International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 75-76   | 1 | 2 |

## Section 1.3.4.2.2.10. Correlation and Patterson functions

G. Bricognea

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#### 1.3.4.2.2.10. Correlation and Patterson functions

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Consider two model electron densities and with the same period lattice and the same space group G. Write their motifs in terms of atomic electron densities (Section 1.3.4.2.2.4) aswhere and label the symmetry-unique atoms placed at positions and , respectively.

To calculate the correlation between and we need the following preliminary formulae, which are easily established: if and f is an arbitrary function on , thenhenceand

The cross correlation between motifs is thereforewhich contains a peak of shape at the interatomic vector for each , , , .

The cross-correlation between the original electron densities is then obtained by further periodizing by .

Note that these expressions are valid for any choice of `atomic' density functions and , which may be taken as molecular fragments if desired (see Section 1.3.4.4.8).

If G contains elements g such that has an eigenspace with eigenvalue 1 and an invariant complementary subspace , while has a nonzero component in , then the Patterson function will contain Harker peaks (Harker, 1936) of the form[where represent the action of g in ] in the translate of by .

### References

Harker, D. (1936). The application of the three-dimensional Patterson method and the crystal structures of proustite, Ag3AsS3, and pyrargyrite, Ag3SbS3. J. Chem. Phys. 4, 381–390.