Representation and Conservation of Angular Momentum in the Born-Oppenheimer Theory of Polyatomic Molecules

TL;DR

This paper investigates how angular momentum operators are represented and conserved within the Born-Oppenheimer approximation for polyatomic molecules, analyzing their transformation laws and proposing a dressing transformation for improved accuracy.

Contribution

It introduces a systematic analysis of angular momentum conservation laws and transformation properties in Born-Oppenheimer theory, and proposes a dressing transformation to replace the standard approximation.

Findings

Angular momentum operators have exact equivalences across representations.

Dressing transformations can replace the Born-Oppenheimer approximation.

Transformation laws are linked to geometric structures in nuclear configuration space.

Abstract

This paper concerns the representation of angular momentum operators in the Born-Oppenheimer theory of polyatomic molecules and the various forms of the associated conservation laws. Topics addressed include the question of whether these conservation laws are exactly equivalent or only to some order of the Born-Oppenheimer parameter $\kappa=(m/M)^{1/4}$, and what the correlation is between angular momentum quantum numbers in the various representations. These questions are addressed both in problems involving a single potential energy surface, and those with multiple, strongly coupled surfaces; and both in the electrostatic model and those for which fine structure and electron spin are important. The analysis leads to an examination of the transformation laws under rotations of the electronic Hamiltonian; of the basis states, both adiabatic and diabatic, along with their phase conventions; of the potential energy matrix; and of the derivative couplings. These transformation laws are placed in the geometrical context of the structures in the nuclear configuration space that are induced by rotations, which include the rotational orbits or fibers, the surfaces upon which the orientation of the molecule changes but not its shape; and the section, an initial value surface that cuts transversally through the fibers. Finally, it is suggested that the usual Born-Oppenheimer approximation can be replaced by a dressing transformation, that is, a sequence of unitary transformations that block-diagonalize the Hamiltonian. When the dressing transformation is carried out, we find that the angular momentum operator does not change. This is a part of a system of exact equivalences among various representations of angular momentum operators in Born-Oppenheimer theory...