Analysis of Rotation--Vibration Relative Equilibria on the Example of a Tetrahedral Four Atom Molecule

Abstract

Abstract. We study relative equilibria (RE) of a nonrigid molecule, which vibrates about a well-defined equilibrium configuration and rotates as a whole. Our analysis unifies the theory of rotational and vibrational RE. We rely on the detailed study of the symmetry group action on the initial and reduced phase space of our system and consider the consequences of this action for the dynamics of the system. We develop our approach on the concrete example of a four-atomic molecule A4 with tetrahedral equilibrium configuration, a dynamical system with six vibrational degrees of freedom. 1. Introduction. This paper unifies modern methods of classical theory of symmetric Hamiltonian dynamical systems and quantum theory of molecules (and other isolated finiteparticle systems). Considerable progress was achieved in both directions in the last decades and deep relations between these seemingly distant theories became evident. Significant effort by mathematicians and molecular physicists to converge the two fields resulted in the qualitative theory of highly excited quantum molecular systems based on recent mathematical developments. We join the two approaches and demonstrate what kind of concrete results can be immediately obtained in molecular systems [1,2,3,4] by applying powerful methods of symmetric Hamiltonian systems [5,6,7,8,9,10,11]. We choose a concrete problem of rotation-vibration of a four-atomic molecule with tetrahedral equilibrium configuration [12,13] in order to explain the details of our approach.

Qualitative Analysis of the Classical and Quantum Manakov Top

“…This result is in fact independent on the concrete form of integrals of motion and relies only on symmetry arguments. Such preliminary symmetry analysis is quite important in the qualitative study of molecular models as it is formulated in our previous works [26,20,34,13].…”

Quantum monodromy and its generalizations and molecular manifestations

“…It is possible to find fractional monodromy in systems with special axial symmetries, such as a nondiagonal S 1 symmetry, which acts on the two subspaces in a way similar the action of the 1:ðÀ2Þ resonant oscillator. Another possibility may occur in the study of localized oscillations about stable nonlinear normal modes (such as local modes) in a system with three or more vibrational degrees of freedom, see [22,[43][44][45][46][47] and references therein.…”

Rotational energy cluster formation in XY3 molecules: Excited vibrational states of BiH3 and SbH3

“…Pavlichenkov and Zhilinskií [7] illustrated how the bifurcation of stationary points on the RES affects the formation of rotational energy clusters. The role of symmetry in the cluster formation process has been also studied in great detail (see, for example, [8]). …”