Table 1.2.7.2

Coppens, P.

doi:10.1107/97809553602060000550

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.2, p. 15

Table 1.2.7.2

P. Coppens^a ^*

^aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260-3000, USA
Correspondence e-mail: coppens@acsu.buffalo.edu

Table 1.2.7.2 | top | pdf |
Index-picking rules of site-symmetric spherical harmonics (Kara & Kurki-Suonio, 1981)

λ, μ and j are integers.

Symmetry	Choice of coordinate axes	Indices of allowed $[y_{lmp}]$ , $[d_{lmp}]$
1	Any	$[{\rm All}\;(l, m, \pm)]$
$[\bar{1}]$	Any	$[(2\lambda, m, \pm)]$
2	$[2\!\!\parallel\!\!z]$	$[(l, 2\mu, \pm)]$
m	$[m\perp z]$	$[(l, l-2j, \pm)]$
	$[2\!\!\parallel\!\! z, m\perp z]$	$[(2\lambda, 2\mu, \pm)]$
222	$[2\!\!\parallel\!\! z, 2\!\!\parallel\!\! y]$	$[(2\lambda, 2\mu, +)]$ , $[(2\lambda + 1, 2\mu, -)]$
mm2	$[2\!\!\parallel\!\! z, m\perp y]$	$[(l, 2\mu, +)]$
mmm	$[m\perp z, m\perp y, m\perp x]$	$[(2\lambda, 2\mu, +)]$
4	$[4\!\!\parallel\!\! z]$	$[(l, 4\mu, \pm)]$
$[\bar{4}]$	$[\bar{4}\!\!\parallel\!\! z]$	$[(2\lambda, 4\mu, \pm)]$ , $[(2\lambda + 1, 4\mu + 2, \pm)]$
	$[4\!\!\parallel\!\! z, m\perp z]$	$[(2\lambda, 4\mu, \pm)]$
422	$[4\!\!\parallel\!\! z, 2\!\!\parallel\!\! y]$	$[(2\lambda, 4\mu, +)]$ , $[(2\lambda + 1, 4\mu, -)]$
4mm	$[4\!\!\parallel\!\! z, m\perp y]$	$[(l, 4\mu, +)]$
$[\bar{4}]$ 2m	$[\bar{4}\!\!\parallel\!\! z, 2\!\!\parallel\!\! x]$	$[(2\lambda, 4\mu, +)]$ , $[(2\lambda + 1, 4\mu + 2, -)]$
	$[m\perp y]$	$[(2\lambda, 4\mu, +)]$ , $[(2\lambda + 1, 4\mu + 2, +)]$
	$[4\!\!\parallel\!\! z, m\perp z, m\perp x]$	$[(2\lambda, 4\mu, +)]$
3	$[3\!\!\parallel\!\! z]$	$[(l, 3\mu, \pm)]$
$[\bar{3}]$	$[\bar{3}\!\!\parallel\!\! z]$	$[(2\lambda, 3\mu, \pm)]$
32	$[3\!\!\parallel\!\! z, 2\!\!\parallel\!\! y]$	$[(2\lambda, 3\mu, +), (2\lambda + 1, 3\mu, -)]$
	$[2\!\!\parallel\!\! x]$	$[(3\mu + 2j, 3\mu, +)]$ ,
		$[(3\mu + 2j + 1, 3\mu, -)]$
3m	$[3\!\!\parallel\!\! z, m \perp y]$	$[(l, 3\mu, +)]$
	$[m \perp x]$	$[(l, 6\mu, +), (l, 6\mu + 3, -)]$
$[\bar{3}m]$	$[\bar{3}\!\!\parallel\!\! z, m \perp y]$	$[(2\lambda, 3\mu, +)]$
	$[m \perp x]$	$[(2\lambda, 6\mu, +), (2\lambda, 6\mu + 3, -)]$
6	$[6\!\!\parallel\!\! z]$	$[(l, 6\mu, \pm)]$
$[\bar{6}]$	$[\bar{6}\!\!\parallel\!\! z]$	$[(2\lambda, 6\mu, \pm), (2\lambda + 1, 6\mu + 3, \pm)]$
	$[6\!\!\parallel\!\! z, m \perp z]$	$[(2\lambda, 6\mu, \pm)]$
622	$[6\!\!\parallel\!\! z, 2\!\!\parallel\!\! y]$	$[(2\lambda, 6\mu, +), (2\lambda + 1, 6\mu, -)]$
6mm	$[6\!\!\parallel\!\! z, m\!\!\parallel\!\! y]$	$[(l, 6\mu, +)]$
$[\bar{6}m2]$	$[\bar{6}\!\!\parallel\!\! z, m \perp y]$	$[(2\lambda, 6\mu, +), (2\lambda + 1, 6\mu + 3, +)]$
	$[m \perp x]$	$[(2\lambda, 6\mu, +), (2\lambda + 1, 6\mu + 3, -)]$
	$[6\!\!\parallel\!\! z, m \perp z, m \perp y]$	$[(2\lambda, 6\mu, +)]$