Atomic-charge log-convexity and radial expectation values

Abstract

Let us denole by p (r) the spherically averaged electron density of an atomic system. First, it is found that there always exbe a parameter mo 3 0 such that the function p(r)/r IS lo arithmically convex for any 01 2 00. Furthermore, 00 3 max{p,O) with to obtain rigorous and wmpact inequalities involving three radial expectation values, which substantially generalize all the similar ones known up to now. n e s e inequalities allow us lo correlate sweral fundamental andlor measurable physical quantities such as the nuclear elslron attraction energy, the Langmin-Pauli diamagnetic susceptibiliry and so on. Thirdly. the logconvexity properly is investigated in a Hanree-Fack framework for all ground-state neutral atoms with Z < 54. In parlicular, it is found that ail the atoms have a lagconvex function p(r)/r6 and only H and He have a logconvex electron density p (r). Finally, the accuracy of the abwe-mentioned inequalities is studied in the same numerical framework.

Seven (Lattice) Paths to Log-Convexity

“…For example, on segment [3,4] the function f is given by f (x) = 7x−1 2(x+2) , and on the segment [4,5] it is given by f (x) = 20x 2 −9x−14 (x+2)(7x −8) . Obviously, the values of function f at positive integers coincide with the elements of the quotient sequence, f (n) = x n .…”

“…Furthermore, f is continuous in all n ∈ N, and f is a rational function on every segment [n, n + 1], for all n ∈ N, n ≥ 2. It is easy to see, by inspection, that f is bounded on [3,4] and on [4,5].…”

“…Moreover, 2 ≤ f (x) ≤ 7/2 on the segment [3,5]. (The upper bound 7/2 is not the best possible, but a tighter upper bound would be more difficult to extend to larger intervals [2, n].)…”

“…Staying in finance, we mention that it was found that log-convexity of the discount function is equivalent to the decreased impatience [37], and that the log-convexity of the residual vulnerability function is the key for the validity of Gordon-Loeb's 1/e rule [10]. Log-convex functions are also important in communication engineering [12], pharmacokinetics [54,55], and atomic physics [3]. We conclude this non-exhaustive list by mentioning the appearance of log-convex sequences in mathematical biology, where they arise as the enumerating sequences of secondary structures of certain biopolymers [23,41,49].…”

Structure of the electron momentum density of atomic systems

“…This theorem was previously applied to the position density ρ(r) of ground-state atoms and used [10,39] to obtain relationships among three position moments r t . Now, the KPB theorem allows us to obtain the following general set of inequalities for momentum moments p t :…”