Errors in the calculated structure factors caused by the free-atom form factor model

The excellent agreement often obtained in least-squares refinement of X-ray data with the free-atom form factor model suggests that deviations from sphericity are small. However, the resulting parameters are affected by bonding effects to a small but significant extent. In the present paper, errors in Fcalc are defined as differences between the free-atom structure factors calculated with the true atomic parameters and those which would be obtained from an exact description of the electron density. An experimental measure of these errors can be obtained from the parameters determined independently by neutron diffraction. The structure factor errors for s-triazine and α-deutero-oxalic acid dideuterate are analysed and contrasted with errors in form factors resulting from neglect of overlap density, prepared state and orbital contraction. The theoretical curves indicate that deviations from the free-atom model persist at values of sin θ/λ larger than 0.6 A−1. Improved models for molecular scattering based on valence bond and molecular orbital theories respectively, are discussed. The importance of errors in neutron scattering lengths is demonstrated by least-squares refinements of α-deutero-oxalic acid with various values of bd. Good agreement between Fobs and Fcalc is obtained at the expense of systematic deviations in the deuteron temperature factors. Finally a summary is given of the errors in X-ray parameters obtained by least-squares adjustment with the free-atom model.