Patterson function

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The Patterson Function, P(u,v,w) is defined as

[P(u,v,w) = \sum\limits_ \left|F_\right|^2 \;e^]

which is essentially the Fourier transform of the intensities rather than the structure factors. But using some quick relations you find that the Patterson function is also equivalent to the electron density convoluted with its inverse:

[P(\vec) = \rho(\vec) * \rho(-\vec)]

Further a Patterson map of N points will have N(N-1) peaks NOT including the central peak and any overlap.

The peaks in the Patterson function are the interatomic distances weighted by the product of the number of electrons in the atoms concerned.

1D Example

Consider the series of delta functions

[f(x) = \delta(x) + 3 \delta(x-2) + \delta(x-5) + 3 \delta(x-8) + 5 \delta(x-10)]

then the Patterson is:

[P(u) = 5 \delta(u+10) + 18 \delta(u+8) + 9 \delta(u+6) + 6 \delta(u+5) + 6 \delta(u+3) + 18 \delta(u+2) + 45 \delta(u) + 18 \delta(u-2) + 6 \delta(u-3) + 6 \delta(u-5) + 9 \delta(u-6) + 18 \delta(u-8) + 5 \delta(u-10)]

Note: the Patterson always has central symmetry.

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