I am a theoretician in the fields of Chemical Physics –Physical Chemistry, the closely related field of Atomic and Molecular Physics and also in NMR Signal Processing. For over fifty three years, often inspired by experiments, I have developed methodologies for computing and/or analyzing from first principles the results of these experiments. In doing so I and my coworkers have proposed physical models, methods of analysis and view points and have developed new computational theories that explained the nature of the systems under experimental study and which in turn became the essence of what was learned from these experiments.

Over the last thirteen years, both before and after my retirement in 2006, I have had two research efforts going. The first which I shall now describe was the study of High Molecular Vibrations. To appreciate our contribution it is well to return to the situation as it existed in the late nineties. A series of beautiful experiments had been reported by Field (the bending spectrum of acetylene), by Treffs (DCO) and by Quack (CHBrClF) that probed the high vibration region where multiple vibrational resonances existed. The problem started with the fact that all previous successful analytical work had been done in the regular single resonant interaction region. Herzberg’s methods as parameterized perturbation theory and wave function inspection were then adequate tools in that they were able to give level assignments with a value of one quasi constant of the motion quantum number per degree of freedom and were able to envision the motions of the nuclei in each state (the Dynamics). In the high vibration region as discussed here multiple resonant interactions appeared and non-linear dynamics taught that the original modes of vibration often disappeared to be replaced by multiple new but unknown motions. Also the quantum chemically computed wave functions were topologically so complex that they defied node counting to give quantum numbers and gave no clue as to the dynamics. Moreover classically in the high region a mix of chaos and regularity existed. Mathematically chaos alone would guarantee the failure of the methods used at lower excitation; for example there is no global convergence of Schrodinger perturbation theory.

Many workers in the field (including myself and my colleague Christophe Jung) tried to overcome these problems by pointing to non-linear dynamics and the need for semi-classical reduced dimension phase space representations for all the classical trajectories and periodic orbits and for the number space quantum wave functions and fitted spectroscopic Hamiltonians obtained from the experimentalist. The latter summarized in a dynamic way the spectrum and the types of resonance interactions underlying the spectrum. Even with all this, until the 1999 effort by Jung and Taylor, aided by the experimentalists Field and his student Jacobson who explained their results to the former, nobody had put it all together and no one was able to contradict Field’s claim based on the use of methods that worked in the lower region, that the bending spectrum of acetylene was “UNASSIGNABLE”. If this were so the promise of these experiments would go unfulfilled and in this sense their value would be reduced. In 1999 J&T resolved this problem. The new idea that J&T added to the above mix came from a somewhat more advanced use of the principles of non-linear dynamics. It taught that the dynamics should be studied in the reduced dimension action angle phase space subspaces, where the constants of the motion (polyad, bending angular momentum) had unique values. There when the excitation was high enough so that the original mode’s anharmonically modified frequencies satisfied rational ratio resonance conditions, the reduced dimension phase space partitioned into, often coexisting in energy, separate regions called resonance zones. Each zone would have a new dynamics and at its center had periodic orbits, one for each resonance that was active in the region; these intern guided the motion in the region. When for one active resonance the periodic orbit or for two periodic orbits, the point of intersection, was transformed back to the full dimensional normal or local displacement coordinate space they would (and did) reveal trajectories that would these exemplify the new motions. Moreover in theory if the motions indicated by the reduced dimension periodic orbits could separately be quantized then each region would yield simple but different assignable progressions or ladders of levels. The levels were simple in the sense that their reduced dimension wave functions exhibited nodes along and locally perpendicular to the periodic orbit, that when counted would give quantum numbers which when combined with the fixed constants of the motion would constitute an assignment in the above sense. Just as in the low energy analysis higher numbers lead to larger excursions of the motion in both the reduced dimensional configuration space and full dimensional displacement coordinate space. Hence the assignment and dynamics problem reduced to the computationally demanding problem of finding the periodic orbits and to the visually simple matching of them to density plots of the reduced dimensional wave functions. Nodal ordering and counting gave the regional ladders and assignments that would have been easily noticed by the experimentalists if they could have divided their observed states and wave functions into ladders. Unfortunately the quantum level steps of the ladders interspersed in energy and no tools existed to sort them. The result was a complex and confusing spectrum. In 2010 our greatly improved skills in illustrating on reduced toroidal angle configuration spaces a given wave function density allowed us to visually sort , count nodes and see the wave function’s structurally guiding elements that well approximated the periodic orbits. This eliminated the “search for periodic orbits” .The analysis was now totally analytic and therefore required no computation. In 2011 it was shown that the assignment, but not the dynamics , could be accomplished without any semi-classics and canonical transformations using only the eigenstates in the number representation as received from the experimentalists. The analysis was now totally quantum. The references to these papers are given on this website under the “Publications Sorted by Subject” link.

The second subject that I have worked on over the last ten years and am still working on grew out of my work on designing new methods for non-linear signal processing. These methods were successfully applied to computing eigenstates in dense spectral regions and to quantizing systems that were classically chaotic. The references for all three of these efforts appear under the “sorted publications” link.The new work is an algorithm for increasing the sensitivity of 1D NMR experiments for systems for which Lorentzian spectra was expected i.e. gases, liquids, solutions and solids measured using magic angle spinning techniques. The aim is to use non –linear signal processing to replace the linear Fourier Transform to reduce the number of transients needed to separate the signal from the noise so as to avoid or minimize the need for enrichment. If one is interested in frequencies, the problem, its solution and the algorithm, which takes little time to learn and employ, is fully explained in the “Handbook” link on this website.

Last but not least I will mention the work that I most value, that being my development of the “Stabilization Method”. The Stabilization Method appeared early in my career (1965-70’s). It was inspired in 1967 by new experiments which for the first time were able to resolve short lived negative ion resonances in electron atom and electron molecule scattering cross sections. For molecules the theories of the day knew that what was seen were the vibration states of some sort of short lived negative ion system. For molecules and in particular diatomic molecules for which most of the data existed no one understood the configurations of these states and the physics of why only certain of these states seemed to be observed. . At that time what made computations on molecular systems seem out of the question was that it was believed (wrongly) by many in the physics community that only true scattering methods , as for example the close coupling method, that specified particular scattering boundary conditions, could be used to compute and calculate such states. Scattering methods of that day for computational reasons could barely study atomic Hydrogen below the n=2 energy level. Calculations on non spherical systems were out of the question. I thought differently. I was at that time a quantum chemist in procession of state of the art programs for small diatomic molecules and ions that I developed jointly with Frank Harris. The programs could compute ground and excited state potential curves and then obtain their vibration levels. These could be used be used to carry out computations on such systems if one big obstacle could be overcome. This was that these short lived negative ions ionized into molecular states of lower energy. As such the Rayleigh-Ritz variation method which underpinned the method’s ability to determine linear and non-linear parameters was no longer valid. If used the method would vary parameters so as to eject the extra electron and then go on to compute the best configuration and energy for the remaining neutral system. I argued and demonstrated that the Rayleigh-Ritz variation principle could be replaced by my Stabilization method which in its first incarnation chose basis configurations and varied their non linear parameters in sequential calculations such that the energy of the state stabilized. Intuitively I argued that this would occur if sequential calculations used more and more diffuse basis functions; physically these changes of basis would not affect the localized resonant states hence the stability, but would cause the energy of non localized continuum states to decrease. Using these ideas I published results for Hydrogen negative molecular ion that reproduced the main observed resonances and predicted others that were soon observed. Simultaneous with these calculations, from the nature of the configurations that stabilized, I was able to develop a physical picture for the nature of such states and an explanation as to why they existed. My “core excited” and “single particle” classifications compared nicely to Feshbach’s classification for nuclear resonances. In 1970, in the now widely known Hazi-Taylor paper, the Stabilization method was shown to be generic to all short lived localized quantum states no matter what the system. The now widely used stabilization diagram was introduced. The method has since been used in the fields of nuclear physics, particle physics, nano-physics (to calculate the states of high transmission in quantum wells and dots),atomic, molecular physics and optical physics where it is used to calculate auto ionization energies, predissociation states and many other phenomena involving decaying states and so called bound states in the continuum. Many variants of stabilization have been published and many attempts by others to derive the method from first principles were published; all failed. It wasn’t until1993 that in collaboration with my associate V.Mandelshtam that we not only derived the method from first principles but showed that a complete theory of scattering phenomena could be based on the stabilization diagram and the discreet state eigenfunctions that were computed in its construction. Papers computing reactive scattering and photoionization cross sections, microcanonical and canonical rates of reaction and dissociative photo absorption cross sections followed so as to demonstrate the generality of the theory

Stabilization’s use is so ubiquitous and accepted that for many years it has been paid the ultimate compliment in that many publications which clearly state that use is made of the method and which often even have a figure with the stabilization diagram do not even reference it. Along with perturbation theory and the variation principle it is, albeit not as often used, now part of basic Quantum Mechanical Theory. Links to this work also appear in the sorted publications list. .