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

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

Section 3.4.7. The cases of [{\bf{\bi n} = 2}] and [{\bf{\bi n} = 3}]

D. E. Williamsa‡

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

3.4.7. The cases of [{\bf{\bi n} = 2}] and [{\bf{\bi n} = 3}]

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If [n = 2] the denominator considered for the limit in the preceding section is linear in |H| so that only one differentiation is needed to obtain the limit by L'Hospital's method. Since a term of the type [\sum q_{j} \exp (2\pi i{\bf H}\cdot {\bf R}_{j})] is always a factor, the requirement that the unit cell have no dipole moment can be relaxed. For [n = 2] the zero-charge condition is still required: [\sum q_{j} = 0]. When [n = 3] the expression becomes determinate and no differentiation is required to obtain a limit. In addition, factoring the [Q_{jk}] sums into [q_{j}] sums is not necessary so that the only remaining requirement for this term to be zero is [\sum \sum Q_{jk} = 0], which is a further relaxation beyond the requirement of cell neutrality.








































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