ISSN:
1434-6079
Keywords:
31.10. + z
;
31.15. + q
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
Abstract A functionf(r) is monotone of orderp if itspth-derivativef (p)(r) fulfils that (−1) p f (p)(r)≧0. So, e.g. the monotonicity properties of orderp=0, 1, 2 describe the non-negativity (p=0), the monotonic decreasing from the origin (p=1) and the convexity (p=2) of the function, respectively. Here, the monotonicity properties of the electron functiong n (r; α)=(−1) n ρ(n) (r)r −α , α≧0, of the ground state of atomic systems are analysed both analytically and numerically. The symbol ρ(r) denotes the spherically averaged electron density. First of all, the condition which specifies, if exists, a value α np such thatg n (r; α np ) be monotone of orderp is obtained. In particular, it is found that α01=max {rρ′(r)/ρ(r)}, α02=max {q 0(r)}, α11=max {rρ″(r)/ρ′(r)} and α12=max {q 1(r)}, whereq 0(r) andq 1(r) are simple combinations of the first few derivatives of ρ(r). Secondly, numerical calculations of the first few values α np in a Hartree-Fock framework for all ground-state atoms with nuclear chargeZ≦54 are performed. In doing so, the pioneering work of Weinstein, Politzer and Srebrenik about the monotonically decreasing behavior of ρ(r) is considerably extended. Also, it is found that Hydrogen and Helium are the only two atoms having the functions ρ(r), −ρ′(r) and ρ″(r) with the property of convexity. Thirdly, it is analytically shown that the charge functionr −α ρ(r) with α≧[(1+4Z 2/I)1/2−1]/2, I being the first ionization potential, is convex everywhere. Finally, the above mentioned monotonicity properties are used to obtain rigorous, simple and universal inequalities involving three radial expectation values which generalize all the similar ones known up to now. These inequalities allow to correlate various statical and dynamical quantities of the atomic system under study, due to the physical meaning of the radial expectation values. It is worth to remember that some of these expectation values may be experimentally measured in experiments of (e, 2e)-type.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01437292
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