Dirac-Kohn-Sham method

Dirac-Hartree-Fock and Dirac-Kohn-Sham methods By an application of an independent-particle approximation with the DC or DCB Hamiltonian, the similar derivation of the non-relativistic Hartree-Fock (HF) method and Kohn-Sham (KS) DFT yields the four-component Dirac-Hartree-Fock (DHF) and Dirac-Kohn-Sham (DKS) methods with large- and small-component spinors. 

As an example of a pseudo-relativistic all-electron DFT method. Table 16.3 lists results from van Wiillen s scalar-relativistic ZORA-DFT study [707]. Leaving the comparatively large error in coe aside, we note that the remaining spectroscopic constants are well reproduced. A subsequent relativistic DFT study on the gold dimer employed a four-component Dirac-Kohn-Sham method and the scalar-relativistic ZORA approach [1114]. Table 16.3 also shows that the scalar-relativistic ZORA approach yields good results compared to the four-component reference. Recall that the accuracy of such DFT calculations is determined by the approximate nature of the exchange-correlation functional employed (see section 8.8). 

Resolution of identity Dirac-Kohn-Sham method using the large component only Calculations of g-tensor and hyperfine tensor. /. Chem. Phys., 124 (2006) 084108. 

Relativistic Electronic Structure Theory Dirac—Hartree-Fock and Dirac-Kohn-Sham Methods for Molecules... 

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. 

We review the Douglas-Kroll-Hess (DKH) approach to relativistic density functional calculations for molecular systems, also in comparison with other two-component approaches and four-component relativistic quantum chemistry methods. The scalar relativistic variant of the DKH method of solving the Dirac-Kohn-Sham problem is an efficient procedure for treating compounds of heavy elements including such complex systems as transition metal clusters, adsorption complexes, and solvated actinide compounds. This method allows routine ad-electron density functional calculations on heavy-element compounds and provides a reliable alternative to the popular approximate strategy based on relativistic effective core potentials. We discuss recent method development aimed at an efficient treatment of spin-orbit interaction in the DKH approach as well as calculations of g tensors. Comparison with results of four-component methods for small molecules reveals that, for many application problems, a two-component treatment of spin-orbit interaction can be competitive with these more precise procedures. 

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 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, ... 

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. 

Heavy-element systems are involved in many important chemical and physical phenomena. However, they still present difficulties to theoretical study, especially in the case of solids containing atoms of heavy elements (with the nuclear charge Z > 50). In this short description of relativistic electronic-structure theory for molecular systems we follow [496] and add a more detailed explanation of the Dirac-Kohn-Sham (DKS) method. For a long time the relativistic effects underlying in heavy atoms had not been regarded as such an important effect for chemical properties because the relativistic effects appear primarily in the core atomic region. However, now the importance of the relativistic effects, which play essential and vital roles in the total natures of electronic structures for heavy-element molecular and periodic qrstems, is recognized [496]. 

The one-electron effective Hamiltonian for the Dirac-Kohn-Sham (DKS) method, as a relativistic extension of the conventional KS approach, takes the form. 

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 DFT concept of calculating the energy of a system from its electron density seems to have arisen in the 1920s with work by Fermi, Dirac, and Thomas. However, this early work was useless for molecular studies, because it predicted molecules to be unstable toward dissociation. Much better for chemical work, but still used mainly for atoms and in solid-state physics, was the Xa method, introduced by Slater in 1951. Nowadays the standard DFT methodology used by chemists is based on the Hohenberg-Kohn theorems and the Kohn-Sham approach... 

The programs described so far use basis-set expansions for the one-electron spinors. The fully numerical approach, which is still a challenging task for general molecules in nonrelativistic theory (Andrae 2001), has also been tested for Dirac-Fock calculations on diatomics (DtisterhOft etal. 1994,1998 Kullie etal. 1999 Sundholm 1987,1994 Sundholm et al. 1987 v. Kopylow and Kolb 1998 v. Kopylow et al. 1998 Yang et al. 1992). The finite-element method (FEM) was tested for Dirac-Fock and Kohn—Sham calculations by Kolb and co-workers in the 1990s. However, this approach has not yet been developed into a general method for systems with more than two atoms only test systems, namely few-electron linear molecules at some fixed intemuclear distance, have been studied with the FEM. Nonetheless, these numerical techniques are able to calculate the Dirac-Fock limit and thus yield reference data for comparisons with more approximate basis-set approaches. The limits of the numerical techniques are at hand ... 

The exchange energy of a UEG can be evaluated analytically by the method of Bloch [103] or Dirac [3]. Details of both approaches are discussed by Gombas [4], Bethe [104], Slater [105], and Parr and Yang [20]. The outline of the derivation is as follows. The Kohn-Sham orbitals for a UEG are plane waves, < (r) = where V is the... 

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... 

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