Exchange-correlation constraints

As a consequence of the size limitations of the ab initio schemes, a large number of more-approximate methods can be found in the literature. Here, we mention only the density functional-based tight binding (DFTB) method, which is a two-center approach to DFT. The method has been successfully applied to the study of proton transport in perov-skites and imidazole (see Section 3.1.1.3). The fundamental constraints of DFT are (i) treatment of excited states and (ii) the ambiguous choice of the exchange correlation function. In many cases, the latter contains several parameters fitted to observable properties, which makes such calculations, in fact, semiempirical. 

The exact energy functional (and the exchange correlation functional) are indeed functionals of the total density, even for open-shell systems [47]. However, for the construction of approximate functionals of closed as well as open-shell systems, it has been advantageous to consider functionals with more flexibility, where the a- and j8-densities can be varied separately, i.e. E[p, p ]. The variational search for a minimum of tire E[p, p ] functional can be carried out by unrestricted and spin-restricted approaches. The two methods differ only by the conditions of constraint imposed in minimization and lead to different sets of Kohn-Sham equations for the spin orbitals. The unrestricted Kohn-Sham approach is the one most commonly used and is implemented in various standard quantum chemistry software packages. However, this method has a major disadvantage, namely a spin contamination problem, and in recent years the alternative spin-restricted Kohn-Sham approach has become a popular contester [48-50]. 

Fig. 12. Calculated binding energy versus internuclear distance for Moj using and C , symmetry constraints and the (a) Xa (Ref. 12) and (b) JMW (Ref. 29) exchange-correlation potentials. (Reproduced from Ref.

The fundamental scaling constraint on the total exchange-correlation energy under uniform scaling for all densities is[19]... 

The orbital form of the trial density ensures that it is n-representable and so we seek a minimum in W[p x) subject to this orthonormality constraint on the orbitals Xi(x). Since the functionals T, V and J are ejl avaiilable explicitly in terms of the orbitals, the variational problem becomes identical to the Hartree-Fock variational problem set up and solved in Chapter 2 except for the problematic exchange-correlation functional Exc which is not known explicitly as a functional of p x) or the orbitals Xt(x). Thus we must simply carry the variation in Exc induced by a variation in p(x) into the differential equation for the optimum orbitals... 

We can say a good deal about the most elementary properties of an exchange-correlation functional by an examination of some of the integral constraints on the densities arising from many-electron wavefunctions. Basically, the normalisation constraints on the wavefunction and its associated one- and two-particle densities generate normalisation conditions on the conditional probability distributions which are involved in the definition of the exchange-correlation functional and these conditions place rather severe constraints on the form of any functional. 

A . A = AfT - AfT = A - AE, etc., using the direct expressions already listed in the table. Comparing the alternative CCI formulas for the change in the electron repulsion energy would then give a number of CCI identities, similar to that of Eq. (95), which provide additional, exact constraints on the exchange-correlation functionals in the subsystem resolution. 

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