Characters of representations for molecular motions

Motion	E	2S4	C2	2C'2	2σd
Cartesian 3N	21	-1	-3	-1	5
Translation (x,y,z)	3	-1	-1	-1	1
Rotation (Rx,Ry,Rz)	3	1	-1	-1	-1
Vibration	15	-1	-1	1	5

Decomposition to irreducible representations

Motion	A1	A2	B1	B2	E	Total
Cartesian 3N	3	1	1	4	6	15
Translation (x,y,z)	0	0	0	1	1	2
Rotation (Rx,Ry,Rz)	0	1	0	0	1	2
Vibration	3	0	1	3	4	11

Molecular parameter

Number of Atoms (N)	7
Number of internal coordinates	15
Number of independant internal coordinates	3
Number of vibrational modes	11

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	B1	B2	E	Total
Linear (IR)	3	0	1	3	4	7 / 4
Quadratic (Raman)	3	0	1	3	4	11 / 0
IR + Raman	- - - -	0	- - - -	3	4	7 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2S4	C2	2C'2	2σd
linear	15	-1	-1	1	5
quadratic	120	0	8	8	20
cubic	680	0	-8	8	60
quartic	3.060	4	36	36	160
quintic	11.628	-4	-36	36	376
sextic	38.760	0	120	120	820

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

Force field	A1	A2	B1	B2	E
linear	3	0	1	3	4
quadratic	23	9	13	19	28
cubic	101	67	71	97	172
quartic	437	339	355	417	756
quintic	1.551	1.345	1.365	1.535	2.916
sextic	5.095	4.625	4.685	5.035	9.660

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 D_2d

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)
..6.	A1A1.	..1.	B1B1.	..6.	B2B2.	..10.	EE.
Subtotal: 23 / 4 / 5
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 10
Total: 23 / 4 / 15

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..10.	A1A1A1.
Subtotal: 10 / 1 / 5
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..3.	A1B1B1.	..18.	A1B2B2.	..30.	A1EE.	..10.	B1EE.	..30.	B2EE.
Subtotal: 91 / 5 / 20
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 10
Total: 101 / 6 / 35

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..15.	A1A1A1A1.	..1.	B1B1B1B1.	..15.	B2B2B2B2.	..90.	EEEE.
Subtotal: 121 / 4 / 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)
..6.	A1A1B1B1.	..36.	A1A1B2B2.	..60.	A1A1EE.	..6.	B1B1B2B2.	..10.	B1B1EE.	..60.	B2B2EE.
Subtotal: 178 / 6 / 10
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
..30.	A1B1EE.	..90.	A1B2EE.	..18.	B1B2EE.
Subtotal: 138 / 3 / 30
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 5
Total: 437 / 13 / 70

Calculate contributions to

A1	A2	B1	B2	E

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