HIGHER-ORDER DERIVATIVES IN DENSITY-FUNCTIONAL THEORY, ESPECIALLY THE HARDNESS DERIVATIVE DELTA-ETA/DELTA-N

Several third derivatives of the ground-state electronic energy E with respect to the electron number N and/or the external potential upsilon, as defined in density-functional theory, are studied. These include the first derivatives of the hardness eta. Legendre transforms of eta[N,upsilon] 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 gamma is studied numerically for atoms and positive ions, and suggested values are displayed and discussed. Recommended values of gamma 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.