Reference formulae for particular values of n

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. 390-391 | 1 | 2 |

Section 3.4.11. Reference formulae for particular values of n

D. E. Williams^a ^‡

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

3.4.11. Reference formulae for particular values of n

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In this section let $[a^{2} = \pi w^{2} |{\bf R}_{k} + {\bf X(d)} - {\bf R}_{j}|^{2}]$ and $[b^{2} = \pi w^{-2} |{\bf H(h)}|^{2}]$ . Let $[T_{0} = \sum Q_{jj} = \sum q_{j}^{2}]$ ; $[T_{1} = \sum \sum Q_{jk} =]$ $[T_{0} + 2 \sum \sum_{j\;\gt\; k} Q_{jk}]$ . If the geometric mean combining law holds, $[T_{1} = ({\textstyle\sum\limits_{j}} q_{j})^{2}]$ ; let $[\eqalign{ T_{2} (h) &= {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}\ Q_{jk} \exp [2 \pi i {\bf H(h)} \cdot ({\bf R}_{k} - {\bf R}_{j})]\cr &= T_{0} + 2 {\lower3pt\hbox{${\sum\sum\atop \scriptstyle j\;\gt\; k}$}}\ Q_{jk} \cos [2 \pi {\bf H(h)} \cdot ({\bf R}_{k} - {\bf R}_{j})].}]$ Then $[T_{2} ({\bf h}) = |F {\bf (h)}|^{2} = |{\textstyle\sum\limits_{j}}\ q_{j} \exp [2 \pi i{\bf H(h)} \cdot {\bf R}_{j}]|^{2} = A {\bf (h)}^{2} + B {\bf (h)}^{2},]$ where $[A {\bf (h)} = {\textstyle\sum\limits_{j}}\ q_{j} \cos [2 \pi {\bf H(h)} \cdot {\bf R}_{j}]]$ and $[B {\bf (h)} = {\textstyle\sum\limits_{j}}\ q_{j} \sin [2 \pi {\bf H(h)} \cdot {\bf R}_{j}].]$ The formulae below describe $[V (n, {\bf R}_{j})]$ in terms of $[T_{0},\ T_{1}]$ and $[T_{2}]$ ; the distance $[|{\bf R}_{k} + {\bf X (d)} - {\bf R}_{j}|]$ is simply represented by $[R_{jkd}]$ . $[\eqalign{ V (1, {\bf R}_{j}) &= (1/2) {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}'\ q_{j}\ q_{k} {\textstyle\sum\limits_{\bf d}} {R}_{jkd}^{-1} \ \hbox{erfc} (a)\cr &\quad + (1/2 \pi V_{d}) {\textstyle\sum\limits_{{\bf h} \neq 0}} T_{2} {\bf (h)} |{\bf H(h)}|^{-2} \exp (-b^{2}) - w T_{0}\cr V (2, {\bf R}_{j}) &= (1/2) {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}'\ Q_{jk} {\textstyle\sum\limits_{\bf d}} {R}_{jkd}^{-2} \exp (-a^{2})\cr &\quad + (\pi / 2 V_{d}) {\textstyle\sum\limits_{{\bf h} \neq 0}} T_{2} {\bf (h)}| {\bf H(h)}|^{-1} \hbox{erfc} (b) - (\pi / 2) w^{2} T_{0}\cr V (3, {\bf R}_{j}) &= (1/2) {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}'\ Q_{jk} {\textstyle\sum\limits_{\bf d}} {R}_{jkd}^{-3} [\hbox{erfc} (a) + 2 a \pi^{-1/2} \exp (-a^{2})]\cr &\quad + (\pi / V_{d}) {\textstyle\sum\limits_{{\bf h} \neq 0}} T_{2} {\bf (h)} E_{1} (b^{2}) - (2 \pi/3) w^{3} T_{0}\cr\noalign{\vskip6pt} V (4, {\bf R}_{j}) &= (1/2) {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}'\ Q_{jk} {\textstyle\sum\limits_{\bf d}} {R}_{jkd}^{-4} (1 + a^{2}) \exp (-a^{2})\cr &\quad + (\pi^{5/2} / V_{d}) {\textstyle\sum\limits_{{\bf h} \neq 0}} T_{2} {\bf (h)} |{\bf H(h)}|\cr &\quad \times [-\pi^{1/2} \hbox{erfc} (b) + b^{-1} \exp (-b^{2})]\cr &\quad + (\pi^{2} / V_{d}) w T_{1} - (\pi^{2}/4) w^{4} T_{0}\cr\noalign{\vskip6pt} {\bi V}(5, {\bf R}_{j}) &= (1/2) {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}'\ Q_{jk} {\textstyle\sum\limits_{\bf d}} R_{jkd}^{-5}\cr &\quad \times [\hbox{erfc} (a) + 2\pi^{-1/2} a(1 + 2a^{2}/3)\exp (-a^{2})]\cr &\quad + (2\pi^{3}/3 V_{d}) {\textstyle\sum\limits_{{\bf h} \neq 0}} T_{2}({\bf h})|{\bf H(h)}|^{2}\cr &\quad \times [b^{-2}\exp (-b^{2}) - E_{1}(b^{2})]\cr &\quad + (2\pi^{2}/3 V_{d}) w^{2}T_{1} - (4\pi^{2}/15) w^{5}T_{0}\cr\noalign{\vskip6pt} {\bi V}(6, {\bf R}_{j}) &= (1/2) {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}'\ Q_{jk} {\textstyle\sum\limits_{\bf d}} R_{jkd}^{-6} [1 + a^{2} + (a^{4}/2)]\exp (-a^{2})\cr &\quad + (\pi^{9/2}/3 V_{d}) {\textstyle\sum\limits_{{\bf h} \neq 0}} T_{2}({\bf h})|{\bf H(h)}|^{3}\cr &\quad \times [\pi^{1/2}\hbox{erfc}(b) + [(1/2b^{3}) - (1/b)]\exp (-b^{2})]\cr &\quad + (\pi^{3}/6 V_{d}) w^{3}T_{1} - (\pi^{3}/12) w^{6}T_{0}\cr}]$ $[\eqalign{ {\bi V}(7, {\bf R}_{j}) &= (1/2) {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}'\ Q_{jk} {\textstyle\sum\limits_{\bf d}} R_{jkd}^{-7}\cr &\quad \times [\hbox{erfc}(a) + 2\pi^{-1/2} a[1 + (2/3)a^{2} + (4/15)a^{4}]\cr &\quad \times \exp (-a^{2})]\cr &\quad + (2\pi^{5}/15 V_{d}) {\textstyle\sum\limits_{{\bf h} \neq 0}} T_{2}({\bf h})|{\bf H(h)}|^{4}\cr &\quad \times [(-b^{-2} + b^{-4})\exp (-b^{2}) + E_{1}(b^{2})]\cr &\quad + (2\pi^{3}/15 V_{d}) w^{4}T_{1} - (8\pi^{3}/105) w^{7}T_{0}\cr\noalign{\vskip6pt} {\bi V}(8, {\bf R}_{j}) &= (1/2) {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}'\ Q_{jk} {\textstyle\sum\limits_{\bf d}} R_{jkd}^{-8}\cr &\quad \times [1 + a^{2} + (a^{4}/2) + (a^{6}/6)]\exp (-a^{2})\cr &\quad + (2\pi^{13/2}/45 V_{d}) {\textstyle\sum\limits_{{\bf h} \neq 0}} T_{2}({\bf h})|{\bf H(h)}|^{5}\cr &\quad \times [-\pi^{1/2}\hbox{erfc}(b) + [(1/b) - (1/2b^{3}) + (3/4b^{5})]\cr &\quad \times \exp (-b^{2})]\cr &\quad + (\pi^{4}/30 V_{d}) w^{5}T_{1} - (\pi^{4}/48) w^{8}T_{0}\cr\noalign{\vskip6pt}{\bi V}(9, {\bf R}_{j}) &= (1/2) {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}'\ Q_{jk} {\textstyle\sum\limits_{\bf d}} R_{jkd}^{-9}\cr &\quad \times [\hbox{erfc}(a) + 2\pi^{-1/2} a[1 + (2/3)a^{2} + (4/15)a^{4}\cr &\quad + (8/105)a^{6}]\exp (-a^{2})]\cr &\quad + (4\pi^{7}/315 V_{d}) {\textstyle\sum\limits_{{\bf h} \neq 0}} T_{2}({\bf h})|{\bf H(h)}|^{6}\cr &\quad \times [(b^{-2} - b^{-4} + 2b^{-6})\exp (-b^{2}) - E_{1}(b^{2})]\cr &\quad + (8\pi^{4}/315 V_{d}) w^{6}T_{1} - (16\pi^{4}/945) w^{9}T_{0}\cr {\bi V}(10, {\bf R}_{j}) &= (1/2) {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}'\ Q_{jk} {\textstyle\sum\limits_{\bf d}} R_{jkd}^{-10}\cr &\quad \times [1 + a^{2} + (a^{4}/2) + (a^{6}/6) + (a^{8}/24)]\exp (-a^{2})\cr &\quad + (\pi^{17/2}/315 V_{d}) {\textstyle\sum\limits_{{\bf h} \neq 0}} T_{2}({\bf h})|{\bf H(h)}|^{7}\cr &\quad \times [\pi^{1/2}\hbox{erfc}(b) + [(-1/b) + (1/2b^{3}) - (3/4b^{5})\cr &\quad + (15/8b^{7})]\exp (-b^{2})]\cr &\quad + (\pi^{5}/168 V_{d}) w^{7}T_{1} - (\pi^{5}/240) w^{10}T_{0}.}]$