Reduction formula for point group D3

Reduction formula for point group D₃

**Characters for molecular motions**
Motion	E	2C₃ (z)	3C'₂
Cartesian 3N	180	0	0
Translation (x,y,z)	3	0	-1
Rotation (R_x,R_y,R_z)	3	0	-1
Vibration	174	0	2

**Decomposition into Irreducible representations**
Motion	A₁	A₂	E	Total
Cartesian 3N	30	30	60	120
Translation (x,y,z)	0	1	1	2
Rotation (R_x,R_y,R_z)	0	1	1	2
Vibration	30	28	58	116

**Molecule Parameter**
Number of Atoms (N)	60
Number of internal coordinates	174
Number of independant internal coordinates	30
Number of vibrational modes	116

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	E	Total
Linear (IR)	30	28	58	86 / 30
Quadratic (Raman)	30	28	58	88 / 28
IR + Raman	-	-	58	58 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2C₃ (z)	3C'₂
linear	1	174	0	2
quadratic	2	15.225	0	89
cubic	3	893.200	58	176
quartic	4	39.524.100	0	4.004
quintic	5	1.407.057.960	0	7.832
sextic	6	41.977.229.140	1.711	121.396

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	E
linear	1	30	28	58
quadratic	2	2.582	2.493	5.075
cubic	3	148.974	148.798	297.714
quartic	4	6.589.352	6.585.348	13.174.700
quintic	5	234.513.576	234.505.744	469.019.320
sextic	6	6.996.266.125	6.996.144.729	13.992.409.143

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 D3

Constructor University Bremen

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