XC potentials

Approximations thus must be introduced that involve modeling both the XC potential and the metric tensor, and a truncation of the space within which to choose the unknown functions v, to finite dimension r < >. The modeling is based on the restt-icted ansatz chosen for the form of states used to determine paths that approximate D (p), D](p) and ). It can be carried... 

In the Kohn-Sham equation above, the Coulomb potential and the XC potential are obtained from their energy counterparts by taking the functional derivative of the latter with respect to the density. Thus... 

Having understood the physical meaning of the XC energy in terms of the corresponding hole, the next step would be to apply the same meaning to the XC potential of Equation 7.4 and calculate it as the electrostatic potential arising from the XC hole. Thus one is tempted to write the XC potential as... 

If we wish to obtain the XC potential of Equation 7.15 as the work done in moving an electron in the field of its Fermi-Coulomb hole pxc(r, r ), we first calculate the field as... 

Our aim here is to apply the differential virial theorem to get an expression for the Kohn-Sham XC potential. To this end, we assume that a noninteracting system giving the same density as that of the interacting system exists. This system satisfies Equation 7.4, i.e., the Kohn-Sham equation. Since the total potential term of Kohn-Sham equation is the external potential for the noninteracting system, application of the differential virial relationship of Equation 7.41 to this system gives... 

Further analysis from the minimum action principle shows that the exchange (xc) potential is then the functional derivative of that quantity in terms of the density ... 

Finally, we note that Eqs. (3.5-3.7) can also be used to extract the behavior of the xc-potential in the asymptotic regime of finite systems. Restricting the analysis to spherically averaged systems, a somewhat tedious analysis (the interested reader is referred to [19]) leads to the relativistic form of the Krieger-Li-lafrate identity for the highest occupied orbital (() , [56],... 

Without going into details we just remark that Eq. (2.45) can be further simplified by applying Wick s theorem to the electronic sector, utilizing the KS propagator (2.25). Taking into account the explicit form (2.8) for it is then possible to eliminate a further class of diagrammatic contributions (the interested reader is referred to [19] for details). Eq. (2.45), which provides an exact representation of in terms of the KS orbitals, the KS eigenvalues and the xc-potential, is the central result... 

Bearing in mind the inherently recursive character of our first construction scheme for Exc, Eqs. (2.44-2.49), one can again choose between two alternative approaches. On the one hand, it is possible to start with an expansion of in powers of e, as outlined in Section 2.3, and subsequently extract the xc-potential order by order. In this case it is important to note, that the lowest order contribution to E, i.e. the exchange energy, only depends on the and k. This allows the calculation of t) as a functional... 

The KS (local) effective potential has three components the external potential, the Hartree potential, and the XC potential. 

It is also important to discuss the effect of the symmetrization on the XC potential, 5E ° /5p. For the exact XCEDF, the symmetric nature of the PCF directly leads to a two-term summation for the XC potential,... 

At large distance from a neutral atom, V2(r) goes to and vi(r) decays exponentially.If a symmetric ansatz for the PCF is employed, the WDA XC potential will be symmetric automatically, just like the exact case above. Additionally, a symmetric XC potential has the exact asymptotic behavior (-1) and the spmious self-interaction effect in the HREDF J[( is mostly removed. Unfortunately, because of the nonsymmetric nature of the ansatz for the PCF in Eq. (113), the XC potential within the present WDA framework has three terms instead. 

The effective Time Dependent Kohn-Sham (TDKS) potential vks p (r>0 is decomposed into several pieces. The external source field vext(r,0 characterizes the excitation mechanism, namely the electromagentic pulse as delivered by a by passing ion or a laser pulse. The term vlon(r,/) accounts for the effect of ions on electrons (the time dependence reflects here the fact that ions are allowed to move). Finally, appear the Coulomb (direct part) potential of the total electron density p, and the exchange correlation potential vxc[p](r,/). The latter xc potential is expressed as a functional of the electronic density, which is at the heart of the DFT description. In practice, the functional form of the potential has to be approximated. The simplest choice consists in the Time Dependent Local Density Approximation (TDLDA). This latter approximation approximation to express vxc[p(r, /)]... 

It has turned out that the computation of metal nuclear shieldings and chemical shifts is much more difficult than the calculation of ligand shifts which were discussed previously. It appears that metal shieldings are more sensitive to the quality of the computed electronic structure and consequently larger influences due to the XC potential are observed. Regarding the 3d metals, it has turned out that hybrid functionals appear to be particularly well suited for NMR computations of the metal shielding constants. This cannot be easily extrapolated to all of the transition-metals, though, since counter examples are known for which nonhybrid functionals perform better. On the other hand, some 4d metals have been treated most successfully with hybrid functionals. 

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