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Journal of Computational Chemistry

Fisher Information Density Functional Theory

Ágnes Nagy

Published Aug 25, 2025• 3 citations
Publisher

Abstract

According to the density functional theory, the density contains sufficient information to compute the value of any observable. It is shown that the Fisher information density also includes this knowledge. The Fisher information density functional theory is constructed. The variational principle is extended to the energy as a functional of the Fisher information density. Hohenberg-Kohn-like theorems are shown to be valid.

DOI
10.1002/jcc.70215

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According to the density functional theory, the density contains sufficient information to compute the value of any observable. It is shown that the Fisher information density also includes this knowledge. The Fisher information density functional theory is constructed. The variational principle is extended to the energy as a functional of the Fisher information density. Hohenberg-Kohn-like theorems are shown to be valid.
A formulation of density functional theory (DFT) is constructed as an application of the method of maximum entropy for an inhomogeneous fluid in thermal equilibrium. The use of entropy as a systematic method to generate optimal approximations is extended from the classical to the quantum domain. This process introduces a family of trial density operators that are parameterized by the particle density. The optimal density operator is that which maximizes the quantum entropy relative to the exact canonical density operator. This approach reproduces the variational principle of DFT and allows a simple proof of the Hohenberg-Kohn theorem at finite temperature. Finally, as an illustration, we discuss the Kohn-Sham approximation scheme at finite temperature.
From the very beginning quantum statistical mechanics can be formulated in terms of a thermodynamic potential, which has the electron density as the basic variable. By employing a well-known procedure in quantum statistical mechanics for the construction of stationarity principles [T. D. Lee and C. N. Yang, Phys. Rev. 113, 1165 (1959); 117, 22 (1960)], the variational principles of Hohenberg, Kohn [P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864 (1964)], and Mermin [N. D. Mermin, Phys. Rev. A 137, 1441 (1965)] are obtained as are generalizations thereof.
Abstract The classical Fisher information concept is generalized to cover the complex probability amplitudes (wave functions) of quantum mechanics. Its contributions due to the probability and probability‐current densities are identified. General properties of this local information measure are examined and the Schrödinger functional for the kinetic energy is interpreted as the average Fisher information. The information current and source densities are introduced in terms of which the relevant continuity equation is formulated, expressing the local balance of the information content. The superposition principles for the particle probabilities, probability currents, and the Fisher information densities are examined. The interference (nonadditive) contributions to these quantities due to the quantum mechanical mixing of individual states are identified. An illustrative application of using the nonadditive Fisher information of the atomic orbital (AO) resolution in indexing the chemical bond within the 2‐AO model is presented. This AO‐phase sensitive index is capable of distinguishing the bonding, nonbonding, and antibonding electronic states, as do the familiar bond‐orders of quantum chemistry. It is interpreted as a measure of an extra delocalization of electrons due to the bond formation. This is in contrast to the so called quadratic bond‐multiplicity measures and the nonprojected entropy/information concepts of the communication theory of the chemical bond, which both loose the memory about the relative AO‐phases in molecular orbitals, thus failing to distinguish between the bonding and antibonding electron configurations. Finally, the Schrödinger variational principle of quantum mechanics and the related Kohn‐Sham principle of Density‐Functional theory are interpreted as constrained variational principles of the Fisher information contained in the electron probability distributions. They are examples of the Extreme Physical Information principle related to the measurement process. The Fisher‐information principle for the adiabatic (Born‐Oppenheimer) approximation of the molecular quantum mechanics is also examined. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008
Properties of the Hohenberg-Kohn functional are considered. In particular, the Hohenberg-Kohn functional should (a) give correct results in the variational principle and should be (b) continuous, (c) convex, and (d) size consistent. All of these properties are satisfied by the Legendre-transform functional (equivalently, the density matrix constrained search functional) and, moreover, this is the only functional that possesses all these properties. Not only that, but the Legendre-transform functional is determined uniquely by requiring (a), (b), and either (c) or (d). This shows how an ``axiomatic'' approach to constructing the Hohenberg-Kohn functional leads naturally to the Legendre-transform functional. Among all functionals consistent with the variational principle, the Legendre-transform functional is the smallest. One corollary to this approach is a simple proof of the equivalence of the Legendre-transform and density-matrix constrained search functionals. For completeness, the Appendix shows that ensemble-vvv-representable densities lie dense in the set of NNN-representable densities.
Abstract Relationship 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.