Exchange and Correlation Holes

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... 

The exchange and correlation holes are important concepts in DFT as many approximate functionals are based on modelling of these functions [11,32-34]. They also help to explain the success of DFT in chemical bonding situations [35,36],... 

Currently, research in our laboratory continues on real-space models of exchange and correlation hole functions in inhomogeneous systems. We anticipate that this work will ultimately generate completely non-empirical parameter-free beyond-LDA density functional theories. The quality of molecular dissociation energies and related properties obtainable with existing semi-empirical gradient-corrected DFTs approaches chemical accuracy, and we hope these future theoretical developments will continue this trend. 

The basic properties of the exact exchange and correlation holes are ... 

Illustrating the exchange and correlation holes for the H2 molecule at the dissociation the reference electron located near nucleus A and the vertical axis representing... 

It is customary to separate into two parts, a pure exchange E and a correlation part Ec, which seems reasonable based on the discussion of the exchange and correlation holes above, and their different scaling properties. It should be noted, however, that only the combined exchange-correlation hole has a physical meaning, and it could be argued that this calls for a combined E c- Early work tended to focus on only one of the components, and subsequently combined these, while the current trend is to construct the two parts in a combined fashion. 

Of the four numbers on the right-hand side of (5.46) Bion is certainly the most accurate, and Bi e is probably the second best. The smallness of the electron-electron interaction term Bg-e confirms the notion of the exchange and correlation hole. The interactions are small because the electrons tend to avoid each other. 

Instead of assuming the LDA-form (Eq. 69), one could introduce a function G(q) which immitates this exchange and correlation hole, and which is to be determined later on from many-body theory ... 

In the last decade however, the dynamics of the exchange and correlation hole became an important topic, in view of the more accurate measurements of the dynamical structure factor at large wave vector, and because sum rules and causality arguments revealed that a static treatment of G[q, w)... 

Of course, as discussed earlier, the determination of G q) remains a still unsolved many-body problem. In fact, it is quite questionable whether the exchange and correlation hole can be describe in such a simple way by a local function. But in practice, one can proceed by using the best known approximations. 

The exchange-hole density of (1.137) can also be spin scaled. Expressions for the exchange and correlation holes for arbitrary and are given in [58]. 

Powerful reasons for the success of LSD are provided by the coupling constant integration of Sect. 1.3.5. Comparison of (1.86) and (1.11) reveals that the LSD approximations for the exchange and correlation holes of an inhomogeneous system are... 

The mixing coefficient a is not equal to or close to 1, because full exact exchange is incompatible with GGA correlation. The exact exchange hole in a molecule can have a highly nonlocal, multi-center character which is largely cancelled by an almost equal-but-opposite nonlocal, multicenter character in the exact correlation hole. The GGA exchange and correlation holes are more local, and more localized aroimd the reference electron. 

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