A Direct, Time Dependent, Lanczos Propagation Method for Non-Orthogonal Basis Sets

We have developed an efficient approach for solving the time-dependent Schroedinger equation for the interaction of a strong laser pulse with a general atom, when the many-electron basis set is non-orthogonal. The propagation equations have the form, iS dC(t) / dt = HC(t) where S and H are respectively the overlap and Hamiltonian matrices in the many-electron space. By a succession of Lanczos orthogonalizations, the Hamiltonian is reduced to tri-diagonal form, but the overlap matrix remains full in the small, Lanczos basis. Thus, we are faced with solving a small, generalized eigenvalue problem at each step of the Lanczos recursion. The approach is still dominated by the need to find an efficient way to multiply the H and S matrix on a vector. Some examples of the new method will be presented in the talk.