Abstract

AbstractRelationship of Fisher information and density functional theory is reviewed. Links between the Fisher and Shannon information, the local wave‐vector and the relative information are displayed. Euler equations for the Fisher and relative Fisher information are presented. A combined information theoretical and thermodynamic view of density functional theory is analyzed. The extremum of the Shannon and Fisher information results a constant temperature and for Coulomb systems a simple relation between the total energy and phase‐space Fisher information. Relations for phase‐spase fidelity, fidelity susceptibility and relative information are also presented.

Highlights

  • Fisher information [1] has proved to be very useful among others in physics and chemistry

  • It has turned out to be valuable in density functional theory (DFT) [2]

  • Its suitability was first emphasized in the fundamental paper of Sears, Parr and Dinur [3] presenting a relationship between the Fisher information and the quantum mechanical kinetic energy functional

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Summary

| INTRODUCTION

Fisher information [1] has proved to be very useful among others in physics and chemistry. The local Shannon information determines every property of a finite Coulomb system both in the ground and excited states [18]. As there is only one Euler equation, while there are several Kohn-Sham equations for a large system, it is worth using orbital-free DFT provided that adequate approximation for the kinetic energy functional is available. Berkowitz and Parr (GBP) [37] proposed thermodynamic interpretation According to it the ground-state DFT can be considered a local thermodynamics and local temperature can be defined. To establish a relationship between the Fisher and the Shannon information [11,12] it is worth to introduce the local wave-number [65] as the ratio of the density gradient to the electron density qðrÞ rρðrÞ ρðrÞ ð12Þ. The Shannon information can be expressed as [9]: S ρShannon ðrÞdr

ÀN þ
The relative Fisher information obtained from the density takes the form
The relative specific Shannon information can be defined as
Therefore the variance of the total momentum can be written as
The difference of the total and the Weizsäcker kinetic energies
Àζ tðrÞdr À Ekin
Ig ðβÞ
For constant inverse temperatures F has the form
| DISCUSSION
AUTHOR BIOGRAPHY
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