Method of calculating ionization potentials.
Proc. Roy. Soc. [London], A 205, 541-552, (1951).      Read it  here! (1.5MB) or, pagewise, here!

8  C. C. J. Roothaan,
New Developments in Molecular Orbital Theory.
Rev. Mod. Phys., 23, 69-89, (1951).      All = 8MB here, otherwiese pagewise here.

9  G. G. Hall,
Proc. Roy. Soc. [London], A 213, 102, (1952);
Trans. Faraday Soc., 50, 319, (1954).

10  S. F. Boys,
Proc. Roy. Soc. [London], A 200, 542, (1950).      Read it pagewise here.

11  G. G. Hall,
Trans. Faraday Soc., 53, 573, (1957).