Hohenberg-Kohn kernel K(r-r’)

Abstract

In a classic paper Hohenberg and Kohn wrote the energy of an electron gas as a functional of the charge density n(r): E[n]= F v(r)n(r)dr+(1/2 F [n(r)n(r’)/‖r-r’‖]dr dr’+G[n]. For a gas of almost constant density, n(r)=$n_0$+ñ(r) with ñ(r)/$n_0$≪1 they expanded G[n]=G[$n_0$]+ F K(r-r’)ñ(r)ñ(r’)dr dr’+.... The kernel K(r) may be written as a sum of kinetic, exchange, and correlation terms, K(r)=$K_s$(r)+$K_x$(r)+$K_c$(r). We present here graphs of $K_s$(r) and $K_x$(r) which are exact to within our numerical accuracy.