Shannon entropy and many‐electron correlations: Theoretical concepts, numerical results, and Collins conjecture

Abstract

In this article, I will discuss the overlap between the concept of Shannon entropy and the concept of electronic correlation. Quantum Monte Carlo numerical results for the uniform electron gas are also presented; these latter on the one hand enhance the hypothesis of a direct link between the two concepts but on the other hand leave a series of open questions which may be used to trace a roadmap for the future research in the field. © 2014 Wiley Periodicals, Inc.

Levy-Lieb-Based Monte Carlo Study of the Dimensionality Behaviour of the Electronic Kinetic Functional

“…In fact in 1D for both spins and spinless systems, due to the break-down of Fermi-liquid behaviour a quadratic power law has been already observed; a similar power-law scaling behaviour in 1D electron gas is also observed experimentally for properties such as conductance, tunneling current I(R), and density of states (DOS) while in 3D the behaviour is logarithmic [25,32,33]. An additional argument in favor of a real physical meaning of our results in 1D is the fact that for the 3D case the logarithmic behaviour of I C [ρ(r)] found with our approach was then qualitatively verified with state of art Quantum Monte Carlo calculations [15]; in addition, in Reference [14], it has been shown that the form of f chosen leads indeed to a first-principle form of electronic correlations, meaning that I C is the average response in energy of the N − 1 electrons to the displacement of the reference electron. It must be also reported that the conclusions of our previous work [12,13] have been strongly criticized by experts of OFDFT [34].…”

“…Some of us have previously proposed a method based on the Levy-Lieb constrained formalism [9,10] to derive a form of the kinetic functional whose non-analytic part can be determined via a Monte Carlo sampling of the electron correlation in space [11][12][13]. For the test case of almost uniform gas in 3D resulted in a kinetic-correlation energy functional which follows the form ρ(r) log ρ(r)dr. Interestingly, the same qualitative behaviour was found also in state-of-the-art Quantum Monte Carlo calculations and opened interesting scenarios where electron correlations may be expressed within the framework of Information Theory [13][14][15]. An interesting question that can be addressed by this method is the following: in general the universal functional of DFT should have a form which is independent of the dimensionality, i.e., the functional behaviour should not depend on the spatial dimensions [16,17].…”

“…Interacting electrons of a gas at uniform density in 1D, 2D and 3D have also been studied extensively using various QMC approaches [23][24][25]; however, a direct comparison between our calculation and QMC calculations in 1D and 2D is not possible at this stage. In fact the quantity we are interested in, i.e., the correlation term of kinetic functional, has not been treated in the QMC studies in 1D and 2D, but only in 3D, where, as mentioned before, there is qualitative agreement with the results of our approach [15]. In this perspective, as underlined before and as will be discussed later on, QMC calculations of the correlation term of the kinetic functional are highly desired, given the results reported here.…”

Shannon Entropy in Atoms: A Test for the Assessment of Density Functionals in Kohn-Sham Theory

“…Thus, we reached the conclusion that the correlation Shannon entropy should also be related to the correlation energy [1]. An interesting discussion about the connection of Shannon entropy and many electron correlation has been given recently by Delle Site [10] (see also Ghiringhelli et al [11]). Trickey et al [12] criticized this view because of the problems connected to the logarithmic density dependence of Shannon entropy.…”

Performance of Shannon-entropy compacted N-electron wave functions for configuration interaction methods

“…This idea underlies several recent studies, including that by Delle Site that also uses Shannon entropic indices albeit directly on the electron density [41][42][43]. For minimising the I C index, we have used the simulated annealing (SA) technique [44,45].…”