Kohn-Sham-Dirac equations

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... 

The Kohn-Sham-Dirac equation (27) introduced in the last section is the basis of most relativistic electronic structure calculations in solid state theory. There are certain aspects which make the numerical solution of this four-component equation more involved than its non-relativistic coimterpart The Hamiltonian of the Kohn-Sham-Dirac equation is, unlike its Schrodinger equivalent and unlike the field-theoretical Hamiltonian (7) with the properly chosen normal order, not bounded below. In the limit of free, non-interacting particles the solutions of the Kohn-Sham-Dirac equation are plane waves with energies e(k) = cVk -I- c, where positive energies correspond to electrons and states with negative energy can be interpreted as positrons. For numerical procedures, which preferably use variational techniques to find electronic solutions, this property of the Dirac operator causes a severe problem, which can be circumvented by certain techniques like the application of a squared Dirac operator or a projection onto the properly chosen electronic states according to their above definition after Eq. (19). 

We start our discussion of the relativistic FPLO method (RFPLO) with the Kohn-Sham-Dirac equation for the crystal... 

The Kohn-Sham-Dirac equation (28) has to be solved self consistently, since the crystal potential and the XC-field depend via the (magnetization) density on its solutions. For a local orbital method it is advantageous to use a strictly local language for all relevant quantities, so that computationally expensive transformations between different numerical representations are avoided during the self consistency cycle. In the (R)FPLO method, the density n(r) and the magnetization density m(r) = m r)z are represented as lattice sums... 

The eigenvectors of the lattice periodic Hamiltonian of the Kohn-Sham-Dirac equation (28) are Bloch states fen) with crystal momentum k and band index n. They are expressed by the ansatz... 

In a next step we insert the Bloch ansatz (32) into the Kohn-Sham-Dirac equation (28) and project it onto the local basis states giving rise to a... 

In eq. (23), Fq is the average potential in the interstitial region of space. In eq. (24), f(fis) and g(Uj) denote the small and large components of the electron wave function at the sphere radius a, respectively, which depend on E. They are obtained by solving the radial Kohn-Sham-Dirac equations for a given E,... 

The basis of the Kohn-Sham-Dirac (KSD) type of equations is a parameterization of the variational four-current in terms of new variational quan-... 

The ground-state wavefunction corresponding to this noninteracting system is then a single Slater determinant i(x) of the lowest occupied orbitals i(x) of the Kohn-Sham differential equation. The Dirac [15] single-particle density matrix y,(r, r ) that results from this Slater determinant is... 

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. 

Relativity affects the kinetic term and the exchange-correlation potential in the Kohn-Sham equation. As investigated in detail for the uranium atom and the cerium atom, the relativistic effect on the exchange correlation potential is rather small and therefore we use /Ac[ ( )] in n relativistic band structure calculation. The relativistic effect on the kinetic term is appreciably large and can be taken into account by adopting the Kohn-Sham-Dirac one-electron equation instead of eq. (3) as follows ... 

Here, C is the gauge constant, / is the boundary of the closed shells n > f indicating the vacant band and the upper continuum electron states matomic core and the states of a negative continuum (accounting for the electron vacuum polarization). The minimization of the functional ImSEninv leads to the Dirac-Kohn-Sham-like equations for the electron density that are numerically solved. Finally an optimal set of the IQP functions results. In concrete calculation it is sufficient to use the simplified procedure, which is reduced to the functional minimization using the variation of the correlation potential parameter b in Eq. 3.11 [20, 32]. The Dirac equations for the radial functions F and G (the large and small Dirac components respectively) are ... 

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. 

The details of implementation of scalar relativity in GTOFF were presented in [41] and reviewed in [75], so we summarize the essential assumptions and methodological features here. First, all practical DFT implementations of relativistic corrections of which we are aware assume the validity (either explicitly or implicitly) of an underlying Dirac-Kohn-Sham four-component equation. We do also. The Hamiltonian is therefore a relativistic free particle Hamiltonian augmented by the usual non-relativistic potentials... 

The computational procedure usually also yields, for given effective potential, the negative-energy solutions for the Dirac equation Eq. (14), for which j <2c. These are not used to construct the Kohn-Sham reference function xs (s Eq. (1)) and likewise, all vacuum... 

The combination of the Dirac-Kohn-Sham scheme with non-relativis-tic exchange-correlation functionals is sometimes termed the Dirac-Slater approach, since the first implementations for atoms [13] and molecules [14] used the Xa exchange functional. Because of the four-component (Dirac) structure, such methods are sometimes called fully relativistic although the electron interaction is treated without any relativistic corrections, and almost no results of relativistic density functional theory in its narrower sense [7] are included. For valence properties at least, the four-component structure of the effective one-particle equations is much more important than relativistic corrections to the functional itself. This is not really a surprise given the success of the Dirac-Coulomb operator in wave function based relativistic ab initio theory. Therefore a major part of the applications of relativistic density functional theory is done performed non-rela-tivistic functionals. 

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