CONVEXITY AND THE TRANSLATIONAL-INVARIANCE CONSTRAINT ON THE EXCHANGE-CORRELATION FUNCTIONAL

Knowledge of the properties of the exchange-correlation functional in the form 1/lambda upsilon(xc)([rho(lambda)],r/lambda), where rho(lambd a)(r)=lambda(3) rho(lambda r), is important when expressing the exchan ge-correlation energy as a line integral E(xc)[rho]=integral(0)(1)d la mbda integral dr1/lambda upsilon(xc)([rho(lambda)],r/lambda)[3 rho(r)r .del rho(r)] [R. van Leeuwen and E. J. Baerends, Phys. Rev. A 51, 17 0 (1995)]. With this in mind, it is shown that in the low-density limi t lim(lambda-->0)integral rho(r)del(2)1/lambda upsilon(xc)([rho(lambda )]d(3)r less than or equal to 4 pi integral rho(r)(2)d(3)r. This inequ ality is violated in the local-density approximation.