On the behavior of Padé approximants in the vicinity of avoided crossings

Journal of Chemical Physics

Volumn 104, Issue 24, 1996, Pages 9870-9875

  Dunn, Martin ^a   Watson, Deborah K ^a   Walkup, John R ^a   Germann, Timothy C ^b  

^a UNIVERSITY OF OKLAHOMA   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0002201149     PISSN: 00219606     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.471751     Document Type: Article

References (29)

1. D. Delande and J.C. Gay, J. Phys. B 19, L173 (1986).
2. L.E. Fried and G.S. Ezra, J. Chem. Phys. 90, 6378 (1989).
3. C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chap. 8.
4. G.A. Baker, Jr. and P. Graves-Morris, Padé Approximants, Part I: Basic Theory, Vol. 13 of Encyclopedia of Mathematics and its Applications (Addison-Wesley, Reading, 1981).
5. See, for example, Ref. 3, Section 7.5, which contains a clear account of why the energy is a multivalued function of the parameters of the theory. For a physically relevant point in parameter space the energy of each state of the system resides on different branches of the energy function, i.e. the energies of different states are analytic continuations of each other. Different branches are usually connected by square-root branch points which, in complex conjugate pairs, can give rise to avoided crossings.
6. B. Barbanis, Astron. J. 71, 415 (1966).
7. E. Fermi, Z. Phys. 71, 250 (1931);
8. G. Herzberg, Molecular Spectra and Molecular Structure V. II (Van Norstrand Reinhold, New York, 1945), p. 215. Fermi resonances occur when the frequencies of zeroth order harmonic states are commensurable and the quanta in the normal modes of a set of states of the same symmetry are such that the energies are degenerate. These states mix strongly even if the perturbation is very small. Even slightly off-resonance, these states cannot mix strongly unless the perturbation is sufficiently large and the separation of the energy levels sufficiently small. Thus if the separation of a set of energy levels narrows as the perturbation is increased, then the energy levels interact (mix) with increasing strength leading to a repulsion between the energy levels that ultimately gives rise to avoided crossings (exchange of character).
9. E.J. Heller, E.B. Stechel, and M.J. Davis, J. Chem. Phys. 71, 4759 (1979);
10. E.J. Heller, E.B. Stechel, and M.J. Davis, J. Chem. Phys. 73, 4720 (1980);
11. D. Farrelly and T. Uzer, J. Chem. Phys. ibid. 85, 308 (1986).
12. M. Dunn, T.C. Germann, D.Z. Goodson, C.A. Traynor, J.D. Morgan III, D.K. Watson, and D.R. Herschbach, J. Chem. Phys. 101, 5987 (1994).
13. M.N. Padé, Sur la représentation approchée d'une fontion par des fractions rationelles, Ann. Sci. Ecole Norm. Sup. 9 (1892), Supplement, pp. 1-8;
14. R.E. Shafer, SIAM J. Numer. Anal. 11, 447 (1972);
15. S. Paszkowski, J. Comput. Appl. Math. 32, 229 (1990).
16. C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Section 3.8.
17. See for example, G.F. Carrier, M. Krook and C.E. Pearson, Functions of a Complex Variable (McGraw-Hill, New York, 1966).
18. Ref. 4, Section 2.2.
19. 1/2 in the Barbanis Hamiltonian ℋ.
20. 2 has no additional branch point singularities than when considered as a function of λ.
21. E.A. Solov'ev, Zh. Éksp. Teor. Fiz. 81, 1681 (1981)
22. [Sov. Phys. JETP 54, 893 (1981)];
23. S. Yu. Ovchinnikov and E.A. Solov'ev, Comments At. Mol. Phys. 22, 69 (1988);
24. T.P. Grozdanov and E.A. Solov'ev, Phys. Rev. A 42, 2703 (1990).
25. 2 where the widths are greater.
26. T.C. Germann, D.R. Herschbach, M. Dunn, and D.K. Watson, Phys. Rev. Lett. 74, 658 (1995).
27. T.C. Germann, Ph.D. thesis, Harvard University, 1995, Chapter 4.
28. T.C. Germann, M. Dunn, J.R. Walkup, D.R. Herschbach, and D.K. Watson (in preparation).
29. J.O. Hirschfelder, Chem. Phys. Lett. 54, 1 (1978), and references therein.