Enhanced matrix spectroscopy: The preconditioned Green-function block Lanczos algorithm

摘要

We present herein the results of a doubly filtered block Lanczos code, which effectively uses shifted simultaneous inverse iteration to make a block tridiagonal representation of H, a Hamiltonian matrix. The first filter preconditions the starting block of Lanczos vectors through one or more applications of p(E), the preconditioning operator. This block is then used to seed the Lanczos recursion, driven with the Green-function filter G(E)=(EI-H)(-1). Each set of successively generated Lanczos vectors is orthogonalized against all previous ones. These steps allow us to converge eigenvalues near E, in the interior region of the eigenspectrum, with extreme accuracy. Degenerate eigenvalues are reported; ''ghost'' eigenvalues (multiple copies of eigenvalues that have already converged) are avoided. For a set number of Lanczos recursions, the use of a preconditioner effectively doubles the number of converged eigenvalues than are resolved without its use. The computation time is increased almost imperceptibly. The 2e(g)<--a(g) torsional transition for the water trimer, (H2O)(3), is examined in an application of our method. We resolve the quantities needed for this calculation is less than one-fifth the time required to directly diagonalize the matrices, with no loss of accuracy.