Computing vibrational energy levels by using mappings to fully exploit the structure of a pruned product basis

Abstract

For the purpose of calculating (ro-)vibrational spectra, rate constants, scattering cross sections, etc. product basis sets are very popular. They, however, have the important disadvantage that they are unusably large for systems with more than four atoms. In this paper we demonstrate that it is possible to efficiently use a basis set obtained by removing, from a product basis set, functions associated with the largest diagonal Hamiltonian matrix elements. This is done by exploiting the fact that for every factor of every term in the Hamiltonian, there is a basis-set order in which the matrix representation of the factor is block diagonal. Due to this block diagonality the Lanczos algorithm can be implemented efficiently. Tests with model Hamiltonians with as many as 32 coordinates illustrate the merit of the ideas.

Two new methods for computing vibrational energy levels

“…[93][94][95][96][97][98][99][100][101] For example, if D = 15 (a seven-atom molecule) and 15 basis functions are used for each coordinate, then the size of the direct product basis is 4 × 10 17 , but by removing all functions n1 1 ͑q 1 ͒ n2 2 ͑q 2 ͒Ê n15 15 ͑q 15 ͒n k ϭ 0,1,Ê for which ͚k n k Ͼ b, the size of the basis is reduced to 7.7 × 10 7 . Pruning will always reduce the spectral range of the matrix and, therefore, decrease the number of Lanczos iterations required to obtain a converged spectrum.…”

Mathematical Methods in Quantum Molecular Dynamics

“…One way to deal with the basis-size problem is to use basis functions that are not products of functions of a single variable [27,28,29,30,23,24,20,31,32,33,21]. Another strategy is to use product functions but only a fraction of the full product basis [34,35,36,37,38,39,40,41,42]. Often many of the functions in a product basis have large zeroth order energies.…”

Perspective: Computing (ro-)vibrational spectra of molecules with more than four atoms

“…97 Even when the basis has no structure, as long as the Hamiltonian is a SOP, it is possible to evaluate MVPs without computing (even non-zero) Hamiltonian matrix elements, by using a mapping. 96 Coupled cluster (CC) methods for vibrational problems 98,99 can also be classified as SOP/I/P. The SOP structure of the Hamiltonian is used to derive an equation for the cluster operator with which one obtains an effective Hamiltonian.…”

“…If the number of terms is large, it is helpful to use a sparse storage format, compressed row storage (CRS) is popular, to store the matrix. 81,96 It is then possible to feed the matrix in the CRS format into ARPACK. 81 When the pruned basis has a structure, it can be used to evaluate MVPs without computing Hamiltonian matrix elements.…”