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Point Group Tables of D6h(6/mmm)

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Character Table of the group D6h(6/mmm) *
D6h(6/mmm)#1632z2h2h'-1-6-3mzmdmvfunctions
Mult.-122133122133
A1gΓ1+111111111111x2+y2,z2
A1uΓ1-111111-1-1-1-1-1-1
A2gΓ2+1111-1-11111-1-1Jz
A2uΓ2-1111-1-1-1-1-1-111z
B1gΓ3+1-11-11-11-11-11-1
B1uΓ3-1-11-11-1-11-11-11
B2gΓ4+1-11-1-111-11-1-11
B2uΓ4-1-11-1-11-11-111-1
E2uΓ6-2-1-1200-211-200
E2gΓ6+2-1-12002-1-1200(x2-y2,xy)
E1uΓ5-21-1-200-2-11200(x,y)
E1gΓ5+21-1-20021-1-200(xz,yz),(Jx,Jy)



Subgroups of the group D6h(6/mmm)
SubgroupOrderIndex
D6h(6/mmm)241
D3h(-62m)122
C6v(6mm)122
D6(622)122
C6h(6/m)122
D3d(-3m)122
C3h(-6)64
C6(6)64
C3v(3m)64
D3(32)64
C3i(-3)64
C3(3)38
D2h(mmm)83
C2v(mm2)46
D2(222)46
C2h(2/m)46
C2(2)212
Cs(m)212
Ci(-1)212
C1(1)124

[ Subduction tables ]

Multiplication Table of irreducible representations of the group D6h(6/mmm)
D6h(6/mmm)A1gA1uA2gA2uB1gB1uB2gB2uE2uE2gE1uE1g
A1gA1gA1uA2gA2uB1gB1uB2gB2uE2uE2gE1uE1g
A1uA1gA2uA2gB1uB1gB2uB2gE2gE2uE1gE1u
A2gA1gA1uB2gB2uB1gB1uE2uE2gE1uE1g
A2uA1gB2uB2gB1uB1gE2gE2uE1gE1u
B1gA1gA1uA2gA2uE1uE1gE2uE2g
B1uA1gA2uA2gE1gE1uE2gE2u
B2gA1gA1uE1uE1gE2uE2g
B2uA1gE1gE1uE2gE2u
E2uA1g+A2g+E2gA1u+A2u+E2uB1g+B2g+E1gB1u+B2u+E1u
E2gA1g+A2g+E2gB1u+B2u+E1uB1g+B2g+E1g
E1uA1g+A2g+E2gA1u+A2u+E2u
E1gA1g+A2g+E2g

[ Note: the table is symmetric ]


Symmetrized Products of Irreps
D6h(6/mmm)A1gA1uA2gA2uB1gB1uB2gB2uE2uE2gE1uE1g
[A1g x A1g]1···········
[A1u x A1u]1···········
[A2g x A2g]1···········
[A2u x A2u]1···········
[B1g x B1g]1···········
[B1u x B1u]1···········
[B2g x B2g]1···········
[B2u x B2u]1···········
[E2u x E2u]1········1··
[E2g x E2g]1········1··
[E1u x E1u]1········1··
[E1g x E1g]1········1··


Antisymmetrized Products of Irreps
D6h(6/mmm)A1gA1uA2gA2uB1gB1uB2gB2uE2uE2gE1uE1g
{A1g x A1g}············
{A1u x A1u}············
{A2g x A2g}············
{A2u x A2u}············
{B1g x B1g}············
{B1u x B1u}············
{B2g x B2g}············
{B2u x B2u}············
{E2u x E2u}··1·········
{E2g x E2g}··1·········
{E1u x E1u}··1·········
{E1g x E1g}··1·········


Irreps Decompositions
D6h(6/mmm)A1gA1uA2gA2uB1gB1uB2gB2uE2uE2gE1uE1g
V···1······1·
[V2]2········1·1
[V3]···2·1·11·2·
[V4]3···1·1··3·2
A··1········1
[A2]2········1·1
[A3]··2·1·1··1·2
[A4]3···1·1··3·2
[V2]xV·1·3·1·12·4·
[[V2]2]5···1·1··4·3
{V2}··1········1
{A2}··1········1
{[V2]2}1·2·1·1··2·3

V ≡ the vector representation
A ≡ the axial representation


IR Selection Rules
IRA1gA1uA2gA2uB1gB1uB2gB2uE2uE2gE1uE1g
A1gxx
A1uxx
A2gxx
A2uxx
B1gxx
B1uxx
B2gxx
B2uxx
E2uxxxx
E2gxxxx
E1uxxxx
E1gxxxx

[ Note: x means allowed ]


Raman Selection Rules
RamanA1gA1uA2gA2uB1gB1uB2gB2uE2uE2gE1uE1g
A1gxxx
A1uxxx
A2gxxx
A2uxxx
B1gxxx
B1uxxx
B2gxxx
B2uxxx
E2uxxxxxx
E2gxxxxxx
E1uxxxxxx
E1gxxxxxx

[ Note: x means allowed ]


Irreps Dimensions Irreps of the point group
Subduction of the rotation group D(L) to irreps of the group D6h(6/mmm)
L2L+1A1gA1uA2gA2uB1gB1uB2gB2uE2uE2gE1uE1g
011···········
13···1······1·
251········1·1
37···1·1·11·1·
491···1·1··2·1
511···1·1·12·2·
6132·1·1·1··2·2
715·1·2·1·12·3·
8172·1·1·1··3·3
919·1·2·2·23·3·
10212·1·2·2··4·3



* 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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