The ideas that determined future research

Upon arriving to Sussex University, Molecular Sciences Department, at the invitation of Prof. J. N. Murrell, I was given two problems to work on. Both concerned three atom molecules, or species, but one was a bound state problem and the other predissociation. In the first I had to calculate bound state energy of three Helium atoms and in the other the Franck-Condon factors for vibrational predissociation of the carbon dioxide molecule. None of them required any physical insight for understanding the problems, but essentially to use a very well established ideas and implementing them on a given system. That, to a great extent, was the status of research in theoretical chemistry, where the basic premise was that the theory is known and one only has to cope with the numerical codes to implement it. That attitude changed in recent times because experimental and computational powers increased beyond anything that could have been envisaged, which made molecular physics/theoretical chemistry much more creative subject. For me this was a shock, graduating from the high energy physics, the forefront of the basic research where the imagination played an important role. The math for the task I was given was also quite elementary, compared to my undergraduate training. However, that shock had an important impact on me because it thought me to see the problems by using pictures, seeing what the structure of the Helium cluster looks and following the path of atoms in the carbon dioxide molecule before it splits up. I also learned that understanding the working of the Nature is not in the sophisticated math, it is us the humans that transformed it from the powerful "slave" into the powerful "master".

The work on the ground state of three Helium atoms was not demanding on the physical understanding of the problem, but definitely very demanding on the numerical part. It was primarily the effort to find an efficient way of solving the quantum equation for the ground state of three atoms. The solution was sought through the use of the variational theorem, but the procedure could be slowly convergent and without much insight into the nature of the solution if the correct basis set of functions is not chosen. There are various ways of improving the convergence of the variational calculation, and one is to remove as many coordinates as possible from the considerations (in the case of three Helium atoms there are six independent coordinates), essentially to isolate only those that are important for determining the bound state. In the case of a single (or two) body problem this is the radial coordinate, where the other two are angular coordinates, which enter the final solution through the spherical harmonics. In the case of three bodies one defines the hyper-spherical coordinates, which are formal extension of the spherical, with the five angles instead of two. These coordinates were investigated in nuclear physics and I tried to use them for the three Helium atoms problem (a general recipe in those days when the inspiration for the study of molecules was drawn from nuclear physics), and my contribution was to select the angular functions with respect to their symmetry properties. On the left is one page from my PhD theses where these coordinates are investigated. It turned out that they are not as useful as one would anticipate, based on their formal properties. In fact a much better convergence was obtained by using the relative separation coordinates. In my later research, when I turned attention to the scattering problems, the coordinates were never used. The reason is simple, their formal properties hide the great inadequacy when, for example, one must match the boundary conditions on the probability amplitude (the wave function).

The other problem that I was given is to analyze the broadening effect of spectral lines in carbon dioxide due to the predissociation. The name predissociation is used for the dissociation if it occurs through a long lived state. In the case of the carbon dioxide molecule it is excited from the ground vibrational state (rotations were neglected in the first instant) into the dissociation channel oxygen-carbon monoxide, but because the energy flows among the vibrational degrees of freedom it takes time before it splits apart. The finite lifetime of this excited species affects the width of the spectral lines for the transition from the ground state; the width is inverse proportional to the lifetime of long lived states. The same study, but only for diatom molecules, was done prior to this example, and the method was perturbation theory, and essentially means to calculate the Franck-Condon factors, the overlap integrals between the initial (ground) and the final state (in the continuum) wave functions. The origin of the long lived states in the diatomics is tunneling in the dissociation channel, in contrast to the system that I was given to investigate. Nevertheless, the suggestion was that the same method is used, the biggest effort being in the calculation of the wave function in the oxygen-carbon monoxide continuum.

Again this was meant to be another routine work, albeit numerically very demanding (much later I realized that this task was beyond our means, both in terms of the knowledge of the theory and the computation power, at that time). The prospect was not something that I aspired to, and my suggestion was to investigate this problem as the "half of the scattering". Roughly speaking, dissociation time is half of time delay in the scattering of O+CO going into O+CO. It was agreed to redefine the project along these lines, but that meant learning the scattering theory that applies to molecules. Coming from high energy physics my first approach was to use the methods of the quantum field theory; to define creation and annihilation operators for these species, and try to find the form factors that would describe the interaction between them. It took me several months to reach important conclusion; the methods of quantum field theory are useful for analyzing systems with poor information content, and therefore for molecules are not applicable. The richness of the information is measured by the ratio of the "wave length of particles" to the range of forces between them and not by the absolute magnitude of energy at which the processes occur. This ratio is favorable for elementary particles and not for molecules, and hence my idea to use the quantum field technique was doomed. That meant that I had to forgo my study of high energy physics and explore new ways, in this particular case non relativistic potential scattering theory. Standard books on this topic were scarce, and dedicated to the subject of nuclear/atomic physics, with the specifics that they implicitly assume. Realizing that came at a later date, but at the time those books were the source of learning the elements of quantum scattering theory, and because of their specifics very often led to "blind alleys". Furthermore, there were only few theoreticians working in this field, mainly in the US and nobody in the UK, or at least I did not know of them (there were few working on the predissociation problems for diatomics, but these could be hardly called scattering problems, despite the fact that the unbound states are involved).

Quantum calculation of time delay for the inelastic oxygen-carbon monoxide collision was at that time very difficult to do (even nowadays it would be a major undertaking despite very powerful calculating machines) and so a compromise was made. Classical calculation, even for a more complex system than this, could have been done with the relative ease and so it was decided to try this pathway. The rationale was that at least this would give an estimate of the lifetime. However, the use of classical mechanics in quantum systems was almost a heresy, but the feeling was that molecules are "nearly classical" objects. Agreeing on taking this approach was not of much use unless there are rules how to apply classical ideas that would be in accord with the quantum results, and at that time very little was known about this. Physicists abandoned classical mechanics long time before that and to the chemists it was a rather remote subject, and so there ware practically no ideas how to apply classical mechanics.

The results that came out of this study greatly determined my future interests.

1. I was intrigued by the finding that classical mechanics is not a heretical subject; in fact it can give good results provided one knows how to implement it. At that time there was no clue how it could be done, but for me it was a challenge for future research, which I pursued ever since.

2. Typical feature of classical studies is the onset of chaos. I also encountered that problem, but we did not have a clue what was all about. Typical finding is shown in the down figure (a copy from my thesis) where the final vibrational number of CO is shown against the initial phase of the same diatomic. There are smooth lines, but also scattered points, and no matter how "close" they are there was no way to find a smooth line that connects them. As a general rule the points from the smooth lines were associated with short lived states whilst the scattered with very long lived. For a long time I blamed numerical code (read computer) for the problems, but after finding that it is valid I tried to give the rationale for this "chaotic" behavior. In fact I rediscovered the Lyapunov theorem, which for me was the relief that the numerics is valid but also indicated a serious problem that will much later become one of the major field of research, the theory of chaos. We did not give it any name, simply the problem of numerical instabilities. This is perhaps why I never worked in the field of chaos research, because my experience was that nothing much could be done in terms of getting meaningful information from it. Whether I was right or wrong is the matter of opinion, and would not want to pass any judgment on that.

   

3. The problem of physical substance that intrigued me was the role of long lived states in collisions. Their impact on the cross sections and time delay, but also their relationship to the resonances. The latter have a rather unique property that their lifetime is inverse proportional to the energy width within which they contribute in the cross sections. Yet my observation was that in the classical study they are formed within a wide range of energy, which contradicts their property. One could argue that classical study is not the relevant indication of such discrepancy, and this dilemma stayed with me for a long time after my PhD. Nevertheless I pursued this path in the quiet of room, which eventually led to the surprising results.