Bifurcation Analysis of Higherm:nResonance Spectroscopic Hamiltonian^†

Abstract

The classical phase space structure of a spectroscopic Hamiltonian for two coupled vibrational modes is analyzed using bifurcation theory, classified on catastrophe maps, for a variety of higher order resonances (3:2, 4:2, 5:2, 6:2 and 4:4, 5:4, 6:4), cases not considered in previous work. A type of bifurcation not encountered for lower resonance orders, based on oVerlap of separatrices rather than change in behavior of fixed points, is analyzed, and a procedure is developed to augment the catastrophe map. Energy level patterns are associated with the new resonances, in analogy with the patterns of adjacent level spacings considered earlier for the 2:1 resonance.

Bending Dynamics of Acetylene: New Modes Born in Bifurcations of Normal Modes

“…(As discussed later, in more general systems, the critical points may correspond to higher-dimensional invariant tori rather than one-dimensional modes, but these structures are still expected to organize the phase space.) The critical points approach has been developed and applied to systems with two [5][6][7][8][11][12][13][29][30][31] and three 9,10,14 coupled modes. In the exposition here we build up from these simpler systems to make the generalization to the acetylene bends system as clear as possible.…”

“…It is notable that for single resonance systems, the solutions for the critical points of the polyad Hamiltonian necessarily lie on great circles on the phase space sphere. 13 An important question is whether this generalizes empirically to systems with more degrees of freedom and multiple resonances, where great circle conditions on the critical points such as (32) are not necessary. Satisfaction of the great circle conditions relates to the primacy of certain individual resonances, since the great circles are defined on phase space spheres whose coordinates are defined by individual resonances, as in Figure 2 for the DD-I and l-resonance spheres.…”

“…To determine the dynamics at the singular points, either direct visualization by running trajectories or a coordinate transformation (called "rotating the sphere" in ref 13) is required. Since direct visualization of four-dimensional phase space is nontrivial, we chose the second path to search for critical points on the boundary.…”

“…We have been developing an approach [1][2][3][4][5][6][7][8][9][10][11][12][13] that analyzes the classical version of the effective quantum Hamiltonian used to fit experimental spectra, to get knowledge of "new modes" that describe the motion at high energy. These new modes are born in bifurcations, or branchings associated with the original normal modes.…”

Assigning vibrational polyads using relative equilibria: application to ozone

“…4.1 of Appendix B.4 (Example 3) in [34,36, p. 40], and the semiclassical analysis in [35]. Regrettably, this mathematical work has been as yet of little impact on the molecular community, and the geometry of the m 1 :m 2 polyad spaces is still not clearly understood [37]. Geometry of the three mode polyad spaces in the case of higher resonances can be very complex.…”