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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 74   | 1 | 2 |

Section 1.3.4.2.2.8. Parseval's theorem with crystallographic symmetry

G. Bricognea

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1.3.4.2.2.8. Parseval's theorem with crystallographic symmetry

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The general statement of Parseval's theorem given in Section 1.3.4.2.1.5 may be rewritten in terms of symmetry-unique structure factors and electron densities by means of orbit decomposition.

In reciprocal space,for each l, the summands corresponding to the various are equal, so that the left-hand side is equal to

In real space, the triple integral may be rewritten as(where D is the asymmetric unit) if and are smooth densities, since the set of special positions has measure zero. If, however, the integral is approximated as a sum over a G-invariant grid defined by decimation matrix N, special positions on this grid must be taken into account:where the discrete asymmetric unit D contains exactly one point in each orbit of G in .