Matrix-free application of Hamiltonian operators in Coifman wavelet bases

Abstract

A means of evaluating the action of Hamiltonian operators on functions expanded in orthogonal compact support wavelet bases is developed, avoiding the direct construction and storage of operator matrices that complicate extension to coupled multidimensional quantum applications. Application of a potential energy operator is accomplished by simple multiplication of the two sets of expansion coefficients without any convolution. The errors of this coefficient product approximation are quantified and lead to use of particular generalized coiflet bases, derived here, that maximize the number of moment conditions satisfied by the scaling function. This is at the expense of the number of vanishing moments of the wavelet function (approximation order), which appears to be a disadvantage but is shown surmountable. In particular, application of the kinetic energy operator, which is accomplished through the use of one-dimensional (1D) [or at most two-dimensional (2D)] differentiation filters, then degrades in accuracy if the standard choice is made. However, it is determined that use of high-order finite-difference filters yields strongly reduced absolute errors. Eigensolvers that ordinarily use only matrix-vector multiplications, such as the Lanczos algorithm, can then be used with this more efficient procedure. Applications are made to anharmonic vibrational problems: a 1D Morse oscillator, a 2D model of proton transfer, and three-dimensional vibrations of nitrosyl chloride on a global potential energy surface.