Relativistic implicit functionals

The importance of a selfconsistent treatment of the transverse interaction is examined in Table 2 (note that different nuclear models underly Tables 1 and 

The fully selfconsistent handling of EJ is compared with two perturbative schemes. In the first the Coulomb-Breit interaction is included on the selfconsistent level, while the contributions beyond the Breit interaction (T—B) are added perturbatively. In the second only the Coulomb-interaction is handled selfconsistently. Even for the heaviest atoms the perturbative evaluation of the 

The same comparison is made on a microscopic level in Fig.l, where the importance of the transverse exchange for Vx is analyzed. Three variants of the relativistic of Hg (Coulomb, Coulomb-Breit and complete interaction) are shown together with the norurelativistic Vx. Taking into account that the r-expectation values of the lsj 2 2s orbitals are 0,017 Bohr and 0.069 Bohr, 

A number of atomic ionization potentials (IPs) obtained by taking the difference between the total energies of the ground and the ionized state are given 

Exchange-only ionization potentials of neutral atoms calculated from ground state energy differences For the ROPM the selfconsistent inclusion of the transverse exchange (C-fT) is compared with complete neglect of (C). The RGGA data have been obtained by the relativistic extension of the Becke parameterization [37] (all energies in mHartree [172]). 

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As a fully nonlocal alternative to these explicit density functionals orbital-dependent (implicit) density functionals have been suggested. In addition to the exact exchange [51,52] some approximate correlation functionals are available, both empirical [166] and first-principles forms [57,59,60], As is already clear from Section 3.4 this concept can also be used in the relativistic situation. The status of relativistic implicit functionals [54] will be reviewed in Section 4.1. In particular, the various ingredients of the exact exchange will be analyzed. Subsequently the results obtained with the exact exchange will then serve as reference data for the analysis of the RLDA and RGGA. 

While spin-orbit coupling contributes significantly to the cohesive energy, its effect is too small to explain the differences between GGA and experimental data in Table 4.5. Thus, on the one hand, the results in Tables 4.4 and 4.5 illustrate the role of error cancellation, in particular for the LDA. On the other hand, they indicate the need for fundamentally new concepts for Exc[ri] (such as implicit functionals) in the relativistic regime. 

The presently available explicit approximations for the relativistic xc-energy functional are presented in Section 4. Both implicit functionals (as the exact exchange) and explicit density functionals (as the RLDA and RGGA) are discussed (on the basis of the information on the RHEG in Appendix C and that on the relativistic gradient expansion in Appendix E). Section 4 also contains a number of illustrative results obtained with the various functionals. However, no attempt is made to review the wealth of RDFT applications in quantum chemistry (see e.g.[74-88]) and condensed matter theory (see e.g.[89-l(X)]) as well as the substantial literature on nonrelativistic xc-functionals (see e.g.[l]). In this respect the reader is referred to the original literature. The review is concluded by a brief summary in Section 5. 

There is one further orbital-dependent functional which can be mentioned at this point. In the nonrelativistic context it has been realized rather early [184] that, as a matter of principle, the self-interaction corrected (SIC) LDA of Perdew and Zunger [143] represents an implicit functional for which the 0PM should be used. A relativistic version of the Perdew-Zunger SIC has been proposed by Rieger and Vogl [185] as well as Severin et al. [186,187,46]. This functional, however, has not yet found widespread use, neither within the conventional,... 

First of all, a few words on the scope of this review seem to be appropriate. For simplicity, all explicit formulae in this chapter will be given for spin-saturated systems only. Of course, the complete formalism can be extended to spin-density functional theory (SDFT) and all numerical results for spin-polarized systems given in this paper were obtained by SDFT calculations. In addition, the discussion is restricted to the nonrelativistic formalism - for its relativistic form see Chap. 3. The concept of implicit functionals has also... 

Solutions with positive and negative energies of a one-particle Dirac equation of a molecular system are represented by states where either electronic or positronic contributions of four-component wave functions dominate. With chemical systems in mind, electronic and positronic components are also referred to as large and small components, respectively. However, small components cannot simply be neglected or projected out to arrive at a simpler two-component description because, in an intrinsic fashion they also contribute in a fully relativistic description of a chemical system. Thus, a projection step, in which positronic components are discarded, can only be applied after a suitable decoupling of electron and positron degrees of freedom. Then the effects of the small components are implicitly accounted for. 

The transition metal ions, Cu , Ag and Au", all have a d ( 5) electronic state-configuration, with = 3,4 and 5, respectively. The RCEP used here were generated from Dirac-Fock (DF) all electron (AE) relativistic atomic orbitals, and therefore implicitly include the indirect relativistic effects of the core electron on the valence electrons, which in these metal ion systems are the major radial scaling effect. In these RCEP the s p subshells are included in the valence orbital space together with the d, ( + l)s and ( + l)p atomic orbitals and all must be adequately represented by basis functions. The need for such semi-core or semi-valence electrons to be treated explicitly together with the traditional valence orbitals for the heavier elements has been adequately documented The gaussian function basis set on each metal atom consists of the published 4 P3 distribution which is double-zeta each in the sp and n + l)sp orbital space, and triple-zeta for the nd electrons. 

In (1) the Coulomb interaction between the conduction states is neglected. These states are rather extended (see fig. 1) and the Coulomb integrals are not very large. For such states the local spin density (LSD) approximation of the spin density functional formalism (Kohn and Sham 1965) has been rather successful (von Barth and Williams 1983). In this scheme the electrons are formally treated as independent and correlation effects are included in an effective one-particle potential. By using 6fc s in (1) which are obtained from a LSD approximation or deduced from experiment, we may incorporate some interaction effects implicitly in the Hamiltonian (1). In this way chemical information about the compound considered may also be incorporated. Relativistic effects on the band structure are also included in this approach. 

If one wishes to proceed in this fashion in the relativistic case, one has to provide accurate atomic data. For this purpose, OPM, the optimized potential method [16] (in the present context the relativistic extension, the ROPM) is a valuable tool. The (R)OPM relies on the fact that the functional derivative with respect to the density (or the four-current) can be evaluated with the chain rule for functional derivatives if the dependence on the density is implicit via Kohn-Sham orbitals, E n = E[(fk =... 

The first two terms in the parentheses are the radial kinetic energy operator and the term Wp y stands for an effective valence Coulomb and exchange potential for y>p j. Relativistic effects are implicitly included in Vy since the AE reference calculation explicitly describes these effects. Repeating this procedure for each //-set, the resulting potentials Vy are tabulated on a grid and are usually fitted by means of a least-squares criterion to a linear combination of Gaussian functions according to... 

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