Higher-order derivatives in density-functional theory, especially the hardness derivative ∂η/∂N

Several third derivatives of the ground-state electronic energy E with respect to the electron number N and/or the external potential v, as defined in density-functional theory, are studied. These include the first derivatives of the hardness η. Legendre transforms of ηN,v are constructed and corresponding Maxwell relations derived. Various new functions and relations are found. The derivative of the hardness with respect to the electron number at constant potential γ is studied numerically for atoms and positive ions, and suggested values are displayed and discussed. Recommended values of γ are positive for most spherical atoms, negative otherwise. Power series expansions of E(N) are not recommended for reasons given. Simple analytical representations are recommended instead.