Density functional theory relativistic

THE DOUGLAS-KROLL-HESS FORMALISM IN DENSITY FUNCTIONAL THEORY 

The relativistic extension of DFT [38,39] can be based on quantum electrodynamics (QED) [40-43]. Using atomic units and invoking the Bom-Oppen- 

All fields and physical constants of the QED Hamiltonian have to be taken as renormalized to avoid divergencies of the electron-electron and electron-photon interaction integrals including the interaction with the external field [41]. Renormalization is independent of die external potential Vg, but requires correction terms, among them an energy correction 5E[Ve ]- Expectation values ( TI I T) of the renormalized QED Hamiltonian are finite for arbitrary N- 

To achieve this goal, one can proceed along two paths. Following physical intuition [44], one subtracts the infinite energy expectation value of 

With a suitable relativistic energy functional defined, one can proceed with the Kohn-Sham strategy [50-52]. Restriction of the four-current to the density component permits the Kohn-Sham separation of the energy functional [39]  

Here J°/c is identical to the ordinary electronic charge density p, while the other three components represent the electronic current density j. is the central quantity of relativistic density functional theory. All properties of the system are determined by J.  

This step in particular allows us to derive a local version of nonrelativistic CDFT. A corresponding explicit expression for the corresponding Exc has been given for the first time by Vignale and Rasolt (1988), 

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 

Within the above approximate relativistic CDFT scheme, the Breit interaction has been ignored. This radiative correction accounts for the retardation of the Coulomb interaction and exchange of transversal photons. A more complete version than that included in Equation (5.1) is given by the Hamiltonian (Bethe and Salpeter 1957 PyykkO 1978)  

For further discussion of the connection of this equation with CDFT see below. 

An interesting approach to the quantum mechanical description of many-electron systems such as atoms, molecules, and solids is based on the idea that it should be possible to find a quantum theory that refers solely to observable quantities. Instead of relying on a wave function, such a theory should be based on the electron density. In this section, we introduce the basic concepts of this density functional theory (DFT) from fundamental relativistic principles. The equations that need to be solved within DFT are similar in structure to the SCF one-electron equations. For this reason, the focus here is on selected conceptual issues of relativistic DFT. From a practical and algorithmic point of view, most contemporary DFT variants can be considered as an improved model compared to the Hartree-Fock method, which is the reason why this section is very brief on solution and implementation aspects for the underlying one-electron equations. For elaborate accounts on nonrelativistic DFT that also address the many formal difficulties arising in the context of DFT, we therefore refer the reader to excellent monographs devoted to the subject [383-385]. 

From what has been said already with respect to the variational collapse and the minimax principle, it is clear from the beginning that the standard derivation of the Hohenberg-Kohn theorems [386], which are the fundamental theorems of nonrelativistic DFT and establish a variational principle, must be modified compared to nonrelativistic theory [383-385]. Also, we already know that the electron density is only the zeroth component of the 4-current, and we anticipate that the relativistic, i.e., the fundamental, version of DFT should rest on the 4-current and that different variants may be derived afterwards. The main issue of nonrelativistic DFT for practical applications is the choice of the exchange-correlation energy functional [387], an issue of equal importance in relativistic DFT [388,389] but beyond the scope of this book. 

Although present-day DFT approaches often show a too low accuracy to be of value for accurate atomic calculations, their simple Hartree-Fock-like algorithmic structure makes them an ideal workhorse for molecular calculations which do not require ultimate accuracy (see section 10.4.3 for a list of references to the original research literature). DFT is in particularly widespread 

Electronic Charge and Current Densities for iUlany Eiectrons 

Within Dirac theory for a single electron, we may write the probability 4-current of section 5.2.3 as 

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Relativistic density functional theory can be used for all electron calculations. Relativistic DFT can be formulated using the Pauli formula or the zero-order regular approximation (ZORA). ZORA calculations include only the zero-order term in a power series expansion of the Dirac equation. ZORA is generally regarded as the superior method. The Pauli method is known to be unreliable for very heavy elements, such as actinides. 

Suzumura, T., Nakajima, T. and Hirao, K. (1999) Ground-state properties of MH, MCI, and M2 (M—Cu, Ag, and Au) calculated by a scalar relativistic density functional theory International Journal of Quantum Chemistry, 75, lVJ-1. ... 

Wang, F. and Li, L. (2002) A singularity excluded approximate expansion scheme in relativistic density functional theory. Theoretical Chemistry Accounts, 108, 53-60. 

El-Basil S (1990) Caterpillar (Gutman) Trees in Chemical Graph Theory. 153 273-290 Engel E (1996) Relativistic Density Functional Theory. 181 1-80... 

Relativistic density functional theory (RDFT), including relativistic exchange-correlation functionals [4, Chapter 4]. 

Finally the BERTHA technology has been applied to relativistic density functional theory by Quiney and Belanzoni [36]. This showed that the method works well for closed shell atoms as compared with benchmark calculations using finite difference methods, and there have been promising parallelization studies [37] which should in future greatly extend the range of application of the code. 

Engel, E. Dreizler, R. M. Relativistic density functional theory. In Density Functional Theory IF, vol. 181 Ed. Nalewajski, R. F. Springer Berlin, 1996, 1-80. 

Engel, E. Keller, S. Dreizler, R. M. Phys. Rev. A 1996, 53, 1367-1374 Engel, E. Relativistic density functional theory foundations and basic formalism. In Relativistic Electronic Structure... 

Os). There also exist a number of volatile Ru and Os halides and oxyhalides. The fluorides and oxyfluorides are of importance, but experimentally difficult to handle. Quite naturally, early considerations [81,82] and experimental developments [83-96] for a first Hs chemistry exclusively concentrated on the tetroxides. This strategy is justified, since classical extrapolations [97] as well as fully relativistic density functional theory calculations on the group-8 tetroxides [98] predict the existence of a volatile and very stable HSO4. 

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