Unpaired electrons at the second‐order reduced density matrix level: Covalent bonding, and coulomb and fermi correlations in closed shell systems

Abstract

The usual one‐electron populations in atomic orbitals of closed shell systems are split into unpaired and paired at the (spin‐dependent) second‐order reduced density matrix level. The unpaired electron in an orbital is defined as the “simultaneous occurrence of an electron and an electron hole of opposite spins in the same spatial orbital,” which for simplicity is called “electropon.” The electropon population in a given orbital reveals whether and to what degree the Coulomb correlations, and hence, the chemical bonding between this orbital and the remaining orbitals of the system are globally favorable or unfavorable. The interaction of two electropons in two target orbitals reveals the quality (favorable or unfavorable) and the strength of the covalent bonding between these orbitals; this establish a bridge between the notion of “unpaired electrons” and the traditional covalent structure of valence‐bond (VB) theory. Favorable/unfavorable bonding between two orbitals is characterized by the positive/negative (Coulomb) correlation of two electropons of opposite spins, or alternatively, by the negative/positive (Fermi) correlation of two parallel spin electropons. A spin‐free index is defined, and the relationship between the electropon viewpoint for chemical bonding and the well‐known two‐electron Coulomb and Fermi correlations is established. Benchmark calculations are achieved for ethylene, hexatriene, benzene, pyrrole, methylamine, and ammonia molecules on the basis of physically meaningful natural orbitals. The results, obtained in the framework of both orthogonal and nonorthogonal population analysis methods, provide the same conceptual pictures, which are in very good agreement with elementary chemical knowledge and VB theory. © 2013 Wiley Periodicals, Inc.

To Be or Not to Be: Demystifying the 2nd‐Quantized Picture of Complex Electronic Configuration Patterns in Chemistry with Natural Poly‐Electron Population Analysis

“…As a result, 2nd‐quantized expressions (22) and (23) allow an important generalization of conventional ( Γ (1) ‐based) NPA to the analogous Natural Poly‐Electron Population Analysis (NPEPA) . The latter builds on Moffitt's theorem and allows the probabilities of more complex ne/nh excitation patterns to be evaluated for wave functions of uncorrelated or correlated form, as described in the following sections.…”

Unpaired electrons, spin polarization, and bond orders in radicals from the 2‐RDM in orbital spaces: Basic notions and testing calculations

“…The main scope of the investigation of unpaired electron distribution is to describe the occupancy of local regions of space of a molecular system by single (α‐ or β‐spin) electrons. This quantity is of broad interest, concerning reaction mechanisms, the design of materials having magnetic properties, and some biochemical topics.…”

“…It is possible to overcome these difficulties by introducing an analogous quantity directly from the 2‐RDM (instead of the product of two 1‐RDMs). In this framework, in a recent work, the conventional one‐electron population, , in an atomic orbital (AO) is split into two component parts: where P 2;0 ( ;0) is a two‐electron (and zero‐hole) quantity, representing the number of electron pairs belonging to , and P 1;1 ( ; ) is an one‐electron‐one‐hole quantity (α‐spin electron in spin‐orbital , and β‐spin hole in ). This splitting is shown schematically in Figure for the case of a radical.…”

“…In a closed shell system, the electropon population in a target orbital , P 1;1 ( ; ), provides a global picture for the interactions of an electron in with all opposite spin electrons in all the other orbitals of the system, because it includes cumulatively the sum of Coulomb correlations, as shows relation (13) of Ref. . When it is necessary to obtain more detailed information, and focus our investigations on two target AOs, one must examine the correlation of two electropons in these AOs.…”

“…When it is necessary to obtain more detailed information, and focus our investigations on two target AOs, one must examine the correlation of two electropons in these AOs. In this framework, it has been shown that favorable (or bonding)/unfavorable (or antibonding) chemical bonding is characterized by positive/negative Coulomb correlations, or alternatively, by negative/positive Fermi correlations between two electropons …”