Computational Time-Dependent Two-Electron Theory and Long-Time Propagators

The computational theory for the time-dependent calculation of the solution of the Schrödinger equation for two electrons is reviewed. A discussion of the time-dependent Hartree-Fock theory and the Dirac-Frenkel variational principle is critiqued, and a new time-dependent exchange, correlation theory is presented. The present work is concerned with the solution of the resultant time-dependent orbital equations in terms of long-time propagators. The development of propagators for the solution of the Schrödinger equation is presently a very active area of research. The need for accurate short-time propagators results from the attempt to model atomic and molecular dynamics in rapidly varying time-dependent fields. Long-time propagators are needed for two basic problems: (1) extraction of eigenvalues and eigenvectors for large systems by time-dependent spectral methods; and (2) reaction dynamics, both for time-independent and time-dependent potentials. Various long-time propagators are reviewed including several polynomial methods. Specifically, the Chebyshev and Hermite polynomial methods, and a new synthetic-cartesian-exponential polynomial propagator, are compared and contrasted. This last method is new and is apparently unique among polynomial propagators since it is unitary and conserves the norm. The discrete variable representation, pseudo-spectral method is discussed as the mechanism of choice for performing the operator mapping.