Von Barth

If the external magnetic field B(r), and m(r) have only a nonvanishing Z-component, B(r) = (0,0, B(r)) and m(r) = (0,0, m(r)), the universal functional F[p, m] may then be considered as a functional of the spin densities ps(r) and p(r), F[ps(r), p(r)], because the spin density is proportional to the z-component of the magnetization m(r) = p-bPsW P-b is the electron Bohr magneton. It is of worth mentioning that it is possible to define two spin densities that are the diagonal elements of the density matrix introduced by von Barth and Hedin [3]. These correspond to the spin-up (alpha) electrons density pT(r), and the spin-down (beta) electrons density p (r). In terms of these quantities, the electron and spin densities can be written as... 

Von Barth, U. and Hedin, L. 1972. A local exchange-correlation potential for the spin polarized case I. J. Phys. C Solid State Phys. 5 1629-1642. 

In Eq. [47], epc ( ) and exc (n) are the exchange-correlation energy densities for the nonpolarized (paramagnetic) and fully polarized (ferromagnetic) homogeneous electron gas. The form of both exc(n) and exc(n) has been conveniently parameterized by von Barth and Hedin. Other interpolations have also been proposed24,33 for eKC(n, J ). The results for the homogeneous electron gas can be used to construct an LSDA... 

Almbladh CO, von Barth U (1985) In Dreizler RM, da Providencia J (eds) Density Functional Methods in Physics. NATO ASI Series B123... 

Though the core expansion leads to the appropriate fit, it may not be the proper explanation for the scale factor discrepancy. Hansen et al. (1987) note that the expansion of the core would lead to a decrease of 7.5 eV in the kinetic energy of the core electrons, at variance with the HF band structure calculations of Dovesi et al. (1982), which show the decrease to be only about 1.5 eV. An alternative interpretation by von Barth and Pedroza (1985) is based on the condition of orthogonality of the core and valence wave functions. The orthogonality requirement introduces a core-like cusp in the s-like valence states, but not in the p-states. Because of the promotion of electrons from s - p in Be metal, the high-order form factor for the crystal must be lower than that for the free atom. It is this effect that can be mimicked by the apparent core expansion. 

In order to calculate the magnetic moment it is necessary to perform spin polarized band calculations. The extension of the local density approximation to cover the spin polarized case has been made by von Barth and Hedin and Guimarsson and Lund-qvist ... 

In a more general setting the recipe [91] can be considered as an implementation of another suggestion by Gunnarsson and Lundqvist [99] and von Barth [100] known also at a pretty early stage of the development of the DFT technique of employing different functionals to describe different spin or symmetry states. In other words the simplified model for the data fit Eq.(lO) changes to ... 

---