In a previous work (Ghiringhelli L. M. and Delle Site L., Phys. Rev. B , 77 (2008) 073104) we have proposed a methodological approach to determine the electron correlations contained in the kinetic functional. Its application to the test case of an almost uniform interacting electron gas led to a local density functional of the form ∫ρ( r )lnρ( r )d r . In this work we discuss the interpretation of such a result by performing an analysis of the formal and numerical properties of the functional under the process of coordinates' scaling. On the basis of such an analysis we propose a local analytic form of the kinetic functional of which the previous numerical result is a first-order approximation.
In this paper, we have presented an analytical model of two electron systems consisting of a many particle correlated wave function with some variational parameters α, λ and µ and used it to quantify the electron-electron correlation described by the wave function containing explicitly r12 (inter atomic distance between two electrons) dependent term. The single particle wave functions and the charge densities have been extracted from the said correlated wave function both for the uncorrelated and correlated systems in coordinate space and its momentum analogs have been obtained by taking the Fourier transform of the coordinate analogs. We have computed and presented the results of the numerical values of the theoretic information entropies of the Shannon entropy, Fisher information entropy, Shannon power and the FisherShannon product. The numerical values are consistently found to satisfy the Beckner, Bialynicki-Birula and Mycielski (BBM ) inequality relation; Stam-Cramer-Rao inequalities or Fisher based uncertainty relation and Fisher-Shannon product relation for the uncorrelated and correlated systems in both the coordinate and momentum spaces.
In this work, we investigate the reliability of information-theoretic measures based on the electron-density and shape-function, specifically Shannon and Rényi entropies, as descriptors of electronic correlation. By establishing a rigorous decomposition of these entropic measures into additive and nonadditive contributions, supported on a Mulliken-like atomic partition of molecules, we systematically analyze the asymptotic behavior of the entropies at the infinite-internuclear-distance limit to assess the problem of static correlation and extensivity. Our algebraic and numerical analysis reveals several flaws in the use of these density-based descriptors. We demonstrate that for minimal-basis and different theoretical levels, the Shannon and Rényi entropies fail to encode the amount of static correlation conveyed by the underlying wavefunction. Conversely, shape-function Shannon entropies and Rényi entropies (for
α=1) violate extensivity. In larger basis sets, uncorrelated Hartree-Fock densities consistently overestimate entropy compared to sufficiently correlated (e.g., full-valence-CAS) densities. Moreover, the entropies for insufficiently correlated methods violate extensivity. These findings indicate that electron-density-based measures are insufficient for capturing static correlation, suggesting that robust entropic descriptors should be constructed from higher-dimensional Hilbert-space objects.
In the context of quantum information, both Shannon entropy and the different Jaynes entropy for a two-electron ``entangled artificial'' atom proposed by Moshinsky are studied here. The Jaynes entropy in this model is shown to be intimately related to the correlation kinetic energy. A generalization of the customary Shannon entropy based on the electron density is then proposed in terms of the electron pair correlation function. Both definitions behave in a similar way to the Jaynes entropy when considered as a function of the strength of the model electron-electron interaction.
The influence of correlations of uniform Fermi systems (nuclear matter, electron gas, and liquid
3He) on Shannon's information entropy,
S, is studied.
S is the sum of the information entropies in position and momentum spaces. It is found that, for three different Fermi systems with different particle interactions, the correlated part of
S (Scor) depends on the correlation parameter of the systems or on the discontinuity gap of the momentum distribution through two parameter expressions. The values of the parameters characterize the strength of the correlations. A two parameter expression also holds between
Scor and the mean kinetic energy
(K) of the Fermi system. The study of thermal effects on the uncorrelated electron gas leads to a relation between the thermal part of
S (Sthermal) and the fundamental quantities of temperature, thermodynamical entropy, and the mean kinetic energy. It is found that, in the case of low temperature limit, the expression connecting
Sthermal with
K is the same to the one which connects
Scor with
K. There are only some small differences on the values of the parameters. Thus, regardless of the reason (correlations or thermal) that changes
K,
S takes almost the same value.