Functional exchange-correlation

According to the Kohn-Sham approach the ground-state energy functional Eks is rewritten as 

The exchange-correlation functional depending on n for the uniform electron gas is known to a high precision. The local density approximation is successfully applied for solids because the valence electron density in many bulk materials varies slowly. This approximation fails for atoms and molecules because the electron density in this case cannot be satisfactorily approximated by the uniform electron gas. 

GGA functionals satisfy the uniform density limit. In addition, they satisfy several known, exact properties of the exchange-correlation hole. Two widely used nonempirical functionals that satisfy these properties are the Perdew-Wang (PW) functional [15] and the Perdew-Burke-Emzerhof (PBE) functional [16]. Because GGA functionals include more physical ingredients than the LDA functional, it is often assumed that GGA functionals should be more accurate than the LDA. This is quite often true, but there are exceptions. One example of such a situation is in the calculation of the surface energy of transition metals and oxides. 

The LDA and GGA functionals are both nonempirical functionals. Empirical functionals have also been proposed. Empirical functionals are fitted to selected experimental data and, therefore, contain a number of parameters that have been introduced and adjusted during the fitting process. 

Weighted density approximation (WDA) incorporates more nonlocal information about electron gas by means of a model pair correlation function. Weighted density approximation greatly improves the calculated energies of atoms and bulk properties of soHds. Nonetheless, the weighted density approximation is more computationally demanding than both LDA and GGA. 

In fact, the true form of the exchange-correlation functional whose existence is guaranteed by the Hohenberg-Kohn theorem is simply not known. Fortunately, there is one case where this functional can be derived exactly the uniform electron gas. In this situation, the electron density is constant at all points in space that is, n(r) = constant. This situation may appear to be of limited value in any real material since it is variations in electron density that define chemical bonds and generally make materials interesting. But the uniform electron gas provides a practical way to actually use the Kohn-Sham equations. To do this, we set the exchange-correlation potential at each position to be the known exchange-correlation potential from the uniform electron gas at the electron density observed at that position  

This approximation uses only the local density to define the approximate exchange-correlation functional, so it is called the local density approximation (LDA). The LDA gives us a way to completely define the Kohn-Sham equations, but it is crucial to remember that the results from these equations do not exactly solve the true Schrodinger equation because we are not using the true exchange-correlation functional. 

It should not surprise you that the LDA is not the only functional that has been tried within DFT calculations. The development of functionals that more faithfully represent nature remains one of the most important areas of active research in the quantum chemistry community. We promised at the beginning of the chapter to pose a problem that could win you the Nobel prize. Here it is Develop a functional that accurately represents nature s exact functional and implement it in a mathematical form that can be efficiently solved for large numbers of atoms. (This advice is a little like the Hohenberg-Kohn theorem—it tells you that something exists without providing any clues how to find it.) 

Our description of GGA functionals as including information from the electron density and the gradient of this density suggests that more sophisticated functionals can be constructed that use other pieces of physical information. In fact, a hierarchy of functionals can be constructed that gradually include 

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In Ecjuation (3.47) we have written the external potential in the form appropriate to the interaction with M nuclei. , are the orbital energies and Vxc is known as the exchange-correlation functional, related to the exchange-correlation energy by ... 

The total electron density is just the sum of the densities for the two types of electron. The exchange-correlation functional is typically different for the two cases, leading to a set of spin-polarised Kohn-Sham equations ... 

In this equation Exc is the exchange correlation functional [46], is the partial charge of an atom in the classical region, Z, is the nuclear charge of an atom in the quantum region, is the distance between an electron and quantum atom q, r, is the distance between an electron and a classical atom c is the distance between two quantum nuclei, and r is the coordinate of a second electron. Once the Kohn-Sham equations have been solved, the various energy terms of the DF-MM method are evaluated as... 

There is no systematic way in which the exchange correlation functional Vxc[F] can be systematically improved in standard HF-LCAO theory, we can improve on the model by increasing the accuracy of the basis set, doing configuration interaction or MPn calculations. What we have to do in density functional theory is to start from a model for which there is an exact solution, and this model is the uniform electron gas. Parr and Yang (1989) write... 

The explicit form of the functional Fh is of course unknown and in practical applications has to be approximated. In order to facilitate the aeation of these approximations one decomposes Fh into a sum of other functionals that focuses all the unknowns into one component, the exchange-correlation functional, Fxo... 

Note in particular that the exchange-correlation functional that emCTges here does not involve the kinetic energy. From the perspective of the DFT literature, (3.16) is a formulation of the Hohenberg-Kohn functional that is constructed to ensure that the functional derivatives required for variational minimization actually exist. We return to these issues in Sect. 3.3. Also note that in the time-dependent case the external potential V(r, )is often considered to be explicitly... 

Just as in the unrestricted Hartree-Fock variant, the Slater determinant constructed from the KS orbitals originating from a spin unrestricted exchange-correlation functional is not a spin eigenfunction. Frequently, the resulting (S2) expectation value is used as a probe for the quality of the UKS scheme, similar to what is usually done within UHF. However, we must be careful not to overstress the apparent parallelism between unrestricted Kohn-Sham and Hartree-Fock in the latter, the Slater determinant is in fact the approximate wave function used. The stronger its spin contamination, the more questionable it certainly gets. In... 

As already alluded to above, the analysis of the properties of model hole functions that emerge from approximate exchange-correlation functionals is a major tool for assessing... 

In this section we introduce the model system on which virtually all approximate exchange-correlation functionals are based. At the center of this model is the idea of a hypothetical uniform electron gas. This is a system in which electrons move on a positive background charge distribution such that the total ensemble is electrically neutral. The number of elec-... 

There is one more problem which is typical for approximate exchange-correlation functionals. Consider the simple case of a one electron system, such as the hydrogen atom. Clearly, the energy will only depend on the kinetic energy and the external potential due to the nucleus. With only one single electron there is absolutely no electron-electron interaction in such a system. This sounds so trivial that the reader might ask what the point is. But... 

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