PHASE-SPACE APPROACH FOR OPTIMIZING GRID REPRESENTATIONS - THE MAPPEDFOURIER METHOD

The representation of a quantum system by an evenly spaced Fourier gri d is examined. This grid faithfully represents wave functions whose pr ojection is contained in a rectangular phase space. This is mathematic ally equivalent to a band limited function with finite support. In gen eral, wave packets decay exponentially in classically forbidden region s of phase space. This idea is then used first to optimize the rectang ular shape of the Fourier grid, leading to exponential convergence. Ne vertheless, in most cases the representation is suboptimal. The repres entation efficiency can then be extremely enhanced by mapping the coor dinates. 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 th e coordinates, improve the representation dramatically. A specific sca ling transformation enables the representation of the notoriously diff icult Coulomb potentials. The scaling transformation enables one to ex tract almost as many converged eigenstate energies as there are grid p oints. The method is extendible to more than one dimension, which is d emonstrated by the study of the H-2(+) problem. This scaling transform ation can bridge the gap between quantum chemistry and quantum molecul ar dynamics by enabling the treatment of electronic problems in the vi cinity of Coulomb potentials by grid methods developed for molecular d ynamics.