Entanglement in the Born-Oppenheimer Approximation

TL;DR

This paper investigates how electron-nuclear entanglement affects the validity of the Born-Oppenheimer approximation, revealing that entanglement levels do not directly correlate with the approximation's accuracy.

Contribution

It demonstrates that electron-nuclear entanglement is not a reliable indicator of the Born-Oppenheimer approximation's validity, challenging previous assumptions.

Findings

Entanglement is uncorrelated with BO approximation validity.

Maximally entangled BO states can still be accurate.

BO states do not preserve entanglement properties of exact states.

Abstract

The role of electron-nuclear entanglement on the validity of the Born-Oppenheimer (BO) approximation is investigated. While nonadiabatic couplings generally lead to entanglement and to a failure of the BO approximation, surprisingly the degree of electron-nuclear entanglement is found to be uncorrelated with the degree of validity of the BO approximation. This is because while the degree of entanglement of BO states is determined by their deviation from the corresponding states in the crude BO approximation, the accuracy of the BO approximation is dictated, instead, by the deviation of the BO states from the exact electron-nuclear states. In fact, in the context of a minimal avoided crossing model, extreme cases are identified where an adequate BO state is seen to be maximally entangled, and where the BO approximation fails but the associated BO state remains approximately unentangled. Further, the BO states are found to not preserve the entanglement properties of the exact electron-nuclear eigenstates, and to be completely unentangled only in the limit in which the BO approximation becomes exact.