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Point Group Tables of D3d(-3m)

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Character Table of the group D3d(-3m) *
D3d(-3m)#132-1-3mdfunctions
Mult.-123123
A1gΓ1+111111x2+y2,z2
A2gΓ2+11-111-1Jz
EgΓ3+2-102-10(x2-y2,xy),(xz,yz),(Jx,Jy)
A1uΓ1-111-1-1-1
A2uΓ2-11-1-1-11z
EuΓ3-2-10-210(x,y)



Subgroups of the group D3d(-3m)
SubgroupOrderIndex
D3d(-3m)121
C3v(3m)62
D3(32)62
C3i(-3)62
C3(3)34
C2h(2/m)43
Cs(m)26
C2(2)26
Ci(-1)26
C1(1)112

[ Subduction tables ]

Multiplication Table of irreducible representations of the group D3d(-3m)
D3d(-3m)A1gA1uA2gA2uEuEg
A1gA1gA1uA2gA2uEuEg
A1uA1gA2uA2gEgEu
A2gA1gA1uEuEg
A2uA1gEgEu
EuA1g+A2g+EgA1u+A2u+Eu
EgA1g+A2g+Eg

[ Note: the table is symmetric ]


Symmetrized Products of Irreps
D3d(-3m)A1gA1uA2gA2uEuEg
[A1g x A1g]1·····
[A1u x A1u]1·····
[A2g x A2g]1·····
[A2u x A2u]1·····
[Eu x Eu]1····1
[Eg x Eg]1····1


Antisymmetrized Products of Irreps
D3d(-3m)A1gA1uA2gA2uEuEg
{A1g x A1g}······
{A1u x A1u}······
{A2g x A2g}······
{A2u x A2u}······
{Eu x Eu}··1···
{Eg x Eg}··1···


Irreps Decompositions
D3d(-3m)A1gA1uA2gA2uEuEg
V···11·
[V2]2····2
[V3]·1·33·
[V4]4·1··5
A··1··1
[A2]2····2
[A3]1·3··3
[A4]4·1··5
[V2]xV·2·46·
[[V2]2]6·1··7
{V2}··1··1
{A2}··1··1
{[V2]2}2·3··5

V ≡ the vector representation
A ≡ the axial representation


IR Selection Rules
IRA1gA1uA2gA2uEuEg
A1gxx
A1uxx
A2gxx
A2uxx
Euxxx
Egxxx

[ Note: x means allowed ]


Raman Selection Rules
RamanA1gA1uA2gA2uEuEg
A1gxx
A1uxx
A2gxx
A2uxx
Euxxx
Egxxx

[ Note: x means allowed ]


Irreps Dimensions Irreps of the point group
Subduction of the rotation group D(L) to irreps of the group D3d(-3m)
L2L+1A1gA1uA2gA2uEuEg
011·····
13···11·
251····2
37·1·22·
492·1··3
511·1·24·
6133·2··4
715·2·35·
8173·2··6
919·3·46·
10214·3··7



* George F. Koster, John O. Dimmock, Robert G. Wheeler, Hermann Statz (1963). Properties of the thirty-two point groups. Published by the M.I.T. press, Cambridge, Massachusetts.
* Simon L. Altmann and Peter Herzig (1994). Point-Group Theory Tables. Oxford Science Publications.


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