Pauli exchange repulsion energy

The above analysis of intermolecular stabilizing interactions with its breakdown into coulombic, polarization, and dispersion attractive terms has the advantage of clarifying, at least to some extent, the nature of these attractions in terms of chemically understandable interactions between charge densities. The price that has to be paid is that for a complete description of the interaction, some kind of repulsive effect must also be partitioned out of total energies. As already mentioned, repulsion must account for the scarce compressibility of condensed media - and, indeed, for the very fact that molecules just do not eollapse onto one another, and macroscopic bodies have a finite spatial size. 

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The exchange repulsion energy (2) is an energy component which is always repulsive, i.e., which is responsible for the finite density of molecules. It arises from the Pauli principle repulsion between the electron clouds of the molecules. It should be emphasized that this component is fundamentally quantum mechanical, i.e., unlike the electrostatic component, there is no classical model for this phenomenon. Although classically, the repulsion between the positively charged nuclei would be expected to keep atoms and molecules apart, such a classical repulsion would occur at much shorter distances than that due to the Pauli principle repulsion. The Pauli principle repulsion is much shorter range than typical electrostatic components and is represented in analytical models with functions that depend on the inverse 9th, inverse 12th or exponential power of the atom-atom distances [3]. 

Section 5.3) of steric effects which compared Pauli exchange repulsions for fully relaxed and rigid rotations, we now partition the fully relaxed barrier energy, AE. into partially relaxed energy terms ... 

EX Exchange energy. Part of the interaction energy between static charge clouds of two subunits, resulting from Pauli exchange between them. Similar to steric repulsion for molecular interactions. 

Hiysically, one can think of rather specific second order effects caused by the exdumge forces". For instance, the Pauli exchan repulsion between two closed sheU systems leads to on outward polarizatkm of the electron douds which lowers this exchange rqxilsioa. This energy lowering, which may be called exchange-induction energy, has indeed been found in variational calculations. The mathematical c3q>ression for this effect k not unique, however. 

The exact meaning of the exchange-correlation energy is a difficult one, partly because the DFT definitions of exchange and correlation are not exactly the same as those used in wave-function methods. As mentioned in the previous section, electron correlation arises from the correlated behavior between electrons that is not accounted for in the mean-field Hartree-Fock approach. The exchange energy is the total electron-electron repulsion minus the Coulomb repulsion, and is basically a consequence of the Pauli principle, which states that no two electrons can have the same quantum numbers, i. e. two electrons in the same orbital must have opposite spin. 

A comparison of Equations (2.78) and (2.79) yields that in both approaches, MO (= Hartree-Fock) and VB, the Pauli exchange has been correctly included. The difference between the two is solely given by their amount of electronic correlation. In the MO approach, the electrons are completely uncorrelated (independent), and they may even go into the same atomic orbital, albeit with different spins, thus producing ionic states (H H+). The MO approach therefore does not take care of the energy penalty due to the Coulomb repulsion between the two electrons (see Section 2.9). Because the electronic Coulomb correlation has been completely ignored, the correlation energy may be defined as the difference between the correct energy and that of the Hartree-Fock solution, that is... 

Thus, any realistic deformation of the electron clouds has to take into account simultaneously the exchange interaction (valence repulsion), or the Pauli principle. Because of this, we have introduced what is called the deformation-exchange interaction energy as... 

The electron-electron repulsion contribution is often decomposed into classical electrostatic repulsion energy, plus corrections for the Pauli principle (exchange) and electron correlation... 

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