Electron pair density information measures in atomic systems

Abstract

Shannon entropies of the pair density, conditional entropies, and mutual information are studied in position and in momentum space for ground state neutral atoms and selected excited states at the Hartree-Fock level. We show that the mutual information, a measure of correlation, is larger in position space than in momentum space. This result also holds for a mutual information defined in terms of the exchange density; however, these quantities display much more structure than the corresponding ones based on the pair densities. The interpretation of this behavior is that exchange effects are smaller in momentum space than in position space in these systems.

On the Measurement of Randomness (Uncertainty): A More Informative Entropy

“…w αi p i (32) which shows that the weights w αi pi " 2, ..., nq are increasing in both α and i. In the case of…”

“…A symmetric measure is the so-called Jensen-Shannon divergence (JSD) (e.g., [30][31][32][33]), which can be expressed in terms of the Kullback-Leibler divergence (KLD) as:…”

Information theory and Wigner crystallization: A model perspective

“…One would expect that an S T derived from a multiconfigurational wave‐function, either an un‐restricted, or CI‐type wave‐function, would result in an S T that would fall between these two bounds: the derived formal bound 3(1 + ln π ) and that upper‐bound defined by The position and momentum components of Figure 3 are understood in terms of the localization of the electrons as an increasing function of R , as the Coulombic term begins to dominate. For small R the electrons do not experience each other, with the kinetic energy being dominant; in this region S p > S r . There is a radius R for which S p = S r , but for further subsequent increases in sphere radius R beyond this value, S r > S p . Sagar et al explored the singlet and triplet state members of the Helium isoelectronic series, and see similar behavior of S r , S p as a function of nuclear charge, Z . Specifically, in their work, for small Z , S r > S p . This precedes a transition point for increasing Z where S p = S r (estimated at Z = 2.85), followed by a region wherein S p > S r . [ 50 ] The confining sphere radius, R , in this current work, is equivalent to the attractive nuclear potential, and we see the reciprocal pattern of behavior manifest in the measures of position and momentum Shannon entropy. The onset of S p < 0 in this current work occurs for R ≈ 7 ( a .…”

Statistical Correlation Measures From Higher‐Order Moments in Quantum Oscillator Systems

“…Mutual information and the correlation coefficient have been applied to study correlation in quantum systems. [ 3,4,9,10 ] It has also been shown that the interpretations obtained from these two measures are not always consistent. [ 10 ] Transitions between negative and positive values of the correlation coefficient have been shown to be captured by a minimum in the mutual information.…”