Representation and conservation of angular momentum in the Born–Oppenheimer theory of polyatomic molecules

Abstract

This paper concerns the representation of angular momentum operatorsin the Born-Oppenheimer theory of polyatomic molecules and the variousforms of the associated conservation laws. Topics addressed includethe question of whether these conservation laws are exactly equivalentor only to some order of the Born-Oppenheimer parameter$\kappa=(m/M)^{1/4}$, and what the correlation is between angularmomentum quantum numbers in the various representations. Thesequestions are addressed both in problems involving a single potentialenergy surface, and those with multiple, strongly coupled surfaces;and both in the electrostatic model and those for which fine structureand electron spin are important. The analysis leads to an examinationof the transformation laws under rotations of the electronicHamiltonian; of the basis states, both adiabatic and diabatic, alongwith their phase conventions; of the potential energy matrix; and ofthe derivative couplings. These transformation laws are placed in thegeometrical context of the structures in the nuclear configurationspace that are induced by rotations, which include the rotationalorbits or fibers, the surfaces upon which the orientation of themolecule changes but not its shape; and the section, an initial valuesurface that cuts transversally through the fibers. Finally, it issuggested that the usual Born-Oppenheimer approximation can bereplaced by a dressing transformation, that is, a sequence of unitarytransformations that block-diagonalize the Hamiltonian. When thedressing transformation is carried out, we find that the angularmomentum operator does not change.

Angular Momentum Transfer between a Molecular System and a Continuous Circularly Polarized Light Field within a Semiclassical Born–Oppenheimer Surface Hopping Framework

“…As shown in ref , for linear and angular momentum conservation, BO theory requires the electronic eigenbasis follow a well-defined phase convention that satisfies translational and rotational invariance: ( P̂ n + P̂ e ) false| ψ j ⟩ = 0 ( Ĵ n + Ĵ e ) false| ψ j ⟩ = 0 In words, these phase conventions demand that the electronic wave function should remain invariant under simultaneous translations or rotations of both the nuclei and electrons . These conventions can be generalized to the case with a radiation field in a straightforward manner: ( P̂ n + P̂ e + P̂ p h ) false| ψ j ′ ⟩ = 0 ( Ĵ n + Ĵ e + Ĵ p h ) false| ψ j ′ <...…”