Reduction formula for point group D2d

Reduction formula for point group D_2d

**Characters for molecular motions**
Motion	E	2S₄	C₂ (z)	2C'₂	2_d
Cartesian 3N	36	0	-4	0	8
Translation (x,y,z)	3	-1	-1	-1	1
Rotation (R_x,R_y,R_z)	3	1	-1	-1	-1
Vibration	30	0	-2	2	8

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	E	Total
Cartesian 3N	6	2	2	6	10	26
Translation (x,y,z)	0	0	0	1	1	2
Rotation (R_x,R_y,R_z)	0	1	0	0	1	2
Vibration	6	1	2	5	8	22

**Molecule Parameter**
Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	6
Number of vibrational modes	22

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	E	Total
Linear (IR)	6	1	2	5	8	13 / 9
Quadratic (Raman)	6	1	2	5	8	21 / 1
IR + Raman	-	1	-	5	8	13 / 1

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2S₄	C₂ (z)	2C'₂	2_d
linear	1	30	0	-2	2	8
quadratic	2	465	-1	17	17	47
cubic	3	4.960	0	-32	32	208
quartic	4	40.920	8	152	152	792
quintic	5	278.256	0	-272	272	2.640
sextic	6	1.623.160	-8	952	952	8.008

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂	E
linear	1	6	1	2	5	8
quadratic	2	76	44	53	68	112
cubic	3	676	556	572	660	1.248
quartic	4	5.372	4.900	4.972	5.292	10.192
quintic	5	35.476	34.020	34.156	35.340	69.632
sextic	6	205.252	200.772	201.252	204.780	405.552

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 D2d

Constructor University Bremen

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