Hartree-Fock kinetic energy

Lee, H., Lee, C.,and Parr, R. G. (1991) Conjoint gradient correction to the Hartree-Fock kinetic-and exchange-energy density functionals. Phys. Rev., A44, 768-771. Yang, W. (1986) Gradient correcttion in Thomas-Fermi theory. Phys. Rev., A34, 4575-4585. 

The kinetic energy in the Hartree-Fock scheme is evidently too low, owing to the fact that we have assumed the existence of a simplified uncorrelated motion, whereas the particles in reality have much more complicated movements because of their tendency to avoid each other. The potential energy, on the other hand, comes out much too high in the HF scheme essentially due to the fact that we have compelled a pair of electrons with opposite spins together in the same orbital in space. 

The first two terms are the kinetic energy and the potential energy due to the electron-nucleus attraction. V HF(i) is the Hartree-Fock potential. It is the average repulsive potential experienced by the i th electron due to the remaining N-l electrons. Thus, the complicated two-electron repulsion operator l/r in the Hamiltonian is replaced by the simple one-electron operator VHF(i) where the electron-electron repulsion is taken into account only in an average way. Explicitly, VHF has the following two components ... 

To understand how Kohn and Sham tackled this problem, we go back to the discussion of the Hartree-Fock scheme in Chapter 1. There, our wave function was a single Slater determinant SD constructed from N spin orbitals. While the Slater determinant enters the HF method as the approximation to the true N-electron wave function, we showed in Section 1.3 that 4>sd can also be looked upon as the exact wave function of a fictitious system of N non-interacting electrons (that is electrons which behave as uncharged fermions and therefore do not interact with each other via Coulomb repulsion), moving in the elfective potential VHF. For this type of wave function the kinetic energy can be exactly expressed as... 

The terms on the right-hand side of eq. (11.41) denote the kinetic energy, the electron-nuclear potential energy, the Coulomb (J) and exchange (K) terms respectively. Together J and K describe an effective electron-electron interaction. The prime on the summation in the expression for K exchange term indicates summing only over pairs of electrons of the same spin. The Hartree-Fock equations (11.40) are solved iteratively since the Fock operator / itself depends on the orbitals iff,. 

The density functional theory (DFT) [32] represents the major alternative to methods based on the Hartree-Fock formalism. In DFT, the focus is not in the wavefunction, but in the electron density. The total energy of an n-electron system can in all generality be expressed as a summation of four terms (equation 4). The first three terms, making reference to the noninteracting kinetic energy, the electron-nucleus Coulomb attraction and the electron-electron Coulomb repulsion, can be computed in a straightforward way. The practical problem of this method is the calculation of the fourth term Exc, the exchange-correlation term, for which the exact expression is not known. 

In the quantitative development of the structure in the self-consistent field approximation (S.C.F.) using the Hartree-Fock method the energy Ei is made up of three terms, one for the mean kinetic energy of the electron in one for its mean potential energy in the field of the nuclei, and a... 

The energy of the electron gas is composed of two terms, one Hartree-Fock term (T)hp) and one correlation term (Hartree-Fock term comprises the zero-point kinetic energy density and the exchange contribution (first and second terms on the right in equation 1.148, respectively) ... 

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