Characters of representations for molecular motions

Motion	E	12C5	12(C5)2	20C3	15C2	i	12S10	12(S10)3	20S6	15σd
Cartesian 3N	180	0.000	-0.000	0	0	0	0.000	-0.000	0	4
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	174	-3.236	1.236	0	2	0	0.000	0.000	0	4

Decomposition to irreducible representations

Motion	Ag	T1g	T2g	Gg	Hg	Au	T1u	T2u	Gu	Hu	Total
Cartesian 3N	2	4	4	6	8	1	5	5	6	7	48
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	3	4	6	8	1	4	5	6	7	46

Molecular parameter

Number of Atoms (N)	60
Number of internal coordinates	174
Number of independant internal coordinates	2
Number of vibrational modes	46

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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	3	4	6	8	1	4	5	6	7	4 / 42
Quadratic (Raman)	2	3	4	6	8	1	4	5	6	7	10 / 36
IR + Raman	- - - -	3	4	6	- - - -	1	- - - -	5	6	7	0* / 32

* 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	174	-3.236	1.236	0	2	0	0.000	0.000	0	4
quadratic	15.225	5.854	-0.854	0	89	87	-1.618	0.618	0	95
cubic	893.200	-7.236	-2.764	58	176	0	0.000	0.000	0	360
quartic	39.524.100	5.854	-0.854	0	4.004	3.828	1.618	-0.618	0	4.540
quintic	1.407.057.960	32.764	37.236	0	7.832	0	-0.000	-0.000	0	16.376
sextic	41.977.229.140	-115.498	45.498	1.711	121.396	113.564	-1.000	-1.000	29	145.604

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	3	4	6	8	1	4	5	6	7
quadratic	151	361	359	510	661	126	380	379	504	630
cubic	7.519	22.262	22.263	29.784	37.274	7.429	22.352	22.353	29.784	37.184
quartic	330.468	987.131	987.130	1.317.597	1.648.065	329.269	988.075	988.073	1.317.342	1.646.611
quintic	11.728.516	35.173.426	35.173.427	46.901.925	58.630.441	11.724.422	35.177.520	35.177.521	46.901.925	58.626.347
sextic	349.844.847	1.049.400.171	1.049.400.207	1.399.245.054	1.749.089.031	349.806.544	1.049.430.894	1.049.430.930	1.399.237.473	1.749.043.176

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.	..6.	T1gT1g.	..10.	T2gT2g.	..21.	GgGg.	..36.	HgHg.	..1.	AuAu.	..10.	T1uT1u.	..15.	T2uT2u.	..21.	GuGu.	..28.	HuHu.
Subtotal: 151 / 10 / 10
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Hu)
Subtotal: 0 / 0 / 45
Total: 151 / 10 / 55

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Hu)
..4.	AgAgAg.	..1.	T1gT1gT1g.	..4.	T2gT2gT2g.	..56.	GgGgGg.	..240.	HgHgHg.
Subtotal: 305 / 5 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Hu)
..48.	T1gT1gHg.	..80.	T2gT2gHg.	..168.	GgGgHg.	..12.	AgT1gT1g.	..20.	AgT2gT2g.	..42.	AgGgGg.	..72.	AgHgHg.	..2.	AgAuAu.	..20.	AgT1uT1u.	..30.	AgT2uT2u.
..42.	AgGuGu.	..56.	AgHuHu.	..45.	T1gGgGg.	..84.	T1gHgHg.	..18.	T1gT1uT1u.	..45.	T1gGuGu.	..63.	T1gHuHu.	..60.	T2gGgGg.	..112.	T2gHgHg.	..40.	T2gT2uT2u.
..60.	T2gGuGu.	..84.	T2gHuHu.	..384.	GgHgHg.	..126.	GgGuGu.	..294.	GgHuHu.	..80.	HgT1uT1u.	..120.	HgT2uT2u.	..168.	HgGuGu.	..448.	HgHuHu.
Subtotal: 2.823 / 29 / 90
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Hu)
..72.	T1gT2gGg.	..96.	T1gT2gHg.	..144.	T1gGgHg.	..12.	T1gAuT1u.	..84.	T1gT1uHu.	..90.	T1gT2uGu.	..105.	T1gT2uHu.	..126.	T1gGuHu.	..192.	T2gGgHg.	..20.	T2gAuT2u.
..96.	T2gT1uGu.	..112.	T2gT1uHu.	..140.	T2gT2uHu.	..168.	T2gGuHu.	..36.	GgAuGu.	..120.	GgT1uT2u.	..144.	GgT1uGu.	..168.	GgT1uHu.	..180.	GgT2uGu.	..210.	GgT2uHu.
..252.	GgGuHu.	..56.	HgAuHu.	..160.	HgT1uT2u.	..192.	HgT1uGu.	..224.	HgT1uHu.	..240.	HgT2uGu.	..280.	HgT2uHu.	..672.	HgGuHu.
Subtotal: 4.391 / 28 / 120
Total: 7.519 / 62 / 220

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Hu)
..5.	AgAgAgAg.	..21.	T1gT1gT1gT1g.	..55.	T2gT2gT2gT2g.	..372.	GgGgGgGg.	..2.298.	HgHgHgHg.	..1.	AuAuAuAu.	..55.	T1uT1uT1uT1u.	..120.	T2uT2uT2uT2u.	..372.	GuGuGuGu.	..1.386.	HuHuHuHu.
Subtotal: 4.685 / 10 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Hu)
..40.	T1gT1gT1gT2g.	..60.	T1gT1gT1gGg.	..64.	T1gT1gT1gHg.	..120.	T2gT2gT2gGg.	..160.	T2gT2gT2gHg.	..1.568.	GgGgGgHg.	..100.	T1uT1uT1uT2u.	..120.	T1uT1uT1uGu.	..140.	T1uT1uT1uHu.	..210.	T2uT2uT2uGu.
..280.	T2uT2uT2uHu.	..1.372.	GuGuGuHu.	..2.	AgT1gT1gT1g.	..8.	AgT2gT2gT2g.	..112.	AgGgGgGg.	..480.	AgHgHgHg.	..60.	T1gT2gT2gT2g.	..378.	T1gGgGgGg.	..1.536.	T1gHgHgHg.	..504.	T2gGgGgGg.
..2.048.	T2gHgHgHg.	..4.512.	GgHgHgHg.	..4.	AuT1uT1uT1u.	..10.	AuT2uT2uT2u.	..56.	AuGuGuGu.	..168.	AuHuHuHu.	..140.	T1uT2uT2uT2u.	..504.	T1uGuGuGu.	..1.372.	T1uHuHuHu.	..630.	T2uGuGuGu.
..1.715.	T2uHuHuHu.	..3.066.	GuHuHuHu.
Subtotal: 21.539 / 32 / 90
Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Hu)
..18.	AgAgT1gT1g.	..30.	AgAgT2gT2g.	..63.	AgAgGgGg.	..108.	AgAgHgHg.	..3.	AgAgAuAu.	..30.	AgAgT1uT1u.	..45.	AgAgT2uT2u.	..63.	AgAgGuGu.	..84.	AgAgHuHu.	..120.	T1gT1gT2gT2g.
..297.	T1gT1gGgGg.	..732.	T1gT1gHgHg.	..6.	T1gT1gAuAu.	..138.	T1gT1gT1uT1u.	..180.	T1gT1gT2uT2u.	..297.	T1gT1gGuGu.	..567.	T1gT1gHuHu.	..510.	T2gT2gGgGg.	..1.248.	T2gT2gHgHg.	..10.	T2gT2gAuAu.
..200.	T2gT2gT1uT1u.	..360.	T2gT2gT2uT2u.	..510.	T2gT2gGuGu.	..966.	T2gT2gHuHu.	..4.452.	GgGgHgHg.	..21.	GgGgAuAu.	..510.	GgGgT1uT1u.	..780.	GgGgT2uT2u.	..1.773.	GgGgGuGu.	..3.423.	GgGgHuHu.
..36.	HgHgAuAu.	..1.248.	HgHgT1uT1u.	..1.900.	HgHgT2uT2u.	..4.452.	HgHgGuGu.	..9.352.	HgHgHuHu.	..10.	AuAuT1uT1u.	..15.	AuAuT2uT2u.	..21.	AuAuGuGu.	..28.	AuAuHuHu.	..300.	T1uT1uT2uT2u.
..510.	T1uT1uGuGu.	..966.	T1uT1uHuHu.	..780.	T2uT2uGuGu.	..1.470.	T2uT2uHuHu.	..3.423.	GuGuHuHu.
Subtotal: 42.055 / 45 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Hu)
..216.	T1gT1gT2gGg.	..288.	T1gT1gT2gHg.	..720.	T1gT1gGgHg.	..12.	T1gT1gAuT1u.	..42.	T1gT1gAuHu.	..120.	T1gT1gT1uT2u.	..144.	T1gT1gT1uGu.	..252.	T1gT1gT1uHu.	..270.	T1gT1gT2uGu.	..315.	T1gT1gT2uHu.
..630.	T1gT1gGuHu.	..1.248.	T2gT2gGgHg.	..30.	T2gT2gAuT2u.	..70.	T2gT2gAuHu.	..200.	T2gT2gT1uT2u.	..384.	T2gT2gT1uGu.	..448.	T2gT2gT1uHu.	..300.	T2gT2gT2uGu.	..560.	T2gT2gT2uHu.	..1.092.	T2gT2gGuHu.
..60.	GgGgAuT1u.	..75.	GgGgAuT2u.	..126.	GgGgAuGu.	..147.	GgGgAuHu.	..840.	GgGgT1uT2u.	..1.368.	GgGgT1uGu.	..2.016.	GgGgT1uHu.	..1.710.	GgGgT2uGu.	..2.520.	GgGgT2uHu.	..3.906.	GgGgGuHu.
..112.	HgHgAuT1u.	..140.	HgHgAuT2u.	..384.	HgHgAuGu.	..504.	HgHgAuHu.	..2.720.	HgHgT1uT2u.	..3.936.	HgHgT1uGu.	..5.376.	HgHgT1uHu.	..4.920.	HgHgT2uGu.	..6.720.	HgHgT2uHu.	..11.088.	HgHgGuHu.
..480.	T1uT1uT2uGu.	..560.	T1uT1uT2uHu.	..1.092.	T1uT1uGuHu.	..1.680.	T2uT2uGuHu.	..96.	AgT1gT1gHg.	..160.	AgT2gT2gHg.	..336.	AgGgGgHg.	..288.	T1gT2gT2gGg.	..384.	T1gT2gT2gHg.	..1.728.	T1gGgGgHg.
..2.304.	T2gGgGgHg.	..70.	AuT1uT1uHu.	..105.	AuT2uT2uHu.	..147.	AuGuGuHu.	..600.	T1uT2uT2uGu.	..700.	T1uT2uT2uHu.	..2.016.	T1uGuGuHu.	..2.520.	T2uGuGuHu.	..90.	AgT1gGgGg.	..168.	AgT1gHgHg.
..36.	AgT1gT1uT1u.	..90.	AgT1gGuGu.	..126.	AgT1gHuHu.	..120.	AgT2gGgGg.	..224.	AgT2gHgHg.	..80.	AgT2gT2uT2u.	..120.	AgT2gGuGu.	..168.	AgT2gHuHu.	..768.	AgGgHgHg.	..252.	AgGgGuGu.
..588.	AgGgHuHu.	..160.	AgHgT1uT1u.	..240.	AgHgT2uT2u.	..336.	AgHgGuGu.	..896.	AgHgHuHu.	..504.	T1gT2gGgGg.	..1.632.	T1gT2gHgHg.	..120.	T1gT2gT1uT1u.	..180.	T1gT2gT2uT2u.	..504.	T1gT2gGuGu.
..1.260.	T1gT2gHuHu.	..2.952.	T1gGgHgHg.	..180.	T1gGgT1uT1u.	..450.	T1gGgT2uT2u.	..1.026.	T1gGgGuGu.	..2.268.	T1gGgHuHu.	..384.	T1gHgT1uT1u.	..600.	T1gHgT2uT2u.	..1.728.	T1gHgGuGu.	..3.528.	T1gHgHuHu.
..3.936.	T2gGgHgHg.	..384.	T2gGgT1uT1u.	..360.	T2gGgT2uT2u.	..1.368.	T2gGgGuGu.	..3.024.	T2gGgHuHu.	..512.	T2gHgT1uT1u.	..800.	T2gHgT2uT2u.	..2.304.	T2gHgGuGu.	..4.704.	T2gHgHuHu.	..1.248.	GgHgT1uT1u.
..1.920.	GgHgT2uT2u.	..4.464.	GgHgGuGu.	..9.744.	GgHgHuHu.	..60.	AuT1uGuGu.	..84.	AuT1uHuHu.	..75.	AuT2uGuGu.	..105.	AuT2uHuHu.	..294.	AuGuHuHu.	..840.	T1uT2uGuGu.	..2.100.	T1uT2uHuHu.
..3.024.	T1uGuHuHu.	..3.780.	T2uGuHuHu.
Subtotal: 138.183 / 112 / 360
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Hu)
..144.	AgT1gT2gGg.	..192.	AgT1gT2gHg.	..288.	AgT1gGgHg.	..24.	AgT1gAuT1u.	..168.	AgT1gT1uHu.	..180.	AgT1gT2uGu.	..210.	AgT1gT2uHu.	..252.	AgT1gGuHu.	..384.	AgT2gGgHg.	..40.	AgT2gAuT2u.
..192.	AgT2gT1uGu.	..224.	AgT2gT1uHu.	..280.	AgT2gT2uHu.	..336.	AgT2gGuHu.	..72.	AgGgAuGu.	..240.	AgGgT1uT2u.	..288.	AgGgT1uGu.	..336.	AgGgT1uHu.	..360.	AgGgT2uGu.	..420.	AgGgT2uHu.
..504.	AgGgGuHu.	..112.	AgHgAuHu.	..320.	AgHgT1uT2u.	..384.	AgHgT1uGu.	..448.	AgHgT1uHu.	..480.	AgHgT2uGu.	..560.	AgHgT2uHu.	..1.344.	AgHgGuHu.	..1.728.	T1gT2gGgHg.	..72.	T1gT2gAuGu.
..84.	T1gT2gAuHu.	..480.	T1gT2gT1uT2u.	..576.	T1gT2gT1uGu.	..672.	T1gT2gT1uHu.	..720.	T1gT2gT2uGu.	..840.	T1gT2gT2uHu.	..1.512.	T1gT2gGuHu.	..90.	T1gGgAuT2u.	..108.	T1gGgAuGu.	..126.	T1gGgAuHu.
..720.	T1gGgT1uT2u.	..1.296.	T1gGgT1uGu.	..1.512.	T1gGgT1uHu.	..1.080.	T1gGgT2uGu.	..1.890.	T1gGgT2uHu.	..3.024.	T1gGgGuHu.	..96.	T1gHgAuT1u.	..120.	T1gHgAuT2u.	..144.	T1gHgAuGu.	..168.	T1gHgAuHu.
..960.	T1gHgT1uT2u.	..1.728.	T1gHgT1uGu.	..2.688.	T1gHgT1uHu.	..2.160.	T1gHgT2uGu.	..3.360.	T1gHgT2uHu.	..5.040.	T1gHgGuHu.	..96.	T2gGgAuT1u.	..144.	T2gGgAuGu.	..168.	T2gGgAuHu.	..960.	T2gGgT1uT2u.
..1.152.	T2gGgT1uGu.	..2.016.	T2gGgT1uHu.	..2.160.	T2gGgT2uGu.	..2.520.	T2gGgT2uHu.	..4.032.	T2gGgGuHu.	..128.	T2gHgAuT1u.	..160.	T2gHgAuT2u.	..192.	T2gHgAuGu.	..224.	T2gHgAuHu.	..1.280.	T2gHgT1uT2u.
..2.304.	T2gHgT1uGu.	..3.584.	T2gHgT1uHu.	..2.880.	T2gHgT2uGu.	..4.480.	T2gHgT2uHu.	..6.720.	T2gHgGuHu.	..192.	GgHgAuT1u.	..240.	GgHgAuT2u.	..288.	GgHgAuGu.	..672.	GgHgAuHu.	..2.880.	GgHgT1uT2u.
..4.608.	GgHgT1uGu.	..6.720.	GgHgT1uHu.	..5.760.	GgHgT2uGu.	..8.400.	GgHgT2uHu.	..14.112.	GgHgGuHu.	..120.	AuT1uT2uGu.	..140.	AuT1uT2uHu.	..168.	AuT1uGuHu.	..210.	AuT2uGuHu.	..2.520.	T1uT2uGuHu.
Subtotal: 124.006 / 90 / 210
Total: 330.468 / 289 / 715

Calculate contributions to

Ag	T1g	T2g	Gg	Hg	Au	T1u	T2u	Gu	Hu

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