Chemical potential, hardness, hardness and softness kernel and local hardness in the isomorphic ensemble of density functional theory

New relations among reactivity descriptors are provided within the recently introduced modified isomorphic ensemble of density functional theory. In addition, expressions for the softness and hardness kernel are derived in the canonical, grand canonical, isomorphic, and grand isomorphic ensemble. There results a new definition for the local hardness, η(r)=∂u(r)/∂Nσ=ησ−g(r), where g(r)=∂υ(r)/∂Nσ and σ=ρ/N is the shape factor. This identifies the local hardness as a function measuring the response of the system’s external potential to a perturbation in electron number at a constant shape factor. Furthermore, it is shown that one cannot represent both local softness and local hardness unambiguously with one representation.