Effective external potential

For each X the effective external potential A t is adapted such that the density always equals the density of the fully interacting system. Hence p(r) is independent of the actual value of X. Clearly, for X = 0, we recover the Hamilton operator for the non-interacting reference system, and V ° = Vs, while X = 1 is the regular expression for the real system... 

Equations 24.75 and 24.76 generalize the Dyson equations derived previously [32], The differences between /1 and the frontier orbitals have nice and simple physical interpretations [24,32] as either the variation of the electronic density induced by effective external potentials 8vj ... 

One can be easily convinced that this prescription is correct, if one compares the variation of the functionals minimized in Eqs. (87) and (64) the last minimization being equivalent to that in Eq. (28), solved via Eq. (33). Because the effective external potential (89) depends functionally on n r), an iterative method, leading to self-consistency, must be employed. 

The corrected HF method described above for the exact GS problem was discussed earlier, among others, by Parr and Yang [10] and termed by them the HF-KS method. However, the contribution of uP = SE Idn to the effective external potential (89) was not included there. The consequences of such neglect are discussed below, see the paragraph starting a few lines after Eq. (94). 

Next, when we compare the variation of the functional minimized in Eq. (108) with that in Eq. (101), we conclude that the exact GS problem can be solved by algorithms of the OP method, if an OP effective external potential... 

Finally, an exact t>,(r [hgs]) may be determined in practice by applying the corrected OP method iteratively. In this case the correlation potential, determining the effective external potential in Eq. (110) during (/ -f l)-th iteration, is defined as... 

The effective external potential Ua includes the Hartree-Fock contribution and reads... 

The introduction in 1965 by Kohn and Sham7 of a practical computational scheme may, therefore, be considered to be the next major milestone in the development of formal DFT. The essential ingredient in this approach is the postulation of a reference system of N noninteracting electrons, moving in an effective external potential vs(r), the so-called Kohn-Sham potential, instead of the electrostatic potential v(r) of the nuclei ... 

In the special case of the interaction of nitrogen with two graphitic slabs (i.e. within a carbon slit), the Steele 10-4-3 potential has been used to obtain the effective external potential (Lastoskie et al., 1993). It is this system which has so far received most attention and generally the surface is assumed homogeneous so that the potential is constant in the xy plane. 

It goes beyond the scope of this chapter to discuss the matter of time scale where such a density function appears justified nevertheless, one should recognize that the effects of zero-point energy and the associated vibrations modify the role of the effective external potential experienced by the actual electron density, which also involves the role of time. Whereas electron density changes and fluctuations can carry information, all such effects on the molecular information are also dependent on the ground-state electron density. 

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