Maximum-entropy analysis of one-particle densities in atoms

Abstract

The Maximum-Entropy formalism is used to obtain approximations to the spherically averaged charge and momentum densities. The only information required is the first few radial expectation values. Analytical and numerical approximations to the central values of the densities are calculated. Moreover, the unused or unknown radial expectation values are estimated by means of the moments of these Maximum-Entropy densities. As illustration, the accuracy of these approximations are numerically studied in a Hartree-Fock framework. This method is complementary to the one which makes use of the Stieltjes-Chebyshev inequalities and leads to the least biased approximate densities compatible with the information we use.

Minimum cross‐entropy estimation of internally folded densities from Compton profiles

“…However, many numerical calculations and estimations 4, 11 on γ(0) have been carried out. Among them, let us notice those based on maximum‐entropy approximations on γ( p ) 5, 12. Additionally, different lower bounds to γ(0) in terms of radial expectation values are also known 13, 14.…”

Maximum-entropy and Padé-like approximations to atomic scattering factors

“…On the other hand, due to the tightness of the ME approximations, this method has been widely used in the literature in a great variety of fields (see e.g. [16][17][18][19][20][21][22][23][24][25][26][27]). It is worthy to remark that analytical ME solutions are only known for the case M = 1, namely…”

Inverse atomic densities and inequalities among density functionals

“…For atomic systems, the expectation values ͗r n ͘ and ͗ln r͘ have been extensively used to bound and/or estimate other global quantities 6,8,9 and the density itself. [10][11][12] Among those quantities, let us remark the so-called frequency moments of both (r) and (r), n ϵ ͵ n ͑ r͒d D r, ͑3͒ n ϵ ͵ n ͑ r ͒d D rϭ⍀ D ͵ 0 ϱ r DϪ1 n ͑ r ͒dr. ͑4͒…”