Electron repulsion energy

To see how and under what conditions stability is enhanced or diminished, we need to consider the symmetry of the orbital (9-32), Flectrons in the antisymmetric orbital r r have a 7ero probability of occurring at the node in u where U] = rj. Electron mutual avoidance of the node due to spin correlation reduces the total energy of the system because it reduces electron repulsion energy due to charge... 

The correlation of electron motion in molecular systems is responsible for many important effects, but its theoretical treatment has proved to be very difficult. Thus many quantum valence calculations use wave functions which are adjusted to optimize kinetic energy effects and the potential energy of interaction of nuclei and electrons but which do not adequately allow for electron correlation and hence yield excessive electron repulsion energy. This problem may be subdivided into cases of overlapping and nonoverlapping electron distributions. Both are very important but we shall concern ourselves here with only the nonoverlapping case. 

In many electron atoms the maximum contributions to the polarizability and to London forces arise from configurations with more than one electron contributing to the net dipole moment of the atom. But in such configurations the electronic repulsion is especially high. The physical meaning to be attributed to the Qkl terms is just the additional electron repulsive energy which these configurations require. 

Table 1.1 The inter-electronic repulsion energies, rep(f"), and the ionization energies, l[f), off configurations according to the theory of atomic spectra.

This gives an overall arrangement in which none of the electrons are paired and the electron repulsion energy is less than in the Lewis structure, hence is the preferred arrangement. All together there are four bonding electrons—three of a spin and one... 

These three objectives could also be achieved if, in the localization sum of Eq. (28), the function r were replaced by any other monotonically varying function of r. The choice of rjj has a further advantage, however. According to Eqs. (12), (14) and (15), the electron repulsion energy can be written as... 

The electron-nuclear attraction energy Ven[/o,/ ] and electron-electron repulsion energy Feel/. / ] in equation (19) are given, respectively, as... 

In 1979, an elegant proof of the existence was provided by Levy [10]. He demonstrated that the universal variational functional for the electron-electron repulsion energy of an A -representable trial 1-RDM can be obtained by searching all antisymmetric wavefunctions that yield a fixed D. It was shown that the functional does not require that a trial function for a variational calculation be associated with a ground state of some external potential. Thus the v-representability is not required, only Al-representability. As a result, the 1-RDM functional theories of preceding works were unified. A year later, Valone [19] extended Levy s pure-state constrained search to include all ensemble representable 1-RDMs. He demonstrated that no new constraints are needed in the occupation-number variation of the energy functional. Diverse con-strained-search density functionals by Lieb [20, 21] also afforded insight into this issue. He proved independently that the constrained minimizations exist. 

Coulomb Energy. The electron-electron repulsion energy according to Coulomb s law. 

A final point about basis functions concerns the way in which their radial parts are represented mathematically. The AOs, obtained from solutions of the Schrbdin-ger equation for one-electron atoms, fall-off exponentially with distance. Unfoitu-nately, if exponentials are used as basis functions, computing the integrals that are required for obtaining electron repulsion energies between electrons is mathematically very cumbersome. Perhaps the most important software development in wave function based calculations came from the realization by Frank Boys that it would be much easier and faster to compute electron repulsion integrals if Gaussian-type functions, rather than exponential functions, were used to represent AOs. 

The energetics of the dihydrogen molecule-ion, H2+, and the dihydrogen molecule were outlined to give a calculation of the inter-electronic repulsion energy in the latter. 

As will be elaborated later, to deal with the spectra of transition metal ions with more than one electron in the empty (or hole in the filled) d10 shell, it is necessary to take into account inter-electronic repulsion energies within the d manifold. On the basis of CFT, of course, the parameters... 

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