Structure of Fukui matrices

Abstract

Fukui matrices considered as the generalization of the concept of Fukui densities are decomposed into their pairing and unpairing contributions within the theory of the reduced density matrices.Their algebraic structure become clear from this decomposition providing their relationships with the spin density matrices and the irreducible part of the second-order reduced density matrix cumulant, that is, the explicit contributions of the many-body or correlation effects. The uncorrelated state function approximation is a simple way to emphasize the physical meaning of these matrices and represents the appropriate starting point for the treatment of a quasi-analytical model to denote the occurrence of correlation effects. K E Y W O R D S Fukui matrices, pairing densities, reduced density matrices, unpairing densities 1 | T HE ORE TI CA L S CE NA R I O Fukui densities (or Fukui functions) are central to the reactivity concept in chemical physics. [1] The natural scenario in which they has beenproperly defined as the first derivative of the electron density respect to the number of particles at fixed external potential is conceptual density functional theory (DFT). [1,2] Its matrix formulation has been reported by use of the first-order reduced density matrices at the DFT level of approximation [3] and also it has been generalized for any type of molecular state functions. [4][5][6][7] The aim of this report is to use the rigorous version of PPLB proposal [8,9] based on the representation of the grand-canonical ensemble to properly express these magnitudes from which the concept of systems with a non-integer number of particles N (N 5N6m) is introduced in a natural manner. Thus, it allows the number of particles to be a mathematically continuous variable to derive the Fukui like density matrices [7] from the first-order reduced density matrices 1 D N6m (1-RDM), with N 2 N o ; m 2 R, and 0 < m < 1. [9] N o and R stand for the set of non-negative integer and real numbers, respectively. This type of systems may be identified with open systems, that is, an atom, a functional groups or a moiety, as a domain within a given molecular structure. [8] The convex structure of the energy in atomic or molecular systems driven by Coulomb interactions, [8] enables the ground state 1 D N6m , obtained by contraction mapping [10] of two integer M-particle system 1-RDMs, to be expressed by the simple convex expansion [8,9] 1 D N6m 5 m 1 D N61 1 ð12mÞ 1 D N (1)where 1 D N and 1 D N61 stand for the 1-RDMs of the systems with N and N 6 1 particles, that is, the neutral and ionic species, respectively. During the course of this work, the states of the neutral species will be considered as singlet closed shells. Equation 1 is the meaningful physical expression for the 1-RDM of a system possessing a non-integer number of electrons that cannot be described neither by a pure nor a canonical ensemble state. [9] Its trace N 6 m may be interpreted as an average of the number of electrons of the systems involved in the expansion. [8,9] The Fukui mat...

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“…The whole space integration of ρ called the trace operation, that is, tr ( 1 D ) ≡ ∫ d r 1 D ( r | r ) = ∫ d r ρ ( r ) = N results in the number of electrons in the system . The matrix u ( r | r ′ ) = 2 1 D ( r | r ′ ) − 1 D 2 ( r | r ′ ) with 1 D 2 ( r | r ′ ) = ∫ d r ″ 1 D ( r | r ′′ ) 1 D ( r ′′ | r ′ ) is the effectively unpairing density matrix whose physical meaning is the measure of the removal of the double occupancy of the orbitals (molecular or natural) because of the spin of the system (difference of spin up and spin down electrons defining the spin state) and the spatial electronic split of electrons because of correlation effects …”

“…is the effectively unpairing density matrix [39,41,42] whose physical meaning is the measure of the removal of the double occupancy of the orbitals (molecular or natural) because of the spin of the system (difference of spin up and spin down electrons defining the spin state) and the spatial electronic split of electrons because of correlation effects. [41,43,44]…”