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.