Errors in coordinates, bond lengths, and bond angles

Standard deviations in bond lengths and bond angles are related to standard deviations in atomic coordinates according to published equations [Cruickshank (1959). International Tables for X-ray Crystallography. Vol. II, pp. 331- 332]. These equations were derived for an idealized model in which the distribution of coordinate errors is isotropic. Tests show that typical structures exhibit only moderate deviations from this model, and so the calculated standard deviations are accurate. Furthermore, the standard deviation in a bond angle φ (in degrees) can be well approximated by the expression σ(φ) ~ 81 [σ(R)/R]_r.m.s., where the quantity in square brackets is the root-mean-square value of σ(R)/R for the two bonds forming the angle. Tests on typical published results show that this equation usually estimates σ(φ) to within the round-off error in the reported value.