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	108	0	0	-4	0	0	0	36	0	4
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	102	-4	2	-2	2	0	0	36	0	4

**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	6	6	2	4	6	12	2	4	6	6	12	6	72
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	6	5	2	4	5	12	2	3	6	6	11	6	68

**Molecular parameter**
Number of Atoms (N)	36
Number of internal coordinates	102
Number of independant internal coordinates	6
Number of vibrational modes	68

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)	6	5	2	4	5	12	2	3	6	6	11	6	14 / 54
Quadratic (Raman)	6	5	2	4	5	12	2	3	6	6	11	6	23 / 45
IR + Raman	- - - -	5	2	4	- - - -	- - - -	2	- - - -	6	6	- - - -	6	0^* / 31

* 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	102	-4	0	2	-2	2	0	0	0	36	0	4
quadratic	5.253	8	0	53	53	53	51	0	0	699	51	59
cubic	182.104	-10	34	104	-104	104	0	12	0	9.624	0	216
quartic	4.780.230	8	0	1.430	1.430	1.430	1.326	0	0	104.790	1.326	1.750
quintic	101.340.876	-4	0	2.756	-2.756	2.756	0	0	0	956.592	0	5.936
sextic	1.807.245.622	19	595	26.182	26.182	26.182	23.426	89	17	7.590.842	23.426	34.874

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	6	5	2	4	5	12	2	3	6	6	11	6
quadratic	280	226	188	190	380	504	190	191	244	242	488	379
cubic	8.023	7.969	7.132	7.238	14.362	15.983	7.165	7.219	7.990	7.988	15.964	14.381
quartic	204.400	202.916	194.752	194.858	389.612	407.314	194.788	194.842	203.480	203.374	406.856	389.628
quintic	4.263.251	4.261.767	4.181.133	4.183.995	8.365.127	8.525.019	4.182.051	4.183.535	4.262.333	4.262.227	8.524.559	8.365.587
sextic	75.634.146	75.606.480	74.984.112	74.986.974	149.970.960	151.240.446	74.985.031	74.986.515	75.617.604	75.614.742	151.232.184	149.971.419

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)
..21.	A_1gA_1g.	..15.	A_2gA_2g.	..3.	B_1gB_1g.	..10.	B_2gB_2g.	..15.	E_1gE_1g.	..78.	E_2gE_2g.	..3.	A_1uA_1u.	..6.	A_2uA_2u.	..21.	B_1uB_1u.	..21.	B_2uB_2u.
..66.	E_1uE_1u.	..21.	E_2uE_2u.
Subtotal: 280 / 12 / 12
Irrep combinations (i,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
Subtotal: 0 / 0 / 66
Total: 280 / 12 / 78

**Contributions to nonvanishing cubic force field constants**
Irrep combinations (i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(E_2u)
..56.	A_1gA_1gA_1g.	..364.	E_2gE_2gE_2g.
Subtotal: 420 / 2 / 12
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
..180.	E_1gE_1gE_2g.	..90.	A_1gA_2gA_2g.	..18.	A_1gB_1gB_1g.	..60.	A_1gB_2gB_2g.	..90.	A_1gE_1gE_1g.	..468.	A_1gE_2gE_2g.	..18.	A_1gA_1uA_1u.	..36.	A_1gA_2uA_2u.	..126.	A_1gB_1uB_1u.	..126.	A_1gB_2uB_2u.
..396.	A_1gE_1uE_1u.	..126.	A_1gE_2uE_2u.	..50.	A_2gE_1gE_1g.	..330.	A_2gE_2gE_2g.	..275.	A_2gE_1uE_1u.	..75.	A_2gE_2uE_2u.	..792.	E_2gE_1uE_1u.	..252.	E_2gE_2uE_2u.
Subtotal: 3.508 / 18 / 132
Irrep combinations (i,j,k) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ pos(E_2u)
..40.	A_2gB_1gB_2g.	..30.	A_2gA_1uA_2u.	..180.	A_2gB_1uB_2u.	..120.	B_1gE_1gE_2g.	..24.	B_1gA_1uB_1u.	..36.	B_1gA_2uB_2u.	..132.	B_1gE_1uE_2u.	..240.	B_2gE_1gE_2g.	..48.	B_2gA_1uB_2u.	..72.	B_2gA_2uB_1u.
..264.	B_2gE_1uE_2u.	..110.	E_1gA_1uE_1u.	..165.	E_1gA_2uE_1u.	..180.	E_1gB_1uE_2u.	..180.	E_1gB_2uE_2u.	..330.	E_1gE_1uE_2u.	..144.	E_2gA_1uE_2u.	..216.	E_2gA_2uE_2u.	..792.	E_2gB_1uE_1u.	..792.	E_2gB_2uE_1u.
Subtotal: 4.095 / 20 / 220
Total: 8.023 / 40 / 364

**Contributions to nonvanishing quartic force field constants**
Irrep combinations (i,i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(E_2u)
..126.	A_1gA_1gA_1gA_1g.	..70.	A_2gA_2gA_2gA_2g.	..5.	B_1gB_1gB_1gB_1g.	..35.	B_2gB_2gB_2gB_2g.	..120.	E_1gE_1gE_1gE_1g.	..3.081.	E_2gE_2gE_2gE_2g.	..5.	A_1uA_1uA_1uA_1u.	..15.	A_2uA_2uA_2uA_2u.	..126.	B_1uB_1uB_1uB_1u.	..126.	B_2uB_2uB_2uB_2u.
..2.211.	E_1uE_1uE_1uE_1u.	..231.	E_2uE_2uE_2uE_2u.
Subtotal: 6.151 / 12 / 12
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
..2.184.	A_1gE_2gE_2gE_2g.	..1.820.	A_2gE_2gE_2gE_2g.	..70.	B_1gE_1gE_1gE_1g.	..140.	B_2gE_1gE_1gE_1g.	..112.	A_1uE_2uE_2uE_2u.	..168.	A_2uE_2uE_2uE_2u.	..1.716.	B_1uE_1uE_1uE_1u.	..1.716.	B_2uE_1uE_1uE_1u.
Subtotal: 7.926 / 8 / 132
Irrep combinations (i,i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
..315.	A_1gA_1gA_2gA_2g.	..63.	A_1gA_1gB_1gB_1g.	..210.	A_1gA_1gB_2gB_2g.	..315.	A_1gA_1gE_1gE_1g.	..1.638.	A_1gA_1gE_2gE_2g.	..63.	A_1gA_1gA_1uA_1u.	..126.	A_1gA_1gA_2uA_2u.	..441.	A_1gA_1gB_1uB_1u.	..441.	A_1gA_1gB_2uB_2u.	..1.386.	A_1gA_1gE_1uE_1u.
..441.	A_1gA_1gE_2uE_2u.	..45.	A_2gA_2gB_1gB_1g.	..150.	A_2gA_2gB_2gB_2g.	..225.	A_2gA_2gE_1gE_1g.	..1.170.	A_2gA_2gE_2gE_2g.	..45.	A_2gA_2gA_1uA_1u.	..90.	A_2gA_2gA_2uA_2u.	..315.	A_2gA_2gB_1uB_1u.	..315.	A_2gA_2gB_2uB_2u.	..990.	A_2gA_2gE_1uE_1u.
..315.	A_2gA_2gE_2uE_2u.	..30.	B_1gB_1gB_2gB_2g.	..45.	B_1gB_1gE_1gE_1g.	..234.	B_1gB_1gE_2gE_2g.	..9.	B_1gB_1gA_1uA_1u.	..18.	B_1gB_1gA_2uA_2u.	..63.	B_1gB_1gB_1uB_1u.	..63.	B_1gB_1gB_2uB_2u.	..198.	B_1gB_1gE_1uE_1u.	..63.	B_1gB_1gE_2uE_2u.
..150.	B_2gB_2gE_1gE_1g.	..780.	B_2gB_2gE_2gE_2g.	..30.	B_2gB_2gA_1uA_1u.	..60.	B_2gB_2gA_2uA_2u.	..210.	B_2gB_2gB_1uB_1u.	..210.	B_2gB_2gB_2uB_2u.	..660.	B_2gB_2gE_1uE_1u.	..210.	B_2gB_2gE_2uE_2u.	..3.000.	E_1gE_1gE_2gE_2g.	..45.	E_1gE_1gA_1uA_1u.
..90.	E_1gE_1gA_2uA_2u.	..315.	E_1gE_1gB_1uB_1u.	..315.	E_1gE_1gB_2uB_2u.	..2.530.	E_1gE_1gE_1uE_1u.	..780.	E_1gE_1gE_2uE_2u.	..234.	E_2gE_2gA_1uA_1u.	..468.	E_2gE_2gA_2uA_2u.	..1.638.	E_2gE_2gB_1uB_1u.	..1.638.	E_2gE_2gB_2uB_2u.	..13.926.	E_2gE_2gE_1uE_1u.
..4.266.	E_2gE_2gE_2uE_2u.	..18.	A_1uA_1uA_2uA_2u.	..63.	A_1uA_1uB_1uB_1u.	..63.	A_1uA_1uB_2uB_2u.	..198.	A_1uA_1uE_1uE_1u.	..63.	A_1uA_1uE_2uE_2u.	..126.	A_2uA_2uB_1uB_1u.	..126.	A_2uA_2uB_2uB_2u.	..396.	A_2uA_2uE_1uE_1u.	..126.	A_2uA_2uE_2uE_2u.
..441.	B_1uB_1uB_2uB_2u.	..1.386.	B_1uB_1uE_1uE_1u.	..441.	B_1uB_1uE_2uE_2u.	..1.386.	B_2uB_2uE_1uE_1u.	..441.	B_2uB_2uE_2uE_2u.	..3.597.	E_1uE_1uE_2uE_2u.
Subtotal: 50.248 / 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)
..60.	E_1gE_1gA_1uA_2u.	..180.	E_1gE_1gA_1uE_2u.	..270.	E_1gE_1gA_2uE_2u.	..360.	E_1gE_1gB_1uB_2u.	..990.	E_1gE_1gB_1uE_1u.	..990.	E_1gE_1gB_2uE_1u.	..396.	E_2gE_2gA_1uA_2u.	..936.	E_2gE_2gA_1uE_2u.	..1.404.	E_2gE_2gA_2uE_2u.	..2.376.	E_2gE_2gB_1uB_2u.
..5.148.	E_2gE_2gB_1uE_1u.	..5.148.	E_2gE_2gB_2uE_1u.	..1.080.	A_1gE_1gE_1gE_2g.	..900.	A_2gE_1gE_1gE_2g.	..792.	A_1uE_1uE_1uE_2u.	..1.188.	A_2uE_1uE_1uE_2u.	..300.	A_1gA_2gE_1gE_1g.	..1.980.	A_1gA_2gE_2gE_2g.	..1.650.	A_1gA_2gE_1uE_1u.	..450.	A_1gA_2gE_2uE_2u.
..4.752.	A_1gE_2gE_1uE_1u.	..1.512.	A_1gE_2gE_2uE_2u.	..3.960.	A_2gE_2gE_1uE_1u.	..1.260.	A_2gE_2gE_2uE_2u.	..80.	B_1gB_2gE_1gE_1g.	..528.	B_1gB_2gE_2gE_2g.	..440.	B_1gB_2gE_1uE_1u.	..120.	B_1gB_2gE_2uE_2u.	..780.	B_1gE_1gE_2gE_2g.	..660.	B_1gE_1gE_1uE_1u.
..210.	B_1gE_1gE_2uE_2u.	..1.560.	B_2gE_1gE_2gE_2g.	..1.320.	B_2gE_1gE_1uE_1u.	..420.	B_2gE_1gE_2uE_2u.	..330.	A_1uA_2uE_1uE_1u.	..90.	A_1uA_2uE_2uE_2u.	..1.980.	B_1uB_2uE_1uE_1u.	..540.	B_1uB_2uE_2uE_2u.	..1.386.	B_1uE_1uE_2uE_2u.	..1.386.	B_2uE_1uE_2uE_2u.
Subtotal: 49.912 / 40 / 660
Irrep combinations (i,j,k,l) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(E_2u)
..240.	A_1gA_2gB_1gB_2g.	..180.	A_1gA_2gA_1uA_2u.	..1.080.	A_1gA_2gB_1uB_2u.	..720.	A_1gB_1gE_1gE_2g.	..144.	A_1gB_1gA_1uB_1u.	..216.	A_1gB_1gA_2uB_2u.	..792.	A_1gB_1gE_1uE_2u.	..1.440.	A_1gB_2gE_1gE_2g.	..288.	A_1gB_2gA_1uB_2u.	..432.	A_1gB_2gA_2uB_1u.
..1.584.	A_1gB_2gE_1uE_2u.	..660.	A_1gE_1gA_1uE_1u.	..990.	A_1gE_1gA_2uE_1u.	..1.080.	A_1gE_1gB_1uE_2u.	..1.080.	A_1gE_1gB_2uE_2u.	..1.980.	A_1gE_1gE_1uE_2u.	..864.	A_1gE_2gA_1uE_2u.	..1.296.	A_1gE_2gA_2uE_2u.	..4.752.	A_1gE_2gB_1uE_1u.	..4.752.	A_1gE_2gB_2uE_1u.
..600.	A_2gB_1gE_1gE_2g.	..120.	A_2gB_1gA_1uB_2u.	..180.	A_2gB_1gA_2uB_1u.	..660.	A_2gB_1gE_1uE_2u.	..1.200.	A_2gB_2gE_1gE_2g.	..240.	A_2gB_2gA_1uB_1u.	..360.	A_2gB_2gA_2uB_2u.	..1.320.	A_2gB_2gE_1uE_2u.	..550.	A_2gE_1gA_1uE_1u.	..825.	A_2gE_1gA_2uE_1u.
..900.	A_2gE_1gB_1uE_2u.	..900.	A_2gE_1gB_2uE_2u.	..1.650.	A_2gE_1gE_1uE_2u.	..720.	A_2gE_2gA_1uE_2u.	..1.080.	A_2gE_2gA_2uE_2u.	..3.960.	A_2gE_2gB_1uE_1u.	..3.960.	A_2gE_2gB_2uE_1u.	..48.	B_1gB_2gA_1uA_2u.	..288.	B_1gB_2gB_1uB_2u.	..120.	B_1gE_1gA_1uE_2u.
..180.	B_1gE_1gA_2uE_2u.	..660.	B_1gE_1gB_1uE_1u.	..660.	B_1gE_1gB_2uE_1u.	..528.	B_1gE_2gA_1uE_1u.	..792.	B_1gE_2gA_2uE_1u.	..864.	B_1gE_2gB_1uE_2u.	..864.	B_1gE_2gB_2uE_2u.	..1.584.	B_1gE_2gE_1uE_2u.	..240.	B_2gE_1gA_1uE_2u.	..360.	B_2gE_1gA_2uE_2u.
..1.320.	B_2gE_1gB_1uE_1u.	..1.320.	B_2gE_1gB_2uE_1u.	..1.056.	B_2gE_2gA_1uE_1u.	..1.584.	B_2gE_2gA_2uE_1u.	..1.728.	B_2gE_2gB_1uE_2u.	..1.728.	B_2gE_2gB_2uE_2u.	..3.168.	B_2gE_2gE_1uE_2u.	..720.	E_1gE_2gA_1uB_1u.	..720.	E_1gE_2gA_1uB_2u.	..1.320.	E_1gE_2gA_1uE_1u.
..1.080.	E_1gE_2gA_2uB_1u.	..1.080.	E_1gE_2gA_2uB_2u.	..1.980.	E_1gE_2gA_2uE_1u.	..2.160.	E_1gE_2gB_1uE_2u.	..2.160.	E_1gE_2gB_2uE_2u.	..11.880.	E_1gE_2gE_1uE_2u.	..216.	A_1uA_2uB_1uB_2u.	..792.	A_1uB_1uE_1uE_2u.	..792.	A_1uB_2uE_1uE_2u.	..1.188.	A_2uB_1uE_1uE_2u.
..1.188.	A_2uB_2uE_1uE_2u.
Subtotal: 90.163 / 71 / 495
Total: 204.400 / 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