Here, the density is written as a product ρ(r) = Nσ (r) of the particle number N with a shape function σ (r) that integrates to 1. According to the theory of optimization under constraints, the Lagrange multiplier µ at the optimum density ρ N is equal to the partial derivative with the shape function held constant, ∂ E/∂ N for short. But the Hohenberg-Kohn theorem does not define the energy for densities with a noninteger number of electrons. So the energy for a density with noninteger N in a neighbourhood of ρ N is not defined and the derivative does not exist. The typical solution in the theory of variations under constraints is to choose some extension of the functional into the domain where it is not defined. This extension is essentially arbitrary, the only requirement is that the extension obeys continuity properties so that the derivative exists. At this point, the Lagrange multiplier, which is the force of constraint keeping the density at integer N, is undetermined. It is determined by the cho-sen extension of E[ρ] into the noninteger domain. This is not a problem, it is directly related to the well-known gauge freedom of the Kohn-Sham potential, to which an arbitrary constant may be added. In a widely cited paper by Parr et al. (see also contribution (2.5.3)), the Lagrange multiplier µ has been described as "a characteristic constant for a system", but without any proof or arguments. This contradicts the essential arbitrariness of the constant µ, and therefore of ∂ E/∂ N. Atoms and molecules have an ionization energy and an electron affinity -there is no additional physical quantity µ = ∂ E/∂ N.