Exchange energy definition

Starting from the Hartree-Fock framework of exchange energy definition in terms of density matrix (Levy et al., 1996),... 

As mentioned in the start of Chapter 4, the correlation between electrons of parallel spin is different from that between electrons of opposite spin. The exchange energy is by definition given as a sum of contributions from the a and /3 spin densities, as exchange energy only involves electrons of the same spin. The kinetic energy, the nuclear-electron attraction and Coulomb terms are trivially separable. 

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). 

We do not distinguish here this density functional definition of exchange energy from that of Hartree-Fock (HF). This simplification is well-justified, if the HF electron density and the exact electron density differ only slightly [40]. Similarly, the coupling-constant averaged exchange-correlation hole is the usual... 

However, expression (149) may be viewed as a definition of a specific exchange energy, say (being thus somehow different from the KS exchange... 

Let us now consider the exchange energy. Starting from the definition ... 

The two methods have a definite relation which comes from the fact that the structure of the molecule, and particularly its dimensions, is regulated by the value of the exchange energy. We thought it possible to specify this relation by using the essential peculiarity of the free-electron model, that is,. the elimination of all dynamic constants by the LCAO MO method. 

The state with the symmetric spin part is labeled the triplet state while the state with the antisyimnetric spin part is the singlet state. As seen below in the context of the hydrogen molecule, there is an electrostatic energy difference AEs,t between the singlet and triplet states. By definition this energy difference equals the exchange energy. 

Dalton s atomic theory, overview, 1 De Broglie equation, 23 Delocalization energy, definition, 174 Density functional theory chemical potential, 192 computational chemistry, 189-192 density function determination, 189 exchange-correlation potential and energy relationship, 191-192 Hohenberg-Kohn theorem, 189-190 Kohn-Sham equations, 191 Weizsacker correction, 191 Determinism, concept, 4 DFT, see Density functional theory Dipole moment, molecular symmetry, 212-213... 

At the conical intersection the two states are, by definition, degenerate (E, = E ), and the total exchange energy T must vanish. Therefore the following two independent relations must hold ... 

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