  We consider the quantum entanglement of the electronic and vibrational
degrees of freedom in molecules with a tendency towards double welled
potentials using model coupled harmonic diabatic potential-energy surfaces. The
von Neumann entropy of the reduced density matrix is used to quantify the
electron-vibration entanglement for the lowest two vibronic wavefunctions in
such a bipartite system. Significant entanglement is found only in the region
in which the ground vibronic state contains a density profile that is bimodal
(i.e., contains two separate local minima). However, in this region two
distinct types of entanglement are found: (1) entanglement that arises purely
from the degeneracy of energy levels in the two potential wells and which is
destroyed by slight asymmetry, and (2) entanglement that involves strongly
interacting states in each well that is relatively insensitive to asymmetry.
These two distinct regions are termed fragile degeneracy-induced entanglement
and persistent entanglement, respectively. Six classic molecular systems
describable by two diabatic states are considered: ammonia, benzene,
semibullvalene, pyridine excited triplet states, the Creutz-Taube ion, and the
radical cation of the "special pair" of chlorophylls involved in
photosynthesis. These chemically diverse systems are all treated using the same
general formalism and the nature of the entanglement that they embody is
elucidated.
