Point Group D6h

Results for Point Group D_6h

**Characters of representations for molecular motions**
Motion	E	2C₆	C₂	3C'₂	3C''₂	i	2S₃	σ_h	3σ_d	3σ_v
Cartesian 3N	144	0	0	0	0	0	0	0	0	8
Translation (x,y,z)	3	2	-1	-1	-1	-3	-2	1	1	1
Rotation (R_x,R_y,R_z)	3	2	-1	-1	-1	3	2	-1	-1	-1
Vibration	138	-4	2	2	2	0	0	0	0	8

**Decomposition to irreducible representations**
Motion	A_1g	A_2g	B_1g	B_2g	E_1g	E_2g	A_1u	A_2u	B_1u	B_2u	E_1u	E_2u	Total
Cartesian 3N	7	5	5	7	12	12	5	7	7	5	12	12	96
Translation (x,y,z)	0	0	0	0	0	0	0	1	0	0	1	0	2
Rotation (R_x,R_y,R_z)	0	1	0	0	1	0	0	0	0	0	0	0	2
Vibration	7	4	5	7	11	12	5	6	7	5	11	12	92

**Molecular parameter**
Number of Atoms (N)	48
Number of internal coordinates	138
Number of independant internal coordinates	7
Number of vibrational modes	92

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A_1g	A_2g	B_1g	B_2g	E_1g	E_2g	A_1u	A_2u	B_1u	B_2u	E_1u	E_2u	Total
Linear (IR)	7	4	5	7	11	12	5	6	7	5	11	12	17 / 75
Quadratic (Raman)	7	4	5	7	11	12	5	6	7	5	11	12	30 / 62
IR + Raman	- - - -	4	5	7	- - - -	- - - -	5	- - - -	7	5	- - - -	12	0^* / 45

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)
Force field	E	2C₆	2C₃	C₂	3C'₂	3C''₂	i	2S₃	2S₆	σ_h	3σ_d	3σ_v
linear	138	-4	0	2	2	2	0	0	0	0	0	8
quadratic	9.591	8	0	71	71	71	69	0	0	69	69	101
cubic	447.580	-10	46	140	140	140	0	0	0	0	0	640
quartic	15.777.195	8	0	2.555	2.555	2.555	2.415	0	0	2.415	2.415	4.815
quintic	448.072.338	-4	0	4.970	4.970	4.970	0	0	0	0	0	25.752
sextic	10.679.057.389	25	1.081	62.125	62.125	62.125	57.155	23	23	57.155	57.155	148.291

Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field	A_1g	A_2g	B_1g	B_2g	E_1g	E_2g	A_1u	A_2u	B_1u	B_2u	E_1u	E_2u
linear	7	4	5	7	11	12	5	6	7	5	11	12
quadratic	448	370	392	400	794	816	394	401	400	392	794	793
cubic	18.773	18.543	18.568	18.728	37.282	37.307	18.613	18.703	18.728	18.568	37.282	37.307
quartic	659.234	656.149	656.976	657.576	1.314.554	1.315.381	657.024	657.554	657.576	656.976	1.314.554	1.314.576
quintic	18.674.349	18.665.426	18.666.255	18.672.693	37.338.947	37.339.776	18.667.911	18.671.864	18.672.693	18.666.255	37.338.947	37.339.776
sextic	445.009.384	444.926.960	444.946.832	444.969.616	889.916.184	889.936.056	444.948.489	444.968.788	444.969.616	444.946.832	889.916.184	889.917.012

Contributions to nonvanishing force field constants

pos(X) : Position of irreducible representation (irrep) X in character table of D_6h

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_1g) ≤ i ≤ pos(E_2u)
..28.	A_1gA_1g.	..10.	A_2gA_2g.	..15.	B_1gB_1g.	..28.	B_2gB_2g.	..66.	E_1gE_1g.	..78.	E_2gE_2g.	..15.	A_1uA_1u.	..21.	A_2uA_2u.	..28.	B_1uB_1u.	..15.	B_2uB_2u.
..66.	E_1uE_1u.	..78.	E_2uE_2u.
Subtotal: 448 / 12 / 12
Irrep combinations (i,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
Subtotal: 0 / 0 / 66
Total: 448 / 12 / 78

**Contributions to nonvanishing cubic force field constants**
Irrep combinations (i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(E_2u)
..84.	A_1gA_1gA_1g.	..364.	E_2gE_2gE_2g.
Subtotal: 448 / 2 / 12
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
..792.	E_1gE_1gE_2g.	..70.	A_1gA_2gA_2g.	..105.	A_1gB_1gB_1g.	..196.	A_1gB_2gB_2g.	..462.	A_1gE_1gE_1g.	..546.	A_1gE_2gE_2g.	..105.	A_1gA_1uA_1u.	..147.	A_1gA_2uA_2u.	..196.	A_1gB_1uB_1u.	..105.	A_1gB_2uB_2u.
..462.	A_1gE_1uE_1u.	..546.	A_1gE_2uE_2u.	..220.	A_2gE_1gE_1g.	..264.	A_2gE_2gE_2g.	..220.	A_2gE_1uE_1u.	..264.	A_2gE_2uE_2u.	..792.	E_2gE_1uE_1u.	..936.	E_2gE_2uE_2u.
Subtotal: 6.428 / 18 / 132
Irrep combinations (i,j,k) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ pos(E_2u)
..140.	A_2gB_1gB_2g.	..120.	A_2gA_1uA_2u.	..140.	A_2gB_1uB_2u.	..660.	B_1gE_1gE_2g.	..175.	B_1gA_1uB_1u.	..150.	B_1gA_2uB_2u.	..660.	B_1gE_1uE_2u.	..924.	B_2gE_1gE_2g.	..175.	B_2gA_1uB_2u.	..294.	B_2gA_2uB_1u.
..924.	B_2gE_1uE_2u.	..605.	E_1gA_1uE_1u.	..726.	E_1gA_2uE_1u.	..924.	E_1gB_1uE_2u.	..660.	E_1gB_2uE_2u.	..1.452.	E_1gE_1uE_2u.	..720.	E_2gA_1uE_2u.	..864.	E_2gA_2uE_2u.	..924.	E_2gB_1uE_1u.	..660.	E_2gB_2uE_1u.
Subtotal: 11.897 / 20 / 220
Total: 18.773 / 40 / 364

**Contributions to nonvanishing quartic force field constants**
Irrep combinations (i,i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(E_2u)
..210.	A_1gA_1gA_1gA_1g.	..35.	A_2gA_2gA_2gA_2g.	..70.	B_1gB_1gB_1gB_1g.	..210.	B_2gB_2gB_2gB_2g.	..2.211.	E_1gE_1gE_1gE_1g.	..3.081.	E_2gE_2gE_2gE_2g.	..70.	A_1uA_1uA_1uA_1u.	..126.	A_2uA_2uA_2uA_2u.	..210.	B_1uB_1uB_1uB_1u.	..70.	B_2uB_2uB_2uB_2u.
..2.211.	E_1uE_1uE_1uE_1u.	..3.081.	E_2uE_2uE_2uE_2u.
Subtotal: 11.585 / 12 / 12
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
..2.548.	A_1gE_2gE_2gE_2g.	..1.456.	A_2gE_2gE_2gE_2g.	..1.430.	B_1gE_1gE_1gE_1g.	..2.002.	B_2gE_1gE_1gE_1g.	..1.820.	A_1uE_2uE_2uE_2u.	..2.184.	A_2uE_2uE_2uE_2u.	..2.002.	B_1uE_1uE_1uE_1u.	..1.430.	B_2uE_1uE_1uE_1u.
Subtotal: 14.872 / 8 / 132
Irrep combinations (i,i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
..280.	A_1gA_1gA_2gA_2g.	..420.	A_1gA_1gB_1gB_1g.	..784.	A_1gA_1gB_2gB_2g.	..1.848.	A_1gA_1gE_1gE_1g.	..2.184.	A_1gA_1gE_2gE_2g.	..420.	A_1gA_1gA_1uA_1u.	..588.	A_1gA_1gA_2uA_2u.	..784.	A_1gA_1gB_1uB_1u.	..420.	A_1gA_1gB_2uB_2u.	..1.848.	A_1gA_1gE_1uE_1u.
..2.184.	A_1gA_1gE_2uE_2u.	..150.	A_2gA_2gB_1gB_1g.	..280.	A_2gA_2gB_2gB_2g.	..660.	A_2gA_2gE_1gE_1g.	..780.	A_2gA_2gE_2gE_2g.	..150.	A_2gA_2gA_1uA_1u.	..210.	A_2gA_2gA_2uA_2u.	..280.	A_2gA_2gB_1uB_1u.	..150.	A_2gA_2gB_2uB_2u.	..660.	A_2gA_2gE_1uE_1u.
..780.	A_2gA_2gE_2uE_2u.	..420.	B_1gB_1gB_2gB_2g.	..990.	B_1gB_1gE_1gE_1g.	..1.170.	B_1gB_1gE_2gE_2g.	..225.	B_1gB_1gA_1uA_1u.	..315.	B_1gB_1gA_2uA_2u.	..420.	B_1gB_1gB_1uB_1u.	..225.	B_1gB_1gB_2uB_2u.	..990.	B_1gB_1gE_1uE_1u.	..1.170.	B_1gB_1gE_2uE_2u.
..1.848.	B_2gB_2gE_1gE_1g.	..2.184.	B_2gB_2gE_2gE_2g.	..420.	B_2gB_2gA_1uA_1u.	..588.	B_2gB_2gA_2uA_2u.	..784.	B_2gB_2gB_1uB_1u.	..420.	B_2gB_2gB_2uB_2u.	..1.848.	B_2gB_2gE_1uE_1u.	..2.184.	B_2gB_2gE_2uE_2u.	..13.926.	E_1gE_1gE_2gE_2g.	..990.	E_1gE_1gA_1uA_1u.
..1.386.	E_1gE_1gA_2uA_2u.	..1.848.	E_1gE_1gB_1uB_1u.	..990.	E_1gE_1gB_2uB_2u.	..11.737.	E_1gE_1gE_1uE_1u.	..13.926.	E_1gE_1gE_2uE_2u.	..1.170.	E_2gE_2gA_1uA_1u.	..1.638.	E_2gE_2gA_2uA_2u.	..2.184.	E_2gE_2gB_1uB_1u.	..1.170.	E_2gE_2gB_2uB_2u.	..13.926.	E_2gE_2gE_1uE_1u.
..16.524.	E_2gE_2gE_2uE_2u.	..315.	A_1uA_1uA_2uA_2u.	..420.	A_1uA_1uB_1uB_1u.	..225.	A_1uA_1uB_2uB_2u.	..990.	A_1uA_1uE_1uE_1u.	..1.170.	A_1uA_1uE_2uE_2u.	..588.	A_2uA_2uB_1uB_1u.	..315.	A_2uA_2uB_2uB_2u.	..1.386.	A_2uA_2uE_1uE_1u.	..1.638.	A_2uA_2uE_2uE_2u.
..420.	B_1uB_1uB_2uB_2u.	..1.848.	B_1uB_1uE_1uE_1u.	..2.184.	B_1uB_1uE_2uE_2u.	..990.	B_2uB_2uE_1uE_1u.	..1.170.	B_2uB_2uE_2uE_2u.	..13.926.	E_1uE_1uE_2uE_2u.
Subtotal: 141.061 / 66 / 66
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ pos(E_2u)
..1.650.	E_1gE_1gA_1uA_2u.	..3.960.	E_1gE_1gA_1uE_2u.	..4.752.	E_1gE_1gA_2uE_2u.	..1.925.	E_1gE_1gB_1uB_2u.	..5.082.	E_1gE_1gB_1uE_1u.	..3.630.	E_1gE_1gB_2uE_1u.	..1.980.	E_2gE_2gA_1uA_2u.	..4.680.	E_2gE_2gA_1uE_2u.	..5.616.	E_2gE_2gA_2uE_2u.	..2.310.	E_2gE_2gB_1uB_2u.
..6.006.	E_2gE_2gB_1uE_1u.	..4.290.	E_2gE_2gB_2uE_1u.	..5.544.	A_1gE_1gE_1gE_2g.	..3.168.	A_2gE_1gE_1gE_2g.	..3.960.	A_1uE_1uE_1uE_2u.	..4.752.	A_2uE_1uE_1uE_2u.	..1.540.	A_1gA_2gE_1gE_1g.	..1.848.	A_1gA_2gE_2gE_2g.	..1.540.	A_1gA_2gE_1uE_1u.	..1.848.	A_1gA_2gE_2uE_2u.
..5.544.	A_1gE_2gE_1uE_1u.	..6.552.	A_1gE_2gE_2uE_2u.	..3.168.	A_2gE_2gE_1uE_1u.	..3.744.	A_2gE_2gE_2uE_2u.	..1.925.	B_1gB_2gE_1gE_1g.	..2.310.	B_1gB_2gE_2gE_2g.	..1.925.	B_1gB_2gE_1uE_1u.	..2.310.	B_1gB_2gE_2uE_2u.	..4.290.	B_1gE_1gE_2gE_2g.	..3.630.	B_1gE_1gE_1uE_1u.
..4.290.	B_1gE_1gE_2uE_2u.	..6.006.	B_2gE_1gE_2gE_2g.	..5.082.	B_2gE_1gE_1uE_1u.	..6.006.	B_2gE_1gE_2uE_2u.	..1.650.	A_1uA_2uE_1uE_1u.	..1.980.	A_1uA_2uE_2uE_2u.	..1.925.	B_1uB_2uE_1uE_1u.	..2.310.	B_1uB_2uE_2uE_2u.	..6.006.	B_1uE_1uE_2uE_2u.	..4.290.	B_2uE_1uE_2uE_2u.
Subtotal: 145.024 / 40 / 660
Irrep combinations (i,j,k,l) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(E_2u)
..980.	A_1gA_2gB_1gB_2g.	..840.	A_1gA_2gA_1uA_2u.	..980.	A_1gA_2gB_1uB_2u.	..4.620.	A_1gB_1gE_1gE_2g.	..1.225.	A_1gB_1gA_1uB_1u.	..1.050.	A_1gB_1gA_2uB_2u.	..4.620.	A_1gB_1gE_1uE_2u.	..6.468.	A_1gB_2gE_1gE_2g.	..1.225.	A_1gB_2gA_1uB_2u.	..2.058.	A_1gB_2gA_2uB_1u.
..6.468.	A_1gB_2gE_1uE_2u.	..4.235.	A_1gE_1gA_1uE_1u.	..5.082.	A_1gE_1gA_2uE_1u.	..6.468.	A_1gE_1gB_1uE_2u.	..4.620.	A_1gE_1gB_2uE_2u.	..10.164.	A_1gE_1gE_1uE_2u.	..5.040.	A_1gE_2gA_1uE_2u.	..6.048.	A_1gE_2gA_2uE_2u.	..6.468.	A_1gE_2gB_1uE_1u.	..4.620.	A_1gE_2gB_2uE_1u.
..2.640.	A_2gB_1gE_1gE_2g.	..500.	A_2gB_1gA_1uB_2u.	..840.	A_2gB_1gA_2uB_1u.	..2.640.	A_2gB_1gE_1uE_2u.	..3.696.	A_2gB_2gE_1gE_2g.	..980.	A_2gB_2gA_1uB_1u.	..840.	A_2gB_2gA_2uB_2u.	..3.696.	A_2gB_2gE_1uE_2u.	..2.420.	A_2gE_1gA_1uE_1u.	..2.904.	A_2gE_1gA_2uE_1u.
..3.696.	A_2gE_1gB_1uE_2u.	..2.640.	A_2gE_1gB_2uE_2u.	..5.808.	A_2gE_1gE_1uE_2u.	..2.880.	A_2gE_2gA_1uE_2u.	..3.456.	A_2gE_2gA_2uE_2u.	..3.696.	A_2gE_2gB_1uE_1u.	..2.640.	A_2gE_2gB_2uE_1u.	..1.050.	B_1gB_2gA_1uA_2u.	..1.225.	B_1gB_2gB_1uB_2u.	..3.300.	B_1gE_1gA_1uE_2u.
..3.960.	B_1gE_1gA_2uE_2u.	..4.235.	B_1gE_1gB_1uE_1u.	..3.025.	B_1gE_1gB_2uE_1u.	..3.300.	B_1gE_2gA_1uE_1u.	..3.960.	B_1gE_2gA_2uE_1u.	..5.040.	B_1gE_2gB_1uE_2u.	..3.600.	B_1gE_2gB_2uE_2u.	..7.920.	B_1gE_2gE_1uE_2u.	..4.620.	B_2gE_1gA_1uE_2u.	..5.544.	B_2gE_1gA_2uE_2u.
..5.929.	B_2gE_1gB_1uE_1u.	..4.235.	B_2gE_1gB_2uE_1u.	..4.620.	B_2gE_2gA_1uE_1u.	..5.544.	B_2gE_2gA_2uE_1u.	..7.056.	B_2gE_2gB_1uE_2u.	..5.040.	B_2gE_2gB_2uE_2u.	..11.088.	B_2gE_2gE_1uE_2u.	..4.620.	E_1gE_2gA_1uB_1u.	..3.300.	E_1gE_2gA_1uB_2u.	..7.260.	E_1gE_2gA_1uE_1u.
..5.544.	E_1gE_2gA_2uB_1u.	..3.960.	E_1gE_2gA_2uB_2u.	..8.712.	E_1gE_2gA_2uE_1u.	..11.088.	E_1gE_2gB_1uE_2u.	..7.920.	E_1gE_2gB_2uE_2u.	..52.272.	E_1gE_2gE_1uE_2u.	..1.050.	A_1uA_2uB_1uB_2u.	..4.620.	A_1uB_1uE_1uE_2u.	..3.300.	A_1uB_2uE_1uE_2u.	..5.544.	A_2uB_1uE_1uE_2u.
..3.960.	A_2uB_2uE_1uE_2u.
Subtotal: 346.692 / 71 / 495
Total: 659.234 / 197 / 1.365

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Results for Point Group D6h

Force field analysis

Further Reading

Contributions to nonvanishing force field constants

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Results for Point Group D_6h