Abstract
Even with restriction to the longitudinal contribution the treatment of the electron-electron interaction is exceedingly difficult. However, in the case of the ground state any energetic, structural or electronic property of an inhomogeneous electron gas can be viewed as a functional of its local density \(n({\mathbf {x}})\). This scalar function of the position \({\mathbf x}\), in principle, determines all the information of the many-electron wave function. For a given external potential \(V_\mathrm{ext}({\mathbf x})\), for instance that due to the arrangement of the charged nuclei, the proofs of existence of such a functional are given by the Hohenberg-Kohn theorems. The ground state energy is minimized by variation of \(n({\mathbf x})\). Thereby, it decomposes into an external part and a universal Hohenberg-Kohn functional. The latter one fully accounts for the electron-electron interaction, but the theory – the density functional theory – provides no guidance for constructing it. Generalizations of the theory are possible in different directions. The most important one is the spin density functional theory with functionals depending also on the vector of the magnetization density \({\mathbf m}({\mathbf x})\).
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Bechstedt, F. (2015). Density Functional Theory. In: Many-Body Approach to Electronic Excitations. Springer Series in Solid-State Sciences, vol 181. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44593-8_5
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