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Point Group Tables of D2h(mmm)

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Character Table of the group D2h(mmm)*
D2h(mmm)#12z2y2x-1mzmymxfunctions
AgΓ1+11111111x2,y2,z2
B1gΓ3+11-1-111-1-1xy,Jz
B2gΓ2+1-11-11-11-1xz,Jy
B3gΓ4+1-1-111-1-11yz,Jx
AuΓ1-1111-1-1-1-1·
B1uΓ3-11-1-1-1-111z
B2uΓ2-1-11-1-11-11y
B3uΓ4-1-1-11-111-1x



Subgroups of the group D2h(mmm)
SubgroupOrderIndex
D2h(mmm)81
C2v(mm2)42
D2(222)42
C2h(2/m)42
C2(2)24
Cs(m)24
Ci(-1)24
C1(1)18

[ Subduction tables ]

Multiplication Table of irreducible representations of the group D2h(mmm)
D2h(mmm)AgAuB1gB1uB2gB2uB3gB3u
AgAgAuB1gB1uB2gB2uB3gB3u
Au·AgB1uB1gB2uB2gB3uB3g
B1g··AgAuB3gB3uB2gB2u
B1u···AgB3uB3gB2uB2g
B2g····AgAuB1gB1u
B2u·····AgB1uB1g
B3g······AgAu
B3u·······Ag

[ Note: the table is symmetric ]


Symmetrized Products of Irreps
D2h(mmm)AgAuB1gB1uB2gB2uB3gB3u
[Ag x Ag]1·······
[Au x Au]1·······
[B1g x B1g]1·······
[B1u x B1u]1·······
[B2g x B2g]1·······
[B2u x B2u]1·······
[B3g x B3g]1·······
[B3u x B3u]1·······


Antisymmetrized Products of Irreps
D2h(mmm)AgAuB1gB1uB2gB2uB3gB3u
{Ag x Ag}········
{Au x Au}········
{B1g x B1g}········
{B1u x B1u}········
{B2g x B2g}········
{B2u x B2u}········
{B3g x B3g}········
{B3u x B3u}········


Irreps Decompositions
D2h(mmm)AgAuB1gB1uB2gB2uB3gB3u
V···1·1·1
[V2]3·1·1·1·
[V3]·1·3·3·3
[V4]6·3·3·3·
A··1·1·1·
[A2]3·1·1·1·
[A3]1·3·3·3·
[A4]6·3·3·3·
[V2]xV·3·5·5·5
[[V2]2]9·4·4·4·
{V2}··1·1·1·
{A2}··1·1·1·
{[V2]2}3·4·4·4·

V ≡ the vector representation
A ≡ the axial representation


IR Selection Rules
IRAgAuB1gB1uB2gB2uB3gB3u
Ag···x·x·x
Au··x·x·x·
B1g·x···x·x
B1ux···x·x·
B2g·x·x···x
B2ux·x···x·
B3g·x·x·x··
B3ux·x·x···

[ Note: x means allowed ]


Raman Selection Rules
RamanAgAuB1gB1uB2gB2uB3gB3u
Agx·x·x·x·
Au·x·x·x·x
B1gx·x·x·x·
B1u·x·x·x·x
B2gx·x·x·x·
B2u·x·x·x·x
B3gx·x·x·x·
B3u·x·x·x·x

[ Note: x means allowed ]


Irreps Dimensions Irreps of the point group
Subduction of the rotation group D(L) to irreps of the group D2h(mmm)
L2L+1AgAuB1gB1uB2gB2uB3gB3u
011·······
13···1·1·1
252·1·1·1·
37·1·2·2·2
493·2·2·2·
511·2·3·3·3
6134·3·3·3·
715·3·4·4·4
8175·4·4·4·
919·4·5·5·5
10216·5·5·5·


* C. J. Bradley and A. P. Cracknell (1972) The Mathematical Theory of Symmetry in Solids Clarendon Press - Oxford
* Simon L. Altmann and Peter Herzig (1994). Point-Group Theory Tables. Oxford Science Publications.

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