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Point Group Tables of D4h(4/mmm)

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Character Table of the group D4h(4/mmm) *
D4h(4/mmm)#1242h2h'-1mz-4mvmdfunctions
Mult.-1122211222
A1gΓ1+1111111111x2+y2,z2
A2gΓ2+111-1-1111-1-1Jz
B1gΓ3+11-11-111-11-1x2-y2
B2gΓ4+11-1-1111-1-11xy
EgΓ5+2-20002-2000(xz,yz),(Jx,Jy)
A1uΓ1-11111-1-1-1-1-1
A2uΓ2-111-1-1-1-1-111z
B1uΓ3-11-11-1-1-11-11
B2uΓ4-11-1-11-1-111-1
EuΓ5-2-2000-22000(x,y)



Subgroups of the group D4h(4/mmm)
SubgroupOrderIndex
D4h(4/mmm)161
D2d(-42m)82
C4v(4mm)82
D4(422)82
C4h(4/m)82
C4(4)44
S4(-4)44
D2h(mmm)82
C2v(mm2)44
D2(222)44
C2h(2/m)44
Cs(m)28
C2(2)28
Ci(-1)28
C1(1)116

[ Subduction tables ]

Multiplication Table of irreducible representations of the group D4h(4/mmm)
D4h(4/mmm)A1gA1uA2gA2uB1gB1uB2gB2uEuEg
A1gA1gA1uA2gA2uB1gB1uB2gB2uEuEg
A1uA1gA2uA2gB1uB1gB2uB2gEgEu
A2gA1gA1uB2gB2uB1gB1uEuEg
A2uA1gB2uB2gB1uB1gEgEu
B1gA1gA1uA2gA2uEuEg
B1uA1gA2uA2gEgEu
B2gA1gA1uEuEg
B2uA1gEgEu
EuA1g+A2g+B1g+B2gA1u+A2u+B1u+B2u
EgA1g+A2g+B1g+B2g

[ Note: the table is symmetric ]


Symmetrized Products of Irreps
D4h(4/mmm)A1gA1uA2gA2uB1gB1uB2gB2uEuEg
[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·········
[Eu x Eu]1···1·1···
[Eg x Eg]1···1·1···


Antisymmetrized Products of Irreps
D4h(4/mmm)A1gA1uA2gA2uB1gB1uB2gB2uEuEg
{A1g x A1g}··········
{A1u x A1u}··········
{A2g x A2g}··········
{A2u x A2u}··········
{B1g x B1g}··········
{B1u x B1u}··········
{B2g x B2g}··········
{B2u x B2u}··········
{Eu x Eu}··1·······
{Eg x Eg}··1·······


Irreps Decompositions
D4h(4/mmm)A1gA1uA2gA2uB1gB1uB2gB2uEuEg
V···1····1·
[V2]2···1·1··1
[V3]···2·1·13·
[V4]4·1·2·2··3
A··1······1
[A2]2···1·1··1
[A3]··2·1·1··3
[A4]4·1·2·2··3
[V2]xV·1·3·2·25·
[[V2]2]6·1·3·3··4
{V2}··1······1
{A2}··1······1
{[V2]2}1·2·2·2··4

V ≡ the vector representation
A ≡ the axial representation


IR Selection Rules
IRA1gA1uA2gA2uB1gB1uB2gB2uEuEg
A1gxx
A1uxx
A2gxx
A2uxx
B1gxx
B1uxx
B2gxx
B2uxx
Euxxxxx
Egxxxxx

[ Note: x means allowed ]


Raman Selection Rules
RamanA1gA1uA2gA2uB1gB1uB2gB2uEuEg
A1gxxxx
A1uxxxx
A2gxxxx
A2uxxxx
B1gxxxx
B1uxxxx
B2gxxxx
B2uxxxx
Euxxxxx
Egxxxxx

[ Note: x means allowed ]


Irreps Dimensions Irreps of the point group
Subduction of the rotation group D(L) to irreps of the group D4h(4/mmm)
L2L+1A1gA1uA2gA2uB1gB1uB2gB2uEuEg
011·········
13···1····1·
251···1·1··1
37···1·1·12·
492·1·1·1··2
511·1·2·1·13·
6132·1·2·2··3
715·1·2·2·24·
8173·2·2·2··4
919·2·3·2·25·
10213·2·3·3··5



* 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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