Characters of representations for molecular motions

Motion	E	2C4	C2	2σv	2σd
Cartesian 3N	42	2	-2	6	4
Translation (x,y,z)	3	1	-1	1	1
Rotation (Rx,Ry,Rz)	3	1	-1	-1	-1
Vibration	36	0	0	6	4

Decomposition to irreducible representations

Motion	A1	A2	B1	B2	E	Total
Cartesian 3N	8	3	5	4	11	31
Translation (x,y,z)	1	0	0	0	1	2
Rotation (Rx,Ry,Rz)	0	1	0	0	1	2
Vibration	7	2	5	4	9	27

Molecular parameter

Number of Atoms (N)	14
Number of internal coordinates	36
Number of independant internal coordinates	7
Number of vibrational modes	27

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	B1	B2	E	Total
Linear (IR)	7	2	5	4	9	16 / 11
Quadratic (Raman)	7	2	5	4	9	25 / 2
IR + Raman	7	2	- - - -	- - - -	9	16 / 2

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C4	C2	2σv	2σd
linear	36	0	0	6	4
quadratic	666	0	18	36	26
cubic	8.436	0	0	146	84
quartic	82.251	9	171	561	331
quintic	658.008	0	0	1.812	920
sextic	4.496.388	0	1.140	5.552	2.820

Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted

Force field	A1	A2	B1	B2	E
linear	7	2	5	4	9
quadratic	101	70	88	83	162
cubic	1.112	997	1.070	1.039	2.109
quartic	10.528	10.082	10.358	10.243	20.520
quintic	82.934	81.568	82.474	82.028	164.502
sextic	564.284	560.098	562.874	561.508	1.123.812

Further Reading

• 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
  
  
• F. Varga, L. Nemes, J.K.G. Watson. J. Phys. B: At. Mol. Opt. Phys. 10 No. 7, 5043-5048 (1996)
  The number of anharmonic potential constants of the fullerenes C₆₀ and C₇₀

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Contributions to nonvanishing force field constants

pos(X) : Position of irreducible representation (irrep) X in character table of C_4v

Subtotal: <Number of nonvanishing force constants in subsection> / <number of nonzero irrep combinations in subsection> / <number of irrep combinations in subsection>
Total: <Number of nonvanishing force constants in force field> / <number of nonzero irrep combinations in force field> / <number of irrep combinations in force field>

Contributions to nonvanishing quadratic force field constants

Irrep combinations (i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..28.	A1A1.	..3.	A2A2.	..15.	B1B1.	..10.	B2B2.	..45.	EE.
Subtotal: 101 / 5 / 5
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 10
Total: 101 / 5 / 15

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..84.	A1A1A1.
Subtotal: 84 / 1 / 5
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..21.	A1A2A2.	..105.	A1B1B1.	..70.	A1B2B2.	..315.	A1EE.	..72.	A2EE.	..225.	B1EE.	..180.	B2EE.
Subtotal: 988 / 7 / 20
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
..40.	A2B1B2.
Subtotal: 40 / 1 / 10
Total: 1.112 / 9 / 35

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..210.	A1A1A1A1.	..5.	A2A2A2A2.	..70.	B1B1B1B1.	..35.	B2B2B2B2.	..1.530.	EEEE.
Subtotal: 1.850 / 5 / 5
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 20
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..84.	A1A1A2A2.	..420.	A1A1B1B1.	..280.	A1A1B2B2.	..1.260.	A1A1EE.	..45.	A2A2B1B1.	..30.	A2A2B2B2.	..135.	A2A2EE.	..150.	B1B1B2B2.	..675.	B1B1EE.	..450.	B2B2EE.
Subtotal: 3.529 / 10 / 10
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
..504.	A1A2EE.	..1.575.	A1B1EE.	..1.260.	A1B2EE.	..450.	A2B1EE.	..360.	A2B2EE.	..720.	B1B2EE.
Subtotal: 4.869 / 6 / 30
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
..280.	A1A2B1B2.
Subtotal: 280 / 1 / 5
Total: 10.528 / 22 / 70

Calculate contributions to

A1	A2	B1	B2	E

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