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.

Higher-order wavelet reconstruction/differentiation filters and Gibbs phenomena

“…In general, longer wavelets require greater computational effort and pose greater awkwardness in dealing with discontinuous boundaries. On the other hand, there may be applications where longer wavelets remain useful, e.g., applications based on projection using single-point quadrature for Coiflets [40] or generalized Coiflets [7] where the error order is at most M. For our applications, the arguments align solely in favor of using low-L basis functions. This minimizes the number of straddling functions at a boundary and minimizes the length of the multiscale recursions.…”

“…Despite that common sentiment, Keinert and Kwon [5] and Neelov and Goedecker [6] demonstrated that one can beat this limit in reconstruction. Our group has recently generalized the latter results so that the reconstruction error can be tuned just as freely as the projection error [7]. There are particular consequences of this, as will be shown.…”

“…We now build a generalized theory for reconstruction filters accurate beyond the approximation order and even beyond the length L. These include Daubechies and leastasymmetric or symlet families with M = L/2 [1], as well as Coiflet or generalized Coiflet functions with M < L/2 [7]. Consider the scaling function moments…”

“…(3) If  and  are both changed by some integer i, it can be seen from Eqs. (7) and (8) that X +i, +i = X ,  = X 0,  and there is only one X matrix to be calculated. However,  takes non-integer values while  does not.…”

“…For p ≥ L, one can still calculate a p and m p , but they are no longer equal. The results for p = 0 (reconstruction) were quoted and used already in solely filter-based quantum eigenvalue problems by Acevedo, et al [7]. The practical limitation for n is that very large powers p cause loss of precision in the recursion.…”