Two new methods for computing vibrational energy levels

Abstract

I review two new ideas for coping with the size of large product basis sets and large product grids when one computes vibrational energy levels. The first is based on a tensor reduction scheme. It exploits advantages of a sum-of-products potential.The key idea is to use a basis each of whose function is a sum of optimized products and to compress the number of terms in each basis function. When the potential does not have the sum-of-products form, it is usually necessary to use quadrature. The second idea uses a nondirect product grid that has structure and is therefore compatible with efficient matrix-vector products.Résumé : Cet article décrit deux nouvelles approches permettant l'utilisation d'une grande base de produits afin de calculer des niveaux vibrationnels. La première est fondée sur la réduction des tenseurs. Elle exploite les avantages d'un potentiel ayant la forme d'une somme de produits. L'idée clef est d'utiliser une base dont chacune des fonctions est une somme de produits optimisés et de comprimer le nombre de termes dans chaque fonction de la base. Si le potentiel n'est pas une somme de produits, il faut utiliser une quadrature. La seconde approche consiste à utiliser une grille qui n'est pas un produit direct mais qui a assez de structure pour faciliter le calcul de produits matrice-vecteur.Mots-clés : théorie, spectroscopie rotationnelle-vibrationnelle, méthodes itératives, algorithme de Lanczos.

Perspective: Accurate ro-vibrational calculations on small molecules

“…[148,149] One major advantage of this approach is that transition intensities for isotopically substituted species can be computed with some confidence at a similar level of accuracy. Thus, for example, recent calculations have provided transition intensives for the important, radio-active trace species O 14 CO with what can be assumed to be same accuracy as those computed for the main, O 16 CO, isotopologue. [150] The ab initio computation of precise transition intensities requires some adaption to standard procedures used for both electronic stucture and nuclear motion calculations.…”

“…[12] DVR methods are not strictly variational and can show convergence from below, [13] but have proved robust and reliable in practical calculations. Development and improvement of these methods continues apace [14] and the whole area of high accuracy treatment of nuclear motion calculations has been dubbed the fourth age of quantum chemistry. [15] For the purposes of this perspective I will take small molecules to mean ones containing up to five atoms.…”

Electronic and Molecular Dynamics by the Quantum Wave Packet Method

“…There are many studies on the introduction of the sparse grid method in the literature, and their theoretical details are not presented here. With multidimensional sparse grid and hierarchical bases, vibrational states of a molecule with dimensionality of 64 have been obtained [34], electronic structures of small molecules have been solved [35][36][37][38][39][40], and solution of the time-dependent Schrödinger equation of model potentials with split operator method, with hierarchical Fourier basis, have been reported [41].…”