Why Does SHELXL-93 Refine against F-Squared?

WHY DOES SHELXL-93 REFINE AGAINST F2?

Traditionally most crystal structures have been refined against F. For a well-behaved structure the geometrical parameters and their esd's are almost identical for refinement based on all Fo2 values and for an old-fashioned refinement against F ignoring data with Fo less than (say) 3*sigma(Fo). For weakly diffracting crystals and in particular for pseudosymmetry problems the refinement against all data is demonstrably superior. The esd's are reduced because more experimental information is used, and the chance of getting stuck in a local minimum is reduced. In addition, the use of a threshold introduces a systematic error which introduces bias into the displacement parameters Uij. On the other hand, it is impossible to refine on F using ALL data, because it would involve taking the square root of a negative number for reflections with negative Fo2 (i.e. background higher than the peak as a result of statistical fluctuations), and because the estimation of sigma(Fo) from sigma(Fo2) for small or negative Fo2is a difficult statistical problem which requires the assumption of a probability distribution function for the F-values. In the case of pseudosymmetric structures - i.e. the very case where the weak reflections are most important - this distribution function is not known a priori, making it impossible to derive 'correct' sigma(Fo) values and hence correct weights.

The diffraction experiment measures intensities and their standard deviations, which after the various corrections give Fo2 and sigma(Fo2). If your data reduction program only outputs Fo and sigma(Fo), which as explained above involves serious approximations for weak reflections, you MUST CORRECT YOUR DATA REDUCTION PROGRAM, not simply write a routine to square the Fo values or use HKLF 3 to input Fo and sigma(Fo) to SHELXL-93 (although the latter is legal). Note that if an Fo2 value is too large to fit format F8.2, then format F8.0 may be used instead - the decimal point overrides the FORTRAN format specification.

The use of a threshold for ignoring weak reflections may introduce bias which primarily affects the atomic displacement parameters; it is only justified to speed up the early stages of refinement. In the final refinement ALL DATA should be used except for reflections known to suffer from systematic error (i.e. in the final refinement the OMIT instruction may be used to omit specific reflections - although not without good reason - but not ALL reflections below a given threshold). Anyone planning to ignore this advice should read F. L. Hirshfeld and D. Rabinovich, Acta Cryst., A29 (1973) 510-513 and L. Arnberg, S. Hovmoller and S. Westman, Acta Cryst., A35 (1979) 497-499 first. Refinement against F2 also facilitates the treatment of twinned and powder data, and the determination of absolute structure.

One cosmetic disadvantage of refinement against F2 is that R-indices based on F2 are larger than (often about double) those based on F. For comparison with older refinements based on F and an OMIT threshold, a conventional index R1 based on observed F values larger than 4*sigma(Fo) is also printed. The deviation of the Goodness of Fit (S) from unity also tends to be magnified when calculated with F2.

Throughout the output, R indices based on F2 are denoted R2 and those based on F are denoted R1, e.g.

wR2 = (Sum[w(Fo2 - Fc2)2] / Sum[wFo4])1/2

R1 = Sum | |Fo| - |Fc| | / Sum |Fo|

For details of the weights w see WGHT below. The Goodness of Fit (S) is always based on F2:

GooF = S = [ Sigma [ w(Fo2 - Fc2)2 ] / (n-p) ]1/2

where n is the number of reflections and p is the total number of parameters refined. In the 'Restrained Goodness of Fit', Sigma[w(yt-y)2] is added to the numerator and the number of restraints is added to the denominator. This corresponds to treating each restraint as an extra observational equation with weight w = 1/sigma2. y is the quantity (e.g. a bond length) being restrained and yt is its target value. In these expressions, Sigma is written with a capital S to indicate a summation and a small s for an estimated standard deviation (corresponding to the use of capital and small Greek letters for sigma).

In general most statistical quantities are defined as in the I.U.Cr. Commission's report: 'Statistical Descriptors in Crystallography', D. Schwarzenbach et al., Acta Cryst., A45 (1989) 63-75.

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