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Relativistic Density Functional Theory

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Relativistic Methods for Chemists

Part of the book series: Challenges and Advances in Computational Chemistry and Physics ((COCH,volume 10))

Abstract

This review starts with the basics of nonrelativistic density functional theory, followed by the foundation of the relativistic variant. How to formulate a relativistic spin density functional theory is shown through the comparison of the non-collinear and collinear approximations. It is shown where relativistic corrections to the exchange-correlation functionals are important and that they have a sizeable influence on total energies but are not so important for valence properties. After discussing some four-component Dirac–Kohn–Sham implementations, quasirelativistic methods are reviewed with emphasis on the zeroth-order regular approximation and the Douglas–Kroll–Hess method. This review contains 163 references.

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Notes

  1. 1.

    In a nonrelativistic (Galilei-invariant) world, the Lorentz force (exerted by a magnetic field on a moving charge) and the Biot–Savart law (the generation of a magnetic field through currents of moving charges) cannot coexist. In the nonrelativistic limit, we have to choose which one survives, the other being a “relativistic effect”. We follow the SI such that the Lorentz force is independent of the speed of light, and absorb the factor 4πε0 in the definition of the charge, but then \({\mu }_{0} \sim {c}^{{-}^{2} }\), the prefactor in the Biot–Savart law, vanishes in the nonrelativistic limit. Whatever choice is made, the current–current interaction, which involves both laws, is of order \(\mathcal{O}({c}^{-2}\)). In the c.g.s. unit system, frequently used in theoretical physics, this apparent asymmetry is removed and a factor c − 1 is found both in the Lorentz force and in the Biot–Savart law. This causes a lot of confusion when comparing equations from different texts.

  2. 2.

    Spin and magnetization density is not the same. The magnetization density contains an additional prefactor, the Bohr magneton. However, this distinction is not always made in the literature. The two minus signs in Eq. (5-10) imply that not even a one-electron system can be fully spin polarized except in the nonrelativistic limit. While the minus signs follow from the Gordon decomposition, some authors define the spin densities with plus signs here. Numerically, it seems to make little difference.

  3. 3.

    In a speech (witnessed by the present author) given by W. Nieuwpoort at a conference dinner in 1999, the difference between four-component and two-component methods was characterized as the difference between a four-course and a two-course dinner. See also the title of Ref. [159]!

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  1. Fachbereich Chemie and Forschungszentrum OPTIMAS, Technische Universität Kaiserslautern, Erwin-Schrödinger-Straße, 67663, Kaiserslautern, Germany

    Christoph van Wüllen

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Correspondence to Christoph van Wüllen .

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  1. Inst. Chemistry, Dept. Quantum Chemistry, Nicolaus Copernicus University, Gagarin Street 7, Torun, 87-100, Poland

    Maria Barysz

  2. Dept. Chemistry, University of Puerto Rico, San Juan, USA

    Yasuyuki Ishikawa

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van Wüllen, C. (2010). Relativistic Density Functional Theory. In: Barysz, M., Ishikawa, Y. (eds) Relativistic Methods for Chemists. Challenges and Advances in Computational Chemistry and Physics, vol 10. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9975-5_5

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