Characters of representations for molecular motions

Motion	E	2C5	2(C5)2	5C'2	σh	2S5	2(S5)3	5σv
Cartesian 3N	63	1.618	-0.618	-1	1	-0.382	-2.618	5
Translation (x,y,z)	3	1.618	-0.618	-1	1	-0.382	-2.618	1
Rotation (Rx,Ry,Rz)	3	1.618	-0.618	-1	-1	0.382	2.618	-1
Vibration	57	-1.618	0.618	1	1	-0.382	-2.618	5

Decomposition to irreducible representations

Motion	A'1	A'2	E'1	E'2	A''1	A''2	E''1	E''2	Total
Cartesian 3N	4	2	7	6	2	5	6	6	38
Translation (x,y,z)	0	0	1	0	0	1	0	0	2
Rotation (Rx,Ry,Rz)	0	1	0	0	0	0	1	0	2
Vibration	4	1	6	6	2	4	5	6	34

Molecular parameter

Number of Atoms (N)	21
Number of internal coordinates	57
Number of independant internal coordinates	4
Number of vibrational modes	34

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'1	A'2	E'1	E'2	A''1	A''2	E''1	E''2	Total
Linear (IR)	4	1	6	6	2	4	5	6	10 / 24
Quadratic (Raman)	4	1	6	6	2	4	5	6	15 / 19
IR + Raman	- - - -	1	- - - -	- - - -	2	- - - -	- - - -	6	0 / 9

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C5	2(C5)2	5C'2	σh	2S5	2(S5)3	5σv
linear	57	-1.618	0.618	1	1	-0.382	-2.618	5
quadratic	1.653	1.618	-0.618	29	29	0.382	2.618	41
cubic	32.509	-1.000	-1.000	29	29	-1.000	-1.000	165
quartic	487.635	-0.000	0.000	435	435	0.000	0.000	811
quintic	5.949.147	12.000	12.000	435	435	0.000	-0.000	2.791
sextic	61.474.519	-19.416	7.416	4.495	4.495	0.000	-0.000	10.571

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

Force field	A'1	A'2	E'1	E'2	A''1	A''2	E''1	E''2
linear	4	1	6	6	2	4	5	6
quadratic	102	67	168	168	78	84	163	162
cubic	1.675	1.578	3.254	3.254	1.590	1.658	3.248	3.248
quartic	24.715	24.092	48.807	48.807	24.266	24.454	48.720	48.720
quintic	298.288	296.675	594.957	594.957	296.849	298.027	594.870	594.870
sextic	3.077.716	3.070.183	6.147.899	6.147.905	3.071.981	3.075.019	6.147.000	6.147.006

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_5h

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(A'1) ≤ i ≤ pos(E''2)
..10.	A'1A'1.	..1.	A'2A'2.	..21.	E'1E'1.	..21.	E'2E'2.	..3.	A''1A''1.	..10.	A''2A''2.	..15.	E''1E''1.	..21.	E''2E''2.
Subtotal: 102 / 8 / 8
Irrep combinations (i,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
Subtotal: 0 / 0 / 28
Total: 102 / 8 / 36

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E''2)
..20.	A'1A'1A'1.
Subtotal: 20 / 1 / 8
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
..126.	E'1E'1E'2.	..4.	A'1A'2A'2.	..84.	A'1E'1E'1.	..84.	A'1E'2E'2.	..12.	A'1A''1A''1.	..40.	A'1A''2A''2.	..60.	A'1E''1E''1.	..84.	A'1E''2E''2.	..15.	A'2E'1E'1.	..15.	A'2E'2E'2.
..10.	A'2E''1E''1.	..15.	A'2E''2E''2.	..126.	E'1E'2E'2.	..126.	E'1E''2E''2.	..90.	E'2E''1E''1.
Subtotal: 891 / 15 / 56
Irrep combinations (i,j,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E''2)
..8.	A'2A''1A''2.	..60.	E'1A''1E''1.	..120.	E'1A''2E''1.	..180.	E'1E''1E''2.	..72.	E'2A''1E''2.	..144.	E'2A''2E''2.	..180.	E'2E''1E''2.
Subtotal: 764 / 7 / 56
Total: 1.675 / 23 / 120

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E''2)
..35.	A'1A'1A'1A'1.	..1.	A'2A'2A'2A'2.	..231.	E'1E'1E'1E'1.	..231.	E'2E'2E'2E'2.	..5.	A''1A''1A''1A''1.	..35.	A''2A''2A''2A''2.	..120.	E''1E''1E''1E''1.	..231.	E''2E''2E''2E''2.
Subtotal: 889 / 8 / 8
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
..336.	E'1E'1E'1E'2.	..210.	E''1E''1E''1E''2.	..336.	E'1E'2E'2E'2.	..280.	E''1E''2E''2E''2.
Subtotal: 1.162 / 4 / 56
Irrep combinations (i,i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
..10.	A'1A'1A'2A'2.	..210.	A'1A'1E'1E'1.	..210.	A'1A'1E'2E'2.	..30.	A'1A'1A''1A''1.	..100.	A'1A'1A''2A''2.	..150.	A'1A'1E''1E''1.	..210.	A'1A'1E''2E''2.	..21.	A'2A'2E'1E'1.	..21.	A'2A'2E'2E'2.	..3.	A'2A'2A''1A''1.
..10.	A'2A'2A''2A''2.	..15.	A'2A'2E''1E''1.	..21.	A'2A'2E''2E''2.	..666.	E'1E'1E'2E'2.	..63.	E'1E'1A''1A''1.	..210.	E'1E'1A''2A''2.	..780.	E'1E'1E''1E''1.	..666.	E'1E'1E''2E''2.	..63.	E'2E'2A''1A''1.	..210.	E'2E'2A''2A''2.
..465.	E'2E'2E''1E''1.	..1.107.	E'2E'2E''2E''2.	..30.	A''1A''1A''2A''2.	..45.	A''1A''1E''1E''1.	..63.	A''1A''1E''2E''2.	..150.	A''2A''2E''1E''1.	..210.	A''2A''2E''2E''2.	..465.	E''1E''1E''2E''2.
Subtotal: 6.204 / 28 / 28
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E''2)
..120.	E'1E'1A''1A''2.	..252.	E'1E'1A''1E''2.	..504.	E'1E'1A''2E''2.	..630.	E'1E'1E''1E''2.	..120.	E'2E'2A''1A''2.	..210.	E'2E'2A''1E''1.	..420.	E'2E'2A''2E''1.	..630.	E'2E'2E''1E''2.	..504.	A'1E'1E'1E'2.	..126.	A'2E'1E'1E'2.
..180.	A''1E''1E''1E''2.	..360.	A''2E''1E''1E''2.	..60.	A'1A'2E'1E'1.	..60.	A'1A'2E'2E'2.	..40.	A'1A'2E''1E''1.	..60.	A'1A'2E''2E''2.	..504.	A'1E'1E'2E'2.	..504.	A'1E'1E''2E''2.	..360.	A'1E'2E''1E''1.	..126.	A'2E'1E'2E'2.
..126.	A'2E'1E''2E''2.	..90.	A'2E'2E''1E''1.	..540.	E'1E'2E''1E''1.	..756.	E'1E'2E''2E''2.	..80.	A''1A''2E''1E''1.	..120.	A''1A''2E''2E''2.	..210.	A''1E''1E''2E''2.	..420.	A''2E''1E''2E''2.
Subtotal: 8.112 / 28 / 168
Irrep combinations (i,j,k,l) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E''2)
..32.	A'1A'2A''1A''2.	..240.	A'1E'1A''1E''1.	..480.	A'1E'1A''2E''1.	..720.	A'1E'1E''1E''2.	..288.	A'1E'2A''1E''2.	..576.	A'1E'2A''2E''2.	..720.	A'1E'2E''1E''2.	..60.	A'2E'1A''1E''1.	..120.	A'2E'1A''2E''1.	..180.	A'2E'1E''1E''2.
..72.	A'2E'2A''1E''2.	..144.	A'2E'2A''2E''2.	..180.	A'2E'2E''1E''2.	..360.	E'1E'2A''1E''1.	..432.	E'1E'2A''1E''2.	..720.	E'1E'2A''2E''1.	..864.	E'1E'2A''2E''2.	..2.160.	E'1E'2E''1E''2.
Subtotal: 8.348 / 18 / 70
Total: 24.715 / 86 / 330

Calculate contributions to

A'1	A'2	E'1	E'2	A''1	A''2	E''1	E''2

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