Vector parametrization of the N-atom problem in quantum mechanics with non-orthogonal coordinates

Abstract

This article aims to present a general method that enables one to build kinetic energy matrices in getting rid, for the angular coordinates (internal and Eulerian), of the heaviness of differential calculus (for expressing kinetic energy operators) and numerical integration (for calculating matrix elements). Therefore, instead of 3N-3 coordinates, only N-1 radial distances are to be treated as coordinates. In the present formulation, the system is described by any set of n vectors {R_i, i=1, ..., n} and the kinetic energy operator is expressed in term of (n-1) angular momenta {L_i, i=1, ..., n-1} and the total angular momentum J. The formalism proposed is general and gives a remarkably compact expression of the kinetic energy in terms of the angular momenta. This expression allows one to circumvent the seeming angular singularities.

References

1. F. Gatti, C. Iung, M. Menou, Y. Justum and X. Chapuisat, J. Chem. Phys., 1998, 108, 8804.
2. F. Gatti, C. Iung and X. Chapuisat, THEO CHEM., 1998, 430, 201.
3. X. Chapuisat and C. Iung, Phys. Rev. A, 1992, 45, 6217.
4. B. Podolsky, Phys. Rev., 1928, 32, 812.
5. B. R. Johnson, J. Chem. Phys., 1983, 79, 1906.
6. R. T. Pack and G. Parker, J. Chem. Phys., 1987, 87, 3888.
7. R. T. Pack and G. Parker, J. Chem. Phys., 1989, 90, 3511.
8. X. Chapuisat and A. Nauts, Phys. Rev. A, 1991, 44, 1328.
9. Y. Justum, M. Menou, A. Nauts and X. Chapuisat, Chem. Phys., 1997, 223, 211.
10. X. Chapuisat, A. Nauts and J.-P. Brunet, Mol. Phys., 1991, 72, 1.
11. M. J. Bramley, W. H. Green and N. C. Handy, Mol. Phys., 1991, 73, 1183.
12. M. J. Bramley and N. C. Handy, J. Chem. Phys., 1993, 98, 1378.
13. A. G. Csazar and N. C. Handy, J. Chem. Phys., 1995, 102, 3962.
14. V. Aquilanti and S. Cavalli, J. Chem. Phys., 1986, 85, 1355.
15. V. Aquilanti and S. Cavalli, J. Chem. Soc., Faraday Trans., 1997, 93, 801.
16. G. Brocks, A. der Avoird, B. T. Sutcliffe and J. Tennyson, Mol. Phys, 1983, 50, 1025.
17. N. Anderson and B. T. Sutcliffe, Int. J. Quantum Chem., 1996, 60, 37.
18. T. J. Lukka, J. Chem. Phys., 1995, 102, 3945.
19. A. Kuppermann, J. Phys. Chem. A, 1997, 101, 6368.
20. S. Rempe and R. Watts, J. Chem. Phys., 1999, 108, 10095.
21. E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw-Hill, New York, 1955.
22. N. C. Handy, Mol. Phys., 1987, 61, 207.
23. D. W. Jepsen and J. O. Hirschfelder, Proc. Natl. Acad. Sci. USA, 1959, 45, 249.
24. A. Nauts and X. Chapuisat, Mol. Phys., 1985, 55, 1287.
25. F. Gatti, J. Chem. Phys., submitted.
26. R. N. Zare, Angular momentum, Wiley, New York, 1988.
27. J. H. Van Vleck, Rev. Mod. Phys, 1951, 23, 213.
28. A. Messiah, Mecanique quantique, Dunod, Paris, 1995.
29. X. Chapuisat, A. Belafhal, A. Nauts and C. Iung, Mol. Phys., 1992, 77, 947.
30. F. Gatti, C. Iung, C. Leforestier, M. Menou, Y. Justum, A. Nauts and X. Chapuisat, THEOCHEM., 1998, 424, 181.
31. A. S. Davydov, Quantum Mechanics, Pergamon Press, Oxford, 1965.
32. L. C. Biendenharn and J. D. Louck, Angular momentum in Quantum Mechanics, Addison-Wesley, Reading, MA, 1981.
33. G. B. Arfken and H. J. Weber, Mathematical methods for Physicists, 4th edn., Academic Press, London, 1995.
34. S. Wolfram, Mathematica, a system for doing mathematics by computer, 2nd edn., Addison-Wesley, Reading, MA, 1991.
35. C. Leforestier, L. B. Braly, K. Liu, M. Elrod and R. Saykally, J. Chem. Phys., 1997, 106, 8527.
36. F. Gatti, C. Iung, C. Leforestier and X. Chapuisat, J. Chem Phys., in the press.