The Ewald equations presented earlier can be simplified for efficient convergence [43]. One straightforward simplification is to transform the double loop in the reciprocal sum of Eq.(4) into a single sum which is rewritten as follows:
Expanding the term in 
and simplifying yields:
The above equation can be further simplified using a trigonometry identity:
and finally by defining
known by protein crystallographers as the ``structure factor'' [33], we obtain the following:
Therefore, the above manipulation has transformed the double sum over i and j of order 
,
Eq.(11), into two single sums of order N, Eq.(14), and finally into a single sum, Eq.(16), that is much more efficient.
