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	51	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	45	0	1	-1	5

**Decomposition into Irreducible representations**
Motion	A₁	A₂	E	T₁	T₂	Total
Cartesian 3N	3	1	4	5	8	21
Translation (x,y,z)	0	0	0	0	1	1
Rotation (R_x,R_y,R_z)	0	0	0	1	0	1
Vibration	3	1	4	4	7	19

**Molecule Parameter**
Number of Atoms (N)	17
Number of internal coordinates	45
Number of independant internal coordinates	3
Number of vibrational modes	19

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	E	T₁	T₂	Total
Linear (IR)	3	1	4	4	7	7 / 12
Quadratic (Raman)	3	1	4	4	7	14 / 5
IR + Raman	-	1	-	4	7	7 / 5

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	8C₃	3C₂	6S₄	6_d
linear	1	45	0	1	-1	5
quadratic	2	1.035	0	23	1	35
cubic	3	16.215	15	23	-1	135
quartic	4	194.580	0	276	12	580
quintic	5	1.906.884	0	276	-12	1.876
sextic	6	15.890.700	120	2.300	12	6.300

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	E	T₁	T₂
linear	1	3	1	4	4	7
quadratic	2	55	37	92	118	135
cubic	3	717	650	1.352	1.990	2.058
quartic	4	8.290	7.994	16.284	24.146	24.430
quintic	5	79.954	79.022	158.976	237.854	238.798
sextic	6	664.018	660.862	1.324.760	1.984.478	1.987.622

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