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

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Abstract
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 η[N,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.

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