Solution of the Schrödinger equation by a spectral method II: Vibrational energy levels of triatomic molecules

Abstract

The spectral method utilizes numerical solutions to the time-dependent Schrödinger equation to generate the energy eigenvalues and eigenfunctions of the time-independent Schrödinger equation. Accurate time-dependent wave functions ψ(r, t) are generated by the split operator FFT method, and the correlation function 〈ψ(r, 0) ‖ ψ(r, t)〉 is computed by numerical integration. Fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system. Analysis of the location of these peaks reveals the eigenvalues with high accuracy. Additional Fourier transforms of ψ(r, t) with respect to time generate the eigenfunctions. Previous applications of the method were to two-dimensional potentials. In this paper energy eigenvalues and wave functions obtained with the spectral method are presented for vibrational states of three-dimensional Born–Oppenheimer potentials applicable to SO2, O3, and H2O. The energy eigenvalues are compared with results obtained with the variational method. It is concluded that the spectral method is an accurate tool for treating a variety of practical three-dimensional potentials.

Theoretical study of the electronic excited states in ultrathin ionic layers supported on metal surfaces: NaCl/Cu(111)

“…The time propagation given by Eq. (2) is performed numerically using the split-operator technique for the exponential of the Hamiltonian operator 37 and the Fourier-grid pseudospectral approach for the kinetic energy term. 38 A time propagation step t = 0.1 a.u.…”

Design of Optimal Laser Fields to Control Vibrational Excitations in Carboxy-myoglobin

“…The wave function, ψ i (t), and the Lagrange multiplier, χ f (t), are propagated in time using the second order split-operator method [19]. In order to determine the maximum value of the cost functional a line search is performed along the Polak-Ribiere-Polyak search direction calculated using Eq.…”

A novel quantum dynamical approach in electron microscopy combining wave-packet propagation with Bohmian trajectories

“…The approach here follows this scheme. In particular, first the time-dependent Schrödinger equation is numerically solved by means of the split-operator method [28][29][30][31]. This equation can be solved by a variety of numerical methods available in the literature (see, for instance, Refs.…”