A modification to the tangent formula of Karle and Hauptman was made by
Germain *et al* [37], the main difference being the inclusion of a weighting
scheme into the formula. The most likely value, , of the phase , in
this case, is given by

where and are the weights associated with and . The corresponding weight which is assigned to may be calculated using

The corresponding value of given by Eq. is modified to

This weighted Tangent formula has been implemented in the computer program package MULTAN80 [81]. The program takes a multi-solution approach to solving the phase problem, i.e. having identified a `good' triplet phase relationship (the starting set) at which to start calculating phases using other triplet invariants, it generates a number of different solutions starting with different permutations of the values of phases assigned in the starting set. All solutions are then tested by means of three figures of merit (FOM's) which hopefully indicate the correct solution(s).

The program function may be summarised by a series of logical steps, the major points of which will be described in more detail afterwards.

- Input structure factors are normalised to give . The strongest reflections are output along with of the weakest for use in calculation of the FOM's.
- The program searches for a maximum of 8000 triplet phase invariants (, and ). All phase relationships with in Eq. less than 0.6 are rejected.
- The best starting set of reflections is found using the convergence method of
Germain
*et al*[36]. Origin and enantiomorph (handedness) defining reflections are also located at this stage. - Multiple starting sets of phases are then produced using the magic integer approach [116].
- Phases are expanded from each of the starting sets using the weighted Tangent formula and FOM's are calculated.
- The Fourier transforms (E-maps) of and their associated are calculated for the best solutions as determined by the FOM's.
- A peak search program [68] locates the highest peaks in the E-map and outputs their coordinates for chemical interpretation.

Fri Oct 7 15:42:16 MET 1994