A solid understanding of the Lieb functional FL is important because of its centrality in the foundations of electronic density functional theory. A basic question is whether directional derivatives of FL at an ensemble-V-representable density are given by (minus) the potential. A widely accepted purported proof that FL is Gâteaux differentiate at EV-representable densities would say, "yes." But that proof is fallacious, as shown here. F L is not Gâteaux differentiable in the normal sense, nor is it continuous. By means of a constructive approach, however, we are able to show that the derivative of FL at an EV-representable density ρ0 in the direction of ρ1 is given by the potential if ρ0 and ρ1 are everywhere strictly greater than zero, and they and the ground state wave function have square integrable derivatives through second order.
All Science Journal Classification (ASJC) codes
- Atomic and Molecular Physics, and Optics
- Condensed Matter Physics
- Physical and Theoretical Chemistry