Dirac-Kohn-Sham calculations

The discrepancy between theories is indicated and the maximum discrepancy ranges from 23 ppm to 30 ppm for Z = 18-26 and differences are consistent. Our measurements are at this level of uncertainty. A further indication of the uncertainty or accuracy of theory can be gained by considering the two configuration interaction (Cl) calculations of Cheng et al [6,7]. The latest calculations for mid-Z helium-like ions [7] has resulted in new values which have shifted by up to 14 ppm from the earlier calculation [6]. The difference has been attributed to the exclusion of the Latter correction to the Dirac-Kohn-Sham potentials from which the QED corrections are evaluated. The new results are considered to be more reliable [7] and are in closer agreement to the unified calculation... 

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, ... 

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. 

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). 

F. TaranteUi. Recent advances and perspectives in four-component Dirac-Kohn-Sham calculations. Phys. Chem. Chem. Phys., 13 (2011) 12368-12394. 

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

O. L. Malkina, V. G. Malkin. Relativistic four-component calculations of electronic g-tensors in the matrix Dirac-Kohn-Sham framework. Chem. Phys. Lett., 488 (2010) 94-97. 

Restricted magnetically balanced basis has been applied by Malkin and co-workers for relativistic calculations of scalar nuclear spin-spin coupling tensors in the matrix Dirac-Kohn-Sham framework. Benchmark relativistic calculations have been carried out for the H-X and H-H couplings in the XH4 series where X = C, Si, Ge, Sn and Pb. One-bond couplings, X, in the gas-phase have been determined by Antusek et for CH4, /hc= 125.3 Hz, SiH4, /hsi = (-) 201.0 Hz, GeH4, /HOe = (-)96.7 Hz, and calculated theoretically. The calculations have been also performed for whose experimental value in SnH4 has been reported by Laaksonen and Wasylishen. ... 

Based on an empirical correlation between adsorption enthalpies of single atoms on Au surfaces with their sublimation enthalpy (see Fig. 38), for Cn a value of A/is = 39( ( kJ moP (= A.% kcal moP ) results [136]. This value is significantly lower compared to a theoretical prediction based on solid-state theory using relativistic Dirac-Kohn-Sham calculations, which predicted that Cn is a semiconductor with a cohesive energy of about 110 kJ moP [138]. 

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 ... 

Liu W, Peng D. Exact two-component Hamiltonians revisited. J Chem Phys. 2009 131 031104. Nakajima T, Hirao K. The Douglas-Kroll-Hess Approach. Chem Rev. 2011 112 385-402. Belpassi L, Storchi L, Quiney HM, Tarantelli F. Recent advances and perspectives in four-component Dirac-Kohn-Sham calculations. Phys Chem Chem Phys. 2011 13 12368-12394. Peng D, Reiher M. Exact decoupling of the relativistic Fock operator. Theor Chem Acc. 2012 131 1081. 

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 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). 

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