Born-Oppenheimer Invariants Along Nuclear Configuration Paths

Abstract

Whenever a quantum chemist extracts the eigenstate of an electronic Hamiltonian, he makes, consciously or not, a decision concerning the phase of the wavefunction. This is done for each calculated state at each nuclear position. Thus he defines a Born-Oppenheimer (BO) frame of reference. There is no absolute phase just as there is no absolute position or time in mechanics. This leads naturally to the question: what are the quantities which do not depend on the arbitrary phases, i.e. what are the BO invariants? In this article we identify BO invariants with respect to an arbitrary path in nuclear configuration space. We identify invariant electronic states along these paths and their Aharonov-Anandan geometric phases. For closed loops not passing through electronic energy degeneracies these invariant states are the BO adiabatic wavefunctions and the phases are the Berry phases. The results establish rigorous relations between the non-adiabatic couplings and the geometric phases.

Keywords: Physical Chemistry > Theoretical Chemistry, physchem/0203010