Spectral difference methods for solving the differential equations of chemical physics

Abstract

Spectral differences [D. A. Mazziotti, Chem. Phys. Lett. 299, 473 (1999)] is a family of techniques for solving differential equations in which the summation in the numerical derivative is accelerated to produce a matrix representation that is not only exponentially convergent like the discrete variable representation (DVR) and other spectral methods but also sparse like traditional finite differences and finite elements. Building upon important work by Boyd [Comput. Methods Appl. Mech. Eng. 116, 1 (1994)] and Gray and Goldfield [J. Chem. Phys. 115, 8331 (2001)], we explore a new class of spectral difference methods which yields solutions that are more accurate than high-order finite differences by several orders of magnitude. With the generating weight for Gegenbauer polynomials we design a new spectral difference method where the limits of an adjustable parameter α generate both finite differences (α=∞), emphasizing the low Fourier frequencies, and a truncated sinc-DVR (α=0), emphasizing all Fourier frequencies below the aliasing limit of the grid. A range of choices for α∈[0,∞] produces solutions which are significantly better than the equivalent order of finite differences. We compare the Gegenbauer-weighted spectral differences with methods by Boyd as well as Gray and Goldfield which employ a hyperbolic secant and a step function as frequency weights, respectively. The solutions from the Gegenbauer- and the sech-weighted differences are shown to be less sensitive to parameter selection than the step-weighted differences. We illustrate all of the spectral difference methods through vibrational and quantum control calculations with diatomic iodine and the van der Waals cluster NeCO. Spectral differences also have important applications in molecular dynamics and electronic structure as well as other areas of science and engineering.