The Patterson function 
[92] is defined as the autocorrelation function of the
electron density distribution or
where 
and 
are respectively position vectors in Patterson space and real space and 
denotes
convolution.
By substituting in for 
and 
using Eq. 
we find that
The resulting map calculated using this equation is essentially a map of all the cross vectors between each
atom pair in the unit cell. For a structure which contains 
atoms in the unit cell or equivalently has peaks
in its electron density distribution, the Patterson function will contain 
peaks.
of the peaks represent self vectors which are superimposed at the origin, 
, while 
peaks
represent cross vectors and appear elsewhere in the unit cell. Deconvolution of such Patterson maps is a
relatively simple procedure for small but becomes complicated as increases.
Location of the heavy atoms may be achieved by use of suitable coefficients 
in Eq.
which correspond to the Fourier transform of the heavy atom electron density. Ideally one would use the
set of 
's but these are not measurable quantities.
The alternative procedure is to calculate using coefficients which are largely
dependent on the heavy atom structure. If the heavy atoms display significant anomalous scattering
the coefficients 
may be used [97].
The values of 
are measurable and, as can be seen
from Eq. , they are strongly correlated with the imaginary part of the heavy atom structure
factor 
. Patterson maps calculated using coefficients are called
anomalous difference Patterson maps.
If native protein data are available the coefficients 
may also be used
to calculate isomorphous difference Patterson maps; this is in fact the standard procedure in isomorphous
replacement.
