Quantum dynamics calculations using symmetrized, orthogonal Weyl-Heisenberg wavelets with a phase space truncation scheme. III. Representations and calculations

Abstract

In a previous paper [J. Theo. Comput. Chem. 2, 65 (2003)], one of the authors (B.P.) presented a method for solving the multidimensional Schrodinger equation, using modified Wilson-Daubechies wavelets, and a simple phase space truncation scheme. Unprecedented numerical efficiency was achieved, enabling a ten-dimensional calculation of nearly 600 eigenvalues to be performed using direct matrix diagonalization techniques. In a second paper [J. Chem. Phys. 121, 1690 (2004)], and in this paper, we extend and elaborate upon the previous work in several important ways. The second paper focuses on construction and optimization of the wavelength functions, from theoretical and numerical viewpoints, and also examines their localization. This paper deals with their use in representations and eigenproblem calculations, which are extended to 15-dimensional systems. Even higher dimensionalities are possible using more sophisticated linear algebra techniques. This approach is ideally suited to rovibrational spectroscopy applications, but can be used in any context where differential equations are involved.

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

“…[30][31][32][33][34][35][36][37][38] Wavelets are localized in both the position and momentum representations, and are commonly used for spectral analysis, but have also found application to quantum dynamics. 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.…”

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

“…The two sets of 4D SD functions are then combined ͑there are 25 000 000 of them͒ to form an 8D basis from which we retain the functions with the smallest diagonal elements. 41,42 In 16D we use two groups of eight coordinates ͑retaining 5000 functions for each group͒, each of which is formed from bases for two groups of four coordinates ͑retaining 5000 functions for each group͒. A basis of selected EF functions ͑i.e., ␣ c =0͒ is formed in the same way ͑by discarding product functions with larger diagonal elements͒.…”

“…41 By using efficient indexing and sorting algorithms it is possible to identify the product functions to be retained without first building the full direct product basis. 41,42 However, for the purpose of understanding why the method works, it is useful to imagine a huge matrix representing the multidimensional Hamiltonian in the basis of product functions sorted so that the diagonal matrix elements increase from top to bottom. The key to the success of the pruned product approach of Ref.…”

Iterative Methods for Computing Vibrational Spectra

“…Many authors have implemented basis pruning strategies [16,18,19,53,62,63,[70][71][72][73][74][75][76][77][78][79][80][81][82][83][84][85][86]. Although pruning has the obvious advantage that it decreases the size of the vectors one must store and the spectral range of the Hamiltonian matrix, if one uses an iterative method, it complicates the evaluation of MVPs.…”