Reduction formula for point group Td

Reduction formula for point group T_d

**Characters for molecular motions**
Motion	E	8C₃	3C₂	6S₄	6_d
Cartesian 3N	27	0	-1	-1	5
Translation (x,y,z)	3	0	-1	-1	1
Rotation (R_x,R_y,R_z)	3	0	-1	1	-1
Vibration	21	0	1	-1	5

**Decomposition into Irreducible representations**
Motion	A₁	A₂	E	T₁	T₂	Total
Cartesian 3N	2	0	2	2	5	11
Translation (x,y,z)	0	0	0	0	1	1
Rotation (R_x,R_y,R_z)	0	0	0	1	0	1
Vibration	2	0	2	1	4	9

**Molecule Parameter**
Number of Atoms (N)	9
Number of internal coordinates	21
Number of independant internal coordinates	2
Number of vibrational modes	9

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	E	T₁	T₂	Total
Linear (IR)	2	0	2	1	4	4 / 5
Quadratic (Raman)	2	0	2	1	4	8 / 1
IR + Raman	-	0	-	1	4	4 / 1

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	8C₃	3C₂	6S₄	6_d
linear	1	21	0	1	-1	5
quadratic	2	231	0	11	1	23
cubic	3	1.771	7	11	-1	75
quartic	4	10.626	0	66	6	226
quintic	5	53.130	0	66	-6	586
sextic	6	230.230	28	286	6	1.430

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	E	T₁	T₂
linear	1	2	0	2	1	4
quadratic	2	17	5	22	22	33
cubic	3	96	59	148	201	239
quartic	4	509	393	902	1.265	1.375
quintic	5	2.367	2.077	4.444	6.485	6.781
sextic	6	9.997	9.279	19.248	28.387	29.099

Literature

J.K.G. Watson, J. Mol. Spec. 41 229 (1972)
The Numbers of Structural Parameters and Potential Constants of Molecules
X.F. Zhou, P. Pulay. J. Comp. Chem. 10 No. 7, 935-938 (1989)
Characters for Symmetric and Antisymmetric Higher Powers of Representations:
Application to the Number of Anharmonic Force Constants in Symmetrical Molecules

Character tables for chemically important point groups

Character table for point group Td

Constructor University Bremen

Last update November, 13^th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement