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Relativistic Density Functional Theory
[chapter]

Eberhard Engel

2015
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Handbook of Relativistic Quantum Chemistry
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This chapter gives an overview of relativistic density functional theory. Both its foundations, the existence theorem and the Kohn-Sham equations, and its core quantity, the exchangecorrelation (xc) energy functional, are discussed. It is first outlined how a workable relativistic Kohn-Sham scheme can be obtained within the framework of quantum electrodynamics and which alternatives for its implementation are available. Particular emphasis is then placed on the relativistic corrections to the
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... -functional. The modification of its functional form due to the relativistic motion of the electrons and their retarded interaction via exchange of photons is distinguished from the effect resulting from insertion of a relativistic density into the functional. The difference between the relativistic xc-functional and its nonrelativistic form is studied in detail for the case of the exchange functional (which can be handled exactly via the optimized effective potential method). This analysis is complemented by some first-principles results for the correlation functional, relying on a perturbative approach. Finally, the accuracy of approximate relativistic xc-functionals, the local density and the generalized gradient approximation, is assessed on the basis of the exact results. positron pairs to all relevant quantities are neglected No-pair relativistic Hamiltonians: Q4C and X2C. The resulting relativistic Kohn-Sham (RKS) equations differ from their nonrelativistic counterpart by the relativistic form of the kinetic energy operator and the relativistic coupling between the particles and the effective RKS potential. As in the nonrelativistic situation, the total KS potential consists of an external, a Hartree, and an exchange-correlation (xc) component, which is obtained as functional derivative of the core quantity of RDFT, the xc-energy functional E xc . Both the Hartree term and E xc reflect the fact that in QED the electrons interact by the exchange of photons rather than via the instantaneous Coulomb interaction. Together with the relativistic kinematics of the electrons, this retarded interaction leads to a modification of E xc , compared to its nonrelativistic form. It is the primary intention of this chapter to discuss the xc-functional of RDFT, with an emphasis on the role of the relativistic corrections in this functional. The chapter starts with an overview of the foundations of QED-based RDFT, the existence theorem and the resulting Kohn-Sham (KS) formalism. In this approach, the four-current density j is the fundamental variable which is used to represent observables such as the ground state energy. In practice, however, RDFT variants working with the charge and magnetization densities or the relativistic extensions of the nonrelativistic spin densities are utilized. The corresponding KS equations are therefore also summarized. For a rigorous assessment of the relevance of relativistic corrections in E xc , one has to resort to first-principles expressions for E xc . A firstprinciples treatment is in particular possible for the RDFT exchange E x . This functional, while known as an implicit functional of j , is explicitly known only in terms of the KS four spinors. Its self-consistent application in the KS formalism relies on the relativistic optimized potential method, which is outlined next. On this basis, the properties of E xc are analyzed more closely, addressing in particular the transverse interaction. The analysis of the exact E x is complemented by results obtained with an MP2-type correlation functional. Finally, the relativistic extension of the local density approximation (LDA) as well as the generalized gradient approximation (GGA) are discussed. The resulting functionals are used to examine the importance of the relativistic corrections in E xc for bonding/cohesive properties. [The present text is based on Chapter 8 of [1] .] Handbook of Relativistic Quantum Chemistry

doi:10.1007/978-3-642-41611-8_18-1
fatcat:nvhpzxhemzafbor3v7hbxxxv5y