An effective Hamiltonian-based method for mixed quantum-classical dynamics on coupled electronic surfaces

Abstract

We describe an approximate method for treating the mixed quantum-classical (QC) dynamics of many-body systems on N coupled electronic surfaces. The approach is based on calculating N×N reduced Hamiltonian matrices for the classical and quantal degrees of freedom by partial averaging, and then solving the appropriate equations of motion—Hamilton’s equations or the Schrödinger equation—self-consistently. The degrees of freedom requiring a quantum mechanical description are treated using a multistate Schrödinger equation with classically averaged effective time-dependent Hamiltonians and off-diagonal couplings. The classical degrees of freedom are treated by propagating N ensembles of trajectories, one on each electronic surface, using N reduced classical Hamiltonians defined in terms of the expectation value of the full Hamiltonian calculated using the evolving quantum wave functions. An ansatz is proposed to approximately estimate classical off-diagonal density matrix elements required for calculating the classically averaged interactions that couple quantum wave functions on different electronic states. We present the theory and then test it for a simple two-dimensional and two-state model system. Exact quantum and multiconfiguration time-dependent self-consistent-field (MCTDSCF) calculations are carried out to evaluate the QC performance. Good agreement between the MCTDSCF and QC results is obtained for the model considered.