THEOREM FOR FUNCTIONAL-DERIVATIVES IN DENSITY-FUNCTIONAL THEORY

Consider a functional Q[n] that scales homogeneously, Q[n-lambda] = lambda(k)Q[n], where n-lambda-(r) = lambda-3n(lambda-r). It is already known that Q[n] is related to its functional derivative, q([n];r) = delta-Q[n]/delta-n, by kQ[n] = - integral d3r n(r)r.NABLA q([n];r). We here prove that q([n];r) is related to its functional derivative by kq([n];r) = - integral d3r'n(r')r'.NABLA'{delta-q([n];r)/delta-n(r')}-r.NABLA q([n];r) and that there also exists a homogeneouslike scaling relation for q([n];r): q([n-lambda];r) = lambda-(k)q([n];lambda-r). In addition, delta-q([n];r)/delta-n(r') = delta-q([n];r')/delta-n(r) because q([n];r) is the functional derivative of Q[n]. Based upon these exact properties of q([n];r) it is proved that if a trial potential qBAR([n];r) satisfies kQ[n] = - integral d3r n(r)r.NABLA qBAR([n];r) and delta-qBAR([n];r)/delta-n(r') = delta-qBAR([n];r')/delta-n(r), then kq([n];r) = - integral d3r'n(r')r'.NABLA'{delta-qBAR([n];r)/delta-n(r')}-r.NABLA qBAR([n];r. If qBAR([n];r) further satisfies qBAR([n-lambda];r) = lambda-(k)q([n];lambda-r), we prove that q([n];r) = qBAR([n];r), which means that qBAR is exact. Application of the theorem to density-functional theory is discussed.