Electron localization function partition

Distributed Electrostatic Moments Based on the Electron Localization Function Partition... 

Crystallographic electron-density functions exhibit local maxima at the positions of atomic nuclei and, not surprisingly, several efforts to partition the density function into regions that represent individual atoms have been made. 

This mathematical theory provides a partition of the space which is analogous to the more familiar partition made in hydrology in river basins delimited by watersheds. It relies on the study of a local function F(r) called the potential function. The potential function carries the physical or chemical information e.g. the electron density, the ELF (see below), or even the electrostatic potential [56-58]. In the cases treated in the present book, the potential function is required to be defined at any point of a manifold which is either for molecules or the unit cell for periodic systems. Moreover the first and second derivatives with respect to the point coordinates must be defined for any point. Its gradient W(r) forms a vector field bounded on the manifold and determines two kinds of points on the one hand are the wandering points corresponding to W(r ) f 0. and on the other hand are the critical points for which VF(rc) = 0. A critical point is characterized by the index Ip, the number of positive eigenvalues of the second derivatives matrix (the Hessian matrix). There are four kinds of critical points in... 

The ELF approach uses a local function related to the Pauh repulsion to probe the separation of the different electron pairs and from this analysis carries out a partition of the molecular space into basins that correspond to the volumes occupied by core inner shells, bonds, and lone pairs. As in the Lewis model, a valence basin may either belong to a single atomic shell or be shared by several ones. In the first case, the basin is called monosynaptic and corresponds to a lone-pair region, and in the second case, it is polysynaptic and specifically disynaptic for a two-center bond that is of interest in this chapter. 

Y -Hab(Hbb -E)- Hbd Haa -Hab(Hbb -E)"Hh If the Hamiltonian is a one-electron Hamiltonian, for example the Fock operator, the partitioning is done by basis functions, since the latter are usually centered on the atomic nuclei, which belong to donor (d), bridge (b) or acceptor (a). In the Hartree-Fock case, the total wave function is a Slater determinant. There may be problems with symmetry breaking in the symmetric case. Cl that includes the two localized solutions can solve this problem [29-31]. The problem is that the Hartree-Fock method gives energy advantage to a localized state, which holds true also in the unsymmetric case. 

---