Point Group Ih

Results for Point Group I_h

**Characters of representations for molecular motions**
Motion	E	12C₅	12(C₅)²	15C₂	i	12S₁₀	12(S₁₀)³	15σ_d
Cartesian 3N	60	0.000	-0.000	0	0	0.000	-0.000	4
Translation (x,y,z)	3	1.618	-0.618	-1	-3	0.618	-1.618	1
Rotation (R_x,R_y,R_z)	3	1.618	-0.618	-1	3	-0.618	1.618	-1
Vibration	54	-3.236	1.236	2	0	0.000	0.000	4

**Decomposition to irreducible representations**
Motion	A_g	T_1g	T_2g	G_g	H_g	T_1u	T_2u	G_u	H_u	Total
Cartesian 3N	1	1	1	2	3	2	2	2	2	16
Translation (x,y,z)	0	0	0	0	0	1	0	0	0	1
Rotation (R_x,R_y,R_z)	0	1	0	0	0	0	0	0	0	1
Vibration	1	0	1	2	3	1	2	2	2	14

**Molecular parameter**
Number of Atoms (N)	20
Number of internal coordinates	54
Number of independant internal coordinates	1
Number of vibrational modes	14

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A_g	T_2g	G_g	H_g	T_1u	T_2u	G_u	H_u	Total
Linear (IR)	1	1	2	3	1	2	2	2	1 / 13
Quadratic (Raman)	1	1	2	3	1	2	2	2	4 / 10
IR + Raman	- - - -	1	2	- - - -	- - - -	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	15σ_d
linear	54	-3.236	1.236	0	2	0	0.000	0.000	0	4
quadratic	1.485	5.854	-0.854	0	29	27	-1.618	0.618	0	35
cubic	27.720	-7.236	-2.764	18	56	0	0.000	0.000	0	120
quartic	395.010	5.854	-0.854	0	434	378	1.618	-0.618	0	610
quintic	4.582.116	8.764	13.236	0	812	0	0.000	0.000	0	1.856
sextic	45.057.474	-37.833	15.833	171	4.466	3.654	-1.000	-1.000	9	7.134

Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field	A_g	T_1g	T_2g	G_g	H_g	A_u	T_1u	T_2u	G_u	H_u
linear	1	0	1	2	3	0	1	2	2	2
quadratic	21	31	29	50	71	12	38	37	48	60
cubic	255	670	671	928	1.174	225	700	701	928	1.144
quartic	3.426	9.755	9.754	13.179	16.605	3.267	9.889	9.887	13.154	16.421
quintic	38.520	114.220	114.221	152.735	191.255	38.056	114.684	114.685	152.735	190.791
sextic	376.987	1.125.071	1.125.083	1.502.070	1.878.967	375.140	1.126.672	1.126.684	1.501.823	1.876.882

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(A_g) ≤ i ≤ pos(H_u)
..1.	A_gA_g.	..1.	T_2gT_2g.	..3.	G_gG_g.	..6.	H_gH_g.	..1.	T_1uT_1u.	..3.	T_2uT_2u.	..3.	G_uG_u.	..3.	H_uH_u.
Subtotal: 21 / 8 / 10
Irrep combinations (i,j) with indices: pos(A_g) ≤ i ≤ j ≤ pos(H_u)
Subtotal: 0 / 0 / 45
Total: 21 / 8 / 55

**Contributions to nonvanishing cubic force field constants**
Irrep combinations (i,i,i) with indices: pos(A_g) ≤ i ≤ pos(H_u)
..1.	A_gA_gA_g.	..4.	G_gG_gG_g.	..20.	H_gH_gH_g.
Subtotal: 25 / 3 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A_g) ≤ i ≤ j ≤ pos(H_u)
..3.	T_2gT_2gH_g.	..9.	G_gG_gH_g.	..1.	A_gT_2gT_2g.	..3.	A_gG_gG_g.	..6.	A_gH_gH_g.	..1.	A_gT_1uT_1u.	..3.	A_gT_2uT_2u.	..3.	A_gG_uG_u.	..3.	A_gH_uH_u.	..1.	T_2gG_gG_g.
..3.	T_2gH_gH_g.	..1.	T_2gT_2uT_2u.	..1.	T_2gG_uG_u.	..1.	T_2gH_uH_u.	..18.	G_gH_gH_g.	..6.	G_gG_uG_u.	..8.	G_gH_uH_u.	..3.	H_gT_1uT_1u.	..9.	H_gT_2uT_2u.	..9.	H_gG_uG_u.
..18.	H_gH_uH_u.
Subtotal: 110 / 21 / 90
Irrep combinations (i,j,k) with indices: pos(A_g) ≤ i ≤ j ≤ k ≤ pos(H_u)
..6.	T_2gG_gH_g.	..2.	T_2gT_1uG_u.	..2.	T_2gT_1uH_u.	..4.	T_2gT_2uH_u.	..4.	T_2gG_uH_u.	..4.	G_gT_1uT_2u.	..4.	G_gT_1uG_u.	..4.	G_gT_1uH_u.	..8.	G_gT_2uG_u.	..8.	G_gT_2uH_u.
..8.	G_gG_uH_u.	..6.	H_gT_1uT_2u.	..6.	H_gT_1uG_u.	..6.	H_gT_1uH_u.	..12.	H_gT_2uG_u.	..12.	H_gT_2uH_u.	..24.	H_gG_uH_u.
Subtotal: 120 / 17 / 120
Total: 255 / 41 / 220

**Contributions to nonvanishing quartic force field constants**
Irrep combinations (i,i,i,i) with indices: pos(A_g) ≤ i ≤ pos(H_u)
..1.	A_gA_gA_gA_g.	..1.	T_2gT_2gT_2gT_2g.	..11.	G_gG_gG_gG_g.	..63.	H_gH_gH_gH_g.	..1.	T_1uT_1uT_1uT_1u.	..6.	T_2uT_2uT_2uT_2u.	..11.	G_uG_uG_uG_u.	..16.	H_uH_uH_uH_u.
Subtotal: 110 / 8 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A_g) ≤ i ≤ j ≤ pos(H_u)
..2.	T_2gT_2gT_2gG_g.	..24.	G_gG_gG_gH_g.	..2.	T_1uT_1uT_1uT_2u.	..2.	T_1uT_1uT_1uG_u.	..8.	T_2uT_2uT_2uG_u.	..4.	T_2uT_2uT_2uH_u.	..16.	G_uG_uG_uH_u.	..4.	A_gG_gG_gG_g.	..20.	A_gH_gH_gH_g.	..6.	T_2gG_gG_gG_g.
..27.	T_2gH_gH_gH_g.	..94.	G_gH_gH_gH_g.	..4.	T_1uT_2uT_2uT_2u.	..6.	T_1uG_uG_uG_u.	..8.	T_1uH_uH_uH_u.	..12.	T_2uG_uG_uG_u.	..16.	T_2uH_uH_uH_u.	..32.	G_uH_uH_uH_u.
Subtotal: 287 / 18 / 90
Irrep combinations (i,i,j,j) with indices: pos(A_g) ≤ i ≤ j ≤ pos(H_u)
..1.	A_gA_gT_2gT_2g.	..3.	A_gA_gG_gG_g.	..6.	A_gA_gH_gH_g.	..1.	A_gA_gT_1uT_1u.	..3.	A_gA_gT_2uT_2u.	..3.	A_gA_gG_uG_u.	..3.	A_gA_gH_uH_u.	..6.	T_2gT_2gG_gG_g.	..18.	T_2gT_2gH_gH_g.	..2.	T_2gT_2gT_1uT_1u.
..6.	T_2gT_2gT_2uT_2u.	..6.	T_2gT_2gG_uG_u.	..9.	T_2gT_2gH_uH_u.	..87.	G_gG_gH_gH_g.	..6.	G_gG_gT_1uT_1u.	..19.	G_gG_gT_2uT_2u.	..29.	G_gG_gG_uG_u.	..41.	G_gG_gH_uH_u.	..18.	H_gH_gT_1uT_1u.	..57.	H_gH_gT_2uT_2u.
..87.	H_gH_gG_uG_u.	..132.	H_gH_gH_uH_u.	..6.	T_1uT_1uT_2uT_2u.	..6.	T_1uT_1uG_uG_u.	..9.	T_1uT_1uH_uH_u.	..19.	T_2uT_2uG_uG_u.	..28.	T_2uT_2uH_uH_u.	..41.	G_uG_uH_uH_u.
Subtotal: 652 / 28 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A_g) ≤ i ≤ j ≤ k ≤ pos(H_u)
..12.	T_2gT_2gG_gH_g.	..2.	T_2gT_2gT_1uT_2u.	..2.	T_2gT_2gT_1uG_u.	..2.	T_2gT_2gT_1uH_u.	..4.	T_2gT_2gT_2uG_u.	..4.	T_2gT_2gT_2uH_u.	..8.	T_2gT_2gG_uH_u.	..12.	G_gG_gT_1uT_2u.	..14.	G_gG_gT_1uG_u.	..16.	G_gG_gT_1uH_u.
..28.	G_gG_gT_2uG_u.	..32.	G_gG_gT_2uH_u.	..44.	G_gG_gG_uH_u.	..42.	H_gH_gT_1uT_2u.	..48.	H_gH_gT_1uG_u.	..54.	H_gH_gT_1uH_u.	..96.	H_gH_gT_2uG_u.	..108.	H_gH_gT_2uH_u.	..156.	H_gH_gG_uH_u.	..4.	T_1uT_1uT_2uG_u.
..4.	T_1uT_1uT_2uH_u.	..8.	T_1uT_1uG_uH_u.	..28.	T_2uT_2uG_uH_u.	..3.	A_gT_2gT_2gH_g.	..9.	A_gG_gG_gH_g.	..24.	T_2gG_gG_gH_g.	..8.	T_1uT_2uT_2uG_u.	..8.	T_1uT_2uT_2uH_u.	..16.	T_1uG_uG_uH_u.	..32.	T_2uG_uG_uH_u.
..1.	A_gT_2gG_gG_g.	..3.	A_gT_2gH_gH_g.	..1.	A_gT_2gT_2uT_2u.	..1.	A_gT_2gG_uG_u.	..1.	A_gT_2gH_uH_u.	..18.	A_gG_gH_gH_g.	..6.	A_gG_gG_uG_u.	..8.	A_gG_gH_uH_u.	..3.	A_gH_gT_1uT_1u.	..9.	A_gH_gT_2uT_2u.
..9.	A_gH_gG_uG_u.	..18.	A_gH_gH_uH_u.	..48.	T_2gG_gH_gH_g.	..2.	T_2gG_gT_1uT_1u.	..6.	T_2gG_gT_2uT_2u.	..14.	T_2gG_gG_uG_u.	..22.	T_2gG_gH_uH_u.	..3.	T_2gH_gT_1uT_1u.	..12.	T_2gH_gT_2uT_2u.	..24.	T_2gH_gG_uG_u.
..36.	T_2gH_gH_uH_u.	..12.	G_gH_gT_1uT_1u.	..42.	G_gH_gT_2uT_2u.	..66.	G_gH_gG_uG_u.	..108.	G_gH_gH_uH_u.	..12.	T_1uT_2uG_uG_u.	..20.	T_1uT_2uH_uH_u.	..22.	T_1uG_uH_uH_u.	..44.	T_2uG_uH_uH_u.
Subtotal: 1.399 / 59 / 360
Irrep combinations (i,j,k,l) with indices: pos(A_g) ≤ i ≤ j ≤ k ≤ l ≤ pos(H_u)
..6.	A_gT_2gG_gH_g.	..2.	A_gT_2gT_1uG_u.	..2.	A_gT_2gT_1uH_u.	..4.	A_gT_2gT_2uH_u.	..4.	A_gT_2gG_uH_u.	..4.	A_gG_gT_1uT_2u.	..4.	A_gG_gT_1uG_u.	..4.	A_gG_gT_1uH_u.	..8.	A_gG_gT_2uG_u.	..8.	A_gG_gT_2uH_u.
..8.	A_gG_gG_uH_u.	..6.	A_gH_gT_1uT_2u.	..6.	A_gH_gT_1uG_u.	..6.	A_gH_gT_1uH_u.	..12.	A_gH_gT_2uG_u.	..12.	A_gH_gT_2uH_u.	..24.	A_gH_gG_uH_u.	..8.	T_2gG_gT_1uT_2u.	..8.	T_2gG_gT_1uG_u.	..12.	T_2gG_gT_1uH_u.
..24.	T_2gG_gT_2uG_u.	..24.	T_2gG_gT_2uH_u.	..32.	T_2gG_gG_uH_u.	..12.	T_2gH_gT_1uT_2u.	..18.	T_2gH_gT_1uG_u.	..24.	T_2gH_gT_1uH_u.	..36.	T_2gH_gT_2uG_u.	..48.	T_2gH_gT_2uH_u.	..60.	T_2gH_gG_uH_u.	..36.	G_gH_gT_1uT_2u.
..48.	G_gH_gT_1uG_u.	..60.	G_gH_gT_1uH_u.	..96.	G_gH_gT_2uG_u.	..120.	G_gH_gT_2uH_u.	..168.	G_gH_gG_uH_u.	..24.	T_1uT_2uG_uH_u.
Subtotal: 978 / 36 / 210
Total: 3.426 / 149 / 715

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Results for Point Group Ih

Force field analysis

Further Reading

Contributions to nonvanishing force field constants

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Results for Point Group I_h