Differentiation of density functionals that conserves the normalization of the density -: art. no. 022506

The formula for differentiation of functional A[rho] with respect to rho (r) while keeping the normalization integral rho (r)dr of rho (r) fixed is derived, and basic properties arising from it are concerned. The results are then generalized for time-dependent theories. One of the essential consequences of normalization conservation for functional differentiation, namely, that it ruins the symmetry in r of multiple derivatives, is shown to give the resolution of the causality paradox of response functions in time-dependent density-functional theory. The formula for the differentiation of functionals A[rho] that conserves the shape of rho (r) is also presented.