Hartree-Fock theory exchange potential

The exchange-correlation energy density can be split into two parts exchange component Ex n) and correlation component e Cn). The explicit expression for the exchange component is known from Hartree-Fock theory but the correlation component is known only numerically. Several parametrisations exist for the exchange-correlation energy and potential of a homogeneous gas system which can be used for the LDA calculations within DFT. 

It has recently been shown [ 12] that time-dependent or linear-response theory based on local exchange and correlation potentials is inconsistent in the pure exchange limit with the time-dependent Hartree-Fock theory (TDHF) of Dirac [13] and with the random-phase approximation (RPA) [14] including exchange. The DFT-based exchange-response kernel [15] is inconsistent with the structure of the second-quantized Hamiltonian. 

In the 0PM schemes one starts from a Hartree-Fock like exchange energy, but the energy is optimized under the restriction that the effective potential is local. So exchange only 0PM is as close to Hartree-Fock theory as a scheme with a local potential can be. 

Equation (2.37a) shows explicitly the change in the potential in orbital t due to the presence of another electron with opposite spin. The last term in Eq.(2.37b) is due to the presence of the exchange integral in which results in this specific form according to Hartree-Fock theory. The first three terms of Ffj are similar in form to that derived with the Extended-Huckel method. The contribution to the total energy due to the changed electron distributions becomes ... 

The precursor to Kohn-Sham density-functional theory is Slater theory [12], In the latter theory, the nonlocal exchange operator of Hartree-Fock theory [25] is replaced by the Slater local exchange potential Vf(r) defined in terms of the Fermi hole p,(r, r ) as... 

Another use of the electron density was introduced by Slater [140] quite early on, as a method of approximating the nonlocal exchange potential which is so tedious to calculate within the framework of Hartree-Fock theory (see Section 2.11.3). With reference to an electron-gas model of uniform density. Slater replaced the nonlocal exchange potential by the so-called Xot local potential... 

Clearly, J (l) just multiplies xi) by the value of the potential at point JTi due to electrons distributed according to the density function for group S in state s. On the other hand K (l) is an integral operator, of the kind used in Hartree-Fock theory (Sections 6.1 and 6.4). Tliese two operators are the coulomb and exchange operators for an electron in the effective field due to the electrons of group S. It is now possible to write the interaction terms in (14.2.2) in the form... 

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