Agreement factors between calculated and observed integrated intensities

Considering organic compounds, in previous work the simulated annealing algorithm has been applied ignoring the overlap of the peaks, with a cost function based in the fit of the complete diffraction pattern (Rwp) [1],  evaluating correlated individual Bragg intensities from Pawley fits [2] and from a reconstructed pattern [3]. The first one has the advantage that it considers implicitly the overlap of the Bragg reflections in the powder pattern, however it requires more computational time.
In our work, the goodness of the trial solutions (cost function) was first determined through the R agreement factor, defined as:

R = ( Si  | Itrial model, i - ILe Bail, i | ) / ( Si ILe Bail, i  ) x 100

where i ran over the first 40 Bragg reflections of the pattern. In some cases, no overlap correction is necessary for the integrated intensities obtained in the Le Bail fits. The packing in the structures I to VII was found using the first 40 reflection peaks and the validity of the solutions was later confirmed when the models were used in Rietveld refinements of the complete diffraction pattern.

For other materials (like b-haematin, triclinic, Z=2, a=12.196(2) , b=14.684(2), c=8.040(1), a=90.22(1)°, b=96.80(1)°, g=97.92(1)°), the overlap of the reflection peaks makes not possible to solve the structure without any correction. We note that a reasonable profile factor can be approximate from the integrated intensities alone [4]. Thus, we redefined the cost function as follows:

S = ( ILe Bail, total - Itrial model, total )2 / ( ILe Bail, total )2

based in the difference squared of the total Le Bail and trial model integrated intensity of the diffraction pattern. Since we have a set of overlapping individual integrated intensities for the hkl reflections extracted in the Le Bail fit (Ai) and the same set is calculated for each trial structure (Bi), we write:

S = ò2q { (Si Ai fi(2q) - Si Bi fi(2q) }2 /  { ò2q Si Ai fi(2q) }2

where fi are the profile functions for each diffraction peak in the powder pattern (area normalized to 1 after integration). In the integration of the previous equation, the overlap coefficients Fi, i' are defined as follows:

Fi, i' = òd2q f i(2q) f i' (2q)

The integrals can be evaluated analytically. Since Fi, i' = Fi', i the expresion of  S can be reordered as:

S = { Si,i' (Ai - Bi) Fi, i' (Ai' - Bi')} / { SiAi }2

In practice, the overlap coefficients are calculated for the i=5 adjacent reflections in the powder pattern.

References:

[1]-Y. G. Andreev and P. G. Bruce, J. Chem. Soc., Dalton Trans., (1998) 4071-4080.
[2]-W. I. F. David, K. Shankland and N. Shankland, Chem. Comm. (1998) 931-932.
[3]-A. Le Bail, to be published in the proceedings of EPDIC-7.
[4]-S. Pagola, P. W. Stephens, D. S. Bohle, A. D. Kosar and S. K. Madsen, Nature 404 (2000) 307-310.