Quantum dynamics of kinematic invariants in tetra- and polyatomic systems

(Note: The full text of this document is currently only available in the PDF Version )

Robert G. Littlejohn, Kevin A. Mitchell and Vincenzo Aquilanti


Abstract

For the dynamical treatment of polyatomic molecules or clusters as n-body systems, coordinates are conveniently broken up into external (or spatial) rotations, kinematic invariants, and internal (or kinematic) rotations. The kinematic invariants are related to the three principal moments of inertia of the system. At a fixed value of the hyperradius (a measure of the total moment of inertia), the space of kinematic invariants is a certain spherical triangle, depending on the number of bodies, upon which angular coordinates can be imposed. It is shown that this triangle provides the 24-element (group O) octahedral tesselation of the sphere for n=4 and the 48-element (group Oh) octahedral tesselation for n[greater than or equal, slant]5. Eigenfunctions describing the kinematics of systems with vanishing internal and external angular momentum can be obtained in closed form in terms of Bessel functions of the hyperradius and surface spherical harmonics. They can be useful as orthonormal expansion basis sets for the hyperspherical treatment of the n-body particle dynamics.


References

  1. W. Zickendraht, J. Math. Phys., 1969, 10, 30 CrossRef.
  2. Y. Öhrn and J. Linderberg, Mol. Phys., 1983, 49, 53.
  3. X. Chapuisat, J. P. Brunet and A. Nauts, Chem. Phys. Lett., 1987, 136, 153 CrossRef CAS.
  4. X. Chapuisat, Phys. Rev. A, 1992, 45, 4277 CrossRef.
  5. A. Kuppermann, in Advances in Molecular Vibrations and Collision Dynamics, ed. J. Bowman, JAI, Greenwich, CT, 1994, vol. 2B, pp. 117–186 Search PubMed.
  6. A. Kuppermann, J. Phys. Chem. A, 1997, 101, 6368 CrossRef CAS.
  7. V. Aquilanti and S. Cavalli, J. Chem. Soc., Faraday Trans., 1997, 93, 801 RSC.
  8. R. G. Littlejohn, K. A. Mitchell, M. Reinsch, V. Aquilanti and S. Cavalli, Phys. Rev. A, 1998, 58, 3718 CrossRef CAS the simpler case of frames in three-body systems is studied in R. G. Littlejohn, K. A. Mitchell, V. Aquilanti and S. Cavalli, Phys. Rev. A, 1998, 58, 3705 Search PubMed Kinematic rotations are also studied in V. Aquilanti, L. Bonnet and S. Cavalli, Mol. Phys., 1996, 89, 1 CrossRef CAS.
  9. W. Zickendraht, J. Math. Phys., 1971, 12, 1663 CrossRef In this reference, kinematic invariants and external coordinates are introduced to describe collective motions in nuclear dynamicsSee also M. Moshinsky, J. Math. Phys., 1984, 25, 1555 Search PubMed and O. Castaños, A. Frank, E. Chacón, P. Hess and M. Moshinsky, J. Math. Phys., 1982, 25, 2537 Search PubMed.
  10. R. G. Littlejohn and M. Reinsch, Rev. Mod. Phys., 1997, 69, 213 CrossRef.
  11. V. Aquilanti and S. Cavalli, J. Chem. Phys., 1986, 85, 1355 CrossRef CAS.
  12. V. Aquilanti, S. Cavalli and G. Grossi, J. Chem. Phys., 1986, 85, 1362 CrossRef CAS.
  13. R. G. Littlejohn and M. Reinsch, Phys. Rev. A, 1995, 52, 2035 CrossRef CAS.
  14. X. Chapuisat, A. Belafhal and A. Nauts, J. Mol. Spectrosc., 1991, 149, 274 CrossRef CAS.
  15. V. Aquilanti, S. Cavalli and D. De Fazio, J. Chem. Phys., 1998, 109, 3792 CrossRef CAS.
  16. V. Aquilanti, S. Cavalli, D. De Fazio, A. Volpi, A. Aguilar, X. Giménez and J. M. Lucas, J. Chem. Phys., 1998, 109, 3805 CrossRef CAS; Phys. Chem. Chem. Phys., 1999, 1, 1091 Search PubMed.
  17. W. Magnus, Noneuclidean Tesselations and Their Groups, Academic Press, New York, 1974 Search PubMed.
  18. X. Chapuisat, A. Belafhal, A. Nauts and C. Iung, Mol. Phys., 1992, 77, 947 CAS.
  19. A. Nauts and X. Chapuisat, Mol. Phys., 1985, 55, 1287 CAS.
Click here to see how this site uses Cookies. View our privacy policy here.