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 χ. 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 cn=〈χ|Tn(Ĥ)|χ〉, calculation of which requires a minimal storage if the Ĥ 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 Ĥ using the same sequence {cn}. We present an application of the method to a realistic quantum dynamics...

[1]  J. Cullum,et al.  Lanczos algorithms for large symmetric eigenvalue computations , 1985 .

[2]  Stephen K. Gray,et al.  Novel methods for spectral analysis , 1991 .

[3]  Daniel Neuhauser,et al.  Extraction, through filter‐diagonalization, of general quantum eigenvalues or classical normal mode frequencies from a small number of residues or a short‐time segment of a signal. I. Theory and application to a quantum‐dynamics model , 1995 .

[4]  Wei Zhu,et al.  A general time-to-energy transform of wavepackets. Time-independent wavepacket-Schrödinger and wavepacket-Lippmann—Schwinger equations , 1993 .

[5]  S. L. Marple,et al.  Dispersion curves from short-time molecular dynamics simulations. 1. Diatomic chain results , 1988 .

[6]  R. Levine,et al.  An information theoretical approach to inversion problems , 1980 .

[7]  V. Mandelshtam,et al.  Recursion polynomial expansion of the Green’s function with absorbing boundary conditions: Calculations of resonances of HCO by filter diagonalization , 1995 .

[8]  Wei Zhu,et al.  Acceleration of convergence in the polynomial-expanded spectral density approach to bound and resonance state calculations , 1995 .

[9]  T. Carrington,et al.  A time-to-energy Fourier resolution method for calculating bound state energies and wavefunctions. Analysis of the method and application to 2D ArHCl , 1994 .

[10]  V. Mandelshtam,et al.  Spectral projection approach to the quantum scattering calculations , 1995 .

[11]  D. Neuhauser.  Bound state eigenfunctions from wave packets: Time→energy resolution , 1990 .

[12]  J. Tennyson,et al.  On the Meyer–Botschwina–Burton potential energy surface for H3+ , 1993 .

[13]  W. M. Carey,et al.  Digital spectral analysis: with applications , 1986 .

[14]  R. Kosloff,et al.  Large amplitude ground state vibrational coherence induced by impulsive absorption in CsI. A computer simulation , 1989 .

[15]  V. Mandelshtam,et al.  Bound states and resonances of the hydroperoxyl radical HO2: An accurate quantum mechanical calculation using filter diagonalization , 1995 .

[16]  Wei Zhu,et al.  Orthogonal polynomial expansion of the spectral density operator and the calculation of bound state energies and eigenfunctions , 1994 .

[17]  S. Gray.  Wave packet dynamics of resonance decay: An iterative equation approach with application to HCO→H+CO , 1992 .

[18]  Daniel Neuhauser,et al.  Circumventing the Heisenberg principle: A rigorous demonstration of filter‐diagonalization on a LiCN model , 1994 .