Use of Fisher information in quantum chemistry

Abstract

AbstractThe 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