Relative equilibria of D₂H⁺and H₂D⁺

Abstract

Relative equilibria of molecules are classical trajectories corresponding to steady rotations about stationary axes during which the shape of the molecule does not change. They can be used to explain and predict features of quantum spectra at high values of the total angular momentum J in much the same way that absolute equilibria are used at low J . This paper gives a classification of the symmetry types of relative equilibria of AB2 molecules and computes the relative equilibria bifurcation diagrams and normal mode frequencies for D2H+ and HID+. These are then fed into a harmonic quantization procedure to produce a number of predictions concerning the structures of energy level clusters and their rearrangements as J increases. In particular the formation of doublet pairs is predicted for H2D+ from J M 26.

Hamiltonian Systems Near Relative Periodic Orbits

“…For example, generalizations of the Weinstein-Moser theorem show that they are typically present near stable relative equilibria [25,39,43] and can therefore be found in virtually any physical application with a continuous symmetry group. Specific examples for which relative periodic orbits have been discussed or could be found by applying the Weinstein-Moser theorem to stable relative equilibria include rigid bodies [1,31,28,24], deformable bodies [8,27,13], gravitational N-body problems [32,47], molecules [17,19,20,34,48], and point vortices [26,46,38].Existing theoretical work on Hamiltonian relative periodic orbits includes results on their stability [41,42] and on their persistence to nearby energy-momentum levels in the case of compact symmetry groups [33]. However, stability, persistence, and bifurcations are still a long way from being well understood, especially in the presence of actions of noncompact symmetry groups with nontrivial isotropy subgroups.…”

Strand spaces and rank functions:more than distant cousins

“…To see this recall that linearly stable relative equilibria organize regular series of localized quantum states that can be labeled by quantum numbers derived from local integrable harmonic approximations to the Hamiltonian. 33,34 Examples of this include pendular states of rigid molecules in electric fields which are far from the symmetric top or zero field integrable limits. This harmonic approximation can be extended beyond a Hamiltonian Hopf point at which the relative equilibrium loses stability by using a normal form approximation 28 which is again integrable and so introduces approximate conserved quantities and quantum numbers.…”

“…We have shown elsewhere 33,34 that relative equilibria can be used to obtain new approximations for the qualitative analysis of quantum levels. This can be done for any relative equilibrium but the most experimentally relevant for the Stark effect is the P relative equilibrium.…”

“…This is not uncommon: any given rotating molecular system is likely to exhibit several such bifurcations as the angular momentum is increased. 33,34 Of course molecular systems are not integrable. However integrable approximations often provide useful models.…”

“…Relative equilibria have been used as ''organizing centers'' for molecular spectra at high angular momenta in much the same way as equilibria are used for vibrational and low angular momentum spectra. [30][31][32][33][34] They appear in families parametrized by total angular momentum and typically undergo a number of bifurcations as the angular momentum is increased. Hamiltonian Hopf bifurcations occur when two normal mode frequencies become equal and then complex, resulting in loss of stability.…”