Characters of representations for molecular motions

Motion	E	12C5	12(C5)2	20C3	15C2	i	12S10	12(S10)3	20S6	15σd
Cartesian 3N	72	6.472	-2.472	0	0	0	0.000	-0.000	0	8
Translation (x,y,z)	3	1.618	-0.618	0	-1	-3	0.618	-1.618	0	1
Rotation (Rx,Ry,Rz)	3	1.618	-0.618	0	-1	3	-0.618	1.618	0	-1
Vibration	66	3.236	-1.236	0	2	0	0.000	0.000	0	8

Decomposition to irreducible representations

Motion	Ag	T1g	T2g	Gg	Hg	Au	T1u	T2u	Gu	Hu	Total
Cartesian 3N	2	2	0	2	4	0	4	2	2	2	20
Translation (x,y,z)	0	0	0	0	0	0	1	0	0	0	1
Rotation (Rx,Ry,Rz)	0	1	0	0	0	0	0	0	0	0	1
Vibration	2	1	0	2	4	0	3	2	2	2	18

Molecular parameter

Number of Atoms (N)	24
Number of internal coordinates	66
Number of independant internal coordinates	2
Number of vibrational modes	18

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	Ag	T1g	T2g	Gg	Hg	Au	T1u	T2u	Gu	Hu	Total
Linear (IR)	2	1	0	2	4	0	3	2	2	2	3 / 15
Quadratic (Raman)	2	1	0	2	4	0	3	2	2	2	6 / 12
IR + Raman	- - - -	1	0	2	- - - -	0	- - - -	2	2	2	0* / 9

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	12C5	12(C5)2	20C3	15C2	i	12S10	12(S10)3	20S6	15σd
linear	66	3.236	-1.236	0	2	0	0.000	0.000	0	8
quadratic	2.211	4.618	2.382	0	35	33	1.618	-0.618	0	65
cubic	50.116	3.236	-1.236	22	68	0	0.000	0.000	0	352
quartic	864.501	1.000	1.000	0	629	561	1.000	1.000	0	1.809
quintic	12.103.014	14.000	14.000	0	1.190	0	0.000	0.000	0	7.752
sextic	143.218.999	45.305	-17.305	253	7.735	6.545	-0.000	-0.000	11	31.441

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

Force field	Ag	T1g	T2g	Gg	Hg	Au	T1u	T2u	Gu	Hu
linear	2	1	0	2	4	0	3	2	2	2
quadratic	32	44	44	74	106	15	59	58	72	87
cubic	474	1.201	1.200	1.674	2.137	386	1.289	1.288	1.674	2.049
quartic	7.514	21.322	21.322	28.835	36.349	7.052	21.746	21.746	28.798	35.850
quintic	101.979	301.459	301.459	403.431	505.410	100.041	303.397	303.397	403.431	503.472
sextic	1.198.490	3.575.750	3.575.736	4.774.226	5.972.584	1.190.517	3.583.283	3.583.269	4.773.786	5.964.182

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 I_h

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(Ag) ≤ i ≤ pos(Hu)
..3.	AgAg.	..1.	T1gT1g.	..3.	GgGg.	..10.	HgHg.	..6.	T1uT1u.	..3.	T2uT2u.	..3.	GuGu.	..3.	HuHu.
Subtotal: 32 / 8 / 10
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Hu)
Subtotal: 0 / 0 / 45
Total: 32 / 8 / 55

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Hu)
..4.	AgAgAg.	..4.	GgGgGg.	..40.	HgHgHg.
Subtotal: 48 / 3 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Hu)
..4.	T1gT1gHg.	..12.	GgGgHg.	..2.	AgT1gT1g.	..6.	AgGgGg.	..20.	AgHgHg.	..12.	AgT1uT1u.	..6.	AgT2uT2u.	..6.	AgGuGu.	..6.	AgHuHu.	..1.	T1gGgGg.
..6.	T1gHgHg.	..3.	T1gT1uT1u.	..1.	T1gGuGu.	..1.	T1gHuHu.	..32.	GgHgHg.	..6.	GgGuGu.	..8.	GgHuHu.	..24.	HgT1uT1u.	..12.	HgT2uT2u.	..12.	HgGuGu.
..24.	HgHuHu.
Subtotal: 204 / 21 / 90
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Hu)
..8.	T1gGgHg.	..6.	T1gT1uHu.	..4.	T1gT2uGu.	..4.	T1gT2uHu.	..4.	T1gGuHu.	..12.	GgT1uT2u.	..12.	GgT1uGu.	..12.	GgT1uHu.	..8.	GgT2uGu.	..8.	GgT2uHu.
..8.	GgGuHu.	..24.	HgT1uT2u.	..24.	HgT1uGu.	..24.	HgT1uHu.	..16.	HgT2uGu.	..16.	HgT2uHu.	..32.	HgGuHu.
Subtotal: 222 / 17 / 120
Total: 474 / 41 / 220

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Hu)
..5.	AgAgAgAg.	..1.	T1gT1gT1gT1g.	..11.	GgGgGgGg.	..175.	HgHgHgHg.	..21.	T1uT1uT1uT1u.	..6.	T2uT2uT2uT2u.	..11.	GuGuGuGu.	..16.	HuHuHuHu.
Subtotal: 246 / 8 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Hu)
..2.	T1gT1gT1gGg.	..32.	GgGgGgHg.	..20.	T1uT1uT1uT2u.	..20.	T1uT1uT1uGu.	..16.	T1uT1uT1uHu.	..8.	T2uT2uT2uGu.	..4.	T2uT2uT2uHu.	..16.	GuGuGuHu.	..8.	AgGgGgGg.	..80.	AgHgHgHg.
..6.	T1gGgGgGg.	..64.	T1gHgHgHg.	..208.	GgHgHgHg.	..12.	T1uT2uT2uT2u.	..18.	T1uGuGuGu.	..24.	T1uHuHuHu.	..12.	T2uGuGuGu.	..16.	T2uHuHuHu.	..32.	GuHuHuHu.
Subtotal: 598 / 19 / 90
Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Hu)
..3.	AgAgT1gT1g.	..9.	AgAgGgGg.	..30.	AgAgHgHg.	..18.	AgAgT1uT1u.	..9.	AgAgT2uT2u.	..9.	AgAgGuGu.	..9.	AgAgHuHu.	..6.	T1gT1gGgGg.	..30.	T1gT1gHgHg.	..12.	T1gT1gT1uT1u.
..6.	T1gT1gT2uT2u.	..6.	T1gT1gGuGu.	..9.	T1gT1gHuHu.	..150.	GgGgHgHg.	..39.	GgGgT1uT1u.	..19.	GgGgT2uT2u.	..29.	GgGgGuGu.	..41.	GgGgHuHu.	..198.	HgHgT1uT1u.	..96.	HgHgT2uT2u.
..150.	HgHgGuGu.	..226.	HgHgHuHu.	..36.	T1uT1uT2uT2u.	..39.	T1uT1uGuGu.	..57.	T1uT1uHuHu.	..19.	T2uT2uGuGu.	..28.	T2uT2uHuHu.	..41.	GuGuHuHu.
Subtotal: 1.324 / 28 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Hu)
..16.	T1gT1gGgHg.	..6.	T1gT1gT1uT2u.	..6.	T1gT1gT1uGu.	..6.	T1gT1gT1uHu.	..4.	T1gT1gT2uGu.	..4.	T1gT1gT2uHu.	..8.	T1gT1gGuHu.	..36.	GgGgT1uT2u.	..42.	GgGgT1uGu.	..48.	GgGgT1uHu.
..28.	GgGgT2uGu.	..32.	GgGgT2uHu.	..44.	GgGgGuHu.	..216.	HgHgT1uT2u.	..252.	HgHgT1uGu.	..288.	HgHgT1uHu.	..168.	HgHgT2uGu.	..192.	HgHgT2uHu.	..272.	HgHgGuHu.	..36.	T1uT1uT2uGu.
..36.	T1uT1uT2uHu.	..60.	T1uT1uGuHu.	..28.	T2uT2uGuHu.	..8.	AgT1gT1gHg.	..24.	AgGgGgHg.	..32.	T1gGgGgHg.	..24.	T1uT2uT2uGu.	..24.	T1uT2uT2uHu.	..48.	T1uGuGuHu.	..32.	T2uGuGuHu.
..2.	AgT1gGgGg.	..12.	AgT1gHgHg.	..6.	AgT1gT1uT1u.	..2.	AgT1gGuGu.	..2.	AgT1gHuHu.	..64.	AgGgHgHg.	..12.	AgGgGuGu.	..16.	AgGgHuHu.	..48.	AgHgT1uT1u.	..24.	AgHgT2uT2u.
..24.	AgHgGuGu.	..48.	AgHgHuHu.	..84.	T1gGgHgHg.	..12.	T1gGgT1uT1u.	..8.	T1gGgT2uT2u.	..14.	T1gGgGuGu.	..22.	T1gGgHuHu.	..36.	T1gHgT1uT1u.	..16.	T1gHgT2uT2u.	..32.	T1gHgGuGu.
..48.	T1gHgHuHu.	..120.	GgHgT1uT1u.	..56.	GgHgT2uT2u.	..88.	GgHgGuGu.	..144.	GgHgHuHu.	..36.	T1uT2uGuGu.	..60.	T1uT2uHuHu.	..66.	T1uGuHuHu.	..44.	T2uGuHuHu.
Subtotal: 3.166 / 59 / 360
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Hu)
..16.	AgT1gGgHg.	..12.	AgT1gT1uHu.	..8.	AgT1gT2uGu.	..8.	AgT1gT2uHu.	..8.	AgT1gGuHu.	..24.	AgGgT1uT2u.	..24.	AgGgT1uGu.	..24.	AgGgT1uHu.	..16.	AgGgT2uGu.	..16.	AgGgT2uHu.
..16.	AgGgGuHu.	..48.	AgHgT1uT2u.	..48.	AgHgT1uGu.	..48.	AgHgT1uHu.	..32.	AgHgT2uGu.	..32.	AgHgT2uHu.	..64.	AgHgGuHu.	..24.	T1gGgT1uT2u.	..36.	T1gGgT1uGu.	..36.	T1gGgT1uHu.
..16.	T1gGgT2uGu.	..24.	T1gGgT2uHu.	..32.	T1gGgGuHu.	..48.	T1gHgT1uT2u.	..72.	T1gHgT1uGu.	..96.	T1gHgT1uHu.	..48.	T1gHgT2uGu.	..64.	T1gHgT2uHu.	..80.	T1gHgGuHu.	..144.	GgHgT1uT2u.
..192.	GgHgT1uGu.	..240.	GgHgT1uHu.	..128.	GgHgT2uGu.	..160.	GgHgT2uHu.	..224.	GgHgGuHu.	..72.	T1uT2uGuHu.
Subtotal: 2.180 / 36 / 210
Total: 7.514 / 150 / 715

Calculate contributions to

Ag	T1g	T2g	Gg	Hg	Au	T1u	T2u	Gu	Hu

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