A low-storage filter diagonalization method for quantum eigenenergy calculation or for spectral analysis of time signals
- Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482 (United States)
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, {bold H}, and overlap, {bold 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. {bold 102}, 8011 (1995)] {bold H} and {bold S} are computed directly using only the Chebyshev coefficients c{sub n}={l_angle}{chi}{vert_bar}T{sub n}({cflx H}){vert_bar}{chi}{r_angle}, calculation of which requires a minimal storage if the {cflx H} matrix is sparse. The expressions for {bold H} and {bold S} are analytically simple, computationally very inexpensive and stable. The method can be used to obtain all the eigenvalues of {cflx H} using the same sequence {l_brace}c{sub n}{r_brace}. We present an application of the method to a realistic quantum dynamics problem of calculating all bound state energies of H{sub 3}{sup +} molecule. Since the sequence {l_brace}c{sub n}{r_brace} 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{sub n})={summation}{sub k}d{sub k}cos(t{sub n}{omega}{sub k}) generated by a set of N=100000 frequencies and amplitudes, ({omega}{sub k},d{sub k}). It is demonstrated that all the {omega}{sub k}`s and d{sub k}`s can be obtained to very high precision using the minimal information, i.e., 200 000 sampling points. {copyright} {ital 1997 American Institute of Physics.}
- DOE Contract Number:
- FG03-94ER14458
- OSTI ID:
- 495857
- Journal Information:
- Journal of Chemical Physics, Vol. 106, Issue 12; Other Information: PBD: Mar 1997
- Country of Publication:
- United States
- Language:
- English
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