Accelerating the calculation of energy levels and wave functions using an efficient preconditioner with the inexact spectral transform method

In an earlier paper [J. Chem. Phys. 112, 8765 (2000)] our group introduced a preconditioned inexact spectral transform method for calculating energy levels and wave functions. Although we could calculate high-lying levels with far fewer matrix–vector products than with the filter diagonalization method of Mandelshtam and Taylor, even better performance can be achieved with a better preconditioner. In this paper, we develop an extremely efficient preconditioner consisting of two components: (1) transformation to an optimal separable basis, in which off-diagonal elements of the Hamiltonian matrix are minimized; and (2) removal of all off-diagonal coupling near the energies of interest. The new preconditioner works extremely well; it enables us to calculate high-lying vibrational states of H2O with orders of magnitude fewer matrix–vector products than for all other known methods. The new preconditioner should also accelerate the calculation of other quantities, such as photodissociation cross sections and rate constants.