Exact exchange potential

The key to understanding the difference between the Slater potential and the exact exchange potential lies in the explicit dependence of the Fermi hole pjr, r ) on... 

This method is usually thought as an approach allowing one to find the exact exchange potential. It may be considered [17] as an approximation to the exact GS problem, similar to the HF approximation namely, the solution of the optimized potential (OP) approximation - the energy Egg and the wave function Gs - stems from the following minimization problem... 

In this section a number of rigorous statements on the optimized effective potential for finite systems will be derived. For this purpose, the exchange-only potential and the correlation potential have to be treated separately within the OEP scheme. The exact exchange potential of DFT is defined as... 

The fact that the exact exchange potential has similar behaviour for the ensemble of multiplets suggests that approximations might also be similar. Probably, a small change in the ground-state exchange functionals might lead... 

Subsequent to the study of Holas and March summarized above, Levy and March [56] have effected a generalization which permits a formulation as a function of the electron-electron repulsion coupling constant A. As will be discussed below, the purpose of their generalization is to allow Vxc(r) to be separated into exchange and correlation contributions. This is achieved by then developing relations which are associated with each order in A. The first-order development yields a formal expression for the exact exchange potential Vjf(r), given by... 

Fig. 1 Exchange integrals (ki Viexlki) calculated with the AAFEGE, HEEGE, and (ki Vex k2) of exact exchange potentials for cyclopropane as a function of a simultaneous increase in the norm and angle of the ki vector. Both vectors are constrained to lie in the CCC plane. The ki vector is stepwise increased and rotated as shown in the figure. The k2 vector is adjusted to keep the K vector fixed at the value of 1 a.u. and oriented as shown in the figure...

For the exact-exchange potential other derivations have been presented, which are not based on the OEP equation. 

It has already been mentioned that for physical reasons the exact exchange potential of finite systems must as3miptoticaIIy behave as... 

One important property that has been emphasized a number of times now is the — 1/r decay of the exact exchange potential of finite systems. In view of the long-range character of the underlying self-interaction integral, one might ask whether it is possible to reproduce this behavior by some explicit density functional In Sect. 2.1.2 it has been demonstrated that the LDA potential decays exponentially. In the case of the GGA, the second explicit density functional of interest,... 

Fig. 2.12. Exchange potential of Si along [111] direction of diamond structure Exact exchange potential (OEM) versus EDA as well as B88- and PW91-GGA data ( indicates the position of atom). All results were obtained by plane-wave pseudopotential calculations with a local pseudopotential Ecut =25 Ry, 19 special fc-points)...

The problem is intrinsically related to the existence of the Rydberg series in the OPM spectrum, which originates from the —1/r behavior of the exact exchange potential. The conventional MP2 calculation gives quite reasonable... 

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