Hartree-Fock theory, total energy

DFT has come to the fore in molecular calculations as providing a relatively cheap and effective method for including important correlation effects in the initial and final states. ADFT methods have been used, but by far the most popular approach is that based on Slater s half electron transition state theory [73] and its developments. Unlike Hartree-Fock theory, DFT has no Koopmans theorem that relates the orbital energies to an ionisation potential, instead it has been shown that the orbital energy (e,) is related to the gradient of the total energy E(N) of an N-electron system, with respect to the occupation number of the 2th orbital ( , ) [74],... 

The constant p, the chemical potential, is a Lagrange parameter that is introduced to ensure proper normalization, as in Hartree-Fock theory. At this stage, Kohn and Sham noted that Eq. (3.36) is the Euler equation for noninteracting electrons in the external potential V ff. Thus, finding the total energy and the density of the system of electrons subject to the external potential V is equivalent to finding these quantities for a noninteracting system in the potential Veir- Such a problem can in principle be solved exactly, but we have to know E and the potential V c-The one-particle problem can be solved as ... 

Though the theory of the correlation part is available now, the actual calculation of a total bond energy by itself from such an equation as (166) must await the further development of molecular (localized) Hartree-Fock theory. 

Equation (2.37a) shows explicitly the change in the potential in orbital t due to the presence of another electron with opposite spin. The last term in Eq.(2.37b) is due to the presence of the exchange integral in which results in this specific form according to Hartree-Fock theory. The first three terms of Ffj are similar in form to that derived with the Extended-Huckel method. The contribution to the total energy due to the changed electron distributions becomes ... 

To this end, Kohn and Sham assumed that the electrons in these so-called Kohn-Sham orbitals are non-interacting, such that the total electronic wave function can be written as a Slater Determinant. This allows the kinetic energy functional to be split into two parts, one of which, Ts, can be evaluated exactly, in a fashion very similar to the way it is done in Hartree-Fock theory, and a small correction term, which is formally absorbed in the exchange-correlation energy term. Thus, a general DFT energy expression can be written as... 

Table 3. Total ground-state energies of noble gas and closed s-subshell atoms as determined within Slater theory, the Work-interpretation Pauli-correlated approximation, and Hartree-Fock theory. The negative values of the energies in atomic units are quoted...

The k appears as a parameter in the equation similarly to the nuclear positions in molecular Hartree-Fock theory. The solutions are continuous as a function of k, and provide a range of energies called a band, with the total energy per unit cell being calculated by integrating over k space. Fortunately, the variation with k is rather slow for non-metaUic systems, and the integration can be done numerically by including relatively few points. Note that the presence of the phase factors in eq. (3.76) means that the matrices in eq. (3.79) are complex quantities. 

Hartree Fock theory searches for a local minimum by finding the set of coefihcients of the wave function that minimizes the total energy. 

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