Hartree kinetic functional

Kohn and Sham later proved that Slater s intuitively motivated suggestion can be justified theoretically and procedures which combine the orbital-based Hartree kinetic functional with density-based exchange-correlation functionals are now called Kohn-Sham density functional theories. They are shown as family 3 in Figure 1. 

Suppose that we wish to calculate the electronic energy of a large chemical system. The evaluation of the Hartree kinetic functional and most of the functionals in Sections 4 and 5 involves only a complicated integration over all space. In contrast, the evaluation of... 

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 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. 

Theoretical evidence [Hartree-Fock (RHF) calculations and density functional theory] has been obtained for a concerted mechanism of oxirane cleavage and A-ring formation in oxidosqualene cyclization. A common concerted mechanistic pathway has been demonstrated for the acid-catalysed cyclization of 5,6-unsaturated oxiranes in chemical and enzymic systems. For example, the conversion of (24) into (26) proceeds via (25) and not via a discrete carbocation (27). Kinetic studies and other evidence are presented for various systems. 

These preliminary results indicate that the possibility of finding explicit functionals of the one-particle density for arbitrary systems is not too far away. Clearly, these constructive functionals are both symmetry as well as system-size dependent. In this sense, our constructive approach leads to energy density functionals which are not universal. Nevertheless, Eqs. (184) and (196) show (for the Hartree-Fock case) a remarkable structure for the kinetic energy and exchange functionals. A considerable part of these functionals is common to all systems, regardless of their symmetry or size. This property, can perhaps be favorably exploited in the construction of approximate energy density functionals. 

Orbital interaction theory forms a comprehensive model for examining the structures and kinetic and thermodynamic stabilities of molecules. It is not intended to be, nor can it be, a quantitative model. However, it can function effectively in aiding understanding of the fundamental processes in chemistry, and it can be applied in most instances without the use of a computer. The variation known as perturbative molecular orbital (PMO) theory was originally developed from the point of view of weak interactions [4, 5]. However, the interaction of orbitals is more transparently developed, and the relationship to quantitative MO theories is more easily seen by straightforward solution of the Hiickel (independent electron) equations. From this point of view, the theoretical foundations lie in Hartree-Fock theory, described verbally and pictorially in Chapter 2 [57] and more rigorously in Appendix A. 

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