Energy total, from self-consistent electron density

The self-consistent electronic structure calculations for the energy of vacancy formation are based on the local density approximation, equation (8.17). Authors of [25] used the supercells with 27 and 32 lattice sites for bcc and fee metals, respectively. The atoms neighboring the vacancies are not allowed to relax from their perfect lattice position. It was found that the errors due to omission of the lattice relaxation around the vacancy were only of the order of one-tenth of an eV. The total energy of a supercell tot depends on the number of atoms in the supercell N, the number of vacancies v in a volume Q, Etot = E N,v, Q). In the superceU approximation, the vacancy-formation energy is given by... 

Since we now have a one-electron problem, the Kohn-Sham equations (2.4) can be solved in a self-consistent manner. We obtain a set of orbitals and their energies, much as in HF theory. The density function, p(r, can be found as the sum of the squares of the w/, for the occupied orbitals. From p(r) the expectation value of the energy can be found, as well as other one-electron properties. Just as in the HF method, the total electronic energy is equal to the sum of the energies of the occupied orbitals, minus a correction because the electron-electron interactions have been counted twice. 

When solved self-consistently, the electron densities obtained from Eq. [6] can be used in Eq. [5] to calculate the total electronic energy. This is equivalent to the relationship between Eqs. [3] and [4] for the Hartree approach. Unlike the Hartree approximation, however, this expression takes into account exchange and correlation interactions between electrons directly, and requires no other approximations other than the form of the density functional. 

We have now all the ingredients to obtain the electronic density from (6.2). The self-consistency cycle is stopped when some convergence criterion is reached. The two most common criteria are based on the difference of total energies or densities from iteration i and i - 1, i.e., the cycle is stopped when < t]e or fd r ( - ) < where and... 

Since the effective potential is a function of the density, which is obtained from Eq. (2.79) and hence depends on all the single-particle states, we will need to solve these equations by iteration until we reach self-consistency. As mentioned earlier, this is not a significant problem. A more pressing issue is the exact form of [n(r)] which is unknown. We can consider the simplest situation, in which the true electronic system is endowed with only one aspect of electron interactions (beyond Coulomb repulsion), that is, the exchange property. As we saw in the case of the Hartree-Fock approximation, which takes into account exchange explicitly, in a uniform system the contribution of exchange to the total energy is... 

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