It is known that the asymptotic decay of the electron density n(r) outside a molecule is informative about its first ionization potential I0, n(|r| → ∞) ∼ exp(−2 √ 2I0 r). This dictates the orbital energy of the highest occupied Kohn-Sham (KS) molecular orbital (HOMO) to be ǫH = −I0, if the KS potential goes to zero at infinity. However, when the Kohn-Sham HOMO has a nodal plane, the KS density in that plane will decay as exp (−2 √ −2ǫH−1 r). Conflicting proposals exist for the KS potential: from exact exchange calculations it has been found that the KS potential approaches a positive constant in the plane, but from the assumption of isotropic decay of the exact (interacting) density it has been concluded this constant needs to be negative. Here we show that either 1) the exact density decays differently (according to the second ionization potential I1) in the HOMO nodal plane than elsewhere, and the KS potential has a regular asymptotic behavior (going to zero everywhere) provided that ǫH−1 = −I1; or 2) the density does decay like exp(−2 √ 2I0 r) everywhere but the KS potential exhibits strongly irregular if not divergent behavior around (at) the nodal plane.

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The mathematics of K-conserving functional differentiation, with K being the integral of some invertible function of the functional variable, is clarified. The most general form for constrained functional derivatives is derived from the requirement that two functionals that are equal over a restricted domain have equal derivatives over that domain. It is shown that the K-conserving derivative formula is the one that yields no effect of Kconservation on the differentiation of K-independent functionals, which gives the basis for its generalization for multiple constraints. Connections with the derivative with respect to the shape of the functional variable and with the shape-conserving derivative, together with their use in the density-functional theory of many-electron systems, are discussed. Yielding an intuitive interpretation of K-conserving functional derivatives, it is also shown that Kconserving derivatives emerge as directional derivatives along K-conserving paths, which is achieved via a generalization of the Gâteaux derivative for that kind of paths. These results constitute the background for the practical application of K-conserving differentiation.

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