Iterative Methods for Computing Vibrational Spectra

Abstract

I review some computational methods for calculating vibrational spectra. They all use iterative eigensolvers to compute eigenvalues of a Hamiltonian matrix by evaluating matrix-vector products (MVPs). A direct-product basis can be used for molecules with five or fewer atoms. This is done by exploiting the structure of the basis and the structure of a direct product quadrature grid. I outline three methods that can be used for molecules with more than five atoms. The first uses contracted basis functions and an intermediate (F) matrix. The second uses Smolyak quadrature and a pruned basis. The third uses a tensor rank reduction scheme.

On the vibrations of formic acid predicted from first principles

“…A full direct product basis can be employed as a basis in eqn (6) ( e.g. products of weighted classical orthogonal polynomials, 29 harmonic oscillator basis functions 30 ), however, without further approximations, this leads to an ansatz with an exponential scaling with regard to the size of the system treated and therefore a blowup in the number of terms in the expansion. The truncation of the direct product basis according to some condition is therefore common practice.…”

Program Synthesis of Sparse Algorithms for Wave Function and Energy Prediction in Grid-Based Quantum Simulations

“…As such, there has been a large amount of work focused on reducing the computational effort associated with DVR calculations. − , A primary interest has been the development of efficient pruning methods, which seek to remove those grid points in the full direct-product grid, which are predicted to have essentially zero wave function amplitude for a given PES; for example, research by Carrington and co-workers has clearly demonstrated how pruned basis sets can be developed to significantly reduce the computational cost of DVR calculations. , A complementary approach is to seek to reduce the number of grid points required in each DOF, for example by seeking to optimize the choice of underlying DVR basis set to best match the nature of the problem at hand; this is exemplified by the potential-optimized DVR methodology, as well as developments aimed at using nonproduct coordinate grids . Finally, we note another alternative to reducing computational effort, namely, the use of ML strategies to generate global PESs using a reduced number of ab initio PES evaluations.…”