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	36	0	-4	0	4
Translation (x,y,z)	3	0	-1	-1	1
Rotation (R_x,R_y,R_z)	3	0	-1	1	-1
Vibration	30	0	-2	0	4

**Decomposition into Irreducible representations**
Motion	A₁	A₂	E	T₁	T₂	Total
Cartesian 3N	2	0	2	4	6	14
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	3	5	12

**Molecule Parameter**
Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	2
Number of vibrational modes	12

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	E	T₁	T₂	Total
Linear (IR)	2	0	2	3	5	5 / 7
Quadratic (Raman)	2	0	2	3	5	9 / 3
IR + Raman	-	0	-	3	5	5 / 3

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	8C₃	3C₂	6S₄	6_d
linear	1	30	0	-2	0	4
quadratic	2	465	0	17	-1	23
cubic	3	4.960	10	-32	0	72
quartic	4	40.920	0	152	8	256
quintic	5	278.256	0	-272	0	680
sextic	6	1.623.160	55	952	-8	1.904

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	E	T₁	T₂
linear	1	2	0	2	3	5
quadratic	2	27	16	43	50	62
cubic	3	224	188	402	606	642
quartic	4	1.790	1.658	3.448	5.034	5.158
quintic	5	11.730	11.390	23.120	34.646	34.986
sextic	6	68.243	67.295	135.483	202.298	203.254

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