Time-dependent quantum-mechanical methods for molecular dynamics

The basic framework of time-dependent quantum-mechanical methods for molecular dynamics calculations is described. The central problem addressed by computational methods is a discrete representation of phase space. In classical mechanics, phase space is represented by a set of points whereas in quantum mechanics it is represented by a discrete Hilbert space. The discretization described in this paper is based on collocation. Special cases of this method include the discrete variable representation (DVR) and the Fourier method. The Fourier method is able to represent a system in phase space with the efficiency of one sampling point per unit volume in phase space h, so that, with the proper choice of the initial wave function, exponential convergence is obtained in relation to the number of sampling points. The numerical efficiency of the Fourier method leads to the conclusion that computational effort scales semilinearly with the volume in the phase space occupied by the molecular system. Methods of time propagation are described for time-dependent and time-independent Hamiltonians. The time-independent approaches are based on a polynomial expansion of the evolution operator. Two of these approaches, the Chebychev propagation and the Lanczos recurrence, are also compared. Methods to obtain the Raman spectra directly by using the Chebychev propagation method are shown. For time-dependent problems unitary short-time propagators are described: the second-order differencing and the split operator. Consideration of all these methods has led to scaling laws of computation. The conclusion from such scaling laws is that, for simulations of complex molecular systems, approximation techniques have to be employed which reduce the dimensionality of the problem. The time-dependent self-consistent field (TDSCF) is suggested. Finally, a brief description is presented of current applications of the time-dependent method.