Once the space group has been determined (and the reflection indices have been transformed, if necessary, to be consistent with a/the standard setting for the unit cell axes), the Patterson map can be calculated and its peaks used to determine the position of the "heavy" atom by: correctly matching the observed peaks with the expected peaks; and, then calculating the X,Y,Z coordinates of the "heavy" atom from the U,V,W coordinates of the Patterson map peaks.

Example:

1. molecule (formula unit) contains one "heavy" atom
2. crystal system is monoclinic (b-axis unique)
3. reflection conditions indicate space group #14 in the second setting (P2₁/n)
4. unit cell contains four molecules (Z = 4)

From a. and d.: there will be four "heavy" atoms in the unit cell. As the second setting of space group #14 is accepted as a standard setting, all work will be done using the expected Patterson map peak positions and multiplicities for space group P2₁/n.

1. The Patterson map peaks (coordinates, multiplicities) are:

                      Observed:     Expected for P21/n (#14):
  1.    0.00   0.00   0.00   7.                  0, 0, 0   4.
  2.    0.30   0.50   0.20   2.      1/2+2X, 1/2, 1/2+2Z   2.
  3.    0.50   0.10   0.50   2.         1/2, 1/2+2Y, 1/2   2.
  4.    0.20   0.40   0.30   1.               2X, 2Y, 2Z   1.
         etc.

The above matching of the observed peaks with the expected peaks is taken as correct. It is easiest to use the Harker peaks to calculate the X,Y,Z coordinates of the "heavy" atom and then to verify the coordinates by using a general peak.
Using peak 3) as the 1/2, 1/2+2Y, 1/2 expected peak:

          1/2+2Y = 0.10,  2Y = -0.40,  and  Y = -0.20

Using peak 2) as the 1/2+2X, 1/2, 1/2+2Z expected peak:

          1/2+2X = 0.30,  2X = -0.20,  and  X = -0.10
          1/2+2Z = 0.20,  2Z = -0.30,  and  Z = -0.15

The X,Y,Z coordinates determined from peaks 2) and 3) are -0.10, -0.20, -0.15. If true, the expected 2X, 2Y, 2Z peak should be observed at -0.20, -0.40, -0.30 but peak 4) is observed at 0.20, 0.40, 0.30.

As the Patterson map for space group P2₁/n has the symmetry of space group P2/m, for any peak at U,V,W, equivalent peaks will be present at U,-V,W, -U,V,-W, and at -U,-V,-W. As the observed Patterson map peaks are listed by the Fourier computer program only within the asymmetric unit of the unit cell (for P2/m: x: 0 - 1/2, y: 0 - 1/2, z: 0 - 1), the three peaks equivalent to 0.20, 0.40, 0.30 (0.20, -0.40, 0.30; -0.20, 0.40, -0.30; and -0.20, -0.40, -0.30) do exist, but lie outside the asymmetric unit and are not reported. It is seen that an equivalent to peak 4) does match the expected 2X, 2Y, 2Z peak and, thus, the "heavy" atom coordinates are verified.

 
2. The Patterson map peaks (coordinates and multiplicities) are:

                      Observed:     Expected for P21/n (#14):
  1.    0.00   0.00   0.00   7.                  0, 0, 0   4.
  2.    0.50   0.50   0.50   4.      1/2+2X, 1/2, 1/2+2Z   2.
  3.    0.50   0.00   0.50   4.         1/2, 1/2+2Y, 1/2   2.
  4.    0.00   0.50   0.00   4.               2X, 2Y, 2Z   1.
         etc.

The above matching of the observed peaks with the expected peaks is obviously not correct. Now, one must think of x = 0,1/2, y = 0,1/2, z = 0,1/2 and of special positions for the "heavy" atoms. The expected Patterson map peaks for atoms lying in any of the four special positions in P2₁/n are:

                                                 0, 0, 0   2.
                                           1/2, 1/2, 1/2   2.

This explains the observed peaks 1) and 2). As the special positions in space group P2₁/n have multiplicity = 2, the four "heavy" atoms in the unit cell must lie two each in two special positions. Assume two of the "heavy" atoms to be in position "a" (one can be rather arbitrary in choosing the position of the first atom) and note that peaks 3) and 4) are identical to the vectors between position "a" and position "c". If one calculates the vectors between the atoms in position "a" and the atoms in position "c", the 16 vectors superimpose to give the observed peaks 1), 2), 3) and 4), each with multiplicity = 4. This set of observed Patterson map peaks does yield the "heavy" atom coordinates once it is realized that the atoms lie in special positions.