Relativistic Local Density Approximation

After the discovery of the relativistic wave equation for the electron by Dirac in 1928, it seems that all the problems in condensed-matter physics become a matter of mathematics. However, the theoretical calculations for surfaces were not practical until the discovery of the density-functional formalism by Hohenberg and Kohn (1964). Although it is already simpler than the Hartree-Fock formalism, the form of the exchange and correlation interactions in it is still too complicated for practical problems. Kohn and Sham (1965) then proposed the local density approximation, which assumes that the exchange and correlation interaction at a point is a universal function of the total electron density at the same point, and uses a semiempirical analytical formula to represent such universal interactions. The resulting equations, the Kohn-Sham equations, are much easier to handle, especially by using modern computers. This method has been the standard approach for first-principles calculations for solid surfaces. 

Fig. 3. A comparison of the eigenvalues of the outermost valence electrons for Pu using relativistic, semi-relativistic and non-relativistic kinematics and the local density approximation (LSD). Dirac-Fock eigenvalues after Desclaux are also shown. The total energies of the atoms (minus sign omitted), calculated with relativistic and non-relativistic kinematics are also shown. HF means Hartree Fock...

For direct Af-electron variational methods, the computational effort increases so rapidly with increasing N that alternative simplified methods must be used for calculations of the electronic structure of large molecules and solids. Especially for calculations of the electronic energy levels of solids (energy-band structure), the methodology of choice is that of independent-electron models, usually in the framework of density functional theory [189, 321, 90], When restricted to local potentials, as in the local-density approximation (LDA), this is a valid variational theory for any A-electron system. It can readily be applied to heavy atoms by relativistic or semirelativistic modification of the kinetic energy operator in the orbital Kohn-Sham equations [229, 384],... 

All-electron DFT calculations were performed using the DMOL [24] code. These incorporated scalar relativistic corrections and employed the non-local exchange and correlation functional Perdew-Wang91 [25] denoted GGA in the rest of the paper, which is generally found to be superior to the local density approximation (EDA)... 

Most frequently, however, a purely density-dependent version of RDFT is used. In this context we have examined the role of relativistic corrections to the exchange-correlation (xc) energy functional. In view of the limited accuracy of the relativistic local density approximation (RLDA) (Das etal. 1980 Engel etal. 1995a Ramana et... 

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

Yamagami and Hasegawa carried out a self-consistent calculation of the energy band structure by solving the Kohn-Sham-Dirac one-electron equation by the density-functional theory in a local-density approximation (LDA). This self-consistent, symmetrized relativistic APW approach was applied to many lanthanide compounds and proved to give quite accurate results for the Fermi surface. 

In the derivation of the Kohn-Sham equations we have hidden a number of difficulties in the exchange-correlation potential, (r). Indeed, the success of DFT depends on finding an accurate and convenient form of this potential. There is an extensive literature discussing the merits of various potentials, and good accounts of these may be found elsewhere (Koch and Holthausen 2001). Here, we restrict the discussion to the local density approximation (LDA), because it provides a link to another approximation that has been used extensively in relativistic atomic and molecular calculations and which predates the Kohn-Sham equations. 

In section 2, we briefly outline density functional theory and the various approximations employed including the local density approximation (LDA), and discuss generalizations of LDA to incorporate spin (LSDA), orbital, and relativistic effects. We also discuss phenomenological renormalized band schemes based on the slave-boson or Kondo phase-shift methods. 

Usually, self-consistent, all-electron calculations are performed within the relativistic local density approximation (LDA). The general gradient approximation (GGA), also in the relativistic form, RGGA, are then included perturbatively in E p,m). The accuracy depends on the adequate knowledge of the potential, whose exact form is, however, unknown. There is quite a number of these potentials and their choice is dependent on the system. Thus, PBE is usually favored by the physics community, PBEO, BLYP, B3LYP, B88/P86, etc., by the chemical community, while LDA is still used extensively for the solid state. 

B3LYP = Becke s 3-parameter hybrid functional using the nonlocal correlation functional due to Lee, Yang, and Parr BP = nonlocal exchange correlation functional due to Becke and Perdew DF = Dirac-Fock DIIS = direct inversion of iterative subspace KS = Kohn-Sham LDA = local density approximation LSDA = local spin density approximation (R)ECP = (relativistic) effective core potential TM = transition metal. 

In the local density approximation the relativistic potential is as follows ... 

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