Chiral Temporal Structures in Molecular Rotational Dynamics

Abstract

We introduce chiral rotational wavepackets: coherent superpositions of a few low-lying rotational states of an achiral molecule for which the expectation value of a molecular axis traces a three-dimensional chiral curve in the laboratory frame. The handedness is a dynamical property of the quantum state rather than of the equilibrium structure, and we show that it can be imprinted on linear rotors as well as on spherical and symmetric tops. We derive the action of space inversion, motion reversal, and their product on such wavepackets in the Wigner $D^J_{MK}$ basis, and define the rotational enantiomers of a given wavepacket. To quantify the handedness of the orientation trajectory we introduce two derivative-based pseudoscalar measures: a geometrical measure built from the curvature and torsion of the trajectory, which is even under motion reversal, and an origin-referenced measure built from the binormal vector, which is odd under motion reversal. The latter is shown to be proportional to the geodesic curvature of the trajectory projected onto the unit sphere, which links it, via the Gauss--Bonnet theorem, to the solid angle enclosed by the projected trajectory and to the parallel-transport holonomy of a molecule-fixed vector. Finally we show that a nonzero geometrical measure is a sufficient condition for the wavepacket to be truly chiral in the sense of Barron, while wavepackets with real expansion coefficients are only falsely chiral.