Exchange-correlation functionals

we can say that the exchange-correlation energy xc[p] is the functional that contains everything that is unknown. 

Introducing the expression (2.32) into (2,19), we can write down the expression for the energy functional of the real system as  

introducing into this last equation the restriction that the electron density resulting from the summation of the moduli of the squared orbitals cpi exactly equals the ground state density of our real system of interacting electrons. 

In this last equation, the only term that is unknown is the potential Vxc du exchange-correlation energy Exc- This potential is simply defined as the functional derivative of Exc with respect to the electron density  

It is very important to realize that if the exact forms of Exc and Vxc were known, the Kohn-Sham method would provide the exact energy. Unfortunately, this is not the case. Furthermore, since the effective potential depends on the electron density, the Kohn-Sham Eqs. (2.37) have to be solved in an iterative way. For this process, we define a trial electron density from which we can calculate the effective potential through Eq. (2.38). Then, with this effective potential we solve the Kohn-Sham Eqs. (2.37) and obtain the orbitals (pi, which are introduced into the Eq. (2.35) resulting in a new electron density. This proeess is iteratively repeated until the difference between this new electron density and the trial density satisfies the desired convergence criterion. Once this is done, the energy can be easily computed from Eq. (2.34) using the converged electron density. 

These concepts can be illustrated for the H2 molecule for increasing internuclear distances. In wave mechanics, the ground state for H2 has two electrons of opposite spin in the same spatial orbital, and the exchange hole is thus entirely the self-interaction correction (no same-spin exchange). 

Within wave mechanics, the exchange hole is static and delocalized over the whole molecule, while the long-range part of the electron correlation is dynamical and serves to cancel the exchange hole away from the reference electron. 

The difference between various DFT methods is the choice of functional form for the exchange-correlation energy. It can be proven that the exchange-correlation potential is a unique functional, valid for aU systems, but an explicit functional form of this potential has been elusive, except for special cases such as a uniform electron gas. It is 

Illustrating the exchange and correlation holes for the H2 molecule at the dissociation the reference electron located near nucleus A and the vertical axis representing 

The energy functional should be self-interaction-free, i.e. the exchange energy for a one-electron system, such as the hydrogen atom, should exactly cancel the Coulomb energy, and the correlation energy should be zero. Although these seem like obvious requirements, none of the conamon functionals have this property. When the density becomes constant, the uniform electron gas result should be recovered. While this surely is a valid mathematical requirement, and important for applications in solid-state physics, it may not be as important for chemical applications, as molecular densities are relatively poorly described by uniform electron gas methods. 

For scientific theories, being exact in principle seems to be a nice euphemism for being approximate in practice. Density-functional theory suffers from the same fate, and any DFT calculation can only be as reliable as the incorporated parametrization scheme for exchange and correlation. Indeed, the search for reliable exchange-correlation functionals is the greatest challenge to DFT. 

The first and, possibly, most influential and also very successful approximation [144] is given by the local-density approximation (LDA), 

At the same time, the LDA gave an a posteriori justification of the old Xa method by Slater, because the latter is a special LDA variant without correlation. The corresponding spin-dependent version of the LDA is called a local spin-density approximation (LSDA or LSD or just spin-polarized LDA), and even now when people talk of LDA functionals, they always refer to its generalized form for systems with (potentially) unpaired spins. Among the most influential LDA parametrizations, the one of von Barth and Hedin (BH) [154] and the one of Vosko, Wilk and Nusair (VWN) [155] are certainly worth mentioning. The latter is based on the very accurate Monte Carlo-type calculations of Ceperley and Alder [156] for the uniform electron gas, as indicated above. 

At the present time, the (probably) most accurate representation of the correlation energy is due to Perdew and Wang [157]. 

Functionals such as the GGA have enormously popularized the DFT-based methods in the (Hartree-Fock-oriented) quantum-chemical community because the GGA stands for significantly improved energetic results - as well as better geometries - especially when it comes to bond breaking and bond formation. Here, the expectations of chemical accuracy [160] have been rather high and, indeed, some functionals that are part of the GGA route seem to achieve that goal. Historically, the following steps have been taken  

Of course, there is a key difference between HF theory and DFT - as we have derived it so far, DFT contains no approximations it is exact. All we need to know is xc as a function of p. .. Alas, while Hohenberg and Kohn proved diat a functional of the density must exist. their proofs provide no guidance whatsoever as to its fonn. As a result, considerable research effort has gone into dnding functions of die density diat may be expected to reasonably approximate xc, and a discussion of diese is die subject of the next section. We close here by emphasizing that the key contrast between HF and DFT (in the limit of an infinite basis set) is that HF is a deliberately approximate theory, whose development was in part motivated by an ability to solve die relevant equations exactly, while DFT is an exact theory, but the relevant equations must be solved approximately because a key operator has unknown form. 

The Kohn-Sham methodology has many similarities, and a few important differences, to the HF approach. We will, however, delay briefly a full discussion of how exactly to carry out a KS calculation, as it is instructive first to consider how to go about determining Fxc- 

As already emphasized above, in principle Fxc not only accounts for the difference between the classical and quantum mechanical electron-electron repulsion, but it also includes the difference in kinetic energy between the fictitious non-interacting system and the real system. In practice, however, most modem functionals do not attempt to compute this portion explicitly. Instead, they either ignore the term, or they attempt to constmct a hole function that is analogous to that of Eq. (8.6) except that it also incorporates the kinetic energy difference between the interacting and non-interacting systems. Furthermore, in many functionals 

In discussing the nature of various functionals, it is convenient to adopt some of the notation commonly used in the field. For instance, the functional dependence of Exc on the electron density is expressed as an interaction between the electron density and an energy density that is dependent on the electron density, viz. 

The energy density Cxc is always treated as a sum of individual exchange and correlation contributions. Note that there is some potential for nomenclature confusion here because two different kinds of densities are involved the electron density is a per unit volume density, while tlie energy density is a per particle density. In any case, within this formalism, it is clear from inspection of Eq. (8.7) that the Slater exchange energy density, for example, is 

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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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