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We present explicit formulas for arbitrary-order derivatives of the energy, grand potential, electron density, and higher-order response functions with respect to the number of electrons, and the chemical potential for any smooth and differentiable model of the energy versus the number of electrons. The resulting expressions for global reactivity descriptors (hyperhardnesses and hypersoftnesses), local reactivity descriptors (hyperFukui functions and local hypersoftnesses), and nonlocal response functions are easy to evaluate computationally. Specifically, the explicit formulas for global/local/nonlocal hypersoftnesses of arbitrary order are derived using Bell polynomials. Explicit expressions for global and local hypersoftness indicators up to fifth order are presented. [ABSTRACT FROM AUTHOR]

We present spatially and velocity-resolved echelle spectroscopy for NGC 7009 obtained with the UVES spectrograph at the European Southern Observatory's Very Large Telescope. Our objective is to analyze the kinematics of emission lines excited by recombination and collisions with electrons to determine whether similarities or differences could be useful in elucidating the well-known abundance discrepancy derived from them. We construct position–velocity maps for recombination, fluorescence, charge transfer, and collisionally excited lines. We find a plasma component emitting in the C II, N II, O II, and Ne II recombination lines whose kinematics are discrepant: they are incompatible with the ionization structure derived from all other evidence and the kinematics derived from all of these lines are unexpectedly very similar. We find direct evidence for a recombination contribution to [N II] λ5755. Once taken into account, the electron temperatures from [N II], [O III], and [Ne III] agree at a given position and velocity. The electron densities derived from [O II] and [Ar IV] are consistent with direct imaging and the distribution of hydrogen emission. The kinematics of the C II, N II, O II, and Ne ii lines does not coincide with the kinematics of the [O III] and [Ne III] forbidden emission, indicating that there is an additional plasma component to the recombination emission that arises from a different volume from that giving rise to the forbidden emission from the parent ions within NGC 7009. Thus, the chemical abundances derived from either type of line are correct only for the plasma component from which they arise. Apart from [N II] λ5755, we find no anomaly with the forbidden lines usually used to determine chemical abundances in ionized nebulae, so the abundances derived from them should be reliable for the medium from which they arise. [ABSTRACT FROM AUTHOR]