Shannon Entropy and Fisher Information from a Non-Born-Oppenheimer Perspective

We study the Shannon entropy and the Fisher information in a non-Born-Oppenheimer (nBO) regime, where these quantities are constructed from one-particle densities obtained from an exact nBO analytic wave function for a Coulomb-Hooke model of a four-particle system. This model consists of two electrons and two protons with Coulombic interactions between like particles and Hookean interactions otherwise [ Becerra, M. et al. Int. J. Quantum Chem 2013, 113 (10), 1584-1590 ]. In the nBO case, there arise densities for both the nuclei and electrons. Furthermore, these densities vary with respect to a particular point of reference from which they are calculated. We consider, in the present work, electron and nuclear densities calculated from the following reference points: a global center of mass, the geometric center between the electrons, and the geometric center between the protons. A comparison of the nBO Shannon entropy and Fisher information, with respect to their counterparts computed from Born-Oppenheimer densities, suggests that the former quantities provide more insights into the chemical reactivity because of the nonuniqueness nature of the nBO electron density as well as the availability and access to the nBO nuclear density. Finally, some comments are made concerning the nBO vs the BO regimes in relation to this particular chemical reactivity indicator.