Exchange repulsion definition

Conventionally, correlation is explained as dynamic fluctuations in the electronic density. By definition, it is the difference of the exact energy from the HF energy. Correlation energy can be separated into dynamical and static contributions. Notice that exchange repulsion occurs between electrons of equal spin. This implies some correlated motion between these electrons which is absent for electrons of opposite spin in HF. When this correlation is related to intramolecular interactions it is called dispersion, a more difficult quantity to calculate. 

In the EH-CSD approach it is not convenient to decouple electrostatic terms into rigid Coulombic and polarization contributions the effective Hamiltonian leads to compute these two terms together. Exchange repulsive terms are hardly computed when the second partner of the interaction is a liquid they may be obtained with delicate simulation procedures, and it is convenient to decouple them into two contributions, namely the work spent to form a cavity of a suitable shape and an additional repulsion contribution. Dispersion contributions may be kept we shall examine this term in more detail later. Charge-transfer contributions are damped in liquids their inclusion could introduce additional problems in the definition of Vjnt via continuous solvent distributions. It is advisable to neglect them, as it is done in the interaction potentials used in simulations with the present approach it is possible to describe the charge transfer effect by enlarging the solute M —> M-Sn. 

From these definitions, it follows that Kii=Jg. Also note that the Coulomb repulsion between two electrons is independent of their spins while the exchange repulsion vanishes unless their spins are the same.J,j- is repulsive (i.e., positive) and represents the electrostatic repulsion between electron fj, in orbital jfj and electron v in orbital j. It increases with increasing the overlap between the electron densities and... 

Because electron densities are positive-definite, the contribution of the electron gas to interaction energy takes the same sign as the electron gas functional (see Clugston, 1978). Hence, the kinetic energy term is repulsive (cf eq. 1.148 and 1.154), exchange energy is attractive (cf eq. 1.148 and 1.154), and the correlation term is also attractive (cf eq. 1.149, 1.150, and 1.152). 

The definition of the exchange matrix precludes its calculation as a simple matrix product involving F. The K matrix involves the same R matrix but with different electron-repulsion integrals ... 

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. 

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