An Introduction to the Electron Localization Function

These lectures present an introduction to density functionals for non-relativistic Coulomb systems. The reader is assumed to have a working knowledge of quantum mechanics at the level of one-particle wavefunctions (r) [1]. The many-electron wavefunction f (ri,r2,..., rjv) [2] is briefly introduced here, and then replaced as basic variable by the electron density n(r). Various terms of the total energy are defined as functionals of the electron density, and some formal properties of these functionals are discussed. The most widely-used density functionals - the local spin density and generalized gradient... 

The optimum value of c is determined by the variational principle. If c = 1, the UHF wave function is identical to RHF. This will normally be the case near the equilibrium distance. As the bond is stretched, the UHF wave function allows each of the electrons to localize on a nucleus c goes towards 0. The point where the RHF and UHF descriptions start to differ is often referred to as the RHF/UHF instability point. This is an example of symmetry breaking, as discussed in Section 3.8.3. The UHF wave function correctly dissociates into two hydrogen atoms, however, the symmetry breaking of the MOs has two other, closely connected, consequences introduction of electron correlation and spin contamination. To illustrate these concepts, we need to look at the 4 o UHF determinant, and the six RHF determinants in eqs. (4.15) and (4.16) in more detail. We will again ignore all normalization constants. 

NaLS) by copper(II) yields assemblies in which Cu2+ ions constitute the counter ion atmosphere of the micelle (Fig. 4.8). These may be photoreduced to the monovalent state by suitable donor molecules incorporated in the micellar interior. An illustrative example is that where D = N,N -dimethyl 5,11-dihydroindolo 3,3-6 carbazole(DI). When dissolved in NaLS micelles, DI displays an intense fluorescence and the fluorescence lifetime measured by laser techniques is 144 ns. Introduction of Cu2+ as counterion atmosphere induces a 300 fold decrease in the fluorescence yield and lifetime of DI. The detailed laser analysis of this system showed that in Cu(LS) micelles there is an extremely rapid electron transfer from the excited singlet to the Cu2+ ions. This process occurs in less than a nanosecond and hence can compete efficiently with fluorescence and intersystem crossing165. This astonishing result must be attributed to a pronounced micellar enhancement of the rate of the transfer reaction. It is, of course, a consequence of the fact that within such a functional surfactant unit regions with extremely high local concentrations of Cu2+ prevail. (Theoretical estimates predict the counterion concentration in the micellar Stem layer to be between 3 and 6 M). 

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