The exchange-correlation functional

As mentioned above, the nuclei are assumed to be fixed and are thus nothing more than sources of an external electrostatic potential in which the electrons move. If there is no magnetic field external to the molecule under consideration, and if external electric fields are time-independent, we arrive at the so-called electrostatic limit of relativistic density functional theory. Note that most molecular systems fall within this regime. In this case, one can prove the relativistic Hohen-berg-Kohn theorem using the charge density, p(r) = J f), only. This leads to a definition of an exchange-correlation functional -Exc[p( )] 

The parameter which measures the importance of relativistic effects is the ratio P of the Fermi momentum and the momentum of a non-relativistic electron travelling at the speed of light 

The exchange energy of a relativistic homogeneous electron gas is known analjrtically, and an expansion in the parameter p gives 

The non-relativistic limit of the electron-electron interaction is the instantaneous Coulomb repulsion, also called the longitudinal part of the electron-electron interaction. The remainder is called the transversal part and mainly covers retardation and magnetic interactions (this separation is somewhat gauge dependent). The longitudinal and transversal part of the relativistic local exchange functional read 

Hartree (3xl0 eV or 0.003 kJ/mol) has been reported [23]. For closed-shell systems, this contribution vanishes altogether. 

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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... 

Schreckenbach, G., 1999, The 57Fe NMR Shielding in Ferrocene Revisited. A Density-Functional Study of Orbital Energies, Shielding Mechanisms, and the Influence of the Exchange-Correlation Functional , J. Chem. Phys., 110, 11936. 

Rashin et al.45 obtained the dipole moment of 32 molecules of biological relevance by means of the DFT(SVWN) and DFT(B88/P86) calculations. The results showed a rather weak dependence of calculated dipole moments on the functional form of the exchange-correlation functional but a strong dependence on the basis set. 

During the last decade, density-functional theory (DFT)-based approaches [1, 2] have advanced to prominent first-principles quantum chemical methods. As computationally affordable tools apt to treat fairly extended systems at the correlated level, they are also of special interest for applications in medicinal chemistry (as demonstrated in the chapters by Rovira, Raber et al. and Cavalli et al. in this book). Several excellent text books [3-5] and reviews [6] are available as introduction to the basic theory and to the various flavors of its practical realization (in terms of different approximations for the exchange-correlation functional). The actual performance of these different approximations for diverse chemical [7] and biological systems [8] has been evaluated in a number of contributions. 

The advantage over the HF scheme is that whereas in conventional ah initio theory we must resort to costly perturbation theory or configuration interaction expansions, in DFT electron correlation is already included explicitly in the exchange-correlation functional. The key problem is instead to find an appropriate expression for xc. As stated above, when we have the correct functional we should be able to extract the exact energy, the exact ground state density, and all properties for our system. 

The problem is that the exchange correlation functional Exc is unknown. Approximate forms have to be used. The most well-known is the local density approximation (LDA) in which the expressions for a uniform electron gas are... 

Equations 24.21 are very particular cases of a general theorem for the responses demonstrated previously [26]. It is interesting to note that the evaluation of nonlinear hyperpolarizabilities is a stringent test of the validity and robustness of exchange-correlation functionals [35]. Equation 24.21 permits to explain qualitatively why the electrostatic part of energy does not contribute at all to h3, which depends only on the exchange-correlation functional. On the contrary, hi is dominated by the Coulomb propagator. 

The wave functions are expended in a plane wave basis set, and the effective potential of ions is described by ultrasoft pseudo potential. The generalized gradient approximation (GGA)-PW91, and local gradient-corrected exchange-correlation functional (LDA)-CAPZ are used for the exchange-correlation functional. 

As a consequence of the size limitations of the ab initio schemes, a large number of more-approximate methods can be found in the literature. Here, we mention only the density functional-based tight binding (DFTB) method, which is a two-center approach to DFT. The method has been successfully applied to the study of proton transport in perov-skites and imidazole (see Section 3.1.1.3). The fundamental constraints of DFT are (i) treatment of excited states and (ii) the ambiguous choice of the exchange correlation function. In many cases, the latter contains several parameters fitted to observable properties, which makes such calculations, in fact, semiempirical. 

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