Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

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

Section 3.4.6. The case of [{\bf{\bi n} = 1}] (Coulombic lattice energy)

D. E. Williamsa

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

3.4.6. The case of [{\bf{\bi n} = 1}] (Coulombic lattice energy)

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As taken above, the limit of the reciprocal-lattice [{\bf h} = 0] term of [S'(n, {\bf R})] or [S'(n, 0)] existed only if n was greater than 3. The corresponding contributions to [V(n, {\bf R}_{j})] were terms (5) and (9) of Section 3.4.5[link]. To extend the method to [n = 1] we will show in this section that these [{\bf h} = 0] terms vanish if conditions of unit-cell neutrality and zero dipole moment are satisfied.

The integral representation of the term (5) is [\eqalign{ &[1/2\Gamma (n/2)] V_{d}^{-1} \pi^{n-(3/2)} {\textstyle\sum\limits_{j \neq k}}\ Q_{jk} \textstyle\int \delta (0) |{\bf H}|^{n-3}\cr &\quad \times \Gamma [(-n/2) + (3/2), \pi w^{-2} |{\bf H}|^{2}]\cr &\quad \times \exp [2 \pi i {\bf H} \cdot ({\bf R}_{k} - {\bf R}_{j})]\; \hbox{d}{\bf H}}] and for term (9) is [\eqalign{ &[1/2\Gamma (n/2)] V_{d}^{-1} \pi^{n-(3/2)} {\textstyle\sum\limits_{j}}\ Q_{jj} \textstyle\int \delta (0) |{\bf H}|^{n-3}\cr &\quad \times \Gamma [(-n/2) + (3/2), \pi w^{-2} |{\bf H}|^{2}]\; \hbox{d}{\bf H}.}] Combining these two sums of integrals into one integral sum gives [\eqalign{ &[1/2\Gamma (n/2)] V_{d}^{-1} \pi^{n-(3/2)} \textstyle\int \delta (0) |{\bf H}|^{n-3}\cr &\quad \times \Gamma [(-n/2) + (3/2), \pi w^{-2} |{\bf H}|^{2}] {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}\ Q_{jk}\cr &\quad \times \exp [2\pi i {\bf H}\cdot ({\bf R}_{k} - {\bf R}_{j})] \hbox{ d}{\bf H}.}]

For [n = 1], suppose [q_{j}] are net atomic charges so that the geometric combining law holds for [Q_{jk} = q_{j}q_{k}]. Then the double sum over j and k can be factored so that the limit that needs to be considered is [\lim\limits_{|{\bf H}| \rightarrow 0} {{\left[{\textstyle\sum_{k{\phantom j}}} q_{k} \exp (2\pi i {\bf H} \cdot {\bf R}_{k})\right] \left[{\textstyle\sum_{j}}\ q_{j} \exp (-2\pi i {\bf H} \cdot {\bf R}_{j})\right]} \over |{\bf H}|^{2}}.] If the unit cell does not have a net charge, the sum over the q's goes to zero in the limit and this is a 0/0 indeterminate form. Let [|{\bf H}|] approach zero along the polar axis so that [{\bf H}\cdot {\bf R}_{k} = H_{3} R_{3k}], where subscript 3 indicates components along the polar axis. To find the limit with L'Hospital's rule the numerator and denominator are differentiated twice with respect to [H_{3}]. Represent the numerator of the limit by the product [(uv)] and note that [{\hbox{d}^{2} (uv)\over\; \hbox{d}x^{2}} = u {\hbox{d}^{2} v\over\; \hbox{d}x^{2}} + v {\hbox{d}^{2} u\over\; \hbox{d}x^{2}} + 2 {\hbox{d}u\over\; \hbox{d}x} {\hbox{d}v\over\; \hbox{d}x}.] It is seen that in addition to cell neutrality the product of the first derivatives of the sums must exist. These sums are [\left[2\pi i {\textstyle\sum\limits_{k}}\ q_{k} R_{3k} \exp (2\pi i H_{3} R_{3k})\right]] and [\left[-2\pi i {\textstyle\sum\limits_{j}}\ q_{j} R_{3j} \exp (-2\pi i H_{3} R_{3j})\right],] which vanish if the unit cell has no dipole moment in the polar direction, that is, if [\sum q_{j} R_{3j} = 0]. Since the second derivative of the denominator is a constant, the desired limit is zero under the specified conditions. Now the polar direction can be chosen arbitrarily, so the unit cell must not have a dipole moment in any direction for the limit of the numerator to be zero. Thus we have the formula for the Coulombic lattice sum [\eqalign{ V(1, {\bf R}_{j}) &= [ 1/2\Gamma (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}|^{-1}\cr &\quad \times \Gamma (1/2, \pi w^{2} |{\bf R}_{k} + {\bf X(d)} - {\bf R}_{j}|^{2})\cr &\quad + [1/2 \Gamma (1/2) V_{d}^{-1} \pi^{-1/2} {\textstyle\sum\limits_{\bf h}} |{\bf H(h)}|^{-2}\cr &\quad \times \Gamma (1/2, \pi w^{-2} |{\bf H(h)}|^{2}) {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{k}}\ Q_{jk}\cr &\quad \times \exp [2\pi i {\bf H(h)} \cdot ({\bf R}_{k} - {\bf R}_{j})]\cr &\quad - [1/\Gamma (1/2)] \pi^{1/2} w {\textstyle\sum\limits_{j}}\ q_{j}^{2},}] which holds on conditions that the unit cell be electrically neutral and have no dipole moment. If the unit cell has a dipole moment, the limiting value discussed above depends on the direction of H. For methods of obtaining the Coulombic lattice sum where the unit cell does have a dipole moment, the reader is referred to the literature (DeWette & Schacher, 1964[link]; Cummins et al., 1976[link]; Bertaut, 1978[link]; Massidda, 1978[link]).


Bertaut, E. F. (1978). The equivalent charge concept and its application to the electrostatic energy of charges and multipoles. J. Phys. (Paris), 39, 1331–1348.
Cummins, P. G., Dunmur, D. A., Munn, R. W. & Newham, R. J. (1976). Applications of the Ewald method. I. Calculation of multipole lattice sums. Acta Cryst. A32, 847–853.
DeWette, F. W. & Schacher, G. E. (1964). Internal field in general dipole lattices. Phys. Rev. 137, A78–A91.
Massidda, V. (1978). Electrostatic energy in ionic crystals by the planewise summation method. Physica (Utrecht), 95B, 317–334.

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