Repulsive energy, between electrons

The reason the Schroedinger equation for molecules cannot be separated appears in the last term, involving a sum of repulsive energies between electrons. To... 

To put it crudely, this correlation ensures that electrons of the same spin cannot be in the same place at the same time. Therefore this type of correlation makes the Coulombic repulsion energy between electrons of the same spin smaller than that between electrons of opposite spin. This is the reason why Hand s rule states that, an electronic state in which two electrons occupy different orbitals with the same spin is lower in energy than an electronic state in which the electrons occupy the orbitals, but with opposite spins. 

Unfortunately, if a single configuration is used to approximate the many-electron wave function, electrons of opposite spin remain uncorrelated. The tacit assumption that electrons of opposite spin move independently of each other is, of course, physically incorrect, because, in order to minimize their mutual Coulombic repulsion energy, electrons of opposite spin do certainly tend to avoid each other. Therefore, a wave function, T, that consists of only one configuration will overestimate the Coulombic repulsion energy between electrons of opposite spin. 

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 influence and impact of these semi-empirical calculations and absolute reaction rate theory on the thinking of physical organic chemists was profound. It makes clear, for example, the electronic basis for some of Ingold s broad generalizations, e.g. In bimolecular eliminations, E2, in systems H—Cp—Ca—X, where X may be neutral or charged, the ]8-CH electrons, independently of the electrostatic situation, enter the Ca octet on the side remote from X, because repulsive energy between electron-pairs in the transition state can thus be minimized the result is anti-elimination, independently of the structural details of the system (Ingold, 1953). 

The repulsion energy for each pair of electrons contributes to the energy of both electrons. For example, the repulsion energy between electrons 2 and 3 is added into the sum for both bj and E3. Therefore, the total Hartree-Fock energy of the electronic state is computed by this method as... 

This is embodied in the antisymmetrization of wavefunctions for electrons, as in Eq. [2], which provides correlation of the motions of electrons of the same spin, so that they tend to avoid each other. If one plots the probability of finding two electrons, i and /, as a function of their distance, r ,-, the resulting curve goes to exactly zero at r = 0 for electrons of the same spin, but not for those of opposite spin. This so-called Fermi hole leads to a reduced overall Coulombic repulsion energy between electrons of the same spin, compared to those of opposite spin. 

In practice, the harmonic oscillator has limits. In the ideal case, the two atoms can approach and recede with no change in the attractive force and without any repulsive force between electron clouds. In reality, the two atoms will dissociate when far enough apart, and will be repulsed by van der Waal s forces as they come closer. The net effect is the varying attraction between the two in the bond. When using a quantum model, the energy levels would be evenly spaced, making the overtones forbidden. 

V negatively to the first reduction, provided that the supporting electrolytes used were tetraalkylammonium salts. Therefore, these reduction potentials were also correlated with the LUMO energies of the HMO model [3]. It was suggested that the energy difference of 0.55 eV corresponds to the repulsion energy between both electrons in the LUMOs of the dianions [1], despite the differences in their structures. 

The crystal field effect is due primarily to repulsive effects between electron clouds. As we have already seen, the repulsive energy is of opposite sign with respect to coulombic attraction and the dispersive forces that maintain crystal cohesion. An increase in repulsive energy may thus be interpreted as actual destabilization of the compound. 

However, there are two more types of one-electron operators in T. One of them, Jis called the Coulomb operator. The other, K,j, is called the exchange operator. Together, they replace the two-electron operators, e /ra, in FL, which give the Coulombic repulsion energy between each pair of electrons, k and 1. 

Jj — K,j operating on / gives the expression in Hartree-Fock (HF) theory for the effective Coulombic repulsion energy between an electron /,- and the pair of... 

The operator 2Jj computes the Coulombic repulsion energy between an electron in v[/, and the pair of electrons in /,-, assuming that the electrons in v[/, and in /,- move independently of each other. The operator ICj corrects the Coulombic repulsion energy, computed from 2Jj, for the fact that antisymmetrization of results in correlation between an electron in /,- and the electron of the same spin in v[/,. 

In addition, the functional must somehow cancel the fictitious repulsion energy between an electron and itself, which arises if the electron density, due to all the electrons, is used to compute the Coulombic energy of a single electron. As discussed in Section 3.2.1, in HF theory cancellation of the self-repulsion energy results from the presence of the exchange operator in T. If this effect of Kj, in the Fock operator is not mirrored exactly by the functional chosen, the cancellation of the self-repulsion energy will not occur. 

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