Variational Solutions for Resonances by a Finite-Difference Grid Method.

Roie Dann,Guy Elbaz,Jonathan Berkheim,Nimrod Moiseyev,Omri Nitecki,Alan Muhafra,Daniel Wilczynski

doi:10.3390/molecules26175248

Abstract

We demonstrate that the finite difference grid method (FDM) can be simply modified to satisfy the variational principle and enable calculations of both real and complex poles of the scattering matrix. These complex poles are known as resonances and provide the energies and inverse lifetimes of the system under study (e.g., molecules) in metastable states. This approach allows incorporating finite grid methods in the study of resonance phenomena in chemistry. Possible applications include the calculation of electronic autoionization resonances which occur when ionization takes place as the bond lengths of the molecule are varied. Alternatively, the method can be applied to calculate nuclear predissociation resonances which are associated with activated complexes with finite lifetimes.

Highlights

In 1978 Frank Weinhold together with Phil Certain and Nimrod Moiseyev, in their work on the complex variational principle, paved the way for the use of electronic structure computational algorithms to metastable states [1]
We show how the the present finite difference method (FDM) can be utilized to calculate the energies and widths of mestasbale states states, embedded in the continuous part of the spectrum, which are associated with the poles of the scattering matrix
We show that by fixing the grid spacing in the finite difference method one obtains a variational principle within the finite box approximation

Summary

Introduction

In 1978 Frank Weinhold together with Phil Certain and Nimrod Moiseyev, in their work on the complex variational principle (a stationary point rather than an upper bound as in Hermitian QM), paved the way for the use of electronic structure computational algorithms to metastable (resonance) states [1]. In this framework, the energies and inverse lifetimes of atoms and molecules are associated with the real and imaginary parts of the complex eigenvalues of non-Hermitian Hamiltonians, respectively. The origin of this failure can be traced back to the fact that the standard FDM does not satisfy the variational principle

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Conclusion