Using simultaneous diagonalization and trace minimization to make an efficient and simple multidimensional basis for solving the vibrational Schrödinger equation

Abstract

In this paper we improve the product simultaneous diagonalization (SD) basis method we previously proposed [J. Chem. Phys. 122, 134101 (2005)] and applied to solve the Schrodinger equation for the motion of nuclei on a potential surface. The improved method is tested using coupled complicated Hamiltonians with as many as 16 coordinates for which we can easily find numerically exact solutions. In a basis of sorted products of one-dimensional (1D) SD functions the Hamiltonian matrix is nearly diagonal. The localization of the 1D SD functions for coordinate qc depends on a parameter we denote alphac. In this paper we present a trace minimization scheme for choosing alphac to nearly block diagonalize the Hamiltonian matrix. Near-block diagonality makes it possible to truncate the matrix without degrading the accuracy of the lowest energy levels. We show that in the sorted product SD basis perturbation theory works extremely well. The trace minimization scheme is general and easy to implement.

Quantum-mechanical wavepacket propagation in a sparse, adaptive basis of interpolating Gaussians with collocation

“…Hence, already for N \ 5 computer algebra programs like MATHEMATICA have to be employed to obtain numerically usable expressions of the Laplace-Beltrami operator [eqn (25)]. This, in turn, leads to the phenomenon that most programs in this field of research are not universal but need to be re-written for each new system under consideration.…”

“…), hence it is hard to turn this approach into a black-box method. Recently, Dawes and Carrington 22,24,25 have used simultaneous diagonalization to construct DVRs corresponding to pruned, contracted basis functions; this appears to be a promising strategy for timeindependent problems.…”

Dynamical pruning of static localized basis sets in time-dependent quantum dynamics

“…3,39,47 The method was used here to generate near diagonal matrices, rather than completely diagonal matrices, because full diagonalization is not possible for noncommuting matrices.…”

“…Dawes and Carrington have shown that a basis set generated in this way can be very effective for time-independent quantum dynamics. 3,39 In this study, we test the SD technique of Dawes and Carrington, along with dynamical pruning of the basis set, in reduced-dimensional, time-dependent wave packet calculations for reactive scattering of H 2 on a Pt͑211͒ stepped surface. This system has recently been the subject of classical trajectory [40][41][42] and quantum wave packet calculations, 43 and is thus a suitable test bed for new quantum dynamics methods.…”

“…33,34,[36][37][38] Here we use a different approach to generating PSL functions, one that has recently been introduced by Dawes and Carrington. 3,39 They use the method of simultaneous diagonalization ͑SD͒, which seeks a single set of eigenfunctions that diagonalize the position and momentum operator matrices. It can be shown that this is equivalent to simultaneously localizing a set of basis functions in the position and momentum representations.…”