Exchange-correlation relativistic potential

This situation has changed with the advent of nonrelativistic CDFT, developed by Vignale and Rasolt, which describes spontaneous currents by introducing in the Kohn-Sham equations a self-consistent exchange-correlation vector potential Axe, which can be nonzero also in the absence of external magnetic fields and of relativistic effects. 

In subsection 3.1, we will present GGA and LDA calculations for Au clusters with 6first principles method outlined in section 2, which employs the same scalar-relativistic pseudo-potential for LDA and GGA (see Fig 1). These calculations show the crucial relevance of the level of density functional theory (DFT), namely the quality of the exchange-correlation functional, to predict the correct structures of Au clusters. Another, even more critical, example is presented in subsection 3.2, where we show that both approaches, LDA and GGA, predict the cage-like tetrahedral structure of Au2o as having lower energy than amorphous-like isomers, whereas for other Au clusters, namely Auig, Au ... 

Fig. 2 Influence of the choice of the exchange-correlation potentials and of relativistic effects on the 31p-31p two-bond coupling constants in cis- and fra s-M(CO)4(PH3)2, M is Cr, Mo or W. (Graphics courtesy of Kaupp [97])...

We are discussing our manner to calculate the total energy for small molecules within the DV-Xa approximation by using only the monopol part of the potential in the solution of the Poisson equation. A discussion of the relativistic effects, including our results for heavy diatomic molecules, is followed by remarks on the choice of the exchange-correlation potential together with our results of calculations on molecules for the element 106 and their chemical interpretation. We conclude with results on very heavy correlation diagrams for collision systems with a united Z above 110. 

A number of exchange-correlation potentials have been proposed over the years including some based on relativistic treatments. Those reported in Refs 31 and 32 are parametrizations of accurate Monte Carlo calculations for the electron gas and are believed to represent closely the limit of the LSD approximation. 

The Vignale-Rasolt CDFT formalism can be obtained as the weakly relativistic limit of the fully relativistic Kohn-Sham-Dirac equation (5.1). This property has been exploited to set up a computational scheme that works in the framework of nonrelativistic CDFT and accounts for the spin-orbit coupling at the same time (Ebert et al. 1997a). This hybrid scheme deals with the kinematic part of the problem in a fully relativistic way, whereas the exchange-correlation potential terms are treated consistently to first order in 1 /c. In particular, the corresponding modified Dirac equation... 

In the DFT approach with our general DK transformation, the exchange-correlation potential, Vxci is corrected relativistically. The effect on the DK transformation to the exchange-correlation potential was estimated by comparison with the result without the relativistic modification to V c ((no mod. V c) in Table 20.13). Compared with the full DK3-DK3 approach, neglect of the relativistic DK correction to the exchange-correlation potential hardly affects the calculated spectroscopic values its effect merely contributes 0.002 A for R. and 0.006 eV for D. and does not affect (Og and for the At dimer. Thus, it is found that the relativistic correction to the electron-electron interaction contributes mainly to the Coulomb potential, not to the exchange-correlation potential. 

DFT-Based Pseudopotentials. - The model potentials and shape-consistent pseudopotentials as introduced in the previous two sections can be characterized by a Hartree-Fock/Dirac-Hartree-Fock modelling of core-valence interactions and relativistic effects. Now, Hartree-Fock has never been popular in solid-state theory - the method of choice always was density-functional theory (DFT). With the advent of gradient-corrected exchange-correlation functionals, DFT has found a wide application also in molecular physics and quantum chemistry. The question seems natural, therefore Why not base pseudopotentials on DFT rather than HF theory ... 

For both the LDA and GGA procedures, the relativistic form of the exchange-correlation potential has been developed by Engel et al. [112]. 

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( )]... 

A much better choice is to use the length of s to define the spin density . In the non-relativistic limit, the absolute value of the spin density of a one-electron system equals the charge density. While one cannot exactly retain this property in the relativistic case because of the small component contribution (except for single-particle plane waves, where s can be parallel to the momentum everywhere), the length of s equals the charge density for one-electron systems in the weakly relativistic limit and in two-component quasi-relativistic approaches. The same holds if there is one electron outside a closed-shell core. The non-collinear approach is not too difficult to implement [27], it generates a spin-dependent exchange-correlation potential of the form... 

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