View source for Table of Clebsch–Gordan coefficients

{{short description|Used for adding angular momentum values in quantum mechanics}}
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This is a '''table of [[Clebsch–Gordan coefficients]]''' used for adding [[angular momentum]] values in [[quantum mechanics]]. The overall sign of the coefficients for each set of constant <math>j_1</math>, <math>j_2</math>, <math>j</math> is arbitrary to some degree and has been fixed according to the Condon–Shortley and Wigner sign convention as discussed by Baird and [[Lawrence Biedenharn|Biedenharn]].<ref>{{cite journal |last=Baird |first=C.E. |author2=L. C. Biedenharn  |title=On the Representations of the Semisimple Lie Groups. III. The Explicit Conjugation Operation for SU<sub>''n''</sub> |journal=J. Math. Phys. |volume=5 |issue=12 |date=October 1964 |pages=1723–1730 |doi=10.1063/1.1704095 |bibcode=1964JMP.....5.1723B}}</ref>  Tables with the same sign convention may be found in the [[Particle Data Group]]'s ''Review of Particle Properties''<ref>{{cite journal |last=Hagiwara |first=K. |title=Review of Particle Properties |journal=Phys. Rev. D |volume=66 |issue=1 |date=July 2002 |pages=010001 |doi=10.1103/PhysRevD.66.010001 |url=http://pdg.lbl.gov/2002/clebrpp.pdf |accessdate=2007-12-20 |bibcode=2002PhRvD..66a0001H|display-authors=etal}}</ref> and in online tables.<ref>{{cite web |last=Mathar |first=Richard J. |title=SO(3) Clebsch Gordan coefficients |date=2006-08-14 |url=http://www.mpia.de/~mathar/progs/CGord |format=text |accessdate=2012-10-15}}</ref>

==Formulation==
The Clebsch–Gordan coefficients are the solutions to

:<math>
  |j_1,j_2;J,M\rangle = \sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2}
  |j_1,m_1;j_2,m_2\rangle \langle j_1,j_2;m_1,m_2\mid j_1,j_2;J,M\rangle
</math>

Explicitly:

:<math>
\begin{align}
& \langle j_1,j_2;m_1,m_2\mid j_1,j_2;J,M\rangle \\[6pt]
= {} & \delta_{M,m_1+m_2} \sqrt{\frac{(2J+1)(J+j_1-j_2)!(J-j_1+j_2)!(j_1+j_2-J)!}{(j_1+j_2+J+1)!}}\ \times {} \\[6pt]
&\sqrt{(J+M)!(J-M)!(j_1-m_1)!(j_1+m_1)!(j_2-m_2)!(j_2+m_2)!}\ \times {} \\[6pt]
&\sum_k \frac{(-1)^k}{k!(j_1+j_2-J-k)!(j_1-m_1-k)!(j_2+m_2-k)!(J-j_2+m_1+k)!(J-j_1-m_2+k)!}.
\end{align}
</math>

The summation is extended over all integer {{mvar|k}} for which the argument of every factorial is nonnegative.<ref>(2.41), p. 172, ''Quantum Mechanics: Foundations and Applications'', Arno Bohm, M. Loewe, New York: Springer-Verlag, 3rd ed., 1993, {{ISBN|0-387-95330-2}}.</ref>

For brevity, solutions with {{math|''M'' < 0}} and {{math|''j''<sub>1</sub> < ''j''<sub>2</sub>}} are omitted.  They may be calculated using the simple relations

: <math>\langle j_1,j_2;m_1,m_2\mid j_1,j_2;J,M\rangle=(-1)^{J-j_1-j_2}\langle j_1,j_2;-m_1,-m_2\mid j_1,j_2;J,-M\rangle.</math>

and

:<math>\langle j_1,j_2;m_1,m_2\mid j_1,j_2;J,M\rangle=(-1)^{J-j_1-j_2} \langle j_2,j_1;m_2,m_1\mid j_2, j_1;J,M\rangle.</math>

== Specific values ==

The Clebsch–Gordan coefficients for ''j'' values less than or equal to 5/2 are given by:<ref>{{cite book|last=Weissbluth|first=Mitchel|title=Atoms and molecules|url=https://archive.org/details/atomsmolecules0000weis|url-access=registration|year=1978|publisher=ACADEMIC PRESS|isbn=0-12-744450-5|page=[https://archive.org/details/atomsmolecules0000weis/page/28 28]}} Table 1.4 resumes the most common.</ref>

=== {{math|size=100%|1=&nbsp;''j''<sub>2</sub> = 0}} ===

When {{math|size=100%|1=''j''<sub>2</sub> = 0}}, the Clebsch–Gordan coefficients are given by <math>\delta_{j,j_1}\delta_{m,m_1}</math> .

=== {{math|size=100%|1=&nbsp;''j''<sub>1</sub> = {{sfrac|1|2}}, ''j''<sub>2</sub> = {{sfrac|1|2}}}} ===

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 1}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 1
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;{{sfrac|1|2}} 
| <math>1</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = −1}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 1
|----- valign="center"  align="center"
! –{{sfrac|1|2}},&nbsp;–{{sfrac|1|2}} 
| <math>1</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 0}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 1 
! 0
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;–{{sfrac|1|2}} 
| <math>\sqrt{\frac{1}{2}}</math>
| <math>\sqrt{\frac{1}{2}}</math>
|----- valign="center"  align="center"
! –{{sfrac|1|2}},&nbsp;{{sfrac|1|2}} 
| <math>\sqrt{\frac{1}{2}}</math>
| <math>-\sqrt{\frac{1}{2}}</math>
|}

=== {{math|size=100%|1=&nbsp;''j''<sub>1</sub> = 1, ''j''<sub>2</sub> = {{sfrac|1|2}}}} ===

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|3|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|3|2}}
|----- valign="center"  align="center"
! 1,&nbsp;{{sfrac|1|2}} 
| <math>1</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|1|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|3|2}} 
! {{sfrac|1|2}}
|----- valign="center"  align="center"
! 1,&nbsp;–{{sfrac|1|2}} 
| <math>\sqrt{\frac{1}{3}}</math>
| <math>\sqrt{\frac{2}{3}}</math>
|----- valign="center"  align="center"
! 0,&nbsp;{{sfrac|1|2}} 
| <math>\sqrt{\frac{2}{3}}</math>
| <math>-\sqrt{\frac{1}{3}}</math>
|}

=== {{math|size=100%|1=&nbsp;''j''<sub>1</sub> = 1, ''j''<sub>2</sub> = 1}} ===

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 2}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 2
|----- valign="center"  align="center"
! 1,&nbsp;1 
| <math>1</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 1}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 2 
! 1
|----- valign="center"  align="center"
! 1,&nbsp;0 
| <math>\sqrt{\frac{1}{2}}</math>
| <math>\sqrt{\frac{1}{2}}</math>
|----- valign="center"  align="center"
! 0,&nbsp;1 
| <math>\sqrt{\frac{1}{2}}</math>
| <math>-\sqrt{\frac{1}{2}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 0}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 2 
! 1 
! 0
|----- valign="center"  align="center"
! 1,&nbsp;−1 
| <math>\sqrt{\frac{1}{6}}</math>
| <math>\sqrt{\frac{1}{2}}</math> 
| <math>\sqrt{\frac{1}{3}}</math>
|----- valign="center"  align="center"
! 0,&nbsp;0 
| <math>\sqrt{\frac{2}{3}}</math>
| <math>0</math>
| <math>-\sqrt{\frac{1}{3}}</math>
|----- valign="center"  align="center"
! −1,&nbsp;1 
| <math>\sqrt{\frac{1}{6}}</math>
| <math>-\sqrt{\frac{1}{2}}</math>
| <math>\sqrt{\frac{1}{3}}</math>
|}

=== {{math|size=100%|1=&nbsp;''j''<sub>1</sub> = {{sfrac|3|2}}, ''j''<sub>2</sub> = {{sfrac|1|2}}}} ===

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 2}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 2
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;{{sfrac|1|2}} 
| <math>1</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 1}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 2 
! 1
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;–{{sfrac|1|2}} 
| <math>\frac{1}{2}</math>
| <math>\sqrt{\frac{3}{4}}</math>
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;{{sfrac|1|2}} 
| <math>\sqrt{\frac{3}{4}}</math>
| <math>-\frac{1}{2}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 0}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 2 
! 1
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;–{{sfrac|1|2}} 
| <math>\sqrt{\frac{1}{2}}</math>
| <math>\sqrt{\frac{1}{2}}</math>
|----- valign="center"  align="center"
! –{{sfrac|1|2}},&nbsp;{{sfrac|1|2}} 
| <math>\sqrt{\frac{1}{2}}</math>
| <math>-\sqrt{\frac{1}{2}}</math>
|}

=== {{math|size=100%|1=&nbsp;''j''<sub>1</sub> = {{sfrac|3|2}}, ''j''<sub>2</sub> = 1}} ===

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|5|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|5|2}}
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;1 
| <math>1</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|3|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|5|2}} 
! {{sfrac|3|2}}
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;0 
| <math>\sqrt{\frac{2}{5}}</math>
| <math>\sqrt{\frac{3}{5}}</math>
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;1 
| <math>\sqrt{\frac{3}{5}}</math>
| <math>-\sqrt{\frac{2}{5}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|1|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|5|2}} 
! {{sfrac|3|2}} 
! {{sfrac|1|2}}
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;−1
| <math>\sqrt{\frac{1}{10}}</math>
| <math>\sqrt{\frac{2}{5}}</math> 
| <math>\sqrt{\frac{1}{2}}</math>
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;0 
| <math>\sqrt{\frac{3}{5}}</math>
| <math>\sqrt{\frac{1}{15}}</math>
| <math>-\sqrt{\frac{1}{3}}</math>
|----- valign="center"  align="center"
! –{{sfrac|1|2}},&nbsp;1
| <math>\sqrt{\frac{3}{10}}</math>
| <math>-\sqrt{\frac{8}{15}}</math>
| <math>\sqrt{\frac{1}{6}}</math>
|}

=== {{math|size=100%|1=&nbsp;''j''<sub>1</sub> = {{sfrac|3|2}}, ''j''<sub>2</sub> = {{sfrac|3|2}}}} ===

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 3}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 3
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;{{sfrac|3|2}} 
| <math>1</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 2}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 3 
! 2
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;{{sfrac|1|2}} 
| <math>\sqrt{\frac{1}{2}}</math>
| <math>\sqrt{\frac{1}{2}}</math>
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;{{sfrac|3|2}} 
| <math>\sqrt{\frac{1}{2}}</math>
| <math>-\sqrt{\frac{1}{2}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 1}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 3 
! 2 
! 1
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;–{{sfrac|1|2}} 
| <math>\sqrt{\frac{1}{5}}</math>
| <math>\sqrt{\frac{1}{2}}</math>
| <math>\sqrt{\frac{3}{10}}</math>
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;{{sfrac|1|2}} 
| <math>\sqrt{\frac{3}{5}}</math>
| <math>0</math>
| <math>-\sqrt{\frac{2}{5}}</math>
|----- valign="center"  align="center"
! –{{sfrac|1|2}},&nbsp;{{sfrac|3|2}} 
| <math>\sqrt{\frac{1}{5}}</math>
| <math>-\sqrt{\frac{1}{2}}</math>
| <math>\sqrt{\frac{3}{10}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 0}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 3 
! 2 
! 1 
! 0
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;–{{sfrac|3|2}}
| <math>\sqrt{\frac{1}{20}}</math>
| <math>\frac{1}{2}</math>
| <math>\sqrt{\frac{9}{20}}</math>
| <math>\frac{1}{2}</math>
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;–{{sfrac|1|2}}
| <math>\sqrt{\frac{9}{20}}</math>
| <math>\frac{1}{2}</math>
| <math>-\sqrt{\frac{1}{20}}</math>
| <math>-\frac{1}{2}</math>
|----- valign="center"  align="center"
! –{{sfrac|1|2}},&nbsp;{{sfrac|1|2}}
| <math>\sqrt{\frac{9}{20}}</math>
| <math>-\frac{1}{2}</math>
| <math>-\sqrt{\frac{1}{20}}</math>
| <math>\frac{1}{2}</math>
|----- valign="center"  align="center"
! –{{sfrac|3|2}},&nbsp;{{sfrac|3|2}}
| <math>\sqrt{\frac{1}{20}}</math>
| <math>-\frac{1}{2}</math>
| <math>\sqrt{\frac{9}{20}}</math>
| <math>-\frac{1}{2}</math>
|}

=== {{math|size=100%|1=&nbsp;''j''<sub>1</sub> = 2, ''j''<sub>2</sub> = {{sfrac|1|2}}}} ===

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|5|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|5|2}}
|----- valign="center"  align="center"
! 2,&nbsp;{{sfrac|1|2}} 
| <math>1</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|3|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|5|2}} 
! {{sfrac|3|2}}
|----- valign="center"  align="center"
! 2,&nbsp;–{{sfrac|1|2}} 
| <math>\sqrt{\frac{1}{5}}</math>
| <math>\sqrt{\frac{4}{5}}</math>
|----- valign="center"  align="center"
! 1,&nbsp;{{sfrac|1|2}} 
| <math>\sqrt{\frac{4}{5}}</math>
| <math>-\sqrt{\frac{1}{5}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|1|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|5|2}} 
! {{sfrac|3|2}}
|----- valign="center"  align="center"
! 1,&nbsp;–{{sfrac|1|2}} 
| <math>\sqrt{\frac{2}{5}}</math>
| <math>\sqrt{\frac{3}{5}}</math>
|----- valign="center"  align="center"
! 0,&nbsp;{{sfrac|1|2}} 
| <math>\sqrt{\frac{3}{5}}</math>
| <math>-\sqrt{\frac{2}{5}}</math>
|}

=== {{math|size=100%|1=&nbsp;''j''<sub>1</sub> = 2, ''j''<sub>2</sub> = 1}} ===

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 3}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 3
|----- valign="center"  align="center"
! 2,&nbsp;1 
| <math>1</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 2}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 3 
! 2
|----- valign="center"  align="center"
! 2,&nbsp;0 
| <math>\sqrt{\frac{1}{3}}</math>
| <math>\sqrt{\frac{2}{3}}</math>
|----- valign="center"  align="center"
! 1,&nbsp;1 
| <math>\sqrt{\frac{2}{3}}</math>
| <math>-\sqrt{\frac{1}{3}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 1}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 3 
! 2 
! 1
|----- valign="center"  align="center"
! 2,&nbsp;−1
| <math>\sqrt{\frac{1}{15}}</math>
| <math>\sqrt{\frac{1}{3}}</math> 
| <math>\sqrt{\frac{3}{5}}</math>
|----- valign="center"  align="center"
! 1,&nbsp;0
| <math>\sqrt{\frac{8}{15}}</math>
| <math>\sqrt{\frac{1}{6}}</math>
| <math>-\sqrt{\frac{3}{10}}</math>
|----- valign="center"  align="center"
! 0,&nbsp;1 
| <math>\sqrt{\frac{2}{5}}</math>
| <math>-\sqrt{\frac{1}{2}}</math>
| <math>\sqrt{\frac{1}{10}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 0}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 3 
! 2 
! 1
|----- valign="center"  align="center"
! 1,&nbsp;−1 
| <math>\sqrt{\frac{1}{5}}</math>
| <math>\sqrt{\frac{1}{2}}</math>
| <math>\sqrt{\frac{3}{10}}</math>
|----- valign="center"  align="center"
! 0,&nbsp;0 
| <math>\sqrt{\frac{3}{5}}</math>
| <math>0</math>
| <math>-\sqrt{\frac{2}{5}}</math>
|----- valign="center"  align="center"
! −1,&nbsp;1 
| <math>\sqrt{\frac{1}{5}}</math>
| <math>-\sqrt{\frac{1}{2}}</math>
| <math>\sqrt{\frac{3}{10}}</math>
|}

=== {{math|size=100%|1=&nbsp;''j''<sub>1</sub> = 2, ''j''<sub>2</sub> = {{sfrac|3|2}}}} ===

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|7|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|7|2}}
|----- valign="center"  align="center"
! 2,&nbsp;{{sfrac|3|2}} 
| <math>1</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|5|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|7|2}} 
! {{sfrac|5|2}}
|----- valign="center"  align="center"
! 2,&nbsp;{{sfrac|1|2}} 
| <math>\sqrt{\frac{3}{7}}</math>
| <math>\sqrt{\frac{4}{7}}</math>
|----- valign="center"  align="center"
! 1,&nbsp;{{sfrac|3|2}} 
| <math>\sqrt{\frac{4}{7}}</math>
| <math>-\sqrt{\frac{3}{7}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|3|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|7|2}} 
! {{sfrac|5|2}} 
! {{sfrac|3|2}}
|----- valign="center"  align="center"
! 2,&nbsp;–{{sfrac|1|2}} 
| <math>\sqrt{\frac{1}{7}}</math>
| <math>\sqrt{\frac{16}{35}}</math>
| <math>\sqrt{\frac{2}{5}}</math>
|----- valign="center"  align="center"
! 1,&nbsp;{{sfrac|1|2}} 
| <math>\sqrt{\frac{4}{7}}</math>
| <math>\sqrt{\frac{1}{35}}</math>
| <math>-\sqrt{\frac{2}{5}}</math>
|----- valign="center"  align="center"
! 0,&nbsp;{{sfrac|3|2}} 
| <math>\sqrt{\frac{2}{7}}</math>
| <math>-\sqrt{\frac{18}{35}}</math>
| <math>\sqrt{\frac{1}{5}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|1|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|7|2}} 
! {{sfrac|5|2}} 
! {{sfrac|3|2}} 
! {{sfrac|1|2}}
|----- valign="center"  align="center"
! 2,&nbsp;–{{sfrac|3|2}}
| <math>\sqrt{\frac{1}{35}}</math>
| <math>\sqrt{\frac{6}{35}}</math>
| <math>\sqrt{\frac{2}{5}}</math> 
| <math>\sqrt{\frac{2}{5}}</math>
|----- valign="center"  align="center"
! 1,&nbsp;–{{sfrac|1|2}}
| <math>\sqrt{\frac{12}{35}}</math>
| <math>\sqrt{\frac{5}{14}}</math>
| <math>0</math>
| <math>-\sqrt{\frac{3}{10}}</math>
|----- valign="center"  align="center"
! 0,&nbsp;{{sfrac|1|2}}
| <math>\sqrt{\frac{18}{35}}</math>
| <math>-\sqrt{\frac{3}{35}}</math>
| <math>-\sqrt{\frac{1}{5}}</math>
| <math>\sqrt{\frac{1}{5}}</math>
|----- valign="center"  align="center"
! −1,&nbsp;{{sfrac|3|2}}
| <math>\sqrt{\frac{4}{35}}</math>
| <math>-\sqrt{\frac{27}{70}}</math>
| <math>\sqrt{\frac{2}{5}}</math>
| <math>-\sqrt{\frac{1}{10}}</math>
|}

=== {{math|size=100%|1=&nbsp;''j''<sub>1</sub> = 2, ''j''<sub>2</sub> = 2}} ===

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 4}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 4
|----- valign="center"  align="center"
! 2,&nbsp;2 
| <math>1</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 3}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 4 
! 3
|----- valign="center"  align="center"
! 2,&nbsp;1 
| <math>\sqrt{\frac{1}{2}}</math>
| <math>\sqrt{\frac{1}{2}}</math>
|----- valign="center"  align="center"
! 1,&nbsp;2 
| <math>\sqrt{\frac{1}{2}}</math>
| <math>-\sqrt{\frac{1}{2}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 2}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 4 
! 3 
! 2
|----- valign="center"  align="center"
! 2,&nbsp;0
| <math>\sqrt{\frac{3}{14}}</math>
| <math>\sqrt{\frac{1}{2}}</math> 
| <math>\sqrt{\frac{2}{7}}</math>
|----- valign="center"  align="center"
! 1,&nbsp;1 
| <math>\sqrt{\frac{4}{7}}</math>
| <math>0</math>
| <math>-\sqrt{\frac{3}{7}}</math>
|----- valign="center"  align="center"
! 0,&nbsp;2
| <math>\sqrt{\frac{3}{14}}</math>
| <math>-\sqrt{\frac{1}{2}}</math>
| <math>\sqrt{\frac{2}{7}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 1}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 4 
! 3 
! 2 
! 1
|----- valign="center"  align="center"
! 2,&nbsp;−1
| <math>\sqrt{\frac{1}{14}}</math>
| <math>\sqrt{\frac{3}{10}}</math>
| <math>\sqrt{\frac{3}{7}}</math> 
| <math>\sqrt{\frac{1}{5}}</math>
|----- valign="center"  align="center"
! 1,&nbsp;0 
| <math>\sqrt{\frac{3}{7}}</math>
| <math>\sqrt{\frac{1}{5}}</math>
| <math>-\sqrt{\frac{1}{14}}</math>
| <math>-\sqrt{\frac{3}{10}}</math>
|----- valign="center"  align="center"
! 0,&nbsp;1 
| <math>\sqrt{\frac{3}{7}}</math>
| <math>-\sqrt{\frac{1}{5}}</math>
| <math>-\sqrt{\frac{1}{14}}</math>
| <math>\sqrt{\frac{3}{10}}</math>
|----- valign="center"  align="center"
! −1,&nbsp;2
| <math>\sqrt{\frac{1}{14}}</math>
| <math>-\sqrt{\frac{3}{10}}</math>
| <math>\sqrt{\frac{3}{7}}</math>
| <math>-\sqrt{\frac{1}{5}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 0}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 4 
! 3 
! 2 
! 1
! 0
|----- valign="center"  align="center"
! 2,&nbsp;-2
| <math>\sqrt{\frac{1}{70}}</math>
| <math>\sqrt{\frac{1}{10}}</math>
| <math>\sqrt{\frac{2}{7}}</math> 
| <math>\sqrt{\frac{2}{5}}</math>
| <math>\sqrt{\frac{1}{5}}</math>
|----- valign="center"  align="center"
! 1,&nbsp;−1
| <math>\sqrt{\frac{8}{35}}</math>
| <math>\sqrt{\frac{2}{5}}</math>
| <math>\sqrt{\frac{1}{14}}</math>
| <math>-\sqrt{\frac{1}{10}}</math>
| <math>-\sqrt{\frac{1}{5}}</math>
|----- valign="center"  align="center"
! 0,&nbsp;0
| <math>\sqrt{\frac{18}{35}}</math>
| <math>0</math>
| <math>-\sqrt{\frac{2}{7}}</math>
| <math>0</math> 
| <math>\sqrt{\frac{1}{5}}</math>
|----- valign="center"  align="center"
! −1,&nbsp;1
| <math>\sqrt{\frac{8}{35}}</math>
| <math>-\sqrt{\frac{2}{5}}</math>
| <math>\sqrt{\frac{1}{14}}</math>
| <math>\sqrt{\frac{1}{10}}</math>
| <math>-\sqrt{\frac{1}{5}}</math>
|----- valign="center"  align="center"
! -2,&nbsp;2
| <math>\sqrt{\frac{1}{70}}</math>
| <math>-\sqrt{\frac{1}{10}}</math>
| <math>\sqrt{\frac{2}{7}}</math>
| <math>-\sqrt{\frac{2}{5}}</math>
| <math>\sqrt{\frac{1}{5}}</math>
|}

=== {{math|size=100%|1=&nbsp;''j''<sub>1</sub> = {{sfrac|5|2}}, ''j''<sub>2</sub> = {{sfrac|1|2}}}} ===

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 3}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 3
|----- valign="center"  align="center"
! {{sfrac|5|2}},&nbsp;{{sfrac|1|2}} 
| <math>1</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 2}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 3 
! 2
|----- valign="center"  align="center"
! {{sfrac|5|2}},&nbsp;–{{sfrac|1|2}} 
| <math>\sqrt{\frac{1}{6}}</math>
| <math>\sqrt{\frac{5}{6}}</math>
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;{{sfrac|1|2}} 
| <math>\sqrt{\frac{5}{6}}</math>
| <math>-\sqrt{\frac{1}{6}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 1}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 3 
! 2
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;–{{sfrac|1|2}} 
| <math>\sqrt{\frac{1}{3}}</math>
| <math>\sqrt{\frac{2}{3}}</math>
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;{{sfrac|1|2}} 
| <math>\sqrt{\frac{2}{3}}</math>
| <math>-\sqrt{\frac{1}{3}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 0}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 3 
! 2
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;–{{sfrac|1|2}} 
| <math>\sqrt{\frac{1}{2}}</math>
| <math>\sqrt{\frac{1}{2}}</math>
|----- valign="center"  align="center"
! –{{sfrac|1|2}},&nbsp;{{sfrac|1|2}} 
| <math>\sqrt{\frac{1}{2}}</math>
| <math>-\sqrt{\frac{1}{2}}</math>
|}

=== {{math|size=100%|1=&nbsp;''j''<sub>1</sub> = {{sfrac|5|2}}, ''j''<sub>2</sub> = 1}} ===

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|7|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|7|2}}
|----- valign="center"  align="center"
! {{sfrac|5|2}},&nbsp;1 
| <math>1</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|5|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|7|2}} 
! {{sfrac|5|2}}
|----- valign="center"  align="center"
! {{sfrac|5|2}},&nbsp;0 
| <math>\sqrt{\frac{2}{7}}</math>
| <math>\sqrt{\frac{5}{7}}</math>
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;1 
| <math>\sqrt{\frac{5}{7}}</math>
| <math>-\sqrt{\frac{2}{7}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|3|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|7|2}} 
! {{sfrac|5|2}} 
! {{sfrac|3|2}}
|----- valign="center"  align="center"
! {{sfrac|5|2}},&nbsp;−1
| <math>\sqrt{\frac{1}{21}}</math>
| <math>\sqrt{\frac{2}{7}}</math> 
| <math>\sqrt{\frac{2}{3}}</math>
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;0
| <math>\sqrt{\frac{10}{21}}</math>
| <math>\sqrt{\frac{9}{35}}</math>
| <math>-\sqrt{\frac{4}{15}}</math>
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;1
| <math>\sqrt{\frac{10}{21}}</math>
| <math>-\sqrt{\frac{16}{35}}</math>
| <math>\sqrt{\frac{1}{15}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|1|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|7|2}} 
! {{sfrac|5|2}} 
! {{sfrac|3|2}}
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;−1 
| <math>\sqrt{\frac{1}{7}}</math>
| <math>\sqrt{\frac{16}{35}}</math>
| <math>\sqrt{\frac{2}{5}}</math>
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;0 
| <math>\sqrt{\frac{4}{7}}</math>
| <math>\sqrt{\frac{1}{35}}</math>
| <math>-\sqrt{\frac{2}{5}}</math>
|----- valign="center"  align="center"
! –{{sfrac|1|2}},&nbsp;1 
| <math>\sqrt{\frac{2}{7}}</math>
| <math>-\sqrt{\frac{18}{35}}</math>
| <math>\sqrt{\frac{1}{5}}</math>
|}

=== {{math|size=100%|1=&nbsp;''j''<sub>1</sub> = {{sfrac|5|2}}, ''j''<sub>2</sub> = {{sfrac|3|2}}}} ===

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 4}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 4
|----- valign="center"  align="center"
! {{sfrac|5|2}},&nbsp;{{sfrac|3|2}} 
| <math>1</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 3}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 4 
! 3
|----- valign="center"  align="center"
! {{sfrac|5|2}},&nbsp;{{sfrac|1|2}} 
| <math>\sqrt{\frac{3}{8}}</math>
| <math>\sqrt{\frac{5}{8}}</math>
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;{{sfrac|3|2}} 
| <math>\sqrt{\frac{5}{8}}</math>
| <math>-\sqrt{\frac{3}{8}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 2}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 4 
! 3 
! 2
|----- valign="center"  align="center"
! {{sfrac|5|2}},&nbsp;–{{sfrac|1|2}}
| <math>\sqrt{\frac{3}{28}}</math>
| <math>\sqrt{\frac{5}{12}}</math>
| <math>\sqrt{\frac{10}{21}}</math>
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;{{sfrac|1|2}}
| <math>\sqrt{\frac{15}{28}}</math>
| <math>\sqrt{\frac{1}{12}}</math>
| <math>-\sqrt{\frac{8}{21}}</math>
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;{{sfrac|3|2}}
| <math>\sqrt{\frac{5}{14}}</math>
| <math>-\sqrt{\frac{1}{2}}</math>
| <math>\sqrt{\frac{1}{7}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 1}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 4 
! 3 
! 2 
! 1
|----- valign="center"  align="center"
! {{sfrac|5|2}},&nbsp;–{{sfrac|3|2}}
| <math>\sqrt{\frac{1}{56}}</math>
| <math>\sqrt{\frac{1}{8}}</math>
| <math>\sqrt{\frac{5}{14}}</math>
| <math>\sqrt{\frac{1}{2}}</math>
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;–{{sfrac|1|2}}
| <math>\sqrt{\frac{15}{56}}</math>
| <math>\sqrt{\frac{49}{120}}</math>
| <math>\sqrt{\frac{1}{42}}</math>
| <math>-\sqrt{\frac{3}{10}}</math>
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;{{sfrac|1|2}}
| <math>\sqrt{\frac{15}{28}}</math>
| <math>-\sqrt{\frac{1}{60}}</math>
| <math>-\sqrt{\frac{25}{84}}</math>
| <math>\sqrt{\frac{3}{20}}</math>
|----- valign="center"  align="center"
! –{{sfrac|1|2}},&nbsp;{{sfrac|3|2}}
| <math>\sqrt{\frac{5}{28}}</math>
| <math>-\sqrt{\frac{9}{20}}</math>
| <math>\sqrt{\frac{9}{28}}</math>
| <math>-\sqrt{\frac{1}{20}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 0}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! 4 
! 3 
! 2 
! 1
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;–{{sfrac|3|2}}
| <math>\sqrt{\frac{1}{14}}</math>
| <math>\sqrt{\frac{3}{10}}</math>
| <math>\sqrt{\frac{3}{7}}</math> 
| <math>\sqrt{\frac{1}{5}}</math>
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;–{{sfrac|1|2}} 
| <math>\sqrt{\frac{3}{7}}</math>
| <math>\sqrt{\frac{1}{5}}</math>
| <math>-\sqrt{\frac{1}{14}}</math>
| <math>-\sqrt{\frac{3}{10}}</math>
|----- valign="center"  align="center"
! –{{sfrac|1|2}},&nbsp;{{sfrac|1|2}} 
| <math>\sqrt{\frac{3}{7}}</math>
| <math>-\sqrt{\frac{1}{5}}</math>
| <math>-\sqrt{\frac{1}{14}}</math>
| <math>\sqrt{\frac{3}{10}}</math>
|----- valign="center"  align="center"
! –{{sfrac|3|2}},&nbsp;{{sfrac|3|2}}
| <math>\sqrt{\frac{1}{14}}</math>
| <math>-\sqrt{\frac{3}{10}}</math>
| <math>\sqrt{\frac{3}{7}}</math>
| <math>-\sqrt{\frac{1}{5}}</math>
|}

=== {{math|size=100%|1=&nbsp;''j''<sub>1</sub> = {{sfrac|5|2}}, ''j''<sub>2</sub> = 2}} ===

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|9|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|9|2}}
|----- valign="center"  align="center"
! {{sfrac|5|2}},&nbsp;2 
| <math>1</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|7|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|9|2}} 
! {{sfrac|7|2}}
|----- valign="center"  align="center"
! {{sfrac|5|2}},&nbsp;1 
| <math>\frac{2}{3}</math>
| <math>\sqrt{\frac{5}{9}}</math>
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;2 
| <math>\sqrt{\frac{5}{9}}</math>
| <math>-\frac{2}{3}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|5|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|9|2}} 
! {{sfrac|7|2}} 
! {{sfrac|5|2}}
|----- valign="center"  align="center"
! {{sfrac|5|2}},&nbsp;0 
| <math>\sqrt{\frac{1}{6}}</math>
| <math>\sqrt{\frac{10}{21}}</math>
| <math>\sqrt{\frac{5}{14}}</math>
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;1 
| <math>\sqrt{\frac{5}{9}}</math>
| <math>\sqrt{\frac{1}{63}}</math>
| <math>-\sqrt{\frac{3}{7}}</math>
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;2
| <math>\sqrt{\frac{5}{18}}</math>
| <math>-\sqrt{\frac{32}{63}}</math>
| <math>\sqrt{\frac{3}{14}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|3|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|9|2}} 
! {{sfrac|7|2}} 
! {{sfrac|5|2}} 
! {{sfrac|3|2}}
|----- valign="center"  align="center"
! {{sfrac|5|2}},&nbsp;−1
| <math>\sqrt{\frac{1}{21}}</math>
| <math>\sqrt{\frac{5}{21}}</math>
| <math>\sqrt{\frac{3}{7}}</math> 
| <math>\sqrt{\frac{2}{7}}</math>
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;0
| <math>\sqrt{\frac{5}{14}}</math>
| <math>\sqrt{\frac{2}{7}}</math>
| <math>-\sqrt{\frac{1}{70}}</math>
| <math>-\sqrt{\frac{12}{35}}</math>
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;1
| <math>\sqrt{\frac{10}{21}}</math>
| <math>-\sqrt{\frac{2}{21}}</math>
| <math>-\sqrt{\frac{6}{35}}</math>
| <math>\sqrt{\frac{9}{35}}</math>
|----- valign="center"  align="center"
! –{{sfrac|1|2}},&nbsp;2
| <math>\sqrt{\frac{5}{42}}</math>
| <math>-\sqrt{\frac{8}{21}}</math>
| <math>\sqrt{\frac{27}{70}}</math>
| <math>-\sqrt{\frac{4}{35}}</math>
|}

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = {{sfrac|1|2}}}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
! {{sfrac|9|2}} 
! {{sfrac|7|2}} 
! {{sfrac|5|2}} 
! {{sfrac|3|2}}
! {{sfrac|1|2}}
|----- valign="center"  align="center"
! {{sfrac|5|2}},&nbsp;-2
| <math>\sqrt{\frac{1}{126}}</math>
| <math>\sqrt{\frac{4}{63}}</math>
| <math>\sqrt{\frac{3}{14}}</math>
| <math>\sqrt{\frac{8}{21}}</math>
| <math>\sqrt{\frac{1}{3}}</math>
|----- valign="center"  align="center"
! {{sfrac|3|2}},&nbsp;−1
| <math>\sqrt{\frac{10}{63}}</math>
| <math>\sqrt{\frac{121}{315}}</math>
| <math>\sqrt{\frac{6}{35}}</math>
| <math>-\sqrt{\frac{2}{105}}</math>
| <math>-\sqrt{\frac{4}{15}}</math>
|----- valign="center"  align="center"
! {{sfrac|1|2}},&nbsp;0
| <math>\sqrt{\frac{10}{21}}</math>
| <math>\sqrt{\frac{4}{105}}</math>
| <math>-\sqrt{\frac{8}{35}}</math>
| <math>-\sqrt{\frac{2}{35}}</math>
| <math>\sqrt{\frac{1}{5}}</math>
|----- valign="center"  align="center"
! –{{sfrac|1|2}},&nbsp;1
| <math>\sqrt{\frac{20}{63}}</math>
| <math>-\sqrt{\frac{14}{45}}</math>
| <math>0</math>
| <math>\sqrt{\frac{5}{21}}</math>
| <math>-\sqrt{\frac{2}{15}}</math>
|----- valign="center"  align="center"
! –{{sfrac|3|2}},&nbsp;2
| <math>\sqrt{\frac{5}{126}}</math>
| <math>-\sqrt{\frac{64}{315}}</math>
| <math>\sqrt{\frac{27}{70}}</math>
| <math>-\sqrt{\frac{32}{105}}</math>
| <math>\sqrt{\frac{1}{15}}</math>
|}

=== {{math|size=100%|1=&nbsp;''j''<sub>1</sub> = {{sfrac|5|2}}, ''j''<sub>2</sub> = {{sfrac|5|2}}}} ===

{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 5}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
!5
|----- valign="center" align="center"
! {{sfrac|5|2}},&nbsp;{{sfrac|5|2}} 
| <math>1</math>
|}
{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 4}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
!5
!4
|----- valign="center" align="center"
! {{sfrac|5|2}},&nbsp;{{sfrac|3|2}} 
| <math>\sqrt{\frac{1}{2}}</math>
|<math>\sqrt{\frac{1}{2}}</math>
|----- valign="center" align="center"
! {{sfrac|3|2}},&nbsp;{{sfrac|5|2}} 
|<math>\sqrt{\frac{1}{2}}</math>
| <math>-\sqrt{\frac{1}{2}}</math>
|}
{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 3}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
!5
!4
!3
|----- valign="center" align="center"
! {{sfrac|5|2}},&nbsp;{{sfrac|1|2}} 
| <math>\sqrt{\frac{2}{9}}</math>
| <math>\sqrt{\frac{1}{2}}</math>
| <math>\sqrt{\frac{5}{18}}</math>
|----- valign="center" align="center"
! {{sfrac|3|2}},&nbsp;{{sfrac|3|2}} 
| <math>\sqrt{\frac{5}{9}}</math>
| <math>0</math>
| <math>-{\frac{2}{3}}</math>
|----- valign="center" align="center"
! {{sfrac|1|2}},&nbsp;{{sfrac|5|2}}
| <math>\sqrt{\frac{2}{9}}</math>
| <math>-\sqrt{\frac{1}{2}}</math>
|<math>\sqrt{\frac{5}{18}}</math>
|}
{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 2}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
!5
!4
!3
!2
|----- valign="center" align="center"
! {{sfrac|5|2}},&nbsp;–{{sfrac|1|2}}
| <math>\sqrt{\frac{1}{12}}</math>
| <math>\sqrt{\frac{9}{28}}</math>
| <math>\sqrt{\frac{5}{12}}</math>
| <math>\sqrt{\frac{5}{28}}</math>
|----- valign="center" align="center"
! {{sfrac|3|2}},&nbsp;{{sfrac|1|2}}
| <math>\sqrt{\frac{5}{12}}</math>
| <math>\sqrt{\frac{5}{28}}</math>
| <math>-\sqrt{\frac{1}{12}}</math>
|<math>-\sqrt{\frac{9}{28}}</math>
|----- valign="center" align="center"
! {{sfrac|1|2}},&nbsp;{{sfrac|3|2}}
|<math>\sqrt{\frac{5}{12}}</math>
|<math>-\sqrt{\frac{5}{28}}</math>
|<math>-\sqrt{\frac{1}{12}}</math>
| <math>\sqrt{\frac{9}{28}}</math>
|----- valign="center" align="center"
! –{{sfrac|1|2}},&nbsp;{{sfrac|5|2}}
|<math>\sqrt{\frac{1}{12}}</math>
|<math>-\sqrt{\frac{9}{28}}</math>
| <math>\sqrt{\frac{5}{12}}</math>
|<math>-\sqrt{\frac{5}{28}}</math>
|}
{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 1}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
!5
! 4 
! 3 
! 2
! 1
|----- valign="center" align="center"
! {{sfrac|5|2}},&nbsp;–{{sfrac|3|2}}
| <math>\sqrt{\frac{1}{42}}</math>
| <math>\sqrt{\frac{1}{7}}</math>
| <math>\sqrt{\frac{1}{3}}</math>
| <math>\sqrt{\frac{5}{14}}</math>
| <math>\sqrt{\frac{1}{7}}</math>
|----- valign="center" align="center"
! {{sfrac|3|2}},&nbsp;–{{sfrac|1|2}}
| <math>\sqrt{\frac{5}{21}}</math>
| <math>\sqrt{\frac{5}{14}}</math>
|<math>\sqrt{\frac{1}{30}}</math>
|<math>-\sqrt{\frac{1}{7}}</math>
|<math>-\sqrt{\frac{8}{35}}</math>
|----- valign="center" align="center"
! {{sfrac|1|2}},&nbsp;{{sfrac|1|2}}
| <math>\sqrt{\frac{10}{21}}</math>
| <math>0</math>
| <math>-\sqrt{\frac{4}{15}}</math>
| <math>0</math>
|<math>\sqrt{\frac{9}{35}}</math>
|----- valign="center" align="center"
! –{{sfrac|1|2}},&nbsp;{{sfrac|3|2}}
| <math>\sqrt{\frac{5}{21}}</math>
| <math>-\sqrt{\frac{5}{14}}</math>
|<math>\sqrt{\frac{1}{30}}</math>
| <math>\sqrt{\frac{1}{7}}</math>
| <math>-\sqrt{\frac{8}{35}}</math>
|----- valign="center" align="center"
! –{{sfrac|3|2}},&nbsp;{{sfrac|5|2}}
| <math>\sqrt{\frac{1}{42}}</math>
|<math>-\sqrt{\frac{1}{7}}</math>
|<math>\sqrt{\frac{1}{3}}</math>
|<math>-\sqrt{\frac{5}{14}}</math>
| <math>\sqrt{\frac{1}{7}}</math>
|}
{| class="wikitable"
|+ {{math|size=100%|1=''m'' = 0}} 
|----- valign="center" 
! {{diagonal split header|{{math|size=100%|''m''<sub>1</sub>,&nbsp;''m''<sub>2</sub>}}|{{mvar|j}}}}
!5
! 4 
! 3 
! 2
! 1
!0
|----- valign="center" align="center"
! {{sfrac|5|2}},&nbsp;–{{sfrac|5|2}}
| <math>\sqrt{\frac{1}{252}}</math>
| <math>\sqrt{\frac{1}{28}}</math>
|<math>\sqrt{\frac{5}{36}}</math>
| <math>\sqrt{\frac{25}{84}}</math>
| <math>\sqrt{\frac{5}{14}}</math>
|<math>\sqrt{\frac{1}{6}}</math>
|----- valign="center" align="center"
! {{sfrac|3|2}},&nbsp;–{{sfrac|3|2}}
|<math>\sqrt{\frac{25}{252}}</math>
| <math>\sqrt{\frac{9}{28}}</math>
|<math>\sqrt{\frac{49}{180}}</math>
|<math>\sqrt{\frac{1}{84}}</math>
| <math>-\sqrt{\frac{9}{70}}</math>
|<math>-\sqrt{\frac{1}{6}}</math>
|----- valign="center" align="center"
! {{sfrac|1|2}},&nbsp;–{{sfrac|1|2}}
|<math>\sqrt{\frac{25}{63}}</math>
| <math>\sqrt{\frac{1}{7}}</math>
| <math>-\sqrt{\frac{4}{45}}</math>
| <math>-\sqrt{\frac{4}{21}}</math>
| <math>\sqrt{\frac{1}{70}}</math>
|<math>\sqrt{\frac{1}{6}}</math>
|----- valign="center" align="center"
! –{{sfrac|1|2}},&nbsp;{{sfrac|1|2}}
| <math>\sqrt{\frac{25}{63}}</math>
|<math>-\sqrt{\frac{1}{7}}</math>
|<math>-\sqrt{\frac{4}{45}}</math>
| <math>\sqrt{\frac{4}{21}}</math>
|<math>\sqrt{\frac{1}{70}}</math>
|<math>-\sqrt{\frac{1}{6}}</math>
|----- valign="center" align="center"
! –{{sfrac|3|2}},&nbsp;{{sfrac|3|2}}
| <math>\sqrt{\frac{25}{252}}</math>
|<math>-\sqrt{\frac{9}{28}}</math>
| <math>\sqrt{\frac{49}{180}}</math>
|<math>-\sqrt{\frac{1}{84}}</math>
|<math>-\sqrt{\frac{9}{70}}</math>
|<math>\sqrt{\frac{1}{6}}</math>
|-
!–{{sfrac|5|2}},&nbsp;{{sfrac|5|2}}
|<math>\sqrt{\frac{1}{252}}</math>
|<math>-\sqrt{\frac{1}{28}}</math>
|<math>\sqrt{\frac{5}{36}}</math>
|<math>-\sqrt{\frac{25}{84}}</math>
|<math>\sqrt{\frac{5}{14}}</math>
|<math>-\sqrt{\frac{1}{6}}</math>
|}

==SU(N) Clebsch–Gordan coefficients==

Algorithms to produce Clebsch–Gordan coefficients for higher values of <math>j_1</math> and <math>j_2</math>, or for the su(N) algebra instead of su(2), are known.<ref>{{cite journal |last=Alex |first=A. |author2=M. Kalus |author3=A. Huckleberry |author4=J. von Delft |title=A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch–Gordan coefficients |journal=J. Math. Phys. |volume=82 |date=February 2011 |pages=023507 |doi=10.1063/1.3521562 |bibcode=2011JMP....52b3507A |arxiv=1009.0437 }}</ref>
A [http://homepages.physik.uni-muenchen.de/~vondelft/Papers/ClebschGordan/ web interface for tabulating SU(N) Clebsch–Gordan coefficients] is readily available.

==References==
<references/>

==External links==
* Online, [[Java (programming language)|Java]]-based [http://personal.ph.surrey.ac.uk/~phs3ps/cgjava.html Clebsch–Gordan Coefficient Calculator] by Paul Stevenson
* [http://functions.wolfram.com/HypergeometricFunctions/ClebschGordan/06/01/ Other formulae] for Clebsch–Gordan coefficients.
*  [http://homepages.physik.uni-muenchen.de/~vondelft/Papers/ClebschGordan/ Web interface for tabulating SU(N) Clebsch–Gordan coefficients]

{{DEFAULTSORT:Table of Clebsch-Gordan coefficients}}
[[Category:Representation theory of Lie groups]]
[[Category:Mathematical tables|Clebsch–Gordan coefficients]]