Abstract
In an earlier article [Found. Phys. 30, 1191 (2000)], a quasiclassical phase space approximation for quantum projection operators was presented, whose accuracy increases in the limit of large basis size (projection subspace dimensionality). In a second paper [J. Chem. Phys. 111, 4869 (1999)], this approximation was used to generate a nearly optimal direct-product basis for representing an arbitrary (Cartesian) quantum Hamiltonian, within a given energy range of interest. From a few reduced-dimensional integrals, the method determines the optimal 1D marginal Hamiltonians, whose eigenstates comprise the direct-product basis. In the present paper, this phase space optimized direct-product basis method is generalized to incorporate non-Cartesian coordinate spaces, composed of radii and angles, that arise in molecular applications. Analytical results are presented for certain standard systems, including rigid rotors, and three-body vibrators.
Similar content being viewed by others
REFERENCES
J. M. Bowman, Acc. Chem. Res. 19,202 (1986).
Z. BačIć and J. C. Light, Annu. Rev. Phys. Chem. 40, 469 (1989).
R. Baer and M. Head-Gordon, Phys. Rev. Lett. 79, 3962 (1997).
S. Goedecker, Rev. Mod. Phys. 71, 1085 (1999).
B. Poirier, Phys. Rev. A 56, 120 (1997).
B. Poirier and J. C. Light, J. Chem. Phys. 111, 4869 (1999).
D. O. Harris, G. G. Engerholm, and W. D. Gwinn, J. Chem. Phys. 43, 1515 (1965).
A. S. Dickinson and P. R. Certain, J. Chem. Phys. 49, 4209 (1968).
J. C. Light, R. M. Whitnell, T. J. Park, and S. E. Choi, in Supercomputer Algorithms for Reactivity, Dynamics and Kinetics of Small Molecules, A. Lagana, ed. (Kluwer Academic, Boston, 1989), pp. 187–214.
D. T. Colbert and W. H. Miller, J. Chem. Phys. 96, 1982 (1992).
J. Echave and D. C. Clary, Chem. Phys. Lett. 190, 225 (1992).
H. Wei and T. Carrington, Jr., J. Chem. Phys. 97, 3029 (1992).
B. Poirier, Found. Phys. 30, 1191 (2000).
N. Bohr, Phil. Mag. (Series 6) 26, 857(1913).
W. Wilson, Phil. Mag. 29, 795 (1915).
A. Sommerfeld, Ann. Phys. (Leipzig) 51, 1 (1916).
E. Fattal, R. Baer, and R. Kosloff, Phys. Rev. E. 53, 1217(1996).
D. W. Noid and R. A. Marcus, J. Chem. Phys. 62, 2119 (1976).
M. J. Davis and E. J. Heller, J. Chem. Phys. 71, 3383 (1979).
I. P. Hamilton and J. C. Light, J. Chem. Phys. 84, 306 (1986).
B. Poirier and J. C. Light, J. Chem. Phys. 114, 6562 (2001).
C. Eckart, Phys. Rev. 46, 383 (1934).
R. T. Pack, in Advances in Molecular Vibrations and Collision Dynamics, Vol. 2A (JAI Press Inc., New York, NY, 1994), pp. 111–145.
R. G. Littlejohn and M. Reinsch, Rev. Mod. Phys. 69, 213 (1997).
A. N. Kolmogorov and S. V. Fomin, in Introductory Real Analysis, Chap. 3 (Dover Publications, New York, NY, 1975), pp. 78–117.
H. Weyl, Z. Phys. 46, 1 (1928).
E. Wigner, Phys. Rev. 40, 749 (1932).
J. E. Moyal, Proc. Cambridge Phil. Soc. 45, 99 (1949).
V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978).
J. J. Morehead, J. Math. Phys. 36, 5431 (1995).
B. Poirier, J. Math. Phys. 40, 6302 (1999).
B. Lesche and T. H. Seligman, J. Phys. A: Math. Gen. 19, 91 (1986).
B. Podolsky, Phys. Rev. 32, 812 (1928).
M. M. Nieto, Phys. Rev. A 17, 1273 (1978).
J. Dai and J. C. Light, J. Chem. Phys. 107, 8432 (1997).
H. Goldstein, Classical Mechanics, 2nd edn. (Addison–Wesley, Reading, MA, 1980).
D. J. Evans, Mol. Phys. 34, 317 (1977).
M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957).
B. Poirier, J. Chem. Phys. 108, 5216 (1998). 1610 Poirier
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Poirier, B. Phase Space Optimization of Quantum Representations: Non-Cartesian Coordinate Spaces. Foundations of Physics 31, 1581–1610 (2001). https://doi.org/10.1023/A:1012642832253
Issue Date:
DOI: https://doi.org/10.1023/A:1012642832253