Hole, exchange

Becke, A. and M. E. Russel. 1989. Exchange holes in inhomogeneous systems A coordinate-space model. Phys. Rev. A 39, 3761. 

Further, if the wave function depends also on the electron spins, spin variables over all electrons should also be integrated we will see this below, in the calculation of exchange hole. The expression in the curly brackets above is exactly the XC hole PxCM(r, r ) defined in Equation 7.17. A comparison with Equation 7.19a shows that adding the hole to the density is similar to subtracting the density of one electron p(r )/N from it. The hole thus represents a deficit of one electron from the density. This is easily verified by integrating p tM(V, r ) over the volume dr, which gives a value of — 1. However, the structure of the hole is not simple and this is because of the motion of different electrons correlated due to the Pauli exclusion principle and the Coulomb interaction between them. Finally we note that the product p(r)p cM(r, r ) is symmetric with respect to an exchange in the variables... 

Significant reverse currents at semiconductor electrodes are not only observed for breakdown but also under illumination. For the latter case a quantum efficiency q, the number of exchanged holes and electrons n e per incident photons np, can be defined ... 

By studying the exchange (i.e., A = 0) hole in Ne, Gunnarson et al. pointed out that the LSD approximation to this quantity was far better than the LSD approximation to the exchange hole prior to spherical-averaging. Then, from Eq. (7), the energy depends only on this spherically-averaged hole, i.e.. 

Fig. 4. System-averaged exchange hole density (in atomic units) in the He atom, in LSD, numerical GGA, and exactly (Cl). The area under each curve is the exchange energy...

Successful density functional approximations such as the PW91 GGA or the self-interaction correction (SIC) [57] to LSD recover [19] LSD values for the on-top hole density and cusp. The weighted density approximation (WDA) [41,42], which recovers the LSD exchange hole density but not the LSD correlation hole density [19] in the limit u -> 0, needs improvement in this respect. 

Note that the exchange hole is not affected by the coupling constant integration as the density and therefore the Kohn-Sham orbitals also remain the same during the integration. The exchange and correlation holes satisfy the important sum rule properties... 

As the Kohn-Sham wave function has a delocalized exchange hole, and therefore lacks the left-right correlation, is expected to display different bond midpoint features to In fact, Vs,km is ro for two-electron systems, so that Tc [p] and Vc reduce in this case to... 

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