Use of Gauss-Hermite quadrature to approximate the asymptotic behavior of vibronic resonance wave functions

Abstract

Wave functions obtained employing a standard complex Hamiltonian matrix diagonalization procedure are square-integrable and therefore cannot accurately describe the asymptotic character of resonance solutions of the Schrodinger equation. The nature of this approximation is investigated by means of explicit calculations which are based on diabatic RKR potentials for the B 1Σ+ — D′ 1Σ+ vibronic resonance states of the CO molecule. It is shown that expanding the basis of complex harmonic oscillator fuctions gradually improves the description of the exact resonance wave functions out to ever larger internuclear distances on the real axis before they take on artificial bound-state characteristics due to the square-integrable character of the basis functions. In order to solve the diagonalization problems for as many as 500 such basis functions, it proves necessary to employ specialized numerical techniques such as Gauss-Hermite quadrature to evaluate the required Hamiltonian matrix elements.