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

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.4.2.2.10. Correlation and Patterson functions

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Consider two model electron densities [\rho\llap{$-\!$}_{1}] and [\rho\llap{$-\!$}_{2}] with the same period lattice [{\bb Z}^{3}] and the same space group G. Write their motifs in terms of atomic electron densities (Section 1.3.4.2.2.4[link]) as[\eqalign{ \rho\llap{$-\!$}_{1}^{0} &= {\textstyle\sum\limits_{j_{1} \in J_{1}}} \left({\textstyle\sum\limits_{\scriptstyle_{\gamma_{j_{1}} \in G/G_{{\bf x}_{j_{1}}}^{(1)}}}} S_{\gamma_{j_{1}}}^{\#} (\tau_{{\bf x}_{j_{1}}^{(1)}} \rho\llap{$-\!$}_{j_{1}}^{(1)})\right),\cr \rho\llap{$-\!$}_{2}^{0} &= {\textstyle\sum\limits_{j_{2} \in J_{2}}} \left({\textstyle\sum\limits_{\scriptstyle{\gamma_{j_{2}} \in G/G_{{\bf x}_{j_{2}}}^{(2)}}} S_{\gamma_{j_{2}}}^{\#} (\tau_{{\bf x}_{j_{2}}^{(2)}} \rho\llap{$-\!$}_{j_{2}}^{(2)})}\right),}]where [J_{1}] and [J_{2}] label the symmetry-unique atoms placed at positions [\{{\bf x}_{j_{1}}^{(1)}\}_{j_{1} \in J_{1}}] and [\{{\bf x}_{j_{2}}^{(2)}\}_{j_{2} \in J_{2}}], respectively.

To calculate the correlation between [\rho\llap{$-\!$}_{1}] and [\rho\llap{$-\!$}_{2}] we need the following preliminary formulae, which are easily established: if [S({\bf x}) = {\bf Rx} + {\bf t}] and f is an arbitrary function on [{\bb R}^{3}], then[(R^{\#} f)\breve{} = R^{\#} \breve{f}, \quad (\tau_{{\bf x}}\, f)\breve{} = \tau_{-{\bf x}} \,\breve{f}, \quad R^{\#} (\tau_{{\bf x}}\, f) = \tau_{{\bf Rx}}\, f,]hence[S^{\#} (\tau_{{\bf x}} \,f) = \tau_{S({\bf x})} R^{\#} f \quad \hbox{and} \quad [S^{\#} (\tau_{{\bf x}} \,f)]\,\breve{} = \tau_{-S({\bf x})} R^{\#} \breve{f}\hbox{\semi}]and[S_{1}^{\#} f_{1} * S_{2}^{\#} f_{2} = S_{1}^{\#} [\,f_{1} * (S_{1}^{-1} S_{2})^{\#} f_{2}] = S_{2}^{\#} [(S_{2}^{-1} S_{1})^{\#} f_{1} * f_{2}].]

The cross correlation [\breve{\rho\llap{$-\!$}}_{1}^{0} * \rho\llap{$-\!$}_{2}^{0}] between motifs is therefore[\eqalign{ \breve{\rho\llap{$-\!$}}_{1}^{0} * \rho\llap{$-\!$}_{2}^{0} &= {\textstyle\sum\limits_{j_{1}}} {\textstyle\sum\limits_{j_{2}}} {\textstyle\sum\limits_{\gamma_{j_{1}}}} {\textstyle\sum\limits_{\gamma_{j_{2}}}} [S_{\gamma_{j_{1}}}^{\#} (\tau_{{\bf x}_{j_{1}}^{(1)}} \rho\llap{$-\!$}_{j_{1}}^{(1)})]\,\breve{} * [S_{\gamma_{j_{2}}}^{\#} (\tau_{{\bf x}_{j_{2}}^{(2)}} \rho\llap{$-\!$}_{j_{2}}^{(2)})]\cr &= {\textstyle\sum\limits_{j_{1}}} {\textstyle\sum\limits_{j_{2}}} {\textstyle\sum\limits_{\gamma_{j_{1}}}} {\textstyle\sum\limits_{\gamma_{j_{2}}}} \tau_{S_{\gamma_{j_{2}}}_{({\bf x}_{j_{2}}^{(2)}) - S_{\gamma_{j_{1}}} ({\bf x}_{j_{1}}^{(1)})}} [(R_{\gamma_{j_{1}}}^{\#} \breve{\rho\llap{$-\!$}}_{j_{1}}^{(1)}) * (R_{\gamma_{j_{2}}}^{\#} \rho\llap{$-\!$}_{j_{2}}^{(2)})]}]which contains a peak of shape [(R_{\gamma_{j_{1}}}^{\#} \breve{\rho\llap{$-\!$}}_{j_{1}}^{(1)}) * (R_{\gamma_{j_{2}}}^{\#} \rho\llap{$-\!$}_{j_{2}}^{(2)})] at the interatomic vector [S_{\gamma_{j_{2}}} ({\bf x}_{j_{2}}^{(2)}) - S_{\gamma_{j_{1}}} ({\bf x}_{j_{1}}^{(1)})] for each [j_{1} \in J_{1}], [j_{2} \in J_{2}], [\gamma_{j_{1}} \in G/G_{{\bf x}_{j_{1}}^{(1)}}], [\gamma_{j_{2}} \in G/G_{{\bf x}_{j_{2}}^{(2)}}].

The cross-correlation [r * \breve{\rho\llap{$-\!$}}_{1}^{0} * \rho\llap{$-\!$}_{2}^{0}] between the original electron densities is then obtained by further periodizing by [{\bb Z}^{3}].

Note that these expressions are valid for any choice of `atomic' density functions [\rho\llap{$-\!$}_{j_{1}}^{(1)}] and [\rho\llap{$-\!$}_{j_{2}}^{(2)}], which may be taken as molecular fragments if desired (see Section 1.3.4.4.8[link]).

If G contains elements g such that [{\bf R}_{g}] has an eigenspace [E_{1}] with eigenvalue 1 and an invariant complementary subspace [E_{2}], while [{\bf t}_{g}] has a nonzero component [{\bf t}_{g}^{(1)}] in [E_{1}], then the Patterson function [r * \breve{\rho\llap{$-\!$}}^{0} * \rho\llap{$-\!$}^{0}] will contain Harker peaks (Harker, 1936[link]) of the form[S_{g} ({\bf x}) - {\bf x} = {\bf t}_{g}^{(1)} \oplus (S_{g}^{(2)} ({\bf x}) - {\bf x})][where [S_{g}^{(s)}] represent the action of g in [E_{2}]] in the translate of [E_{1}] by [{\bf t}_{g}^{(1)}].

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.








































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