Monodromy in the spectrum of a rigid symmetric top molecule in an electric field

Abstract

We show that for rigid symmetric top molecules in electric fields the phenomenon of monodromy arises naturally as a ''defect'' in the lattice of quantum states in the energy-momentum diagram. This makes it impossible to use either the total angular momentum or a pendular quantum number to label the states globally. The monodromy is created or destroyed by classical Hamiltonian Hopf bifurcations from relative equilibria. These phenomena are robust and should be observable in quasi-symmetric top molecules with field strengths E satisfying E/bϾ4.5, where is the dipole moment and b the rotational constant perpendicular to the symmetry axis of the molecule.

Classical Mechanics of Dipolar Asymmetric Top Molecules in Collinear Static Electric and Nonresonant Linearly Polarized Laser Fields: Energy-Momentum Diagrams, Bifurcations and Accessible Configuration Space

“…The rotational constants B i , the field parameter ω, and the energy are all scaled by B 3 . For iodobenzene an electric field of ε S = 25 kVcm −1 [53] gives ω/B 3 = 4.52; for pyridazine a field of strength ε S = 56 kVcm −1 [20] gives ω/B 3 = 19.04; in iodopentafluorobenzene, an electric field of ε S = 25 kVcm −1 gives ω/B 3 = 18.93 [17,21,53].…”

“…In this section we treat as examples three asymmetric top molecules of recent theoretical and experimental interest [17,39]: the near-prolate top iodobenzene (C 6 H 5 I) [21,53], the near-oblate top pyridazine (C 4 H 4 N 2 ) [17,20], and the intermediate case iodopentafluorobenzene [21]. Relevant physical parameters for these molecules are given in Table I.…”

“…The E-m diagram provides a useful global classification of the rotor dynamics, as distinct regions of the E-m plane are associated with different allowed types of motion of the diatomic. For symmetric tops in electric fields similar diagrams can be constructed [17], and analysis of classical symmetric top E-m and E-k diagrams helps understand the organization of the quantum level spectrum [17].…”

“…The theory of rigid body motion also provides the basis for analysis and interpretation of the rotational dynamics and spectra of semi-rigid molecules [2,5,13,14,15,16]. In molecular terms the Euler top is simply a free asymmetric top molecule [5,14,15], the Lagrange top models a symmetric top molecule with a dipole moment in an electric field [15,17], while there does not appear to be an obvious molecular analogue for the Kovalevskaya top. Breaking the symmetry of the moment of inertia tensor in the Lagrange top results in the non-integrable problem of an asymmetric top in a static field.…”

Dynamical manifestation of Hamiltonian monodromy

“…A number of simple classical mechanical systems, including the spherical pendulum, the Lagrange top, and others [2] are known to have this property. Quantum monodromy [3,4] was discovered in many fundamental quantum systems, such as the hydrogen atom in external fields [5][6][7], the H + 2 molecular ion [8], (a) E-mail: jbdelo@wm.edu systems with coupled angular momenta [9], rotating dipolar molecules in external electric field [10,11], rovibrational structure in molecules [12][13][14][15][16][17], collective vibrations of nuclei [18], or of Bose condensates [19] -to name af e w .…”