Many-electron methods 2-particle density matrix

The exchange-correlation potential is the source of both the strengths and the weaknesses of the DF approach. In HF theory, the analytical form of the term equivalent to Vxc, the exchange potential, arises directly during the derivation of the equations, but it depends upon the one-particle density matrix, making it expensive to calculate. In DF theory the analytical form of Vxc must be put into the calculations because it does not come from the derivation of the Kohn-Sham equations. Thus, it is possible to choose forms for Vxc that depend only upon the density and its derivatives and which are cheap to calculate (the so-called local and non-local density approximations). The Vxc factor can also be chosen to account for some of the correlation between the electrons, in contrast to HF methods for which additional calculations must be made. The drawback is that there does not appear to be any systematic way of improving the potential. Indeed, many such terms have been proposed. 

D. A. Mazziotti, Variational reduced-density-matrix method using three-particle N-represent-ability conditions with application to many-electron molecules. Phys. Rev. A 74, 032501 (2006). 

Since many experimental studies of 7-Fe were performed for 7-Fe particles in a Cu matrix (or Cu alloy, including Cu-Al) [113], [114], it is important to probe the electronic structure of the particle-matrix systems. Embedded-cluster methods are ideally taylored to treat small particles of a metal in a host matrix, a system that would require a very large supercell in band-structure calculations. DV calculations were performed for the 14-atom Fe particle in copper shown in Fig. 21 [118]. Spin-density contour maps were obtained to assess the polarization of the Cu matrix by the coherent magnetic 7-Fe particle. Examples are given in Figs. 22 and 23 for a Fe particle in Cu and 7-Fe in Cu with two substitutional Al. If the matrix is a Cu-Al alloy, this element is known to penetrate the Fe particle [114]. 

---