Exchange potential, nonlocal

The presence of the nonlocal exchange potentials in the Hartree-Fock equations greatly complicates their solution and necessitates further approximations. Several of these are discussed in the following subsection. In the evaluation of any calculations, it is important to recognize their common (and imperfect) origin, as well as the seriousness of the particular approximations made in solving the equations. 

The exchange potential of Equation 7.31 is called the Slater potential [12], because it was Slater who had proposed [18] that the nonlocal exchange potential of HF theory can be replaced by the potential... 

The asymptotic structure of the exchange potential vx(r) was derived via the relationship between density functional theory and many-body perturbation theory as established by Sham26. The integral equation relating vxc(r) to the nonlocal exchange-correlation component Exc(r, rf ) of the self-energy (r, r7 >) is... 

Alonso JA, Girifalco LA (1978) Nonlocal approximation to exchange potential and kinetic energy in an inhomogenous electron gas, Phys. Rev. B, 17 3735—3743... 

The SCF-Xa-SW method makes the following major assumptions. First one replaces the nonlocal Hartree-Fock exchange potential with the Xa local exchange potential that corresponds to the average exchange potential of a free electron gas. The exchange potential is related to the local electronic density by... 

Because of the nonlocality of the exchange potential (the terms U(a,/3,r), a / ) Eq. (2) cannot be used to eliminate the small component from Eq. (1) exactly. In the QR approaches the local exchange approximation is used in Eq. (2) and only the second-order equation for the large component is retained. However, if the nonlocal potential is written as the sum of its local and nonlocal parts then the small component can be expressed as... 

The so-called nonlocal exchange potential a component of the HF differen-... 

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... 

This integral equation simply produces the same effect as the localization of the nonlocal Hartree-Fock exchange potential. 

Alonso, J. A., Girifalco, L. A. (1978). Nonlocal approximation to the exchange potential and kinetic energy of an inhomogeneous electron gas. Phys. Rev. BIT, 3735-3743. Al-Oweini, R., El-Rassy, H. (2009). Synthesis and characterization by FUR spectroscopy of silica aerogels prepared using several Si(OR)4 and R00Si(OR0)3 precursors. J. Mol. Struct 919,140-145. 

The calculation of the eigenfunctions of the operator F during the pth iteration is itself a difficult problem that may be solved only approximately. To simplify it, the nonlocal exchange potential K is often replaced by a local potential. The one possible form of the local exchange potential is (especially for crystals)... 

It was empirically found that, in a calculation of the electron structure of crystals with nonlocal exchange, the integration of the Hartree-Fock exchange potential over the direct lattice requires a so-called exchange-interaction radius to be introduced. The exchange-interaction radius cannot be arbitrarily large but must correspond to the used number of points k in the Brillouin zone. 

Fig. 2.8. Exchange potential of Zn Percentage nonlocal contribution in the exact Vx and two GGAs [30,38]. All potentials have been evaluated with the exact x-only density...

It involves the difference between the orbital expectation value of the nonlocal HF-type exchange potential E>j j j and that of v, ... 

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