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	0	0	6
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	6

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	E	Total
Cartesian 3N	6	3	3	6	9	27
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	2	3	5	7	23

**Molecule Parameter**
Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	6
Number of vibrational modes	23

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	E	Total
Linear (IR)	6	2	3	5	7	12 / 11
Quadratic (Raman)	6	2	3	5	7	21 / 2
IR + Raman	-	2	-	5	7	12 / 2

**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	6
quadratic	2	465	1	17	17	33
cubic	3	4.960	0	32	32	128
quartic	4	40.920	8	152	152	456
quintic	5	278.256	0	272	272	1.392
sextic	6	1.623.160	8	952	952	3.976

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂	E
linear	1	6	2	3	5	7
quadratic	2	73	48	56	64	112
cubic	3	664	584	600	648	1.232
quartic	4	5.288	4.984	5.056	5.208	10.192
quintic	5	35.232	34.400	34.536	35.096	69.496
sextic	6	204.248	201.784	202.256	203.768	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