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	120	0.000	-0.000	0	0	0.000	-0.000	8
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	114	-3.236	1.236	2	0	0.000	0.000	8

**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	2	2	2	4	6	4	4	4	4	32
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	2	1	2	4	6	3	4	4	4	30

**Molecular parameter**
Number of Atoms (N)	40
Number of internal coordinates	114
Number of independant internal coordinates	2
Number of vibrational modes	30

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A_g	T_1g	T_2g	G_g	H_g	T_1u	T_2u	G_u	H_u	Total
Linear (IR)	2	1	2	4	6	3	4	4	4	3 / 27
Quadratic (Raman)	2	1	2	4	6	3	4	4	4	8 / 22
IR + Raman	- - - -	1	2	4	- - - -	- - - -	4	4	4	0^* / 19

* 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	114	-3.236	1.236	0	2	0	0.000	0.000	0	8
quadratic	6.555	5.854	-0.854	0	59	57	-1.618	0.618	0	89
cubic	253.460	-7.236	-2.764	38	116	0	0.000	0.000	0	544
quartic	7.413.705	5.854	-0.854	0	1.769	1.653	1.618	-0.618	0	3.669
quintic	174.963.438	20.764	25.236	0	3.422	0	0.000	0.000	0	18.600
sextic	3.470.108.187	-76.666	30.666	741	35.931	32.509	-1.000	-1.000	19	96.957

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	2	1	2	4	6	0	3	4	4	4
quadratic	74	148	146	220	294	51	167	166	216	267
cubic	2.200	6.253	6.254	8.456	10.637	2.064	6.389	6.390	8.456	10.501
quartic	62.475	184.705	184.704	247.178	309.653	61.530	185.540	185.538	247.068	308.598
quintic	1.460.786	4.371.335	4.371.336	5.832.110	7.292.896	1.456.136	4.375.985	4.375.986	5.832.110	7.288.246
sextic	28.934.572	86.736.892	86.736.916	115.671.488	144.605.680	28.909.785	86.759.506	86.759.530	115.669.314	144.578.738

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)
..3.	A_gA_g.	..1.	T_1gT_1g.	..3.	T_2gT_2g.	..10.	G_gG_g.	..21.	H_gH_g.	..6.	T_1uT_1u.	..10.	T_2uT_2u.	..10.	G_uG_u.	..10.	H_uH_u.
Subtotal: 74 / 9 / 10
Irrep combinations (i,j) with indices: pos(A_g) ≤ i ≤ j ≤ pos(H_u)
Subtotal: 0 / 0 / 45
Total: 74 / 9 / 55

**Contributions to nonvanishing cubic force field constants**
Irrep combinations (i,i,i) with indices: pos(A_g) ≤ i ≤ pos(H_u)
..4.	A_gA_gA_g.	..20.	G_gG_gG_g.	..112.	H_gH_gH_g.
Subtotal: 136 / 3 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A_g) ≤ i ≤ j ≤ pos(H_u)
..6.	T_1gT_1gH_g.	..18.	T_2gT_2gH_g.	..60.	G_gG_gH_g.	..2.	A_gT_1gT_1g.	..6.	A_gT_2gT_2g.	..20.	A_gG_gG_g.	..42.	A_gH_gH_g.	..12.	A_gT_1uT_1u.	..20.	A_gT_2uT_2u.	..20.	A_gG_uG_u.
..20.	A_gH_uH_u.	..6.	T_1gG_gG_g.	..15.	T_1gH_gH_g.	..3.	T_1gT_1uT_1u.	..6.	T_1gG_uG_u.	..6.	T_1gH_uH_u.	..12.	T_2gG_gG_g.	..30.	T_2gH_gH_g.	..12.	T_2gT_2uT_2u.	..12.	T_2gG_uG_u.
..12.	T_2gH_uH_u.	..144.	G_gH_gH_g.	..40.	G_gG_uG_u.	..64.	G_gH_uH_u.	..36.	H_gT_1uT_1u.	..60.	H_gT_2uT_2u.	..60.	H_gG_uG_u.	..120.	H_gH_uH_u.
Subtotal: 864 / 28 / 90
Irrep combinations (i,j,k) with indices: pos(A_g) ≤ i ≤ j ≤ k ≤ pos(H_u)
..8.	T_1gT_2gG_g.	..12.	T_1gT_2gH_g.	..24.	T_1gG_gH_g.	..12.	T_1gT_1uH_u.	..16.	T_1gT_2uG_u.	..16.	T_1gT_2uH_u.	..16.	T_1gG_uH_u.	..48.	T_2gG_gH_g.	..24.	T_2gT_1uG_u.	..24.	T_2gT_1uH_u.
..32.	T_2gT_2uH_u.	..32.	T_2gG_uH_u.	..48.	G_gT_1uT_2u.	..48.	G_gT_1uG_u.	..48.	G_gT_1uH_u.	..64.	G_gT_2uG_u.	..64.	G_gT_2uH_u.	..64.	G_gG_uH_u.	..72.	H_gT_1uT_2u.	..72.	H_gT_1uG_u.
..72.	H_gT_1uH_u.	..96.	H_gT_2uG_u.	..96.	H_gT_2uH_u.	..192.	H_gG_uH_u.
Subtotal: 1.200 / 24 / 120
Total: 2.200 / 55 / 220

**Contributions to nonvanishing quartic force field constants**
Irrep combinations (i,i,i,i) with indices: pos(A_g) ≤ i ≤ pos(H_u)
..5.	A_gA_gA_gA_g.	..1.	T_1gT_1gT_1gT_1g.	..6.	T_2gT_2gT_2gT_2g.	..91.	G_gG_gG_gG_g.	..777.	H_gH_gH_gH_g.	..21.	T_1uT_1uT_1uT_1u.	..55.	T_2uT_2uT_2uT_2u.	..91.	G_uG_uG_uG_u.	..175.	H_uH_uH_uH_u.
Subtotal: 1.222 / 9 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A_g) ≤ i ≤ j ≤ pos(H_u)
..2.	T_1gT_1gT_1gT_2g.	..4.	T_1gT_1gT_1gG_g.	..16.	T_2gT_2gT_2gG_g.	..12.	T_2gT_2gT_2gH_g.	..360.	G_gG_gG_gH_g.	..40.	T_1uT_1uT_1uT_2u.	..40.	T_1uT_1uT_1uG_u.	..32.	T_1uT_1uT_1uH_u.	..80.	T_2uT_2uT_2uG_u.	..80.	T_2uT_2uT_2uH_u.
..240.	G_uG_uG_uH_u.	..40.	A_gG_gG_gG_g.	..224.	A_gH_gH_gH_g.	..4.	T_1gT_2gT_2gT_2g.	..40.	T_1gG_gG_gG_g.	..216.	T_1gH_gH_gH_g.	..80.	T_2gG_gG_gG_g.	..432.	T_2gH_gH_gH_g.	..1.312.	G_gH_gH_gH_g.	..60.	T_1uT_2uT_2uT_2u.
..120.	T_1uG_uG_uG_u.	..192.	T_1uH_uH_uH_u.	..160.	T_2uG_uG_uG_u.	..256.	T_2uH_uH_uH_u.	..416.	G_uH_uH_uH_u.
Subtotal: 4.458 / 25 / 90
Irrep combinations (i,i,j,j) with indices: pos(A_g) ≤ i ≤ j ≤ pos(H_u)
..3.	A_gA_gT_1gT_1g.	..9.	A_gA_gT_2gT_2g.	..30.	A_gA_gG_gG_g.	..63.	A_gA_gH_gH_g.	..18.	A_gA_gT_1uT_1u.	..30.	A_gA_gT_2uT_2u.	..30.	A_gA_gG_uG_u.	..30.	A_gA_gH_uH_u.	..6.	T_1gT_1gT_2gT_2g.	..20.	T_1gT_1gG_gG_g.
..63.	T_1gT_1gH_gH_g.	..12.	T_1gT_1gT_1uT_1u.	..20.	T_1gT_1gT_2uT_2u.	..20.	T_1gT_1gG_uG_u.	..30.	T_1gT_1gH_uH_u.	..66.	T_2gT_2gG_gG_g.	..204.	T_2gT_2gH_gH_g.	..36.	T_2gT_2gT_1uT_1u.	..66.	T_2gT_2gT_2uT_2u.	..66.	T_2gT_2gG_uG_u.
..96.	T_2gT_2gH_uH_u.	..1.170.	G_gG_gH_gH_g.	..138.	G_gG_gT_1uT_1u.	..236.	G_gG_gT_2uT_2u.	..372.	G_gG_gG_uG_u.	..532.	G_gG_gH_uH_u.	..423.	H_gH_gT_1uT_1u.	..720.	H_gH_gT_2uT_2u.	..1.170.	H_gH_gG_uG_u.	..1.806.	H_gH_gH_uH_u.
..120.	T_1uT_1uT_2uT_2u.	..138.	T_1uT_1uG_uG_u.	..198.	T_1uT_1uH_uH_u.	..236.	T_2uT_2uG_uG_u.	..336.	T_2uT_2uH_uH_u.	..532.	G_uG_uH_uH_u.
Subtotal: 9.045 / 36 / 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)
..8.	T_1gT_1gT_2gG_g.	..12.	T_1gT_1gT_2gH_g.	..48.	T_1gT_1gG_gH_g.	..12.	T_1gT_1gT_1uT_2u.	..12.	T_1gT_1gT_1uG_u.	..12.	T_1gT_1gT_1uH_u.	..16.	T_1gT_1gT_2uG_u.	..16.	T_1gT_1gT_2uH_u.	..32.	T_1gT_1gG_uH_u.	..168.	T_2gT_2gG_gH_g.
..36.	T_2gT_2gT_1uT_2u.	..48.	T_2gT_2gT_1uG_u.	..48.	T_2gT_2gT_1uH_u.	..48.	T_2gT_2gT_2uG_u.	..64.	T_2gT_2gT_2uH_u.	..112.	T_2gT_2gG_uH_u.	..240.	G_gG_gT_1uT_2u.	..312.	G_gG_gT_1uG_u.	..384.	G_gG_gT_1uH_u.	..416.	G_gG_gT_2uG_u.
..512.	G_gG_gT_2uH_u.	..672.	G_gG_gG_uH_u.	..936.	H_gH_gT_1uT_2u.	..1.116.	H_gH_gT_1uG_u.	..1.296.	H_gH_gT_1uH_u.	..1.488.	H_gH_gT_2uG_u.	..1.728.	H_gH_gT_2uH_u.	..2.400.	H_gH_gG_uH_u.	..144.	T_1uT_1uT_2uG_u.	..144.	T_1uT_1uT_2uH_u.
..240.	T_1uT_1uG_uH_u.	..416.	T_2uT_2uG_uH_u.	..12.	A_gT_1gT_1gH_g.	..36.	A_gT_2gT_2gH_g.	..120.	A_gG_gG_gH_g.	..16.	T_1gT_2gT_2gG_g.	..24.	T_1gT_2gT_2gH_g.	..192.	T_1gG_gG_gH_g.	..384.	T_2gG_gG_gH_g.	..192.	T_1uT_2uT_2uG_u.
..192.	T_1uT_2uT_2uH_u.	..384.	T_1uG_uG_uH_u.	..512.	T_2uG_uG_uH_u.	..12.	A_gT_1gG_gG_g.	..30.	A_gT_1gH_gH_g.	..6.	A_gT_1gT_1uT_1u.	..12.	A_gT_1gG_uG_u.	..12.	A_gT_1gH_uH_u.	..24.	A_gT_2gG_gG_g.	..60.	A_gT_2gH_gH_g.
..24.	A_gT_2gT_2uT_2u.	..24.	A_gT_2gG_uG_u.	..24.	A_gT_2gH_uH_u.	..288.	A_gG_gH_gH_g.	..80.	A_gG_gG_uG_u.	..128.	A_gG_gH_uH_u.	..72.	A_gH_gT_1uT_1u.	..120.	A_gH_gT_2uT_2u.	..120.	A_gH_gG_uG_u.	..240.	A_gH_gH_uH_u.
..40.	T_1gT_2gG_gG_g.	..156.	T_1gT_2gH_gH_g.	..12.	T_1gT_2gT_1uT_1u.	..20.	T_1gT_2gT_2uT_2u.	..40.	T_1gT_2gG_uG_u.	..72.	T_1gT_2gH_uH_u.	..372.	T_1gG_gH_gH_g.	..24.	T_1gG_gT_1uT_1u.	..64.	T_1gG_gT_2uT_2u.	..104.	T_1gG_gG_uG_u.
..168.	T_1gG_gH_uH_u.	..54.	T_1gH_gT_1uT_1u.	..96.	T_1gH_gT_2uT_2u.	..192.	T_1gH_gG_uG_u.	..288.	T_1gH_gH_uH_u.	..744.	T_2gG_gH_gH_g.	..72.	T_2gG_gT_1uT_1u.	..80.	T_2gG_gT_2uT_2u.	..208.	T_2gG_gG_uG_u.	..336.	T_2gG_gH_uH_u.
..108.	T_2gH_gT_1uT_1u.	..192.	T_2gH_gT_2uT_2u.	..384.	T_2gH_gG_uG_u.	..576.	T_2gH_gH_uH_u.	..360.	G_gH_gT_1uT_1u.	..624.	G_gH_gT_2uT_2u.	..1.008.	G_gH_gG_uG_u.	..1.632.	G_gH_gH_uH_u.	..240.	T_1uT_2uG_uG_u.	..432.	T_1uT_2uH_uH_u.
..504.	T_1uG_uH_uH_u.	..672.	T_2uG_uH_uH_u.
Subtotal: 26.350 / 92 / 360
Irrep combinations (i,j,k,l) with indices: pos(A_g) ≤ i ≤ j ≤ k ≤ l ≤ pos(H_u)
..16.	A_gT_1gT_2gG_g.	..24.	A_gT_1gT_2gH_g.	..48.	A_gT_1gG_gH_g.	..24.	A_gT_1gT_1uH_u.	..32.	A_gT_1gT_2uG_u.	..32.	A_gT_1gT_2uH_u.	..32.	A_gT_1gG_uH_u.	..96.	A_gT_2gG_gH_g.	..48.	A_gT_2gT_1uG_u.	..48.	A_gT_2gT_1uH_u.
..64.	A_gT_2gT_2uH_u.	..64.	A_gT_2gG_uH_u.	..96.	A_gG_gT_1uT_2u.	..96.	A_gG_gT_1uG_u.	..96.	A_gG_gT_1uH_u.	..128.	A_gG_gT_2uG_u.	..128.	A_gG_gT_2uH_u.	..128.	A_gG_gG_uH_u.	..144.	A_gH_gT_1uT_2u.	..144.	A_gH_gT_1uG_u.
..144.	A_gH_gT_1uH_u.	..192.	A_gH_gT_2uG_u.	..192.	A_gH_gT_2uH_u.	..384.	A_gH_gG_uH_u.	..144.	T_1gT_2gG_gH_g.	..48.	T_1gT_2gT_1uT_2u.	..48.	T_1gT_2gT_1uG_u.	..48.	T_1gT_2gT_1uH_u.	..64.	T_1gT_2gT_2uG_u.	..64.	T_1gT_2gT_2uH_u.
..96.	T_1gT_2gG_uH_u.	..96.	T_1gG_gT_1uT_2u.	..144.	T_1gG_gT_1uG_u.	..144.	T_1gG_gT_1uH_u.	..128.	T_1gG_gT_2uG_u.	..192.	T_1gG_gT_2uH_u.	..256.	T_1gG_gG_uH_u.	..144.	T_1gH_gT_1uT_2u.	..216.	T_1gH_gT_1uG_u.	..288.	T_1gH_gT_1uH_u.
..288.	T_1gH_gT_2uG_u.	..384.	T_1gH_gT_2uH_u.	..480.	T_1gH_gG_uH_u.	..192.	T_2gG_gT_1uT_2u.	..192.	T_2gG_gT_1uG_u.	..288.	T_2gG_gT_1uH_u.	..384.	T_2gG_gT_2uG_u.	..384.	T_2gG_gT_2uH_u.	..512.	T_2gG_gG_uH_u.	..288.	T_2gH_gT_1uT_2u.
..432.	T_2gH_gT_1uG_u.	..576.	T_2gH_gT_1uH_u.	..576.	T_2gH_gT_2uG_u.	..768.	T_2gH_gT_2uH_u.	..960.	T_2gH_gG_uH_u.	..864.	G_gH_gT_1uT_2u.	..1.152.	G_gH_gT_1uG_u.	..1.440.	G_gH_gT_1uH_u.	..1.536.	G_gH_gT_2uG_u.	..1.920.	G_gH_gT_2uH_u.
..2.688.	G_gH_gG_uH_u.	..576.	T_1uT_2uG_uH_u.
Subtotal: 21.400 / 62 / 210
Total: 62.475 / 224 / 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