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International Journal of Quantum Chemistry

Fisher information and density functional theory

Ágnes Nagy

Published May 7, 2021• 24 citations
Publisher

Abstract

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.

DOI
10.1002/qua.26679

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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.
The entropic perspective on the molecular electronic structure is investigated. Information-theoretic description of electron probabilities is extended to cover the complex amplitudes (wave functions) of quantum mechanics. This analysis emphasizes the entropic concepts due to the phase part of electronic states, which generates the probability current density, thus allowing one to distinguish the information content of states generating the same electron density and differing in their current densities. The classical information measures of Fisher and Shannon, due to the probability/density distributions themselves, are supplemented by the nonclassical terms generated by the wave-function phase or the associated probability current. A complementary character of the Fisher and Shannon information measures is explored and the relationship between these classical information densities is derived. It is postulated to characterize also their nonclassical (phase/current-dependent) contributions. The continuity equations of the generalized information densities are examined and the associated nonclassical information sources are identified. The variational rules involving the quantum-generalized Shannon entropy, which generate the stationary and time-dependent Schrödinger equations from the relevant maximum entropy principles, are discussed and their implications for the system “thermodynamic” equilibrium states are examined. It is demonstrated that the lowest, stationary “thermodynamic” state differs from the true ground state of the system, by exhibiting the space-dependent phase, linked to the modulus part of the wave function, and hence also a nonvanishing probability current.
The Fisher information functional occupies a very special position among the uncertainty measures of quantum systems of any dimensionality. In contrast with (global) entropic measures of Shannon, Renyi or Tsallis types, it contitutes a local measure in the sense that it is sensitive to the oscillatory character of the probability density distribution. Moreover, the Fisher information is closely related to various macroscopic properties of the system described by different density functionals. Here, we review its main analytical properties such as, inequality-based relationships with some radial expectation values and its uncertainty relation. Tighter versions of these inequalities are shown for those systems subject to an arbitrary central potential. Moreover, the Fisher information is computed for D-dimensional hydrogenic systems explicitly in terms of the radial and angular hyperquantum numbers characterizing the quantum states of the system in both position and momentum spaces. Finally, the utility of this quantity is discussed for various physico-chemical processes of recent interest: Abstraction and nucleophilic substitution reactions.
In the context of informational analysis of atomic and molecular systems, the connection between local quantum observables and information measures is of interest. In this paper, analytical relationships for the imaginary part of the total local momentum (coordinate) in terms of information theoretic measures have been established. Moreover, on the basis of another scheme in which the relationship among densities of information energy, Shannon entropy and Fisher information has been proposed [M. Alipour and A. Mohajeri, Mol. Phys. 110, 403 (2012)], the general formulae for the imaginary part of the total local momentum and the corresponding variance are expressed. The presented proof may be viewed in light of the relation between the local wave vector and the information energy density. Through this study, another noteworthy application of information theory is highlighted.
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