The function that we have chosen to minimize is
where 
and 
are the observed and calculated structure factors and
k is a scale factor. Note that no exponential factor is applied to the
's. This will force the thermal factors of the individual atoms to include
any `overall' mismatch between the observed and calculated data sets.
At the beginning of each cycle of refinement the scale factor k is determined by minimizing
where 
and 
is treated as a constant.
Equation (7) includes an overall thermal factor B, which is
necessary to allow for an initial overall discrepancy between the 's
and 's. Although both k and B are treated as variables in the
minimization of (7), only k is substituted in (6).
As the refinement proceeds, the discrepancy represented by B is
absorbed within the thermal factors of the individual atoms and, during
successive cycles, rapidly approaches zero.
Each module of the refinement package is able to calculate both the value and
the gradient of its term. For the crystallographic term the structure factors
are calculated by a space-group-specific FFT (Ten Eyck, 1977). The
gradients are calculated by a modified version of the procedure outlined by
Agarwal (1978). This modified version was devised by A. Lifchitz
(Agarwal, Lifchitz & Dodson, 1981; Isaacs, 1982) and is described in detail in
Appendix A. In outline, the procedure is as follows. An ( 
)
map is calculated for the molecular volume. For each parameter in the model
a convolution, evaluated at the atomic position, is calculated between this map
and the derivative of the calculated atomic electron density function for the
atom involved. Because the extent of the electron cloud of a single atom is
small, the calculation of this convolution is rapid. Usually the calculation
of the convolutions takes about a quarter of the time required to calculate the
difference map.
