Determination of eigenstates via lanczos-based forward substitution and filter-diagonalization

Journal of Computational Physics

Volumn 136, Issue 2, 1997, Pages 494-502

  Chen, Rongqing ^a   Guo, Hua ^a  

^a UNIVERSITY OF TOLEDO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

AND FILTERS; EIGENSTATES; ENERGY DOMAIN; ENERGY RANGES; FILTER DIAGONALIZATION; GREENS FUNCTION; LANCZOS; LANCZOS ITERATIONS; LANCZOS TRIDIAGONALIZATION; LOCALISED;

ITERATIVE METHODS;

EID: 0031572071     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcph.1997.5777     Document Type: Article

References (49)

1. Lanczos C. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Natl. Bur. Stand. 45:1950;255.
2. Golub G. H., O'Leary D. P. Some history of the conjugate gradient and Lanczos algorithms: 1948-1976. SIAM Rev. 31:1989;50.
3. Parlett B. N. The Symmetric Eigenvalue Problem. 1980;Prentice-Hall, Englewood Cliffs.
4. Cullum J. K., Willoughby R. A. Lanczos Algorithms for Large Symmetric Eigenvalue Computations. 1985;Birkhauser, Boston.
5. Saad Y. Numerical Methods for Large Eigenvalue Problems. 1992;Manchester University Press, Manchester.
6. Golub G. H., VanLoan C. F. Matrix Computations. 1996;Johns Hopkins Univ. Press, Baltimore.
7. LeQuere F., Leforestier C. Quantum exact 3D study of the photodissociation of ozone molecule. J. Chem. Phys. 94:1991;1118.
8. McNichols A., Carrington T. Vibrational energy levels of formaldehyde calculated from an internal coordinate Hamiltonian using the Lanczos algorithm. Chem. Phys. Lett. 202:1993;464.
9. 2, Chem. Phys. Lett.
10. Wyatt R. E. The recursive residue generation method. Adv. Chem. Phys. 53:1989;231.
11. Park T. J., Light J. C. Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85:1986;5870.
12. Paige C. C. Computational variants of the Lanczos method for the eigenproblem. J. Inst. Math. Appl. 10:1972;373.
13. Paige C. C. Error analysis of the Lanczos algorithm for tridiagonalizing a symmetric matrix. J. Inst. Math. Appl. 18:1976;341.
14. Cullum J., Willoughby R. A. Computing eigenvalues of very large symmetric matrixeds - An implementation of a Lanczos algorithm with no reorthgonalization. J. Comput. Phys. 44:1981;329.
15. Ericsson T., Ruhe A. The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems. Math. Comput. 35:1980;1251.
16. Kono H. Extraction of eigenstates from an optically prepared state by a time-dependent quantum mechanical method. Chem. Phys. Lett. 214:1993;137.
17. Wyatt R. E. Matrix spectroscopy: Computation of interior eigenstates of large matrices using layered iteration. Phys. Rev. E. 51:1995;3643.
18. 65. J. Chem. Phys. 103:1995;8433.
19. Roy P.-N., Carrington T. An evaluation of methods designed to calculate energy levels in selected range and application to a (one-dimensional) Morse oscillator and (three-dimensional) HCN/HNC. J. Chem. Phys. 103:1995;5600.
20. Neuhauser D. Bound state eigenfunctions from wave packets: Time → energy resolution. J. Chem. Phys. 93:1990;2611.
21. Neuhauser D. Time-dependent reactive scattering in the presence of narrow resonances: avoiding long propagation times. J. Chem. Phys. 95:1991;4927.
22. Neuhauser D. Circumventing the Heisenberg principle: A rigorous demonstration of filter-diagonalization on a LiCN model. J. Chem. Phys. 100:1994;5076.
23. Wang Y., Carrington T., Corey G. C. A time-to-energy Fourier resolution method for calculating bound state energies and wavefunctions. Chem. Phys. Lett. 228:1994;144.
24. 2. J. Chem. Phys. 103:1995;10074.
25. Grozdanov T. P., Mandelshtam V. A., Taylor H. S. Recursion polynomial expansion of the Green's function with absorbing boundary conditions: Calculations of resonances of HCO by filter diagonalization. J. Chem. Phys. 103:1995;7990.
26. Mandelshtam V. A., Taylor H. S. The quantum resonance spectrum of the H+3 molecular ions for J = 0. An accurate calculation using filter diagonalization. J. Chem. Soc. Faraday Trans. 93:1997;847.
27. Chen R., Guo H. A general and efficient filter-diagonalization method without time propagation. J. Chem. Phys. 105:1996;1311.
28. Chen R., Guo H. Evolution of quantum system in order domain of Chebyshev operator. J. Chem. Phys. 105:1996;3569.
29. 2. Chem. Phys. Lett. 261:1996;605.
30. Wilkinson J. H. The Algebraic Eigenvalue Problem. 1965;Clarendon Press, Oxford.
31. Huang Y., Zhu W., Kouri D., Hoffman D. K. A general time-to-energy transform of wavepackets. Time-dependent wavepacket-Schrodinger and wavepacket-Lippmann-Schwinger equations. Chem. Phys. Lett. 206:1993;96.
32. Huang Y., Kouri D. J., Hoffman D. K. General, energy-separable Faber polynomial representation of operator functions: Theory and application in quantum scattering. J. Chem. Phys. 101:1994;10493.
33. Mandelshtam V. A., Taylor H. S. A simple recursion polynomial expansion of the Green's function with absorbing boundary conditions. Application to the reactive scattering. J. Chem. Phys. 103:1995;2903.
34. Mandelshtam V. A., Taylor H. S. Spectral projection approach to the quantum scattering calculations. J. Chem. Phys. 102:1995;7390.
35. Zhu W., Huang Y., Kouri D. J., Chandler C., Hoffman D. K. Orthogonal polynomial expansion of the spectral density operator and the calculation of bound state energies and eigenfunctions. Chem. Phys. Lett. 217:1994;73.
36. Silver R. N., Roeder H., Voter A. F., Kress J. D. Kernel polynomial approximations for densities of states and spectral functions. J. Comput. Phys. 124:1996;115.
37. Kosloff R., Tal-Ezer H. A direct relaxation method for calculating eigenfunctions and eigenvalues of the Schrodinger equation on a grid. Chem. Phys. Lett. 127:1986;223.
38. Webster F., Rossky P. J., Friesner R. A. Nonadiabatic processes in condensed matter: semi-classical theory and implementation. Comput. Phys. Comm. 63:1991;494.
39. Braun M., Solianos S. A., Papageorgiou D. G., Lagaris I. E. An efficient Chebyshev-Lanczos method for obtaining eigensolutions of the Schrodinger equation on a grid. J. Comput. Phys. 126:1996;315.
40. Light J. C., Hamilton I. P., Lill J. V. Generalized discrete variable approximation in quantum mechanics. J. Chem. Phys. 82:1985;1400.
41. Kosloff D., Kosloff R. A Fourier method solution for the time dependent schrodinger equation as a tool in molecular dynamics. J. Comput. Phys. 52:1983;35.
42. Boyd J. P. Chebyshev and Fourier Spectral Methods. 1989;Springer-Verlag, Berlin.
43. Tal-Ezer H., Kosloff R. An accurate and efficient scheme for propagating the time dependent Schrodinger equation. J. Chem. Phys. 81:1984;3967.
44. Wall M. R., Neuhauser D. 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. J. Chem. Phys. 102:1995;8011.
45. Mandelshtam V. A., Taylor H. S. A low-storage filter diagonalization method for quantum eigenenergy calculation or for spectral analysis of time signals. J. Chem. Phys. 106:1997;5085.
46. R. Chen, H. Guo, Calculation of matrix elements in filter diagonalization: A generalized method based on Fourier transform, Chem. Phys. Lett.
47. Fujiwara T. J. Phys. C. 10:1977;5035.
48. Koslowski T., vonNiessen W. Linear combination of Lanczos vectors: A storage efficient algorithm for sparse matrix eigenvector computations. J. Comput. Chem. 14:1993;769.
49. Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P. Numerical Recipes. 1992;Cambridge Univ. Press, Cambridge.