Reduction formula for point group C4v

Reduction formula for point group C_4v

**Characters for molecular motions**
Motion	E	2C₄ (z)	C₂	2_v	2_d
Cartesian 3N	36	4	-4	8	4
Translation (x,y,z)	3	1	-1	1	1
Rotation (R_x,R_y,R_z)	3	1	-1	-1	-1
Vibration	30	2	-2	8	4

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	E	Total
Cartesian 3N	8	2	4	2	10	26
Translation (x,y,z)	1	0	0	0	1	2
Rotation (R_x,R_y,R_z)	0	1	0	0	1	2
Vibration	7	1	4	2	8	22

**Molecule Parameter**
Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	7
Number of vibrational modes	22

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	E	Total
Linear (IR)	7	1	4	2	8	15 / 7
Quadratic (Raman)	7	1	4	2	8	21 / 1
IR + Raman	7	1	-	-	8	15 / 1

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2C₄ (z)	C₂	2_v	2_d
linear	1	30	2	-2	8	4
quadratic	2	465	1	17	47	23
cubic	3	4.960	0	-32	208	72
quartic	4	40.920	8	152	792	256
quintic	5	278.256	16	-272	2.640	680
sextic	6	1.623.160	8	952	8.008	1.904

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂	E
linear	1	7	1	4	2	8
quadratic	2	78	43	66	54	112
cubic	3	686	546	650	582	1.248
quartic	4	5.398	4.874	5.266	4.998	10.192
quintic	5	35.582	33.922	35.234	34.254	69.632
sextic	6	205.494	200.538	204.538	201.486	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 C4v

Constructor University Bremen

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