Inhomogeneous electron gas

In order to prepare the discussion of the relativistic generalization of the HK-theorem in Section 3 we finally consider the renormalization procedure for inhomogeneous systems. As the underlying renormalization program of vacuum QED is formulated within a perturbative framework (see Appendix B) we assume the perturbing potential to be sufficiently weak to allow a power series expansion of all relevant quantities with respect to V. In particular, this allows an explicit derivation of the counterterms required for the field theoretical version of the KS equations, i.e. for the four current and kinetic energy of noninteracting particles. 

The first quantity of interest is the four current 5 r) induced by F (r), 

The induced current (308) is automatically UV-finite if the expansion is based on renormalized response functions, i.e. A(r) just sums up the terms required for the transition from the Xr l to their renormalized counterparts. It is in-structive to analyze the corresponding counterterms for the noninteracting limit of (308) given graphically by 

The energy shift resulting from the perturbing potential can be evaluated by a coupling constant integration with respect to V. Scaling the external potential Hamiltonian by A, 

The actually interesting energy, corresponding to A = 1, can be obtained by a coupling constant integration approach (using a normalized ground state 4 (A)) for all A), 

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Flohenberg P and Kohn W 1964 Inhomogeneous electron gas Phys. RevB 136 864-72... 

Hohenberg P and Kohn W 1964. Inhomogeneous Electron Gas. Physical Review B136 864-871. 

Inhomogeneous Electron Gas P. Hohenberg and W. Kohn Physical Review 136 (1964) B864... 

The existence of the first HK theorem is quite surprising since electron-electron repulsion is a two-electron phenomenon and the electron density depends only on one set of electronic coordinates. Unfortunately, the universal functional is unknown and a plethora of different forms have been suggested that have been inspired by model systems such as the uniform or weakly inhomogeneous electron gas, the helium atom, or simply in an ad hoc way. A recent review describes the major classes of presently used density functionals [10]. 

Becke, A. D., 1988a, Correlation Energy of an Inhomogeneous Electron Gas. A Coordinate Space Model , J. Chem Phys., 88, 1053. 

Perdew, J. P., 1986, Density-Functional Approximation for the Correlation Energy of the Inhomogeneous Electron Gas , Phys. Rev. B, 33, 8822. 

Hohenberg, P. and Kohn, W. (1964) Inhomogeneous electron gas, Phys. Rev., 136, B864-B871. 

Heberle J (1971) The Debye integrals, the thermal shift and the Mossbauer fraction. In Mossbauer Effect Methodology Vol 7. Gruverman IJ (ed), Plenum, p 299-308 Herzberg G (1945) Molecular Spectra and Molecular Structure. II. Infrared and Raman Spectra of Polyatomic Molecules. Von Nostrand Reinhold, New York Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev 136 B864-871... 

Kohn, W., and Vashishta, P. (1983). General density functional theory. In Theory of the Inhomogeneous Electron Gas, edited by S. Lundqvist and N. H. March, Plenum, New York, pp. 79-147. 

S. Lundqvist and N.H. March, Theory of the inhomogeneous electron gas (Plenum Press, New York, 1983). 

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