The Raman tensors for Th and corresponding MPGs

Jump to the bottom Back to the main page. *Note: The second row present the symmetric components of the Raman tensors of each irrep, while the remaining rows correspond to the antisymmetric Raman tensor for the isomorphic MPGs.

Character table and corresponding MPGs

${T}_{{h}}({m}-$ 3)	1	$3^{+}$	$3^{-}$	2	-1	$-3^{+}$	$-3^{-}$	m	Basis	MPG
Mult.	1	4	4	3	1	4	4	3	-
${A}_{{g}}$	1	1	1	1	1	1	1	1	${x}^2+{y}^2+{z}^2$	$m-3.1$
${ }^1 {E}_{{g}}$	1	$\omega$	$\omega^2$	1	1	$\omega$	$\omega^2$	1	$\left(2 {z}^2-{x}^2-{y}^2\right.$ ${x}^2-{y}^2$ )
${ }^2 {E}_{{g}}$	1	$\omega^2$	$\omega$	1	1	$\omega^2$	$\omega$	1	$^1{E}_g$
${T}_{{g}}$	3	0	0	-1	3	0	0	-1	$\begin{aligned} & ({xz}, {yz}, {xy}), \\ & \left({J}_{{x}}, {J}_{{y}}, {J}_{{z}}\right) \end{aligned}$
${A}_u$	1	1	1	1	-1	-1	-1	-1		$m'-3'$
${ }^1 {E}_{{u}}$	1	$\omega$	$\omega^2$	1	-1	-ω	0	-1
${ }^2 {E}_{{u}}$	1	$\omega^2$	$\omega$	1	-1	0	$-\omega$	-1
${T}_{{u}}$	3	0	0	-1	-3	0	0	1	(x, y, z)

$\omega=\exp (2 \mathrm{i} \pi / 3)$

Direct product table for IRREPs

${m} \overline{3}\left({~T}_{{h}}\right)$	${A}_{{g}}$	${A}_{{u}}$	${ }^1 {E}_{{g}}$	${ }^2 {E}_{{g}}$	${ }^1 {E}_{{u}}$	${ }^2 {E}_{{u}}$	${T}_{{g}}$	${T}_{{u}}$
${A}_{{g}}$	${A}_{{g}}$	${A}_{{u}}$	${ }^1 {E}_{{g}}$	${ }^2 {E}_{{g}}$	${ }^1 {E}_u$	${ }^2 {E}_{{u}}$	${T}_{{g}}$	${T}_{{u}}$
${A}_{{u}}$	${A}_{{u}}$	${A}_{{g}}$	${ }^1 {E}_{{u}}$	${ }^2 {E}_u$	${ }^1 {E}_{{g}}$	${ }^2 {E}_{{g}}$	${T}_{{u}}$	${T}_{{g}}$
${ }^1 {E}_{{g}}$	${ }^1 {E}_{{g}}$	${ }^1 {E}_{{u}}$	${ }^2 {E}_{{g}}$	${A}_{{g}}$	${ }^2 {E}_{{u}}$	${A}_{{u}}$	${T}_{{g}}$	${T}_{{u}}$
${ }^2 {E}_{{g}}$	${ }^2 {E}_{{g}}$	${ }^2 {E}_u$	${A}_{{g}}$	${ }^1 {E}_{{g}}$	${A}_{{u}}$	${ }^1 E_u$	${T}_{{g}}$	${T}_{{u}}$
${ }^1 {E}_{{u}}$	${ }^1 {E}_{{u}}$	${ }^1 {E}_{{g}}$	${ }^2 {E}_u$	${A}_{{u}}$	${ }^2 {E}_{{g}}$	${A}_{{g}}$	${T}_{{u}}$	${T}_{{g}}$
${ }^2 {E}_{{u}}$	${ }^2 {E}_{{u}}$	${ }^2 {E}_{{g}}$	${A}_{{u}}$	${ }^1 {E}_u$	${A}_{{g}}$	${ }^1 {E}_{{g}}$	${T}_{{u}}$	${T}_{{g}}$
${T}_{{g}}$	${T}_{{g}}$	${T}_{{u}}$	${T}_{{g}}$	${T}_{{g}}$	${T}_{{u}}$	${T}_{{u}}$	${A}_{{g}}+{ }^1 {E}_{{g}}+{ }^2 {E}_{{g}}+2 {~T}_{{g}}$	$A_u+{ }^1 E_u+{ }^2 E_u+2 T_u$
${T}_{{u}}$	${T}_{{u}}$	${T}_{{g}}$	${T}_{{u}}$	${T}_{{u}}$	${T}_{{g}}$	${T}_{{g}}$	$A_u+{ }^1 E_u+{ }^2 E_u+2 T_u$	$A_g+{ }^1 E_g+{ }^2 E_g+2 T_g$

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