Exchange energy for two closed shells generated by a bare Coulomb potential energy −Ze ²/r in the limit of large Z, in two dimensions

Abstract

Two-dimensional (2D) inhomogeneous electron assemblies are becoming increasingly important in Condensed Matter and associated technologies. Here, therefore, we contribute to the Density Functional Theory of such 2D electronic systems by calculating, analytically, (i) the idempotent Dirac density matrix γ(r, r′) generated by two closed shells for the bare Coulomb potential −Ze ²/r and (ii) the exchange energy density ${\varepsilon_x({\bf r})}$ . Some progress is also possible concerning the exchange potential V _x (r), one non-local approximation being the Slater potential ${2\varepsilon_x(r)/n(r)}$ , with n(r) the ground state electron density. However, to complete the theory of V _x (r), the functional derivative of the single-particle kinetic energy per unit area δt(s)/δn(r) is still required.