On the Quantum‐Mechanical Kinetic Energy as a Measure of the Information in a Distribution

Abstract

Standard concepts of information theory, including Shannon's entropy, Fisher's information, Jaynes' principle of entropy maximization, Fisher's locality information matrix, and Kullback and Leibler's information measure, are described and extended to many dimensions as appropriate, to establish precise connections between the many body quantum‐mechanical kinetic energy functional T[Ψ] and information measures. Implications for density functional theory of electronic structure are discussed, and elementary examples are displayed to illustrate the argument. Among the several exact relations obtained, one of special interest is the identity where the first term is the intrinsic accuracy or Fisher's information for locality of the one‐particle density (normalized to 1), \documentclass{article}\pagestyle{empty}\begin{document}$ \rho \left( 1 \right) = \int { \cdot \cdot \cdot \int {|{\rm \psi |}^{\rm 2} d_{{\rm T}_2 } \cdot \cdot \cdot } } \,d_{{\rm T}_{\rm N} ,} $\end{document}, and the second term is the average over the one‐particle density of Fisher's information associated with the conditional density \documentclass{article}\pagestyle{empty}\begin{document}$ f\left( {2,\;3,\; \cdot \cdot \cdot ,\;N|1} \right) \equiv \,|\psi |^2 /\rho \left( 1 \right) $\end{document} that is, the second term is the average over the marginal distribution ρ(1) of the trace of Fisher's information matrix for the distribution f(2,3, …, N|1). Because of this formula, the quantum mechanical variation principle may be precisely stated as a principle of minimal information.

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

“…The inner minimisation of the universal functional is carried out with respect to the wavefunctions integrating to density ρ(r) and the outer minimisation is done on all densities, preserving the N-representability (i.e., integrating to N). In an alternative representation, N-electron wavefunction can be substituted by its corresponding 3N−dimensional probability density and expressed in terms of the one particle density ρ(r 1 ) and the N − 1, conditional electron density [11,18]:…”

Atomic Density Functions: Atomic Physics Calculations Analyzed with Methods from Quantum Chemistry

“…The first applications of information theoretical concepts in quantum chemistry, can be found in the literature of the early eighties. The pioneering work of Sears, Parr and Dinur [54] quickly lead to more novel ideas and publications. Since then, many applications of information theoretical concepts to investigate wave functions and density functions, have been reported.…”

Interacting electrons, spin statistics, and information theory

“…This can be achieved by a formalism in which the N-electron wave function ͑squared͒ is expressed in terms of the one-electron ͑probability͒ density p͑r͒ = p͑r 1 ͒ = ͉͐ ͑r 1 , ... ,r N ͉͒ 2 d 3 r 2 ...d 3 r N ͓so that ͑r͒ = Np͑r͔͒ and the N − 1 conditional electron density f͑r 2 , ... . ,r N ͉ r 1 ͒ as ⌿ 2 a͒ = p · f. 15 Thus, for a fixed configuration of one of the electrons, f͑r 2 , ... . ,r N ͉ r 1 ͒ is the probability of finding an N −1 electron configuration of the other electrons.…”