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

Differential Shannon Entropies and Mutual Information in a Curve Crossing System: Adiabatic and Diabatic Decompositions

The differential Shannon entropy provides a measure for the localization of a wave function. We regard the vibrational wave packet motion in a curve crossing system and calculate time-dependent entropies. Using a numerical example, we analyze how localization inside diabatic and adiabatic states can be accessed and discuss the differences between these two representations. In order to do so, we extend the usual entropy definition and introduce novel state-selective entropies. These quantities contain information on the form of the nuclear density components on the one hand and on the state population on the other, and it is shown how the contribution of the population can be removed. Having the state-selective entropies at hand, two additional functions derived from these, namely, the conditional entropy and the mutual information, are determined and compared. We find that these quantities relate closely to correlation effects rooted in different electronic properties of the system.

Differential Shannon entropies and correlation measures for Born-Oppenheimer electron-nuclear dynamics: numerical results and their analytical interpretation

We study the Born-Oppenheimer dynamics within a model for a coupled electron-nuclear motion. Differential Shannon entropies are calculated from the time-dependent probability densities of the combined system and, using single particle densities, entropies for the electronic and nuclear degrees of freedom are derived. These functions provide information on details of the wave packet motion. From the entropies, we determine the mutual information which characterizes particle correlations. This quantity is compared to other measures of electron-nuclear entanglement. Numerical results are interpreted within an analytically solvable approach, and it is documented how these functions depend on properties of the Born-Oppenheimer wave function and, in particular, how dynamical effects like wave packet focusing and dispersion influence the correlation between the particles.

Differential Shannon Entropies Characterizing Electron-Nuclear Dynamics and Correlation: Momentum-Space Versus Coordinate-Space Wave Packet Motion

We calculate differential Shannon entropies derived from time-dependent coordinate-space and momentum-space probability densities. This is performed for a prototype system of a coupled electron-nuclear motion. Two situations are considered, where one is a Born-Oppenheimer adiabatic dynamics, and the other is a diabatic motion involving strong non-adiabatic transitions. The information about coordinate- and momentum-space dynamics derived from the total and single-particle entropies is discussed and interpreted with the help of analytical models. From the entropies, we derive mutual information, which is a measure for the electron-nuclear correlation. In the adiabatic case, it is found that such correlations are manifested differently in coordinate- and momentum space. For the diabatic dynamics, we show that it is possible to decompose the entropies into state-specific contributions.

Information Theoretical Approach to Coupled Electron-Nuclear Wave Packet Dynamics: Time-Dependent Differential Shannon Entropies

We study differential Shannon entropies determined from position-space quantum probability densities in a coupled electron-nuclear system. In calculating electronic and nuclear entropies, one gains information about the localization of the respective particles and also about the correlation between them. For Born-Oppenheimer dynamics, the correlation decreases at times when the wave packet reaches the classical turning points of its motion. If a strong non-adiabtic coupling is present, leading to a large population transfer between different electronic states, the electronic entropy is approximately constant. Then the time dependence of the entropy reflects the information on the nucleus alone, and the correlation is absent. A decomposition of the entropy into contributions from the participating electronic states reveals insight into the state-specific population and nuclear wave packet localization.

N-Derivatives of Shannon entropy density as response functions

The exact first and second order partial derivatives of Shannon entropy density with respect to the number of electrons at constant external potential are introduced as new descriptors for prediction of the active sites of a molecule. The derivatives, which are a measure of the inhomogeneity of electron density, are calculated both exactly (from analytical forms) and approximately (using the finite difference method) for some molecular systems. According to the maximum entropy principle, the extreme value of the first order derivative on the surface of a given molecule should determine the active sites of the molecule in electrophilic and nucleophilic attack. The second order derivative indicates where the Shannon entropy is more concentrated or depleted during the electron exchange. Although these derivatives on the surfaces of helium and neon atoms are uniform, the corresponding values for argon, krypton and xenon atoms are not. This could explain the greater tendency of heavy noble gas atoms to form stable compounds. A dual descriptor is also defined as the difference between the left and right first order derivatives of Shannon entropy density, which allows one to simultaneously predict the preferable sites for electrophilic and nucleophilic attack over the system at point r. Therefore, the reactivity of an atom in a molecule requires the non-uniformity of the first and second order derivatives of Shannon entropy density on the surface of that atom.