Table 1.2.7.3

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, pp. 16-17

Table 1.2.7.3

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 1.2.7.3 | top | pdf |
`Kubic Harmonic' functions

(a) Coefficients in the expression $[K_{lj} = {\textstyle\sum\limits_{mp}} k_{mpj}^{l} y_{lmp}]$ with normalization $[{\textstyle\int_{0}^{\pi}} {\textstyle\int_{0}^{2\pi}} |K_{lj}|^{2} \sin \theta\;\hbox{d}\theta \;\hbox{d}\varphi = 1]$ (Kara & Kurki-Suonio, 1981).

Even l		mp
l	j	0+	2+	4+	6+	8+	10+
0	1	1
4	1	$[\textstyle{1 \over 2}\left({7 \over 3}\right)^{1/2}]$		$[\textstyle{1 \over 2}\left({5 \over 3}\right)^{1/2}]$
4	1	0.76376		0.64550
6	1	$[\textstyle{1 \over 2}\left({1 \over 2}\right)^{1/2}]$		$[\textstyle-{1 \over 2}\left({7 \over 2}\right)^{1/2}]$
6	1	0.35355		−0.93541
6	2		$[\textstyle{1 \over 4}11^{1/2}]$		$[\textstyle- {1 \over 4} 5^{1/2}]$
6	2		0.82916		−0.55902
8	1	$[\textstyle{1 \over 8}33^{1/2}]$		$[\textstyle{1 \over 4}\left({7 \over 3}\right)^{1/2}]$		$[\textstyle{1 \over 8}\left({65 \over 3}\right)^{1/2}]$
8	1	0.71807		0.38188		0.58184
10	1	$[\textstyle{1 \over 8}\left({65 \over 6}\right)^{1/2}]$		$[\textstyle- {1 \over 4}\left({11 \over 2}\right)^{1/2}]$		$[\textstyle- {1 \over 8}\left({187 \over 6}\right)^{1/2}]$
10	1	0.41143		−0.58630		−0.69784
10	2		$[\textstyle{1 \over 8}\left({247 \over 6}\right)^{1/2}]$		$[\textstyle{1 \over 16}\left({19 \over 3}\right)^{1/2}]$		$[\textstyle{1 \over 16}85^{1/2}]$
10	2		0.80202		0.15729		0.57622
l	j		2−	4−	6−	8−
3	1		1
7	1		$[\textstyle{1 \over 2}\left({13 \over 6}\right)^{1/2}]$		$[\textstyle{1 \over 2}\left({11 \over 16}\right)^{1/2}]$
7	1		0.73598		0.41458
9	1		$[\textstyle{1 \over 4}3^{1/2}]$		$[\textstyle- {1 \over 4} 13^{1/2}]$
9	1		0.43301		−0.90139
9	2		$[\textstyle{1 \over 2}\left({17 \over 6}\right)^{1/2}]$		$[\textstyle- {1 \over 2}\left({7 \over 6}\right)^{1/2}]$
9	2		0.84163		−0.54006

(b) Coefficients $[k_{mpj}^{l}]$ and density normalization factors $[N_{lj}]$ in the expression $[K_{lj} = N_{lj} {\textstyle\sum\limits_{mp}} k_{mpj}^{l} u_{lmp}]$ where $[u_{lm \pm} =P_{l}^{m} (\cos \theta)^{\cos m\varphi}_{\sin m\varphi}]$ (Su & Coppens, 1994).

Even l		$[N_{lj}]$	mp
l	j		0+	2+	4+	6+	8+	10+
0	1	$[1/4\pi = 0.079577]$	1
4	1	0.43454	1
6	1	0.25220	1
6	2	0.020833		1
8	1	0.56292	1		1/5940		$[\textstyle{1 \over 672} \times {1 \over 5940}]$
10	1	0.36490	1		1/5460		$[\textstyle{1 \over 4320} \times {1 \over 5460}]$
10	2	0.0095165	1					$[\textstyle- {1 \over 456} \times {1 \over 43680}]$
l	j			2−	4−	6−	8−
3	1	0.066667		1
7	1	0.014612		1
9	1	0.0059569		1
9	2	0.00014800			1

(c) Density-normalized Kubic harmonics as linear combinations of density-normalized spherical harmonic functions. Coefficients in the expression $[K_{lj} = {\textstyle\sum\limits_{mp}} k^{''l}_{mpj} d_{lmp}]$ . Density-type normalization is defined as $[{\textstyle\int_{0}^{\pi}} {\textstyle\int_{0}^{2\pi}} |K_{lj}| \sin \theta\ \hbox{d} \theta\ \hbox{d} \varphi = 2 - \delta_{l0}]$ .

Even l		mp
l	j	0+	2+	4+	6+	8+	10+
0	1	1
4	1	0.78245		0.57939
6	1	0.37790		−0.91682
6	2		0.83848		−0.50000
l	j	2−	4−	6−	8−
3	1	1
7	1	0.73145		0.63290

(d) Index rules for cubic symmetries (Kurki-Suonio, 1977; Kara & Kurki-Suonio, 1981).

l	j	23	$[m\bar{3}]$	432	$[\bar{4}3m]$	$[m\bar{3}m]$
l	j	T	$[T_{h}]$	O	$[T_{d}]$	$[O_{h}]$
0	1	×	×	×	×	×
3	1	×			×
4	1	×	×	×	×	×
6	1	×	×	×	×	×
6	2	×	×
7	1	×			×
8	1	×	×	×	×	×
9	1	×			×
9	2	×		×
10	1	×	×	×	×	×
10	2	×	×