Electron Density and Hole Functions

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... 

Since the coiTelation between opposite spins has both intra- and inter-orbital contributions, it will be larger than the correlation between electrons having the same spin. The Pauli principle (or equivalently the antisymmetry of the wave function) has the consequence that there is no intraorbital conelation from electron pairs with the same spin. The opposite spin correlation is sometimes called the Coulomb correlation, while the same spin correlation is called the Fermi correlation, i.e. the Coulomb correlation is the largest contribution. Another way of looking at electron correlation is in terms of the electron density. In the immediate vicinity of an electron, here is a reduced probability of finding another electron. For electrons of opposite spin, this is often referred to as the Coulomb hole, the corresponding phenomenon for electrons of the same spin is the Fermi hole. 

Figure 11-11. Comparison of measured and calculated current density as a function of bias for an electron-only Sm/MEH-PPV/Ca structure (upper panel) and a hole-only AI/MEH-DSB/Au structure (lower panel). The main parts of the figure are linear-linear plots and the insets are log-linear plots.

Figure 11-18. Calculated current (solid line) and iccuinbiiuuion current (dashed line) density as a function of voltage bias for a single-layer structure, a two-layer structure with a hole-blocking layer and a two-layer structure in which the hole-blocking layer also serves as an electron transport layer.

Let us introduce another early example by Slater, 1951, where the electron density is exploited as the central quantity. This approach was originally constructed not with density functional theory in mind, but as an approximation to the non-local and complicated exchange contribution of the Hartree-Fock scheme. We have seen in the previous chapter that the exchange contribution stemming from the antisymmetry of the wave function can be expressed as the interaction between the charge density of spin o and the Fermi hole of the same spin... 

Fe—S dimers, 38 443-445 map, four-iron clusters, 38 458 -functional theory, 38 423-467 a and b densities, 38 440 broken symmetry method, 38 425 conservation equation, 38 437 correlation for opposite spins and Coulomb hole, 38 439-440 electron densities, 38 436 exchange energy and Fermi hole, 38 438-439... 

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