Fattal, E. ; Baer, R. ; Kosloff, R. Phase space approach for optimizing grid representations: The mapped Fourier method.
Physical Review E 1996,
53, 1217.
AbstractThe representation of a quantum system by an evenly spaced Fourier grid is examined. This grid faithfully represents wave functions whose projection is contained in a rectangular phase space. This is mathematically equivalent to a band limited function with finite support. In general, have packets decay exponentially in classically forbidden regions of phase space. This idea is then used first to optimize the rectangular shape of the Fourier grid, leading to exponential convergence. Nevertheless, in most cases the representation is suboptimal. The representation efficiency can then be extremely enhanced by mapping the coordinates. The mapping procedure reshapes the wave function to fit into the rectangular Fourier shape such that the wasted phase space area is minimal. It is shown that canonical transformations, which rescale the coordinates, improve the representation dramatically. A specific scaling transformation enables the representation of the notoriously difficult Coulomb potentials. The scaling transformation enables one to extract almost as many converged eigenstate energies as there are grid points. The method is extendible to more than one dimension, which is demonstrated by the study of the H + 2 problem. This scaling transformation can bridge the gap between quantum chemistry and quantum molecular dynamics by enabling the treatment of electronic problems in the vicinity of Coulomb potentials by grid methods developed for molecular dynamics.
fattal1996.pdf Citri, O. ; Baer, R. ; Kosloff, R. The role of non adiabatic mechanisms in the dissociation dynamics of O2 on silver surfaces.
Surf. Sci. 1996,
351, 24–42.
AbstractThe dissociation dynamics of oxygen on silver surfaces is studied theoretically. The method is based on a quantum-mechanical time-dependent non-adiabatic picture. A universal functional form for the potential energy surfaces is employed. The diabatic potentials describing the sequence of events leading to dissociation begin from the physisorption potential crossing over to a charged molecular chemisorption potential and crossing over again to the dissociated atomic-surface potential. Within such a potential surface topology, two different surfaces leading to dissociation are studied: the empirical potential of Spruit and the ab-initio potential of Nakatsuji. It is found that the system is captured by the molecular chemisorption well for a considerable length of time, long enough for thermalization. Thus the calculation is split into two parts: the calculation of “direct” dissociation probability and the calculation of nonadiabatic dissociative tunneling rate from the thermalized chemisorbed molecular state. For the direct probabilities, the Fourier method with the Chebychev polynomial expansion of the evolution operator is used to solve the time-dependent Schrödinger equation. For the tunneling rate calculation, a similar expansion of Green's operator is used. The output of the direct-reaction calculation is the dissociation probability as a function of the initial energy content, while the tunneling calculation yields the dissociation rate. The dependence of the direct dissociation probability on the initial kinetic energy is found to be non-monotonic. A strong isotope effect has been found, favoring the dissociation of the light species.