Electron Localization-Delocalization Matrix

Timm MJ, Matta CF, Massa L, Huang L (2014) The localization-delocalization matrix and the electron density-weighted connectivity matrix of a finite graphene flake reconstructed from kernel fragments. J Phys Chem A 118 11304-11316... 

Superexchange is another mechanism of electron transfer over relatively large distances in which the solvent or matrix acts as a bridge between the donor molecule D and the acceptor molecule A. It differs from electron hopping in that the electron is at no time actually localized on a molecule of the medium there is an interaction between the orbitals of the molecules A, B and D which form a sort of very loose supermolecule over which the electron is delocalized (Figure 4.10). This mechanism seems plausible when the relevant orbitals of A, B and D are rather close in energy. This is similar to the requirement for the interaction of atomic orbitals to form a molecule. 

The electrostatic interaction is apparently insufficient to bind both 0 to the cation vacancy While one 0 remains with the vacancy, though locally delocalized (22.), the other becomes unbound (22). An unbound 0 is a mobile charge carrier moving through the host matrix by a succession of electron jumps from 02 to 02". Because the 0 states are positively charged with respect to 02 they have been called "positive holes" (22). 0 states repel each other in the bulk and tend to diffuse towards the surface causing the surface to become positively charged. 

The LDM codes for more than one aspect of the electron distribution in the molecule (one-electron density and pair density) as described in Sect. 3.1 above. Equations (3.5-3.10) show that an LDM contains information on atomic populations, atomic charges, the total number of electrons in the molecule (and their localized and delocalized subpopulations), and also two-electron information derived from the pair density, that is, the full atom-atom delocalization matrix of the system. It is thus expected that the LDM codes strongly for aromaticity by virtue of the first Hohenberg-Kohn theorem. The core question is how to get from LDMs to a description of aromaticity ... 

Charge Localized vs. Delocalized Wavefunctions. In the spirit of the Condon approximation discussed above, we do not include the full dependence of the purely electronic matrix elements on q n, but rather evaluate them where the diabatic curves... 

The neglect of the electronic coupling in the calculation of the ECWD (assumption 1) was adopted in the original Marcus and Elush formulation. " Within this framework, the ET matrix element does not strongly affect the nuclear fluctuations, although a nonzero value of Hab is required for electronic transitions to occur. In other words, the transferred electron is assumed to be fully localized in the calculation of the ECWD. To classify electronic delocalization, Robin and Day distinguished between three classes of symmetrical (APo = 0) systems. 

In order to find the local density of states at the surface, one requires a formalism not in terms of delocalized orbitals as we have used so far, but in terms of local quantities. Such a formalism is provided by the Green s function method, which Is also a very convenient tool for the study of chemisorption. Here we will introduce its use by applying it to the open and the closed chain problems discussed before. Within the LCAO basis set description and mean field electron approximation (see section 2.7.3.4), the Green s function is a matrix element that satisfies the following set of equations. 

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