Characters of representations for molecular motions

Motion	E	12C5	12(C5)2	20C3	15C2	i	12S10	12(S10)3	20S6	15σd
Cartesian 3N	240	0.000	-0.000	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	234	-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	3	5	5	8	11	1	7	7	8	9	64
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	3	4	5	8	11	1	6	7	8	9	62

Molecular parameter

Number of Atoms (N)	80
Number of internal coordinates	234
Number of independant internal coordinates	3
Number of vibrational modes	62

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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)	3	4	5	8	11	1	6	7	8	9	6 / 56
Quadratic (Raman)	3	4	5	8	11	1	6	7	8	9	14 / 48
IR + Raman	- - - -	4	5	8	- - - -	1	- - - -	7	8	9	0* / 42

* 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	234	-3.236	1.236	0	2	0	0.000	0.000	0	8
quadratic	27.495	5.854	-0.854	0	119	117	-1.618	0.618	0	149
cubic	2.162.940	-7.236	-2.764	78	236	0	0.000	0.000	0	1.024
quartic	128.154.195	5.854	-0.854	0	7.139	6.903	1.618	-0.618	0	10.839
quintic	6.100.139.682	44.764	49.236	0	14.042	0	0.000	0.000	0	65.880
sextic	242.988.897.333	-154.331	60.331	3.081	287.861	273.819	-1.000	-1.000	39	517.787

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	3	4	5	8	11	1	6	7	8	9
quadratic	264	658	656	920	1.184	225	689	688	912	1.137
cubic	18.194	53.915	53.916	72.112	90.267	17.938	54.171	54.172	72.112	90.011
quartic	1.070.257	3.201.781	3.201.780	4.272.036	5.342.293	1.067.432	3.204.146	3.204.144	4.271.576	5.339.008
quintic	50.844.497	152.493.506	152.493.507	203.337.980	254.182.477	50.828.027	152.509.976	152.509.977	203.337.980	254.166.007
sextic	2.025.010.976	6.074.628.544	6.074.628.592	8.099.639.568	10.124.648.984	2.024.876.953	6.074.744.300	6.074.744.348	8.099.621.300	10.124.496.732

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)
..6.	AgAg.	..10.	T1gT1g.	..15.	T2gT2g.	..36.	GgGg.	..66.	HgHg.	..1.	AuAu.	..21.	T1uT1u.	..28.	T2uT2u.	..36.	GuGu.	..45.	HuHu.
Subtotal: 264 / 10 / 10
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Hu)
Subtotal: 0 / 0 / 45
Total: 264 / 10 / 55

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Hu)
..10.	AgAgAg.	..4.	T1gT1gT1g.	..10.	T2gT2gT2g.	..120.	GgGgGg.	..572.	HgHgHg.
Subtotal: 716 / 5 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Hu)
..110.	T1gT1gHg.	..165.	T2gT2gHg.	..396.	GgGgHg.	..30.	AgT1gT1g.	..45.	AgT2gT2g.	..108.	AgGgGg.	..198.	AgHgHg.	..3.	AgAuAu.	..63.	AgT1uT1u.	..84.	AgT2uT2u.
..108.	AgGuGu.	..135.	AgHuHu.	..112.	T1gGgGg.	..220.	T1gHgHg.	..60.	T1gT1uT1u.	..112.	T1gGuGu.	..144.	T1gHuHu.	..140.	T2gGgGg.	..275.	T2gHgHg.	..105.	T2gT2uT2u.
..140.	T2gGuGu.	..180.	T2gHuHu.	..968.	GgHgHg.	..288.	GgGuGu.	..648.	GgHuHu.	..231.	HgT1uT1u.	..308.	HgT2uT2u.	..396.	HgGuGu.	..990.	HgHuHu.
Subtotal: 6.762 / 29 / 90
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Hu)
..160.	T1gT2gGg.	..220.	T1gT2gHg.	..352.	T1gGgHg.	..24.	T1gAuT1u.	..216.	T1gT1uHu.	..224.	T1gT2uGu.	..252.	T1gT2uHu.	..288.	T1gGuHu.	..440.	T2gGgHg.	..35.	T2gAuT2u.
..240.	T2gT1uGu.	..270.	T2gT1uHu.	..315.	T2gT2uHu.	..360.	T2gGuHu.	..64.	GgAuGu.	..336.	GgT1uT2u.	..384.	GgT1uGu.	..432.	GgT1uHu.	..448.	GgT2uGu.	..504.	GgT2uHu.
..576.	GgGuHu.	..99.	HgAuHu.	..462.	HgT1uT2u.	..528.	HgT1uGu.	..594.	HgT1uHu.	..616.	HgT2uGu.	..693.	HgT2uHu.	..1.584.	HgGuHu.
Subtotal: 10.716 / 28 / 120
Total: 18.194 / 62 / 220

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Hu)
..15.	AgAgAgAg.	..55.	T1gT1gT1gT1g.	..120.	T2gT2gT2gT2g.	..1.066.	GgGgGgGg.	..7.777.	HgHgHgHg.	..1.	AuAuAuAu.	..231.	T1uT1uT1uT1u.	..406.	T2uT2uT2uT2u.	..1.066.	GuGuGuGu.	..3.600.	HuHuHuHu.
Subtotal: 14.337 / 10 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Hu)
..100.	T1gT1gT1gT2g.	..160.	T1gT1gT1gGg.	..220.	T1gT1gT1gHg.	..280.	T2gT2gT2gGg.	..440.	T2gT2gT2gHg.	..5.016.	GgGgGgHg.	..392.	T1uT1uT1uT2u.	..448.	T1uT1uT1uGu.	..630.	T1uT1uT1uHu.	..672.	T2uT2uT2uGu.
..1.008.	T2uT2uT2uHu.	..4.104.	GuGuGuHu.	..12.	AgT1gT1gT1g.	..30.	AgT2gT2gT2g.	..360.	AgGgGgGg.	..1.716.	AgHgHgHg.	..140.	T1gT2gT2gT2g.	..1.152.	T1gGgGgGg.	..5.324.	T1gHgHgHg.	..1.440.	T2gGgGgGg.
..6.655.	T2gHgHgHg.	..15.224.	GgHgHgHg.	..20.	AuT1uT1uT1u.	..35.	AuT2uT2uT2u.	..120.	AuGuGuGu.	..330.	AuHuHuHu.	..504.	T1uT2uT2uT2u.	..1.728.	T1uGuGuGu.	..4.374.	T1uHuHuHu.	..2.016.	T2uGuGuGu.
..5.103.	T2uHuHuHu.	..8.472.	GuHuHuHu.
Subtotal: 68.225 / 32 / 90
Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Hu)
..60.	AgAgT1gT1g.	..90.	AgAgT2gT2g.	..216.	AgAgGgGg.	..396.	AgAgHgHg.	..6.	AgAgAuAu.	..126.	AgAgT1uT1u.	..168.	AgAgT2uT2u.	..216.	AgAgGuGu.	..270.	AgAgHuHu.	..300.	T1gT1gT2gT2g.
..888.	T1gT1gGgGg.	..2.310.	T1gT1gHgHg.	..10.	T1gT1gAuAu.	..510.	T1gT1gT1uT1u.	..560.	T1gT1gT2uT2u.	..888.	T1gT1gGuGu.	..1.566.	T1gT1gHuHu.	..1.360.	T2gT2gGgGg.	..3.520.	T2gT2gHgHg.	..15.	T2gT2gAuAu.
..630.	T2gT2gT1uT1u.	..1.050.	T2gT2gT2uT2u.	..1.360.	T2gT2gGuGu.	..2.385.	T2gT2gHuHu.	..14.564.	GgGgHgHg.	..36.	GgGgAuAu.	..1.932.	GgGgT1uT1u.	..2.604.	GgGgT2uT2u.	..5.456.	GgGgGuGu.	..9.792.	GgGgHuHu.
..66.	HgHgAuAu.	..4.983.	HgHgT1uT1u.	..6.699.	HgHgT2uT2u.	..14.564.	HgHgGuGu.	..28.611.	HgHgHuHu.	..21.	AuAuT1uT1u.	..28.	AuAuT2uT2u.	..36.	AuAuGuGu.	..45.	AuAuHuHu.	..1.176.	T1uT1uT2uT2u.
..1.932.	T1uT1uGuGu.	..3.375.	T1uT1uHuHu.	..2.604.	T2uT2uGuGu.	..4.536.	T2uT2uHuHu.	..9.792.	GuGuHuHu.
Subtotal: 131.752 / 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)
..640.	T1gT1gT2gGg.	..880.	T1gT1gT2gHg.	..2.288.	T1gT1gGgHg.	..36.	T1gT1gAuT1u.	..90.	T1gT1gAuHu.	..420.	T1gT1gT1uT2u.	..480.	T1gT1gT1uGu.	..864.	T1gT1gT1uHu.	..896.	T1gT1gT2uGu.	..1.008.	T1gT1gT2uHu.
..1.872.	T1gT1gGuHu.	..3.520.	T2gT2gGgHg.	..70.	T2gT2gAuT2u.	..135.	T2gT2gAuHu.	..630.	T2gT2gT1uT2u.	..1.200.	T2gT2gT1uGu.	..1.350.	T2gT2gT1uHu.	..840.	T2gT2gT2uGu.	..1.575.	T2gT2gT2uHu.	..2.880.	T2gT2gGuHu.
..168.	GgGgAuT1u.	..196.	GgGgAuT2u.	..288.	GgGgAuGu.	..324.	GgGgAuHu.	..3.024.	GgGgT1uT2u.	..4.800.	GgGgT1uGu.	..6.912.	GgGgT1uHu.	..5.600.	GgGgT2uGu.	..8.064.	GgGgT2uHu.	..11.808.	GgGgGuHu.
..330.	HgHgAuT1u.	..385.	HgHgAuT2u.	..968.	HgHgAuGu.	..1.188.	HgHgAuHu.	..10.626.	HgHgT1uT2u.	..14.784.	HgHgT1uGu.	..19.602.	HgHgT1uHu.	..17.248.	HgHgT2uGu.	..22.869.	HgHgT2uHu.	..35.640.	HgHgGuHu.
..2.016.	T1uT1uT2uGu.	..2.268.	T1uT1uT2uHu.	..4.104.	T1uT1uGuHu.	..5.544.	T2uT2uGuHu.	..330.	AgT1gT1gHg.	..495.	AgT2gT2gHg.	..1.188.	AgGgGgHg.	..800.	T1gT2gT2gGg.	..1.100.	T1gT2gT2gHg.	..5.632.	T1gGgGgHg.
..7.040.	T2gGgGgHg.	..189.	AuT1uT1uHu.	..252.	AuT2uT2uHu.	..324.	AuGuGuHu.	..2.352.	T1uT2uT2uGu.	..2.646.	T1uT2uT2uHu.	..6.912.	T1uGuGuHu.	..8.064.	T2uGuGuHu.	..336.	AgT1gGgGg.	..660.	AgT1gHgHg.
..180.	AgT1gT1uT1u.	..336.	AgT1gGuGu.	..432.	AgT1gHuHu.	..420.	AgT2gGgGg.	..825.	AgT2gHgHg.	..315.	AgT2gT2uT2u.	..420.	AgT2gGuGu.	..540.	AgT2gHuHu.	..2.904.	AgGgHgHg.	..864.	AgGgGuGu.
..1.944.	AgGgHuHu.	..693.	AgHgT1uT1u.	..924.	AgHgT2uT2u.	..1.188.	AgHgGuGu.	..2.970.	AgHgHuHu.	..1.440.	T1gT2gGgGg.	..5.060.	T1gT2gHgHg.	..420.	T1gT2gT1uT1u.	..560.	T1gT2gT2uT2u.	..1.440.	T1gT2gGuGu.
..3.420.	T1gT2gHuHu.	..9.856.	T1gGgHgHg.	..672.	T1gGgT1uT1u.	..1.568.	T1gGgT2uT2u.	..3.200.	T1gGgGuGu.	..6.624.	T1gGgHuHu.	..1.584.	T1gHgT1uT1u.	..2.156.	T1gHgT2uT2u.	..5.632.	T1gHgGuGu.	..10.692.	T1gHgHuHu.
..12.320.	T2gGgHgHg.	..1.440.	T2gGgT1uT1u.	..1.120.	T2gGgT2uT2u.	..4.000.	T2gGgGuGu.	..8.280.	T2gGgHuHu.	..1.980.	T2gHgT1uT1u.	..2.695.	T2gHgT2uT2u.	..7.040.	T2gHgGuGu.	..13.365.	T2gHgHuHu.	..5.016.	GgHgT1uT1u.
..6.776.	GgHgT2uT2u.	..14.432.	GgHgGuGu.	..29.304.	GgHgHuHu.	..168.	AuT1uGuGu.	..216.	AuT1uHuHu.	..196.	AuT2uGuGu.	..252.	AuT2uHuHu.	..648.	AuGuHuHu.	..3.024.	T1uT2uGuGu.	..7.182.	T1uT2uHuHu.
..9.936.	T1uGuHuHu.	..11.592.	T2uGuHuHu.
Subtotal: 449.011 / 112 / 360
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Hu)
..480.	AgT1gT2gGg.	..660.	AgT1gT2gHg.	..1.056.	AgT1gGgHg.	..72.	AgT1gAuT1u.	..648.	AgT1gT1uHu.	..672.	AgT1gT2uGu.	..756.	AgT1gT2uHu.	..864.	AgT1gGuHu.	..1.320.	AgT2gGgHg.	..105.	AgT2gAuT2u.
..720.	AgT2gT1uGu.	..810.	AgT2gT1uHu.	..945.	AgT2gT2uHu.	..1.080.	AgT2gGuHu.	..192.	AgGgAuGu.	..1.008.	AgGgT1uT2u.	..1.152.	AgGgT1uGu.	..1.296.	AgGgT1uHu.	..1.344.	AgGgT2uGu.	..1.512.	AgGgT2uHu.
..1.728.	AgGgGuHu.	..297.	AgHgAuHu.	..1.386.	AgHgT1uT2u.	..1.584.	AgHgT1uGu.	..1.782.	AgHgT1uHu.	..1.848.	AgHgT2uGu.	..2.079.	AgHgT2uHu.	..4.752.	AgHgGuHu.	..5.280.	T1gT2gGgHg.	..160.	T1gT2gAuGu.
..180.	T1gT2gAuHu.	..1.680.	T1gT2gT1uT2u.	..1.920.	T1gT2gT1uGu.	..2.160.	T1gT2gT1uHu.	..2.240.	T1gT2gT2uGu.	..2.520.	T1gT2gT2uHu.	..4.320.	T1gT2gGuHu.	..224.	T1gGgAuT2u.	..256.	T1gGgAuGu.	..288.	T1gGgAuHu.
..2.688.	T1gGgT1uT2u.	..4.608.	T1gGgT1uGu.	..5.184.	T1gGgT1uHu.	..3.584.	T1gGgT2uGu.	..6.048.	T1gGgT2uHu.	..9.216.	T1gGgGuHu.	..264.	T1gHgAuT1u.	..308.	T1gHgAuT2u.	..352.	T1gHgAuGu.	..396.	T1gHgAuHu.
..3.696.	T1gHgT1uT2u.	..6.336.	T1gHgT1uGu.	..9.504.	T1gHgT1uHu.	..7.392.	T1gHgT2uGu.	..11.088.	T1gHgT2uHu.	..15.840.	T1gHgGuHu.	..240.	T2gGgAuT1u.	..320.	T2gGgAuGu.	..360.	T2gGgAuHu.	..3.360.	T2gGgT1uT2u.
..3.840.	T2gGgT1uGu.	..6.480.	T2gGgT1uHu.	..6.720.	T2gGgT2uGu.	..7.560.	T2gGgT2uHu.	..11.520.	T2gGgGuHu.	..330.	T2gHgAuT1u.	..385.	T2gHgAuT2u.	..440.	T2gHgAuGu.	..495.	T2gHgAuHu.	..4.620.	T2gHgT1uT2u.
..7.920.	T2gHgT1uGu.	..11.880.	T2gHgT1uHu.	..9.240.	T2gHgT2uGu.	..13.860.	T2gHgT2uHu.	..19.800.	T2gHgGuHu.	..528.	GgHgAuT1u.	..616.	GgHgAuT2u.	..704.	GgHgAuGu.	..1.584.	GgHgAuHu.	..11.088.	GgHgT1uT2u.
..16.896.	GgHgT1uGu.	..23.760.	GgHgT1uHu.	..19.712.	GgHgT2uGu.	..27.720.	GgHgT2uHu.	..44.352.	GgHgGuHu.	..336.	AuT1uT2uGu.	..378.	AuT1uT2uHu.	..432.	AuT1uGuHu.	..504.	AuT2uGuHu.	..9.072.	T1uT2uGuHu.
Subtotal: 406.932 / 90 / 210
Total: 1.070.257 / 289 / 715

Calculate contributions to

Ag	T1g	T2g	Gg	Hg	Au	T1u	T2u	Gu	Hu

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