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	75	2	-1	-1	-1	-3	-2	1	1	9
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	69	-2	1	1	1	-3	-2	1	1	9

**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	4	2	2	4	6	6	2	5	4	2	7	6	50
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	4	1	2	4	5	6	2	4	4	2	6	6	46

**Molecular parameter**
Number of Atoms (N)	25
Number of internal coordinates	69
Number of independant internal coordinates	4
Number of vibrational modes	46

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)	4	1	2	4	5	6	2	4	4	2	6	6	10 / 36
Quadratic (Raman)	4	1	2	4	5	6	2	4	4	2	6	6	15 / 31
IR + Raman	- - - -	1	2	4	- - - -	- - - -	2	- - - -	4	2	- - - -	6	0^* / 21

* 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	69	-2	0	1	1	1	-3	-2	0	1	1	9
quadratic	2.415	2	0	35	35	35	39	2	0	35	35	75
cubic	57.155	-1	23	35	35	35	-109	-1	-1	35	35	435
quartic	1.028.790	0	0	630	630	630	774	0	0	630	630	2.310
quintic	15.020.334	0	0	630	630	630	-2.034	0	0	630	630	10.422
sextic	185.250.786	12	276	7.770	7.770	7.770	10.434	12	12	7.770	7.770	43.738

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	4	1	2	4	5	6	2	4	4	2	6	6
quadratic	128	83	94	104	199	210	94	104	104	94	198	198
cubic	2.449	2.314	2.326	2.426	4.746	4.758	2.338	2.438	2.438	2.338	4.770	4.770
quartic	43.476	42.426	42.636	43.056	85.692	85.902	42.624	43.044	43.044	42.624	85.668	85.668
quintic	627.354	624.276	624.486	626.934	1.251.420	1.251.630	624.708	627.156	627.156	624.708	1.251.864	1.251.864
sextic	7.728.272	7.711.510	7.714.096	7.723.088	15.437.118	15.439.704	7.713.874	7.722.866	7.722.866	7.713.874	15.436.674	15.436.674

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)
..10.	A_1gA_1g.	..1.	A_2gA_2g.	..3.	B_1gB_1g.	..10.	B_2gB_2g.	..15.	E_1gE_1g.	..21.	E_2gE_2g.	..3.	A_1uA_1u.	..10.	A_2uA_2u.	..10.	B_1uB_1u.	..3.	B_2uB_2u.
..21.	E_1uE_1u.	..21.	E_2uE_2u.
Subtotal: 128 / 12 / 12
Irrep combinations (i,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
Subtotal: 0 / 0 / 66
Total: 128 / 12 / 78

**Contributions to nonvanishing cubic force field constants**
Irrep combinations (i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(E_2u)
..20.	A_1gA_1gA_1g.	..56.	E_2gE_2gE_2g.
Subtotal: 76 / 2 / 12
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
..90.	E_1gE_1gE_2g.	..4.	A_1gA_2gA_2g.	..12.	A_1gB_1gB_1g.	..40.	A_1gB_2gB_2g.	..60.	A_1gE_1gE_1g.	..84.	A_1gE_2gE_2g.	..12.	A_1gA_1uA_1u.	..40.	A_1gA_2uA_2u.	..40.	A_1gB_1uB_1u.	..12.	A_1gB_2uB_2u.
..84.	A_1gE_1uE_1u.	..84.	A_1gE_2uE_2u.	..10.	A_2gE_1gE_1g.	..15.	A_2gE_2gE_2g.	..15.	A_2gE_1uE_1u.	..15.	A_2gE_2uE_2u.	..126.	E_2gE_1uE_1u.	..126.	E_2gE_2uE_2u.
Subtotal: 869 / 18 / 132
Irrep combinations (i,j,k) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ pos(E_2u)
..8.	A_2gB_1gB_2g.	..8.	A_2gA_1uA_2u.	..8.	A_2gB_1uB_2u.	..60.	B_1gE_1gE_2g.	..16.	B_1gA_1uB_1u.	..16.	B_1gA_2uB_2u.	..72.	B_1gE_1uE_2u.	..120.	B_2gE_1gE_2g.	..16.	B_2gA_1uB_2u.	..64.	B_2gA_2uB_1u.
..144.	B_2gE_1uE_2u.	..60.	E_1gA_1uE_1u.	..120.	E_1gA_2uE_1u.	..120.	E_1gB_1uE_2u.	..60.	E_1gB_2uE_2u.	..180.	E_1gE_1uE_2u.	..72.	E_2gA_1uE_2u.	..144.	E_2gA_2uE_2u.	..144.	E_2gB_1uE_1u.	..72.	E_2gB_2uE_1u.
Subtotal: 1.504 / 20 / 220
Total: 2.449 / 40 / 364

**Contributions to nonvanishing quartic force field constants**
Irrep combinations (i,i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(E_2u)
..35.	A_1gA_1gA_1gA_1g.	..1.	A_2gA_2gA_2gA_2g.	..5.	B_1gB_1gB_1gB_1g.	..35.	B_2gB_2gB_2gB_2g.	..120.	E_1gE_1gE_1gE_1g.	..231.	E_2gE_2gE_2gE_2g.	..5.	A_1uA_1uA_1uA_1u.	..35.	A_2uA_2uA_2uA_2u.	..35.	B_1uB_1uB_1uB_1u.	..5.	B_2uB_2uB_2uB_2u.
..231.	E_1uE_1uE_1uE_1u.	..231.	E_2uE_2uE_2uE_2u.
Subtotal: 969 / 12 / 12
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
..224.	A_1gE_2gE_2gE_2g.	..56.	A_2gE_2gE_2gE_2g.	..70.	B_1gE_1gE_1gE_1g.	..140.	B_2gE_1gE_1gE_1g.	..112.	A_1uE_2uE_2uE_2u.	..224.	A_2uE_2uE_2uE_2u.	..224.	B_1uE_1uE_1uE_1u.	..112.	B_2uE_1uE_1uE_1u.
Subtotal: 1.162 / 8 / 132
Irrep combinations (i,i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
..10.	A_1gA_1gA_2gA_2g.	..30.	A_1gA_1gB_1gB_1g.	..100.	A_1gA_1gB_2gB_2g.	..150.	A_1gA_1gE_1gE_1g.	..210.	A_1gA_1gE_2gE_2g.	..30.	A_1gA_1gA_1uA_1u.	..100.	A_1gA_1gA_2uA_2u.	..100.	A_1gA_1gB_1uB_1u.	..30.	A_1gA_1gB_2uB_2u.	..210.	A_1gA_1gE_1uE_1u.
..210.	A_1gA_1gE_2uE_2u.	..3.	A_2gA_2gB_1gB_1g.	..10.	A_2gA_2gB_2gB_2g.	..15.	A_2gA_2gE_1gE_1g.	..21.	A_2gA_2gE_2gE_2g.	..3.	A_2gA_2gA_1uA_1u.	..10.	A_2gA_2gA_2uA_2u.	..10.	A_2gA_2gB_1uB_1u.	..3.	A_2gA_2gB_2uB_2u.	..21.	A_2gA_2gE_1uE_1u.
..21.	A_2gA_2gE_2uE_2u.	..30.	B_1gB_1gB_2gB_2g.	..45.	B_1gB_1gE_1gE_1g.	..63.	B_1gB_1gE_2gE_2g.	..9.	B_1gB_1gA_1uA_1u.	..30.	B_1gB_1gA_2uA_2u.	..30.	B_1gB_1gB_1uB_1u.	..9.	B_1gB_1gB_2uB_2u.	..63.	B_1gB_1gE_1uE_1u.	..63.	B_1gB_1gE_2uE_2u.
..150.	B_2gB_2gE_1gE_1g.	..210.	B_2gB_2gE_2gE_2g.	..30.	B_2gB_2gA_1uA_1u.	..100.	B_2gB_2gA_2uA_2u.	..100.	B_2gB_2gB_1uB_1u.	..30.	B_2gB_2gB_2uB_2u.	..210.	B_2gB_2gE_1uE_1u.	..210.	B_2gB_2gE_2uE_2u.	..780.	E_1gE_1gE_2gE_2g.	..45.	E_1gE_1gA_1uA_1u.
..150.	E_1gE_1gA_2uA_2u.	..150.	E_1gE_1gB_1uB_1u.	..45.	E_1gE_1gB_2uB_2u.	..780.	E_1gE_1gE_1uE_1u.	..780.	E_1gE_1gE_2uE_2u.	..63.	E_2gE_2gA_1uA_1u.	..210.	E_2gE_2gA_2uA_2u.	..210.	E_2gE_2gB_1uB_1u.	..63.	E_2gE_2gB_2uB_2u.	..1.107.	E_2gE_2gE_1uE_1u.
..1.107.	E_2gE_2gE_2uE_2u.	..30.	A_1uA_1uA_2uA_2u.	..30.	A_1uA_1uB_1uB_1u.	..9.	A_1uA_1uB_2uB_2u.	..63.	A_1uA_1uE_1uE_1u.	..63.	A_1uA_1uE_2uE_2u.	..100.	A_2uA_2uB_1uB_1u.	..30.	A_2uA_2uB_2uB_2u.	..210.	A_2uA_2uE_1uE_1u.	..210.	A_2uA_2uE_2uE_2u.
..30.	B_1uB_1uB_2uB_2u.	..210.	B_1uB_1uE_1uE_1u.	..210.	B_1uB_1uE_2uE_2u.	..63.	B_2uB_2uE_1uE_1u.	..63.	B_2uB_2uE_2uE_2u.	..1.107.	E_1uE_1uE_2uE_2u.
Subtotal: 10.597 / 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)
..80.	E_1gE_1gA_1uA_2u.	..180.	E_1gE_1gA_1uE_2u.	..360.	E_1gE_1gA_2uE_2u.	..80.	E_1gE_1gB_1uB_2u.	..360.	E_1gE_1gB_1uE_1u.	..180.	E_1gE_1gB_2uE_1u.	..120.	E_2gE_2gA_1uA_2u.	..252.	E_2gE_2gA_1uE_2u.	..504.	E_2gE_2gA_2uE_2u.	..120.	E_2gE_2gB_1uB_2u.
..504.	E_2gE_2gB_1uE_1u.	..252.	E_2gE_2gB_2uE_1u.	..360.	A_1gE_1gE_1gE_2g.	..90.	A_2gE_1gE_1gE_2g.	..252.	A_1uE_1uE_1uE_2u.	..504.	A_2uE_1uE_1uE_2u.	..40.	A_1gA_2gE_1gE_1g.	..60.	A_1gA_2gE_2gE_2g.	..60.	A_1gA_2gE_1uE_1u.	..60.	A_1gA_2gE_2uE_2u.
..504.	A_1gE_2gE_1uE_1u.	..504.	A_1gE_2gE_2uE_2u.	..126.	A_2gE_2gE_1uE_1u.	..126.	A_2gE_2gE_2uE_2u.	..80.	B_1gB_2gE_1gE_1g.	..120.	B_1gB_2gE_2gE_2g.	..120.	B_1gB_2gE_1uE_1u.	..120.	B_1gB_2gE_2uE_2u.	..210.	B_1gE_1gE_2gE_2g.	..210.	B_1gE_1gE_1uE_1u.
..210.	B_1gE_1gE_2uE_2u.	..420.	B_2gE_1gE_2gE_2g.	..420.	B_2gE_1gE_1uE_1u.	..420.	B_2gE_1gE_2uE_2u.	..120.	A_1uA_2uE_1uE_1u.	..120.	A_1uA_2uE_2uE_2u.	..120.	B_1uB_2uE_1uE_1u.	..120.	B_1uB_2uE_2uE_2u.	..504.	B_1uE_1uE_2uE_2u.	..252.	B_2uE_1uE_2uE_2u.
Subtotal: 9.244 / 40 / 660
Irrep combinations (i,j,k,l) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(E_2u)
..32.	A_1gA_2gB_1gB_2g.	..32.	A_1gA_2gA_1uA_2u.	..32.	A_1gA_2gB_1uB_2u.	..240.	A_1gB_1gE_1gE_2g.	..64.	A_1gB_1gA_1uB_1u.	..64.	A_1gB_1gA_2uB_2u.	..288.	A_1gB_1gE_1uE_2u.	..480.	A_1gB_2gE_1gE_2g.	..64.	A_1gB_2gA_1uB_2u.	..256.	A_1gB_2gA_2uB_1u.
..576.	A_1gB_2gE_1uE_2u.	..240.	A_1gE_1gA_1uE_1u.	..480.	A_1gE_1gA_2uE_1u.	..480.	A_1gE_1gB_1uE_2u.	..240.	A_1gE_1gB_2uE_2u.	..720.	A_1gE_1gE_1uE_2u.	..288.	A_1gE_2gA_1uE_2u.	..576.	A_1gE_2gA_2uE_2u.	..576.	A_1gE_2gB_1uE_1u.	..288.	A_1gE_2gB_2uE_1u.
..60.	A_2gB_1gE_1gE_2g.	..8.	A_2gB_1gA_1uB_2u.	..32.	A_2gB_1gA_2uB_1u.	..72.	A_2gB_1gE_1uE_2u.	..120.	A_2gB_2gE_1gE_2g.	..32.	A_2gB_2gA_1uB_1u.	..32.	A_2gB_2gA_2uB_2u.	..144.	A_2gB_2gE_1uE_2u.	..60.	A_2gE_1gA_1uE_1u.	..120.	A_2gE_1gA_2uE_1u.
..120.	A_2gE_1gB_1uE_2u.	..60.	A_2gE_1gB_2uE_2u.	..180.	A_2gE_1gE_1uE_2u.	..72.	A_2gE_2gA_1uE_2u.	..144.	A_2gE_2gA_2uE_2u.	..144.	A_2gE_2gB_1uE_1u.	..72.	A_2gE_2gB_2uE_1u.	..64.	B_1gB_2gA_1uA_2u.	..64.	B_1gB_2gB_1uB_2u.	..120.	B_1gE_1gA_1uE_2u.
..240.	B_1gE_1gA_2uE_2u.	..240.	B_1gE_1gB_1uE_1u.	..120.	B_1gE_1gB_2uE_1u.	..144.	B_1gE_2gA_1uE_1u.	..288.	B_1gE_2gA_2uE_1u.	..288.	B_1gE_2gB_1uE_2u.	..144.	B_1gE_2gB_2uE_2u.	..432.	B_1gE_2gE_1uE_2u.	..240.	B_2gE_1gA_1uE_2u.	..480.	B_2gE_1gA_2uE_2u.
..480.	B_2gE_1gB_1uE_1u.	..240.	B_2gE_1gB_2uE_1u.	..288.	B_2gE_2gA_1uE_1u.	..576.	B_2gE_2gA_2uE_1u.	..576.	B_2gE_2gB_1uE_2u.	..288.	B_2gE_2gB_2uE_2u.	..864.	B_2gE_2gE_1uE_2u.	..240.	E_1gE_2gA_1uB_1u.	..120.	E_1gE_2gA_1uB_2u.	..360.	E_1gE_2gA_1uE_1u.
..480.	E_1gE_2gA_2uB_1u.	..240.	E_1gE_2gA_2uB_2u.	..720.	E_1gE_2gA_2uE_1u.	..720.	E_1gE_2gB_1uE_2u.	..360.	E_1gE_2gB_2uE_2u.	..3.240.	E_1gE_2gE_1uE_2u.	..64.	A_1uA_2uB_1uB_2u.	..288.	A_1uB_1uE_1uE_2u.	..144.	A_1uB_2uE_1uE_2u.	..576.	A_2uB_1uE_1uE_2u.
..288.	A_2uB_2uE_1uE_2u.
Subtotal: 21.504 / 71 / 495
Total: 43.476 / 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