The Raman tensors for C6h 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

${C}_{6 {h}}$ $( 6 / {m} )$	1	$6^{+}$	$3^{+}$	2	$3^-$	$6^{-}$	-1	$-6^{+}$	$-3^{+}$	m	$-3^{-}$	$-6^{-}$	Basis	MPG
${A}_{{g}}$	1	1	1	1	1	1	1	1	1	1	1	1	${x}^2+{y}^2$ ${z}^2, {J}_{{z}}$	6/m. 1
${A}_{{u}}$	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	z	$6 / {m}^{\prime}$
${B}_{{g}}$	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1		$6 ^\prime/{m}^{\prime}$
${B}_{{u}}$	1	-1	1	-1	1	-1	-1	1	-1	1	-1	1		$6'/{{m}}$
${ }^1 {E}_{2 {g}}$	1	$\omega$	$\omega^2$	1	$\omega$	$\omega^2$	1	$\omega$	$\omega^2$	1	$\omega$	$\omega^2$	$( {x}^2-{y}^2$ $xy$ )
${ }^2 {E}_{2 {g}}$	1	$\omega^2$	$\omega$	1	$\omega^2$	$\omega$	1	$\omega^2$	$\omega$	1	$\omega^2$	$\omega$	$( {x}^2-{y}^2$ $xy$ )
${ }^1 {E}_{2 {u}}$	1	$\omega$	$\omega^2$	1	$\omega$	$\omega^2$	-1	-ω	0	-1	$\omega$	0
${ }^2 {E}_{2 {u}}$	1	$\omega^2$	$\omega$	1	$\omega^2$	$\omega$	-1	0	$-\omega$	-1	0	$\omega$
${ }^2 {E}_{1 {g}}$	1	0	$\omega$	-1	$\omega^2$	$-\omega$	1	0	$\omega$	-1	$\omega^2$	$\omega$	$({xz}, {yz})$ $\left({J}_{{x}}, {J}_{{y}}\right)$
${ }^1 {E}_{1 {g}}$	1	-ω	$\omega^2$	-1	$\omega$	0	1	$\omega$	$\omega^2$	-1	$\omega$	0	$({xz}, {yz})$ $\left({J}_{{x}}, {J}_{{y}}\right)$
${ }^2 {E}_{1 {u}}$	1	0	$\omega$	-1	$\omega^2$	$-\omega$	-1	$\omega^2$	$-\omega$	1	0	$\omega$	$(x, y)$
${ }^1 {E}_{1 {u}}$	1	-ω	$\omega^2$	-1	$\omega$	0	-1	$\omega$	0	1	-ω	$\omega^2$	$(x, y)$

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

Direct product table for IRREPs

$6 / {m}\left({C}_{6 {h}}\right)$	${A}_{g}$	${A}_{u}$	${B}_{g}$	${B}_{u}$	$^1 {E}_{1 g}$	$^2 {E}_{1 g}$	$^1 {E}_{1 u}$	$^2 {E}_{1 u}$	$^1 {E}_{2 g}$	$^2 {E}_{2 g}$	$^1 {E}_{2 u}$	$^2 {E}_{2 u}$
${A}_{g}$	${A}_{g}$	${A}_{u}$	${B}_{g}$	${B}_{u}$	$^1 {E}_{1 g}$	$^2 {E}_{1 g}$	$^1 {E}_{1 u}$	$^2 {E}_{1 u}$	$^1 E_{2 { g }}$	$^2 {E}_{2 g}$	$^1 E_{2 u}$	$^2 {E}_{2 u}$
${A}_{u}$	${A}_{u}$	${A}_{g}$	${B}_{u}$	${B}_{g}$	$^1 E_{1 u}$	$^2 {E}_{1 u}$	$^1 E_{1 { g }}$	$^2 {E}_{1 g}$	$^1 E_{2 u}$	$^2 E_{2 u}$	$^1 E_{2 g}$	$^2 {E}_{2 g}$
${B}_{g}$	${B}_{g}$	${B}_{u}$	${A}_{g}$	${A}_{u}$	$^1 E_{2 { g }}$	$^2 {E}_{2 g}$	$^1 E_{2 u}$	$^2 E_{2 u}$	$^1 E_{1 g}$	$^2 {E}_{1 g}$	$^1 E_{1 u}$	$^2 {E}_{1 u}$
${B}_{u}$	${B}_{u}$	${B}_{g}$	${A}_{u}$	${A}_{g}$	$^1 E_{2 u}$	$^2 {E}_{2 u}$	$^1 E_{2 { g }}$	$^2 {E}_{2 g}$	$^1 E_{1 u}$	$^2 {E}_{1 u}$	$^1 E_{1 { g }}$	$^2 {E}_{1 g}$
$^1 {E}_{1 g}$	$^1 {E}_{1 g}$	$^1 E_{1 u}$	$^1 {E}_{2 { g }}$	$^1 E_{2 u}$	$^2 {E}_{1 { g }}$	${A}_{g}$	$^2 {E}_{1 u}$	${A}_{u}$	$^2 {E}_{2 g}$	${B}_{g}$	$^2 E_{2 u}$	${B}_{u}$
$^2 {E}_{1 g}$	$^2 {E}_{1 g}$	$^2 {E}_{1 u}$	$^2 {E}_{2 g}$	$^2 E_{2 u}$	${A}_{g}$	$^1 E_{1 { g }}$	${A}_{u}$	$^1 E_{1 u}$	${B}_{g}$	$^1 E_{2 { g }}$	${B}_{u}$	$^1 {E}_{2 u}$
$^1 {E}_{1 u}$	$^1 {E}_{1 u}$	$^1 E_{1 { g }}$	$^1 E_{2 u}$	$^1 E_{2 { g }}$	$^2 {E}_{1 u}$	${A}_{u}$	$^2 E_{1 { g }}$	${A}_{g}$	$^2 E_{2 u}$	${B}_{u}$	$^2 {E}_{2 g}$	${B}_{g}$
$^2 {E}_{1 u}$	$^2 {E}_{1 u}$	$^2 {E}_{1 g}$	$^2 E_{2 u}$	$^2 {E}_{2 g}$	${A}_{u}$	$^1 E_{1 u}$	${A}_{g}$	$^1 E_{1 { g }}$	${B}_{u}$	$^1 E_{2 u}$	${B}_{g}$	$^1 {E}_{2 g}$
$^1 {E}_{2 g}$	$^1 {E}_{2 g}$	$^1 E_{2 u}$	$^1 E_{1 { g }}$	$^1 E_{1 u}$	$^2 {E}_{2 g}$	${B}_{g}$	$^2 E_{2 u}$	${B}_{u}$	$^2 {E}_{1 g}$	${A}_{g}$	$^2 {E}_{1 u}$	${A}_{u}$
$^2 {E}_{2 g}$	$^2 {E}_{2 g}$	$^2 E_{2 u}$	$^2 {E}_{1 g}$	$^2 E_{1 u}$	${B}_{g}$	$^1 E_{2 { g }}$	${B}_{u}$	$^1 E_{2 u}$	${A}_{g}$	$^1 E_{1 { g }}$	${A}_{u}$	$^1 {E}_{1 u}$
$^1 {E}_{2 u}$	$^1 {E}_{2 u}$	$^1 E_{2 { g }}$	$^1 {E}_{1 u}$	$^1 E_{1 { g }}$	$^2 E_{2 u}$	${B}_{u}$	$^2 {E}_{2 g}$	${B}_{g}$	$^2 {E}_{1 u}$	${A}_{u}$	$^2 {E}_{1 g}$	${A}_{g}$
$^2 {E}_{2 u}$	$^2 {E}_{2 u}$	$^2 {E}_{2 g}$	$^2 {E}_{1 u}$	$^2 {E}_{1 g}$	${B}_{u}$	$^1 E_{2 u}$	${B}_{g}$	$^1 {E}_{2 g}$	${A}_{u}$	$^1 {E}_{1 u}$	${A}_{g}$	$^1 {E}_{1 g}$

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