We present a review of rigorous mathematical results about non– adiabatic transitions in molecular systems that are associated with avoided crossings of electron energy level surfaces. We then present a novel numerical technique for studying these transitions that is based on expansions in semiclassical wavepackets.
We propose a new algorithm for solving the semiclassical time– dependent Schrödinger equation. The algorithm is based on semiclassical wave-packets. The focus of the analysis is only on the time discretization: convergence is proved to be quadratic in the time step and linear in the semiclassical parameter ε.
We propose an alternative to the usual time–independent Born–Oppenheimer approximation that is specifically designed to describe molecules with non–symmetrical hydrogen bonds. In our approach, the masses of the hydrogen nuclei are scaled differently from those of the heavier nuclei, and we employ a specialized form for the electron energy level surface. As… (More)
We review mathematical results concerning exponentially small corrections to adiabatic approximations and Born–Oppenheimer approximations .
We propose an alternative to the usual time–independent Born–Oppenheimer approximation that is specifically designed to describe molecules with symmetrical Hydrogen bonds. In our approach, the masses of the Hydrogen nuclei are scaled differently from those of the heavier nuclei, and we employ a specialized form for the electron energy level surface.… (More)
Although real, normalized Gaussian wave packets minimize the product of position and momentum uncertainties, generic complex normalized Gaussian wave packets do not. We prove they minimize an alternative product of uncertainties that correspond to variables that are phase space rotations of position and momentum.
Dedicated to the memory of our friend and colleague Pierre Duclos. Abstract We study the time behavior of wave functions involved in tunneling through a smooth potential barrier in one dimension in the semiclassical limit. We determine the leading order component of the wave function that tunnels. It is exponentially small in 1/. For a wide variety of… (More)
The standard Born–Oppenheimer approximation for a diatomic molecule yields an expansion in powers of for the bound state associated with a given electron energy level, a fixed vibrational quantum number n, and a fixed angular momentum quantum number l. The expansion parameter is the fourth root of the ratio of the electron mass divided by the mean nuclear… (More)