Lanczos subspace filter diagonalization: Homogeneous recursive filtering and a low-storage method for the calculation of matrix elements

We develop a new iterative filter diagonalization (FD) scheme based on Lanc zos subspaces and demonstrate its application to the calculation of bound-s tate and resonance eigenvalues. The new scheme combines the Lanczos three-t erm vector recursion for the generation of a tridiagonal representation of the Hamiltonian with a three-term scalar recursion to generate filtered sta tes within the Lanczos representation. Eigenstates in the energy windows of interest can then be obtained by solving a small generalized eigenvalue pr oblem in the subspace spanned by the filtered states. The scalar filtering recursion is based on the homogeneous eigenvalue equation of the tridiagona l representation of the Hamiltonian, and is simpler and more efficient than our previous quasi-minimum-residual filter diagonalization (QMRFD) scheme (H. G. Yu and S. C. Smith, Chem. Phys. Lett., 1998, 283, 69), which was bas ed on solving for the action of the Green operator via an inhomogeneous equ ation. A low-storage method for the construction of Hamiltonian and overlap matrix elements in the filtered-basis representation is devised, in which contributions to the matrix elements are computed simultaneously as the rec ursion proceeds, allowing coefficients of the filtered states to be discard ed once their contribution has been evaluated. Application to the HO2 syste m shows that the new scheme is highly efficient and can generate eigenvalue s with the same numerical accuracy as the basic Lanczos algorithm.