Hole function

Thus, the information about T[p] - Ts[p] must be somehow folded into the corresponding hole functions. To do this, imagine that we connect the two systems central for the KS scheme (i. e. the non-interacting reference with no 1/r- electron-electron interaction and the real one where this interaction is operative with full strength) by gradually increasing... 

As already alluded to above, the analysis of the properties of model hole functions that emerge from approximate exchange-correlation functionals is a major tool for assessing... 

In 1989, Rebek and co-workers reported a simple system based on Kemp s triacid that served as a mimic of an enolizing enzyme [86]. This early mimic, however, had the enolizing substrate covalently attached to the triacid skeleton. In addition, the mimic did not possess any oxyanion hole functionalities. However, 2 years later the Rebek group reported a true enolizing catalyst that hosted a carboxylic acid as the oxyanion hole component (Scheme 4.8) [87]. The rate of enolization of the quinuclidone substrate was enhanced by a factor of 10 in the presence of 2.5 mM of the receptor (R = n-Pr). 

One large approximation is the use of Eq. (8.4) for the interelectronic repulsion, since it ignores the energetic effects associated with correlation and exchange. It is useful to introduce the concept of a hole function , which is defined so that it corrects for the energetic errors introduced by assuming classical behavior. In particular, we write... 

The Lh.s. of Eq. (8.6) is the exact QM interelectronic repulsion. The second term on the r.h.s. corrects for the errors in the first term (the classical expression) using the hole function h associated with p (the notation //(rf, r2) emphasizes that the hole is centered on the position of electron 1, and is evaluated from there as a function of the remaining spatial coordinates defining r2 note, then, that not only does the value of h vary as a function of r2 for a given value of Fi, but the precise/orm of h itself can vary as a function of rQ. 

The simplest way to gain a better appreciation for tlie hole function is to consider the case of a one-electron system. Obviously, the Lh.s. of Eq. (8.6) must be zero in that case. However, just as obviously, the first term on the r.h.s. of Eq. (8.6) is not zero, since p must be greater than or equal to zero throughout space. In die one-electron case, it should be clear that h is simply the negative of the density, but in die many-electron case, the exact form of the hole function can rarely be established. Besides die self-interaction error, hole functions in many-electron systems account for exchange and correlation energy as well. 

As already emphasized above, in principle Fxc not only accounts for the difference between the classical and quantum mechanical electron-electron repulsion, but it also includes the difference in kinetic energy between the fictitious non-interacting system and the real system. In practice, however, most modem functionals do not attempt to compute this portion explicitly. Instead, they either ignore the term, or they attempt to constmct a hole function that is analogous to that of Eq. (8.6) except that it also incorporates the kinetic energy difference between the interacting and non-interacting systems. Furthermore, in many functionals... 

Finally, it seems clear that routes to further improve DFT must be associated with better defining hole functions in arbitrary systems. In particular, the current generation of functionals has reached a point where finding efficient algorithms for correction of the classical self-interaction error are likely to have the largest qualitative (and quantitative) impact. 

Fig. 24. Relative state energies (for hole functions) of the Kramers doublets in azidoferrihemoglobin. The spin-orbit coupling constant A is ca. 435 cm-1 in the free iron(III) ion, and might be smaller in the porphyrin complexes (95a)...

An atom capturing both an electron and a hole functions a s a recombination center. In the case of the 100 surface, however, it is conceivable that the surface atoms acquire the sp2 configuration (2). Such interactions do not necessarily lead to surface stabilization. For simplicity, interactions of the dangling bonds and the structural rearrangements of the surface atoms will not be given further consideration. 

Furthermore, since the 2-integration in Equation 4 extracts only the spherical average of the hole function with respect to the reference point 1, details of its angular dependence are unimportant. Spherically symmetric hole-function models are therefore perfectly justified. These constraints, among others discussed elsewhere (4), are satisfied in any many-electron system, whether a helium atom, a transition-metal cluster, a uniform electron gas, etc.. To the extent that properties such as these contain the essential physics of exchange and correlation phenomena, hole-function models provide a simple and convenient alternative to traditional ab initio technology. 

A system of fundamental theoretical importance in many-body theory is the uniform-density electron gas. After decades of effort, exchange-correlation effects in this special though certainly not trivial system are by now well understood. In particular, sophisticated Monte Carlo simulations have provided very useful information (5) and have been conveniently parametrized by several authors (6). If the exchange-correlation hole function at a given reference point r in an atomic or molecular system is approximated by the hole function of a uniform electron gas with spin-densities given by the local values of p (r) and Pp(C obtain an... 

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. 

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