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.

Time-Dependent Expectation Values from Integral Equations for Quantum Flux and Probability Densities.

We compare the calculation of time-dependent quantum expectation values performed in different ways. In one case, they are obtained from an integral over a function of the probability density, and in the other case, the integral is over a function of the probability flux density. The two kinds of coordinate-dependent integrands are very different in their appearance, but integration yields identical results, if the exact wave function enters into the computation. This can be different, if one applies approximations to the wave function. For illustration, we treat one- and two-dimensional dynamics in coupled electron-nuclear systems. Using the adiabatic expansion of the total wave function, the expectation values are decomposed into different contributions. This allows us to discuss the validity of the Born-Oppenheimer (BO) approximation applied to the calculation of the expectation values from probability density- and flux density- integrals. Choosing force- and torque operators as examples, we illustrate the different spatiotemporal characteristics of the various integrands.