A low-storage filter diagonalization method for quantum eigenenergy calculation or for spectral analysis of time signals

A new version of the filter diagonalization method of diagonalizing large real symmetric Hamiltonian matrices is presented. Our previous version would first produce a small set of adapted basis functions by applying the Chebyshev polynomial expansion of the Green's function on a generic initial vector chi. The small Hamiltonian, H, and overlap, S, matrices would then be evaluated in this adapted basis and the corresponding generalized eigenvalue problem would be solved yielding the desired spectral information. Here in analogy to a recent work by Wall and Neuhauser [J. Chem. Phys. 102, 8011 (1995)]H and S are computed directly using only the Chebyshev coefficients c(n) = [chi/T-n((H) over cap)/chi], calculation of which requires a minimal storage if the (H) over cap matrix is sparse. The expressions for H and S are analytically simple, computationally very inexpensive and stable. The method can be used to obtain all the eigenvalues of (H) over cap using the same sequence {c(n)}. We present an application of the method to a realistic quantum dynamics problem of calculating all bound state energies of H-3(+) molecule. Since the sequence {c(n)} is the only input required to obtain all the eigenenergies, the present method can be reformulated for the problem of spectral analysis of a real symmetric time signal defined on an equidistant time grid. The numerical example considers a model signal C(t(n)) = Sigma(k)d(k) cos(t(n) omega(k)) generated by a set of N = 100 000 frequencies and amplitudes, (omega(k),d(k)). It is demonstrated that all the omega(k)'s and d(k)'s can be obtained to very high precision using the minimal information, i.e., 200 000 sampling points. (C) 1997 American Institute of Physics.