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	21	0	-1	-3	0	3
Translation (x,y,z)	3	0	-1	-3	0	1
Rotation (R_x,R_y,R_z)	3	0	-1	3	0	-1
Vibration	15	0	1	-3	0	3

**Decomposition into Irreducible representations**
Motion	A_1g	A_2g	E_g	A_1u	A_2u	E_u	Total
Cartesian 3N	2	1	3	1	3	4	14
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	0	2	1	2	3	10

**Molecule Parameter**
Number of Atoms (N)	7
Number of internal coordinates	15
Number of independant internal coordinates	2
Number of vibrational modes	10

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A_1g	A_2g	E_g	A_1u	A_2u	E_u	Total
Linear (IR)	2	0	2	1	2	3	5 / 5
Quadratic (Raman)	2	0	2	1	2	3	4 / 6
IR + Raman	-	0	-	1	-	-	0^* / 1

* 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	15	0	1	-3	0	3
quadratic	2	120	0	8	12	0	12
cubic	3	680	5	8	-28	-1	28
quartic	4	3.060	0	36	72	0	72
quintic	5	11.628	0	36	-144	0	144
sextic	6	38.760	15	120	300	3	300

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	0	2	1	2	3
quadratic	2	16	6	22	8	10	18
cubic	3	64	46	108	55	65	117
quartic	4	288	234	522	240	258	498
quintic	5	1.002	912	1.914	954	1.008	1.962
sextic	6	3.363	3.153	6.507	3.162	3.252	6.408

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