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

International Tables for Crystallography (2010). Vol. B, ch. 1.2, pp. 16-17

Table 1.2.7.3 

P. Coppensa*

aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260–3000, USA
Correspondence e-mail: coppens@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\,{\rm d}\theta\,{\rm d}\varphi = 1] (Kara & Kurki-Suonio, 1981[link]).

Even lmp
lj0+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}]      
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}]      
0.35355   −0.93541      
6 2   [\textstyle{1 \over 4}11^{1/2}]   [\textstyle- {1 \over 4} 5^{1/2}]    
  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}]  
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}]  
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}]
  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}]  
  0.73598   0.41458  
9 1   [\textstyle{1 \over 4}3^{1/2}]   [\textstyle- {1 \over 4} 13^{1/2}]  
  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}]  
  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[link]).

Even l[N_{lj}]mp
lj 0+2+4+6+8+10+
0 1 [1/4\pi = 0.079577] 1          
4 1 0.43454 1   [+1/168]      
6 1 0.25220 1   [-1/360]      
6 2 0.020833   1   [-1/792]    
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     [1/43680]   [\textstyle- {1 \over 456} \times {1 \over 43680}]
l j     2− 4− 6− 8−
3 1 0.066667   1      
7 1 0.014612   1   [1/1560]  
9 1 0.0059569   1   [1/2520]  
9 2 0.00014800     1   [-1/4080]

(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 lmp
lj0+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[link]; Kara & Kurki-Suonio, 1981[link]).

lj23[m\bar{3}]432[\bar{4}3m][m\bar{3}m]
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 × ×