Characters of representations for molecular motions

Motion	E	2C3	3σv
Cartesian 3N	63	0	5
Translation (x,y,z)	3	0	1
Rotation (Rx,Ry,Rz)	3	0	-1
Vibration	57	0	5

Decomposition to irreducible representations

Motion	A1	A2	E	Total
Cartesian 3N	13	8	21	42
Translation (x,y,z)	1	0	1	2
Rotation (Rx,Ry,Rz)	0	1	1	2
Vibration	12	7	19	38

Molecular parameter

Number of Atoms (N)	21
Number of internal coordinates	57
Number of independant internal coordinates	12
Number of vibrational modes	38

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	E	Total
Linear (IR)	12	7	19	31 / 7
Quadratic (Raman)	12	7	19	31 / 7
IR + Raman	12	7	19	31 / 7

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3σv
linear	57	0	5
quadratic	1.653	0	41
cubic	32.509	19	165
quartic	487.635	0	811
quintic	5.949.147	0	2.791
sextic	61.474.519	190	10.571

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

Force field	A1	A2	E
linear	12	7	19
quadratic	296	255	551
cubic	5.507	5.342	10.830
quartic	81.678	80.867	162.545
quintic	992.920	990.129	1.983.049
sextic	10.251.102	10.240.531	20.491.443

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_3v

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)
..78.	A1A1.	..28.	A2A2.	..190.	EE.
Subtotal: 296 / 3 / 3
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 296 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..364.	A1A1A1.	..1.330.	EEE.
Subtotal: 1.694 / 2 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..336.	A1A2A2.	..2.280.	A1EE.	..1.197.	A2EE.
Subtotal: 3.813 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 5.507 / 5 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..1.365.	A1A1A1A1.	..210.	A2A2A2A2.	..18.145.	EEEE.
Subtotal: 19.720 / 3 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..15.960.	A1EEE.	..9.310.	A2EEE.
Subtotal: 25.270 / 2 / 6
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..2.184.	A1A1A2A2.	..14.820.	A1A1EE.	..5.320.	A2A2EE.
Subtotal: 22.324 / 3 / 3
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
..14.364.	A1A2EE.
Subtotal: 14.364 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 81.678 / 9 / 15

Calculate contributions to

A1	A2	E

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