Derivation of the accelerated convergence formula via the Patterson function

Williams, D. E.

doi:10.1107/97809553602060000562

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

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International Tables for Crystallography (2006). Vol. B. ch. 3.4, pp. 389-390 | 1 | 2 |

Section 3.4.8. Derivation of the accelerated convergence formula via the Patterson function

D. E. Williams^a ^‡

^aDepartment of Chemistry, University of Louisville, Louisville, Kentucky 40292, USA

3.4.8. Derivation of the accelerated convergence formula via the Patterson function

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The structure factor with generalized coefficients $[q_{j}]$ is defined by $[F[{\bf H(h)}] = {\textstyle\sum\limits_{j}}\ q_{j} \exp [2\pi i {\bf H(h)} \cdot {\bf R}_{j}].]$ The corresponding Patterson function is defined by $[P({\bf X}) = V_{d}^{-1} {\textstyle\sum\limits_{\bf h}} |F[{\bf H(h)}]|^{2} \exp [2\pi i {\bf H(h)}\cdot {\bf X}].]$ The physical interpretation of the Patterson function is that it is nonzero only at the intersite vector points $[{\bf R}_{k} + {\bf X(h)} - {\bf R}_{j}]$ . If the origin point is removed, the lattice sum may be expressed as an integral over the Patterson function. This origin point in the Patterson function corresponds to intersite vectors with [j = k] and $[{\bf H(h)} = 0]$ : $[S_{n} = (1/2 V_{d}) \textstyle\int |{\bf X}|^{-n} [P({\bf X}) - P({\bf X}) \delta ({\bf X})] \hbox{ d}{\bf X}.]$ Using the incomplete gamma function as a convergence function, this formula expands into two integrals $[\eqalign{ S_{n} &= [1/2 V_{d} \Gamma (n/2)] \textstyle\int |{\bf X}|^{-n}\cr &\quad \times [P ({\bf X}) - P({\bf X}) \delta ({\bf X})] \Gamma (n/2, \pi w^{2} |{\bf X}|^{2})\; \hbox{d}{\bf X}\cr &\quad + [1/2 V_{d} \Gamma (n/2)] \textstyle\int |{\bf X}|^{-n}\cr &\quad \times [P({\bf X}) - P({\bf X}) \delta ({\bf X})] \gamma (n/2, \pi w^{2} |{\bf X}|^{2})\; \hbox{d}{\bf X}.}]$ The first integral is shown only for a consistent representation; actually it will be reconverted to a sum and evaluated in direct space. The first part of the second integral will be evaluated with Parseval's theorem and the second part in the limit as $[|{\bf X}|]$ approaches zero: $[\eqalign{ &[1/2 V_{d} \Gamma (n/2)] \textstyle\int FT_{3} [P({\bf X})]\cr &\quad \times FT_{3} [|{\bf X}|^{-n} \gamma (n/2, \pi w^{2} |{\bf X}|^{2})] \hbox{ d}{\bf H}\cr &\quad -\lim\limits_{{\bf X} \rightarrow 0} [1/2 V_{d} \Gamma (n/2)] [P(0) |{\bf X}|^{-n} \gamma (n/2, \pi w^{2} |{\bf X}|^{2})].}]$ The first Fourier transform (of the Patterson function) is the set of amplitudes of the structure factors and the second Fourier transform has already been discussed above; the method for obtaining the limit (for n equal to or greater than 1) was also discussed above. The result obtained is $[\eqalign{ &[1/2 V_{d} \Gamma (n/2)] \pi^{n-(3/2)} \textstyle\int |F [{\bf H (h)}]|^{2} |{\bf H}|^{n-3}\cr &\quad \times \Gamma [(-n/2) + (3/2), \pi w^{-2} |{\bf H}|^{2}] \hbox{ d}{\bf H}\cr &\quad -[1/2 V_{d} \Gamma (n/2)] |F(0)|^{2} 2 \pi^{n/2}w^{n} n^{-1}.}]$ The integral can be converted into a sum, since $[|F[{\bf H(h)}]|]$ is nonzero only at the reciprocal-lattice points: $[\eqalign{ &[1/2 V_{d} \Gamma (n/2)] \pi^{n-(3/2)} {\textstyle\sum\limits_{\bf h}} |F[{\bf H(h)}]|^{2} |{\bf H(h)}|^{n-3}\cr &\quad \times \Gamma [(-n/2) + (3/2), \pi w^{-2} |{\bf H(h)}|^{2}].}]$ The term with $[{\bf H(h)} = 0]$ is evaluated in the limit, for n greater than 3, as $[[\Gamma (n/2)]^{-1} V_{d}^{-1} \pi^{n/2} w^{n-3} (n-3)^{-1} |F(0)|^{2}.]$ Since $[|F(0)|^{2} = \sum\sum q_{j} q_{k}]$ , this term is identical with the third term of $[V(n, {\bf R}_{j})]$ as derived earlier. The case of [n = 1] is handled in the same way as previously discussed, where the limit of this term is zero provided the unit cell has no net charge or dipole moment.