The Electron Pair

Abstract

Eighty years have elapsed since Lewis introduced the concept of an electron pair into chemistry where it has continued to play a dominant rôle to this day. The pairing of electrons is a consequence of the Pauli exclusion principle and is the result of the localization of one electron of each spin to a given region of space. It is the purpose of this paper to demonstrate that all manifestations of the spatial localization of an electron of a given spin are a result of corresponding localizations of its Fermi hole. The density of the Fermi hole determines how the charge of a given electron is spread out in the space occupied by a second same-spin electron, thereby excluding an amount of same-spin density equivalent to one electronic charge. The Fermi hole is an electron's doppelgängerit goes where the electron goes and vice versa: if the hole is localized, so is the electron. The topologies of two fields have been shown to provide information about the spatial localization of electronic charge: the negative of the Laplacian of the electron density, referred to here as L(r), and the electron localization function ELF or η(r). The measure provided by L(r) is empirical. It is based upon the remarkable correspondence exhibited by its topology with the number and arrangement of the localized electron domains assumed in the VSEPR model of molecular geometry. η(r) is based upon the local behavior of the same-spin probability, and it is shown that the picture of electron localization that its topology provides is a consequence of a corresponding localization of the Fermi hole density. This paper provides a complete determination and comparison of the topologies of L(r) and η(r) for molecules covering a wide spectrum of atomic interactions. The structures of the two fields are summarized and compared in terms of the characteristic polyhedra that their critical points define for a central atom interacting with a set of ligands. In general, the two fields are found to be homeomorphic in terms of the number and arrangement of electron localization domains that they define. The complementary information provided by the similarities in and differences between these two fields extends our understanding of the origin of electron pairing and its physical consequences.

How Malonaldehyde Bonds Change during Proton Transfer

“…6 It is has been shown that the Laplacian of the charge density and ELF perform partitions of the molecular space which are consistent with the Lewis description and with the VSEPR model. 39,[42][43][44][45][46] The topology of the charge density Laplacina has been studied 38 51 and D h (r) is the kinetic energy of the homogeneous electron gas of density F(r). The excess local kinetic energy defined by eq 2 is the difference between the definite positive kinetic energy density T s (r) ) 1 / 2 ∇∇′F(r,r′)| r′)r and the von Weizsäcker 52 functional T W (r) ) 1 / 8 ∇F(r,r′)| 2 /F(r,r), where F(r,r′) is the first-order density matrix.…”

Electron localisation function in current-density-functional theory

“…The function f ELF is a continuous scalar field and can therefore be subjected to a topological analysis similar to that used for the electron density by Bader [16]. This approach has been extensively used to highlight features of the ELF related to the chemical bonding in systems at zero field.…”

“…The ELF has been widely applied to understand bonding in atoms [3], molecules [4][5][6][7][8][9], clusters [10][11][12][13], and solid-state structures [14,15]. A topological analysis of ELF in the spirit of Bader's atoms-in-molecules approach [16] yields further information on bonding in a given system [2,[17][18][19][20], although some caution should be exercised in interpretations based on this analysis [21].…”

Identification of carbonium and carbenium ions by QTAIM

“…The delocalization index (DI) is a measure of number of electrons that are shared or exchanged between two atomic basins from integration of the Fermi hole density through a pair density matrix. 47,[50][51][52][53][54] The B3LYP functional has an exchange-correlation contribution from Slater determinant and the exchange-correlation potential-energy density. 41 Then, the density matrices of the density functional theory (DFT) yield similar results to post-Hartree-Fock theories like MP2 (see Supplementary Information).…”

“…It is also important to remark that the sum of all localization indexes and half DI´s (because each atomic pair has the same DI) yields the total population of the molecular system. 47,[50][51][52][53][54] The delocalization index of very weak CC interactions lies between 10 -2 and 10 -3 order of magnitude. The DI value of 10 -1 order of magnitude for CC bonds is characteristic of moderate CC interactions (See Supplementary Information).…”