Functional relations for the density-functional exchange and correlation functionals connecting functionals at three densities

It is shown that the density-functional-theory exchange and correlation functionals satisfy 0 = gamma E-hx [rho(N)] + 2E(c)(gamma)[rho(N)] - gamma Ehx[rho(gamma)(N-1)] - 2E(c)(gamma)[rho(gamma)(N-1)] + 2 integral d(3)r'[rho(0)(N-1)(r) - rho(gamma)(N-1)(r)] v(0)([rho(N)]; r) + integral d(3)r[rho(0)(N-1)(r) - rho(gamma)(N-1)(r)] r center dot del upsilon(0) )[rho(N)]; r) + integral d(3)r'rho(N)(r)r . del upsilon(gamma)(c)([rho(N)]; r) - integral d(3)r'[rho(0)(N-1)(r) . del upsilon(gamma)(c) ([rho(N)]; r) - integral d(3)r'f'gamma (r) r. del upsilon(gamma)(hxc)([rho(N)]; r) - 2 integral d(3)r'f'(gamma) upsilon(gamma)(hxc)([rho(N)]; r). In the derivation of this equation the adiabatic connection formulation is used, where the ground-state density of an N-electron system rho(N) is kept constant independent of the electron-electron coupling strength gamma. Here E-hx [rho] is the Hartree plus exchange energy, E-c(gamma)[rho] is the correlation energy, v. hxc [.] is the Hartree plus exchange-correlation potential, v(c)[rho] is the correlation potential, and v(hxc)(gamma)[rho] is the Kohn-Sham potential. The charge densities rho(N) and rho(gamma)(N-1) are the N- and (N - 1)-electron ground-state densities of the same Hamiltonian at electron-electron coupling strength gamma. f(gamma)(r) = rho(N)(r) - rho(N-1)gamma (r) rho(gamma)(N-1)(r) is the Fukui function. This equation can be useful in testing the internal self- consistency of approximations to the exchange and correlation functionals. As an example the identity is tested on the analytical Hooke's atom charge density for some frequently used approximate functionals.