Relativistic Kohn-Sham Equations

In Equations 10.1, 10.4, and 10.5, the standard implicit sum on the index (x is assumed. The relativistic Kohn-Sham equations are obtained by minimization of the Equation 10.4 with respect to the orbitals. In contrast to the nonrelativistic case, the variational procedure gives rise to an infinite set of coupled equations (see the summation restrictions in Equation 10.3) that have to be solved in a self-consistent manner ... 

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. 

QR Method. The first relativistic method is the so-called quasi-relativistic (QR) method. It has been developed by Snijders, Ziegler and co-workers (13). In this approach, a Pauli Hamiltonian is included into the self-consistent solution of the Kohn-Sham equations of DFT. The Pauli operator is in a DFT framework given by... 

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],... 

The last term in Equation (6.1) represents the exchange-correlation energy Exc. The total energy can then be obtained by the solution of the relativistic Kohn-Sham equations (RKS)... 

Extended states kn) fulfill a scalar-relativistic Kohn-Sham-equation... 

A self-consistent scalar-relativistic (SR) version of the LCGTO-DF method has also been developed recently." "The SR variant employs a unitary second-order Douglas-Kroll-Hess (DKH) "" transformation for decoupling large and small components of the full four-component spinor solutions to the Dirac-Kohn-Sham equation. The approximate DKH transformation, very appropriate and efficient for molecular calculations, has been implemented this variant utilizes nuclear potential-based projectors and leaves the electron-electron interaction untransformed. 

The relativistic form of the one-electron Schrodinger equation is the Dirac equation. One can do relativistic Hartree-Fock calculations using the Dirac equation to modify the Fock operator, giving a type of calculation called Dirac-Fock (or Dirac-Hartree-Fock). Likewise, one can use a relativistic form of the Kohn-Sham equations (15.123) to do relativistic density-functional calculations. (Relativistic Xa calculations are called Dirac-Slater or Dirac-Xa calculations.) Because of the complicated structure of the relativistic KS equations, relatively few all-electron fully relativistic KS molecular calculations that go beyond the Dirac-Slater approach have been done. [For relativistic DFT, see E. Engel and R. M. Dreizler, Topics in Current Chemistry, 181,1 (1996).]... 

The relativistic correction for the kinetic energy in the Dirac equation is naturally applicable to the Kohn-Sham equation. This relativistic Kohn-Sham equation is called the Dirac-KohnSham equation (Rajagopal 1978 MacDonald and Vosko 1979). The Dirac-Kohn-Sham equation is founded on the Rajagopal-Callaway theorem, which is the relativistic expansion of the Hohenberg-Kohn theorem on the basis of QED (Rajagopal and Callaway 1973). In this theorem, two theorems are contained The first theorem proves that the four-component external potential, which is the vector-potential-extended external potential, is determined by the four-component current density, which is the current-density-extended electron density. On the other hand, the second theorem establishes the variational principle for every four-component current density. See Sect. 6.5 for vector potential and current density. Consequently, the solution of the Dirac-Kohn-Sham equation is represented by the four-component orbital. This four-component orbital is often called a molecular spinor. However, this name includes no indication of orbital, which is the solution of one-electron SCF equations moreover, the targets of the calculations are not restricted to molecules. Therefore, in this book, this four-component orbital is called an orbital spinor. The Dirac-Kohn-Sham wavefunction is represented by the Slater determinant of orbital spinors (see Sect. 2.3). Following the Roothaan method (see Sect. 2.5), orbital spinors are represented by a linear combination of the four-component basis spinor functions, Xp, ... 

The huge computational time of the Dirac-Kohn-Sham equation calculations is attributable to the use of the small-component wavefunction, producing the coupling of the large-component and small-component wavefunctions. Two-component relativistic approximations have been suggested to solve this problem and have now become mainstream. Breit applied the expansion of K for electronic motions sufficiently slower than the speed of light, as usual. 

In this paper we discussed the status of quantum mechanical calculations focusing on solids and surfaces. In the quantum mechanics section DFT was presented with respect to the alternative approaches such as wave function based methods or many-body physics. For the solution of the DFT Kohn Sham equations we use an adapted augmented plane wave method implemented in our WIEN2k code, which can be shortly summarized as a fiiU-potential, all electron and relativistic code that is one of the most accurate for solids and is used worldwide by more than 1,850 groups in academia and industry. 

S. Komorovsy, M. Repisky, O. L. Malkina, V. G. Malkin, 1. Malkin, M. Kaupp. A fully relativistic method for calculation of nudear magnetic shielding tensors with a restricted magnetically balanced basis in the framework of the matrix Dirac-Kohn-Sham equation. /. Chem. Phys., 128 (2008) 104101. 

Although the DFT method has been extensively applied to nonrelativistic calculations, the four-component DFT approaches have only recently appeared (see the book [335] and the review [499] and references therein). Relativistic versions of the Kohn-Sham equations have been developed based on the relativistic extension of the Hohenberg-Kohn theory [500]. 

Foundation containing some comments on the relativistic Hohenberg-Kohn theorem and indicating how the exact (but not easily solvable) relativistic Kohn-Sham equations (containing radiative corrections and all that) can be reduced to the standard approximate variant. 

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 ... 

Modern DFT is founded on the Hohenberg-Kohn theorems and the Kohn-Sham equations. These are presented in detail in textbooks as well as in the review literature on the subject (Parr and Yang 1989, Koch and Holthausen 2001, Eschrig 1996, Gross and Kurth 1994, Salahub et al. 1994). However, to set the stage for a discussion of the relativistic case, a brief summary of the nonrelativistic foundations serves as a convenient starting point. 

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. 

From the minimum principle we may go on to obtain the relativistic Kohn-Sham equations... 

Starting from the Dirac-Coulomb approximation, a set of Dirac-Kohn-Sham equations may again be derived. In chapter 8, a spinor-rotation procedure was used to derive the relativistic Fock operator. A similar procedure applied to the present case shows that the gradient of the energy has elements the form... 

For the relativistic case [85, 86] in the non-coUinear spin-polarize (SP) formalism, the Kohn-Sham equation for the total energy lying in the basis of the calculational algorithms is... 

For this kind of confinement, the solution of the non-relativistic time independent Schrodinger equation has been tackled by different techniques. For confined many-electron atoms the density functional theory [6], using the Kohn-Sham model [7], has given some estimations of the non-classical effects [8-11], through the exchange-correlation functional. An elementary review of this subject can be found in Ref [12], where numerical techniques are discussed to solve the Kohn-Sham equations. Furthermore, in this reference, some chemical predictors are analyzed as a function of the confinement radii. 

Frequently, noncollinearity is due to spin-orbit coupling. Although spin-orbit terms can be added as a perturbation to the equations of SDFT, a complete description requires a relativistic formulation. A generalization of DFT that does account for spin-orbit coupling and other relativistic effects is rdft. ° Here the fundamental variable is the relativistic four-component current and the Kohn-Sham equation is now of the form of the single-particle Dirac equation, instead of the Schrodinger equation. 

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. 

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