Exchange Slater model

The process of equilibration of the atmosphere with the ocean is called gas exchange. Several models are available however, the simplest model for most practical problems is the one-layer stagnant boundary layer model (Fig. 9-18). This model assumes that a well-mixed atmosphere and a weU-mixed surface ocean are separated by a film on the liquid side of the air-water interface through which gas transport is controlled by molecular diffusion. [A similar layer exists on the air side of the interface that can be neglected for most gases. SO2 is a notable exception (Liss and Slater, 1974).]... 

Koopmans Theorem applies to Hartree-Fock theory by virtue of the particular method for evaluating the quantum mechanical exchange interaction. In Density Functional Theory, a different method is employed. Hence, HF orbitals are not the same as DFT orbitals and Koopmans Theorem does not apply. This can be illustrated with reference to Slater s Xu (i.e. DFT exchange only) model [15]. 

Approximation to Dirac-Hartree-Fock method, using Slater exchange to model the exchange term. 

Just to remind you, the electron density and therefore the exchange potential are both scalar fields they vary depending on the position in space r. We often refer to models that make use of such exchange potentials as local density models. The disagreement between Slater s and Dirac s numerical coefficients was quickly resolved, and authors began to write the exchange potential as... 

The general idea of using different orbitals for different spins" seems thus to render an important extension of the entire framework of the independent-particle model. There seem to be essential physical reasons for a comparatively large orbital splitting depending on correlation, since electrons with opposite spins try to avoid each other because of their mutual Coulomb repulsion, and, in systems with unbalanced spins, there may further exist an extra exchange polarization of the type emphasized by Slater. 

In this section we will approach the question which is at the very heart of density functional theory can we possibly replace the complicated N-electron wave function with its dependence on 3N spatial plus N spin variables by a simpler quantity, such as the electron density After using plausibility arguments to demonstrate that this seems to be a sensible thing to do, we introduce two early realizations of this idea, the Thomas-Fermi model and Slater s approximation of Hartree-Fock exchange defining the X(/ method. The discussion in this chapter will prepare us for the next steps, where we will encounter physically sound reasons why the density is really all we need. 

This simplified model of electronic polarization may be used within a KS like formalism to determine the electron density p(r). For instance, if we place the model within the Hartree-Fock-Slater X — a approximation [33], the exchange-correlation potential reduces to ... 

Here ip is an orbital of an electron with Mg = 1/2(t), e is its one-electron energy, is the classical Coulomb potential (including electron self-interaction terms), and represents the effects of electron exchange. In Slater s model, this is related to p h, the local density of electrons of the same spin... 

The distinction is probably best indicated by example. Following from Eq. (8.7) and the discussion in Section 8.1.2, the exchange energy for the uniform electron gas can be computed exactly, and is given by Eq. (8.23) with the constant a equal to. However, the Slater approach takes a value for a of 1, and the Xa model most typically uses j. All of these models have the same local dependence on the density, but only the first is typically referred to as LDA, while the other two are referred to by name as Slater (S) and Xa. ... 

The so-called Hartree-Fock-Slater method is much more widely utilized, and is a hybrid of the Hartree and Thomas-Fermi-Dirac methods. In this method the direct part of the potential is calculated using the Hartree-Fock approach, whereas the exchange part is approximated by some statistical expression of the model of free electrons. The Slater potential is given by ... 

The Xa (X = exchange, a is a parameter in the Xa equation) method gives much better results [23, 24]. It can be regarded as a more accurate version of the Thomas-Fermi model, and is probably the first chemically useful DFT method. It was introduced in 1951 [25] by Slater, who regarded it [26] as a simplification of the Hartree-Fock (Section 5.2.3) approach. The Xa method, which was developed mainly for atoms and solids, has also been used for molecules, but has been replaced by the more accurate Kohn-Sham type (Section 7.2.3) DFT methods. 

Expanding the wave function in a linear combination of pure spin functions could yield the correct secular equations and thus correct eigenvalues. However, such spin-only wave functions could not be considered complete since complete wave functions must describe both the spatial and spin motions of electrons and must be antisymmetric under exchange of any two electrons. It would be better to rewrite the VB model (18) in the second quantization form as given in Eq. (20), in which its eigenstates can be taken as a linear combination of Slater determinants or neutral VB structures. Then... 

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