Relativistic homogeneous electron gas

In this appendix we summarise some properties of the relativistic homogeneous electron gas (RHEG) in order to illustrate the renormalisation of ground state energies (indicated in Section 2) and to provide the details for the RLDA and the relativistic gradient expansion (RGE), which are discussed in Section 4 and Appendix D, respectively. For simplicity we restrict the discussion to the unpolarised RHEG. 

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

The simplest approximation is the local density approximation (LDA), which is obtained from the energy density of the relativistic homogeneous electron gas (RHEG)... 

As in the TFD method, this equation has to be solved with the boundary conditions (t)(0) = 1 andXcCt) (Xc) = < Xc) where = bXc is the cutoff point where the pressure of the electron gas becomes zero. The value of ( )(Xc)/Xcnecessary for this differs from the non-relativistic case and can be found from similar grounds [27], just by making zero the pressure of a homogeneous electron gas, given by... 

In the present set of calculations we have used the functional proposed by Becke (7), which adopts a non-local correction to the HFS exchange, and treats correlation between electrons of different spins at the local density functional level. All calculations presented here were based on the LCAO-HFS program system due to Baerends et al, (2) or its relativistic extension due to Snijders et al.(3), with minor modifications to allow for Becke s non-local exchange correction as well as the correlation between electrons of different spins in the formulation by Stoll et al, (4) based on Vosko s parametrization (5) from homogeneous electron gas data. Bond energies were evaluated by the Generalized Transition State method (6), or its relativistic extensions (7). 

QMC methods were first applied to the case of the electron gas by Ceper-ley in the late 1970s,and the results have been widely used in density functional theory. Only recently have these early calculations been extended by others to provide greater detail. Pickett and Broughton carried out VQMC calculations for the spin-polarized gas. Ortiz and Ballone used both VQMC and fixed-node DQMC for the spin-polarized gas in the density range most important to density functional theory. Kenny et al. performed VQMC and DQMC calculations for the nonpolarized homogeneous electron gas, incorporating relativistic effects via first-order perturbation theory. 

It is impossible to develop a current-dependent relativistic exchange-correlation functional, which is computationally tractable and reduces to spin-density functional theory in the weakly relativistic limit. One reason is that there is no local approximation to such a functional since j vanishes for any homogeneous system. This means that the relativistic electron gas cannot serve as a starting point. More insight is gained from a Gordon decomposition of the current density j (see e.g. Refs. [7, 24]), which shows that j consists of an orbital part and the curl of a magnetisation density fh. 

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