Extension of the method to a composite lattice

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. 388-389 | 1 | 2 |

Section 3.4.5. Extension of the method to a composite lattice

D. E. Williams^a ^‡

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

3.4.5. Extension of the method to a composite lattice

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Define a general lattice sum over direct-space points $[{\bf R}_{j}]$ which interact with pairwise coefficients $[Q_{jk}]$ , where $[Q_{jk} = Q_{kj}]$ : $[V (n, {\bf R}_{j}) = (1/2) {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}'\ Q_{jk} {\textstyle\sum\limits_{\bf d}} | {\bf R}_{k} + {\bf X(d)} - {\bf R}_{j}|^{-n},]$ where the prime indicates that when $[{\bf d} = 0]$ the self-terms with [j = k] are omitted. For convenience the terms may be divided into three groups: the first group of terms has $[{\bf d} = 0]$ , where j is unequal to k; the second group has d not zero and j not equal to k; and the third group had d not zero and [j = k] . (A possible fourth group with $[{\bf d} = 0]$ and [j = k] is omitted, as defined.) $[\eqalign{ V (n, {\bf R}_{j}) &= (1/2) {\textstyle\sum\limits_{j \neq k}}\ Q_{jk} |{\bf R}_{k} - {\bf R}_{j}|^{-n}\cr &\quad + (1/2) {\textstyle\sum\limits_{j \neq k}}\ Q_{jk} S' (n, |{\bf R}_{j} - {\bf R}_{k}|) + (1/2) {\textstyle\sum\limits_{j}}\ Q_{jj} S' (n, 0).}]$ By expanding this expression we obtain $[\eqalignno{ V (n, {\bf R}_{j}) &= (1/2) {\textstyle\sum\limits_{j \neq k}}\ Q_{jk} |{\bf R}_{k} - {\bf R}_{j}|^{-n} &(1)\cr &\quad + \left\{[1/2 \Gamma (n/2)] {\textstyle\sum\limits_{j \neq k}}\ Q_{jk} {\textstyle\sum\limits_{{\bf d} \neq 0}} |{\bf R}_{k} + {\bf X(d)} - {\bf R}_{j}|^{-n}\right.\cr &\quad \left.\times\ \Gamma (n/2, \pi w^{2} |{\bf R}_{k} + {\bf X(d)} - {\bf R}_{j}|^{2})\vphantom{\sum_{d}}\right\} &(2)\cr &\quad - \left\{[1/2 \Gamma (n/2)] {\textstyle\sum\limits_{j \neq k}}\ Q_{jk} |{\bf R}_{k} - {\bf R}_{j}|^{-n}\right.\cr &\quad \left. \times\ \gamma (n/2, \pi w^{2} |{\bf R}_{k} - {\bf R}_{j}|^{2})\vphantom{\sum_{d}}\right\} &(3)\cr &\quad + \left\{[1/2 \Gamma (n/2)] V_{d}^{-1} \pi^{n-(3/2)} {\textstyle\sum\limits_{j \neq k}}\ Q_{jk} {\textstyle\sum\limits_{{\bf h} \neq 0}} |{\bf H(h)}|^{n-3}\right.\cr &\quad \times\ \Gamma [(-n/2) + (3/2), \pi w^{-2} |{\bf H(h)}|^{2}]\cr &\quad \left.\times \exp [2 \pi i {\bf H(h)} \cdot ({\bf R}_{k} - {\bf R}_{j})]\vphantom{\sum_{d}}\right\} &(4)\cr&\quad + [1/2 \Gamma (n/2)] V_{d}^{-1} \pi^{n/2} w^{n-3} 2(n - 3)^{-1} {\textstyle\sum\limits_{j \neq k}}\ Q_{jk} &(5)\cr &\quad + \left\{[1/2 \Gamma (n/2)] {\textstyle\sum\limits_{j}}\ Q_{jj} {\textstyle\sum\limits_{{\bf d} \neq 0}} |{\bf X(d)}|^{-n}\right.\cr &\quad \left.\times\ \Gamma (n/2, \pi w^{2} |{\bf X(d)}|^{2})\vphantom{\sum_{d}}\right\} &(6)\cr &\quad - [1 / \Gamma (n/2)] \pi^{n/2} w^{n} n^{-1} {\textstyle\sum\limits_{j}}\ Q_{jj} &(7)\cr &\quad + \left\{[1/2 \Gamma (n/2)] V_{d}^{-1} \pi^{n-(3/2)} {\textstyle\sum\limits_{j}}\ Q_{jj} {\textstyle\sum\limits_{{\bf h} \neq 0}} |{\bf H(h)}|^{n-3}\right.\cr &\quad \left.\times\ \Gamma [(- n/2) + (3/2), \pi w^{-2} |{\bf H(h)}|^{2}]\vphantom{\sum_{d}}\right\} &(8)\cr &\quad + [1/2 \Gamma (n/2)] V_{d}^{-1} \pi^{n/2} w^{n-3} 2(n - 3)^{-1} {\textstyle\sum\limits_{j}}\ Q_{jj}. &(9)}]$ This expression for V has nine terms, which are numbered on the right-hand side. Term (3) can be expressed in terms of Γ rather than γ: $[\eqalign{ (3) &= -(1/2) {\textstyle\sum\limits_{j \neq k}}\ Q_{jk} |{\bf R}_{k} - {\bf R}_{j}|^{-n}\cr &\quad + [1 / \Gamma (n/2)] {\textstyle\sum\limits_{j \neq k}}\ Q_{jk} |{\bf R}_{k} - {\bf R}_{j}|^{-n} \Gamma (n/2, \pi w^{2}| {\bf R}_{k} - {\bf R}_{j}|^{2}).}]$ It is seen that cancellation occurs with term (1) so that $[\eqalign{ (1) + (3) &= [1 / \Gamma (n/2)] {\textstyle\sum\limits_{j \neq k}}\ Q_{jk} |{\bf R}_{k} - {\bf R}_{j}|^{-n}\cr &\quad \times \Gamma (n/2, \pi w^{2} |{\bf R}_{k} - {\bf R}_{j}|^{2}),}]$ which is the $[{\bf d} = 0]$ , j unequal to k portion of the treated direct-lattice sum. The d unequal to 0, j unequal to k portion corresponds to term (2) and the d unequal to 0, [j = k] portion corresponds to term (6). The direct-lattice terms may be consolidated as $[\eqalign{ (1)\! +\! (2)\! +\! (3)\! +\! (6) &= [1/2 \Gamma (n/2)] {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}'\ Q_{jk} {\textstyle\sum\limits_{\bf d}} |{\bf R}_{k} + {\bf X(d)}- {\bf R}_{j}|^{-n}\cr &\quad \times \Gamma [n/2, \pi w^{2} |{\bf R}_{k} + {\bf X(d)} - {\bf R}_{j}|^{2}].}]$ Now let us combine terms (4) and (8), carrying out the h summation first: $[\eqalign{ (4) + (8) &= [1/2 \Gamma (n/2)] V_{d}^{-1} \pi^{n-(3/2)} {\textstyle\sum\limits_{\bf h}} |{\bf H(h)}|^{n-3}\cr &\quad \times \Gamma [(-n/2) + (3/2), \pi w^{-2} |{\bf H(h)}|^{2}]\cr &\quad \times {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}\ Q_{jk} \exp [2 \pi i {\bf H(h)} \cdot ({\bf R}_{k} - {\bf R}_{j})].}]$ Terms (5) and (9) may be combined: $[(5) + (9) = [\Gamma (n/2)]^{-1} V_{d}^{-1} \pi^{n/2} w^{n-3} (n-3)^{-1} \left({\textstyle\sum\limits_{j}}\ Q_{ij} + {\textstyle\sum\limits_{j \neq k}}\ Q_{jk}\right).]$ The final formula is shown below. The significance of the four terms is: (1) the treated direct-lattice sum; (2) a correction for the difference resulting from the removal of the origin term in direct space; (3) the reciprocal-lattice sum, except $[{\bf h} = 0]$ ; and (4) the $[{{\bf h} = 0}]$ term of the reciprocal-lattice sum. $[\eqalignno{ &V (n, {\bf R}_{j})\cr &\quad = [1/2 \Gamma (n/2)] {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}'\ Q_{jk} {\textstyle\sum\limits_{\bf d}} |{\bf R}_{k} + {\bf X(d)} - {\bf R}_{j}|^{-n}\cr &\qquad \times \Gamma (n/2, \pi w^{2} |{\bf R}_{k} + {\bf X(d)} - {\bf R}_{j}|^{2}) &(1)\cr &\qquad - [1 / \Gamma (n/2)] \pi^{n/2} w^{n} n^{-1} {\textstyle\sum\limits_{j}}\ Q_{jj} &(2)\cr &\qquad + [1/2 \Gamma (n/2)] V_{d}^{-1} \pi^{n-(3/2)} {\textstyle\sum\limits_{\bf h}} |{\bf H(h)}|^{n-3}\cr &\qquad \times \Gamma [(-n/2) + (3/2), \pi w^{-2} |{\bf H(h)}|^{2}]\cr &\qquad \times {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}\ Q_{jk} \exp [2 \pi i {\bf H(h)} \cdot ({\bf R}_{k} - {\bf R}_{j})] &(3)\cr &\qquad + [\Gamma (n/2)]^{-1} V_{d}^{-1} \pi^{n/2} w^{n-3} (n - 3)^{-1} \left({\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}\ Q_{jk}\right). &(4)}]$