Rovibrational spectroscopy calculations of neon dimer using a phase space truncated Weyl-Heisenberg wavelet basis

Spectral convergence of the quadrature discretization method in the solution of the Schrödinger and Fokker-Planck equations: Comparison with sinc methods

Spectral methods based on nonclassical polynomials and Fourier basis functions or sinc interpolation techniques are compared for several eigenvalue problems for the Fokker-Planck and Schrodinger equations. A very rapid spectral convergence of the eigenvalues versus the number of quadrature points is obtained with the quadrature discretization method (QDM) and the appropriate choice of the weight function. The QDM is a pseudospectral method and the rate of convergence is compared with the sinc method reported by Wei [J. Chem. Phys., 110, 8930 (1999)]. In general, sinc methods based on Fourier basis functions with a uniform grid provide a much slower convergence. The paper considers Fokker-Planck equations (and analogous Schrodinger equations) for the thermalization of electrons in atomic moderators and for a quartic potential employed to model chemical reactions. The solution of the Schrodinger equation for the vibrational states of I2 with a Morse potential is also considered.

Improving the accuracy of Weyl-Heisenberg wavelet and symmetrized Gaussian representations using customized phase-space-region operators

A particular basis set method developed by one of the authors, involving maximally localized orthogonal Weyl-Heisenberg wavelets (or "weylets") and a phase space truncation scheme, has been successfully applied to exact quantum calculations for many degrees of freedom (DOF's) [B. Poirier and A. Salam, J. Chem. Phys. 121, 1740 (2004)]. However, limitations in accuracy arise in the many-DOF case, owing to memory limits on conventional computers. This paper addresses this accuracy limitation by introducing phase space region operators (PSRO's) that customize individual weylet basis functions for the problem of interest. The construction of the PSRO's is straightforward, and does not require a priori knowledge of the desired eigenstates. The PSRO, when applied to weylets, as well as to simple phase space Gaussian basis functions, exhibits remarkable improvements in accuracy, reducing computed eigenvalue errors by orders of magnitude. The method is applied to various model systems at varying DOF's.