Reduction formula for point group D3d

Reduction formula for point group D_3d

**Characters for molecular motions**
Motion	E	2C₃	3C'₂	i	2S₆	3_d
Cartesian 3N	36	0	-4	0	0	0
Translation (x,y,z)	3	0	-1	-3	0	1
Rotation (R_x,R_y,R_z)	3	0	-1	3	0	-1
Vibration	30	0	-2	0	0	0

**Decomposition into Irreducible representations**
Motion	A_1g	A_2g	E_g	A_1u	A_2u	E_u	Total
Cartesian 3N	2	4	6	2	4	6	24
Translation (x,y,z)	0	0	0	0	1	1	2
Rotation (R_x,R_y,R_z)	0	1	1	0	0	0	2
Vibration	2	3	5	2	3	5	20

**Molecule Parameter**
Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	2
Number of vibrational modes	20

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A_1g	A_2g	E_g	A_1u	A_2u	E_u	Total
Linear (IR)	2	3	5	2	3	5	8 / 12
Quadratic (Raman)	2	3	5	2	3	5	7 / 13
IR + Raman	-	3	-	2	-	-	0^* / 5

* Center of inversion: Mutual Exclusion Principle

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2C₃	3C'₂	i	2S₆	3_d
linear	1	30	0	-2	0	0	0
quadratic	2	465	0	17	15	0	15
cubic	3	4.960	10	-32	0	0	0
quartic	4	40.920	0	152	120	0	120
quintic	5	278.256	0	-272	0	0	0
sextic	6	1.623.160	55	952	680	5	680

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A_1g	A_2g	E_g	A_1u	A_2u	E_u
linear	1	2	3	5	2	3	5
quadratic	2	48	32	80	38	37	75
cubic	3	407	423	825	407	423	825
quartic	4	3.488	3.352	6.840	3.408	3.392	6.800
quintic	5	23.120	23.256	46.376	23.120	23.256	46.376
sextic	6	135.738	134.922	270.630	135.283	135.147	270.405

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 D3d

Constructor University Bremen

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