Section 1.3.4.4.4. Structure factors from model atomic parameters

An atomic model of a crystal structure consists of a list of symmetry-unique atoms described by their positions, their thermal agitation and their chemical identity (which can be used as a pointer to form-factor tables). Form factors are usually parameterized as sums of Gaussians, and thermal agitation by a Gaussian temperature factor or tensor. The formulae given in Section 1.3.4.2.2.6 for Gaussian atoms are therefore adequate for most purposes. High-resolution electron-density studies use more involved parameterizations.

Early calculations were carried out by means of Bragg–Lipson charts (Bragg & Lipson, 1936) which gave a graphical representation of the symmetrized trigonometric sums Ξ of Section 1.3.4.2.2.9. The approximation of form factors by Gaussians goes back to the work of Vand et al. (1957) and Forsyth & Wells (1959). Agarwal (1978) gave simplified expansions suitable for medium-resolution modelling of macromolecular structures.

This method of calculating structure factors is expensive because each atom sends contributions of essentially equal magnitude to all structure factors in a resolution shell. The calculation is therefore of size for N atoms and reflections. Since N and are roughly proportional at a given resolution, this method is very costly for large structures.

Two distinct programming strategies are available (Rollett, 1965) according to whether the fast loop is on all atoms for each reflection, or on all reflections for each atom. The former method was favoured in the early times when computers were unreliable. The latter was shown by Burnett & Nordman (1974) to be more amenable to efficient programming, as no multiplication is required in calculating the arguments of the sine/cosine terms: these can be accumulated by integer addition, and used as subscripts in referencing a trigonometric function table.