• The structure is 'routine'
  • There are ample data at 3sigma(I), (where sigma(I) for area detectors may need to be multiplied by a factor of 3 to 5 to bring it into the same sale as serial diffractometers - see Kirschbaum et al below).
  • The structure is not pseudo-centric, pseudo-centred, does not have a marginal Flack enantiopole parameter, etc.
  • In the case of routine structures, the final atomic parameters will be substantially the same whether the refinement is with F or Fsq, and whether the weak reflections are included or rejected. Older crystallographers may recall that lowering the sigma threshold to get more reflections into a refinement usually had no impact on the parameters values, though the sus (esds) usually rose a little. Fsq refinement may have a wider radius of convergence (though modern Direct Methods usually give models well within convergence for F or Fsq), but is sensitive to bad observations (though modern diffractometers rarely yield totally spurious observations). In principle the standard uncertainty of Fsq is readily calculated, though in practice weights derived from it need to be well massaged, as do the weights for F refinement.
    
• The structure is non-routine, and suffers from one or more of the above problems.
In this case no single recipe will resolve the problem. Simply refining on Fsq is unlikely to provide a clear answer. Adding in large numbers of essentially 'unobserved' reflections will contribute nothing to finding a solution, particularly if the weak reflections are systematically over estimated (check the mean value of the systematic absences). Imagine collecting the full silver-radiation sphere for a crystal of a soft organic small molecule, in P 21. The fact that the 0 1 0, 0 3 0, 0 5 0, 0 7 0 are very weak is significant. Nothing can be gleaned from the observation that the 0 39 0, 0 40 0 and 0 41 0 are also unobserved. Dumping masses of weak reflections into a refinement (something now very easy to do with the advent of area detectors) not only serves no useful purpose, it actually degrades the usefulness of one-parameter estimators (such as the Hamilton weighted R factor), unless the weight of these refections is insignificant. Prince (1994) has suggested how the important reflections (strong or weak) may be identified, but few programs systems include this calculation.

If the question of F or Fsq, 3 sigma or -ve reflections becomes significant to a structure analysis, there are certainly more fundamental questions to be answered.

Other interesting reading is:

1. J.S.Rollett, Crystallographic Computing Techniques, (1976), ed F.R.Ahmed et al, Munksgaard, pp 414.
   
   • 'We have considered the relationship between M (Fsq refinement) and N (Fo refinement). The minima of these functions can be made to coincide ........and both functions, properly weighted, give the same true minimum.'
2. J.S.Rollett, Crystallographic Computing Techniques, (1976), ed F.R.Ahmed et al, Munksgaard, pp 414.
   
   • 'This led us to predict that N might have false minima not present in M, and we have found a case in which this was so.'
3. E.Prince, Mathematical Techniques in Crystallography and Material Science, (1994), Springer-Verlag, pp 125
   
   • 'Therefore the results (of refinement) will be substantially identical whether Fsq or F is used as the data value, and which is used is largely a matter of local fashion.'
4. A.J.Wilson, (1976), Acta Cryst A32, 994-996.
   
   • 'Refinement in R1 (against Fo) can be regarded as a special case of refinement in R2 with weights
     w2 = w1 (sqrt(Io)+sqrt(Ic))**-2
     where the w1's are the weights that would have been used in R1.'
     (put otherwise, w1 = w2 (sqrt(Io)+sqrt(Ic))**2)
5. F.L.Hirshfeld & D.Rabinovich, (1973), Acta Cryst A29, 510-513.
   
   • (referring to neglect of negative observations) 'our limited experience indicates that in real situations the effect of biased data (distributions) on the structurally interesting parameters is rarely large enough to matter'
6. J.D.Dunitz, X-ray Analysis and the Structure of Organic Molecules, (1979), Cornell University Press, pp 208-209.
   
   • 'The best way to decide whether the structure is really or only nearly centrosymmetric is to scrutinize the few reflections that are most sensitive.... The sensitive reflections are the ones, generally weak, for which the calculated value of A (real part of the structure factor) is close to zero'
7. E.Prince & W.L.Nicholson, in Structure & Statistics in Crystallography, (1984), ed A.J.C.Wilson, Adenine Press, pp 191
   
   • 'Actually, omitting a weak reflection, or any reflection, cannot bias the parameter estimates, as may be shown ....'
8. K.Kirschbaum, A.Martin & A.A.Pinkerton, �/2 Contamination in CCD area- detector data, (1997), J.Appl.Cryst. 30, pp 515-516.
   
   • 'Some reduction in the number of 'observed' systematic absences is noted, .... The improvement in the data is shown to be insignificant for routine data collection'