The maximum entropy distribution consistent with observed structure amplitudes

About the reliability of the Maximum Entropy Method in reconstructing electron density: the case of MgO

The reliability of the Maximum Entropy Method (MEM) to reconstruct finite temperature electron density (ED) is here discussed, investigating the case of periclase (MgO). A theoretical electron density has been generated by quantum mechanic calculations and folded with a function simulating atomic thermal motion, in order to produce a reference errorless ED [ρ(r)REF]. The Fourier coefficients of ρ(r)REF have been calculated, and used as “observed” diffraction intensities to reconstruct via MEM the original ED. The electron density attained by MEM [ρ(r)MEM] and ρ(r)REF have been compared with each other (pixel-by-pixel and critical points) to assess the ability of MEM to retrieve EDs, on the basis of a set of observed structure factors. We have carried out our study varying the number of observed structure factors [i.e. sin (θ)/λ cut-off], the nature of the prior-density [uniform density and procrystal-like model] and the way in which the prior-density is treated during MEM maximization [fixed or free to change]. We observe that (i) it is recommendable to use the prior-density as a start point only, and allow it to change during maximization; (ii) the closer is the prior-density to ρ(r)REF, the easier one attains by MEM a correct ED; (iii) if the prior-density is varied and a sufficient large number of observed structure factors used, then MEM tends to yield converging EDs, regardless of the prior-density chosen as a start point.