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	36	0	0	-4	0	0	0	12	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	30	-4	2	-2	2	0	0	12	0	4

**Decomposition to irreducible representations**
Motion	A_1g	A_2g	B_2g	E_1g	E_2g	A_2u	B_1u	B_2u	E_1u	E_2u	Total
Cartesian 3N	2	2	2	2	4	2	2	2	4	2	24
Translation (x,y,z)	0	0	0	0	0	1	0	0	1	0	2
Rotation (R_x,R_y,R_z)	0	1	0	1	0	0	0	0	0	0	2
Vibration	2	1	2	1	4	1	2	2	3	2	20

**Molecular parameter**
Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	2
Number of vibrational modes	20

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A_1g	A_2g	B_2g	E_1g	E_2g	A_2u	B_1u	B_2u	E_1u	E_2u	Total
Linear (IR)	2	1	2	1	4	1	2	2	3	2	4 / 16
Quadratic (Raman)	2	1	2	1	4	1	2	2	3	2	7 / 13
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	2C₆	2C₃	C₂	3C'₂	3C''₂	i	2S₃	2S₆	σ_h	3σ_d	3σ_v
linear	30	-4	0	2	-2	2	0	0	0	12	0	4
quadratic	465	8	0	17	17	17	15	0	0	87	15	23
cubic	4.960	-10	10	32	-32	32	0	4	0	472	0	72
quartic	40.920	8	0	152	152	152	120	0	0	2.112	120	256
quintic	278.256	-4	0	272	-272	272	0	0	0	8.184	0	680
sextic	1.623.160	7	55	952	952	952	680	13	5	28.336	680	1.904

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	2	1	0	2	1	4	0	1	2	2	3	2
quadratic	34	16	14	16	32	48	16	17	22	20	44	31
cubic	237	219	170	204	370	455	179	197	228	226	448	377
quartic	1.890	1.720	1.598	1.632	3.232	3.608	1.610	1.628	1.798	1.764	3.564	3.236
quintic	12.031	11.861	11.089	11.395	22.483	23.893	11.179	11.349	11.941	11.907	23.847	22.529
sextic	69.448	68.326	66.290	66.596	132.876	137.754	66.381	66.551	68.902	68.596	137.484	132.921

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)
..3.	A_1gA_1g.	..1.	A_2gA_2g.	..3.	B_2gB_2g.	..1.	E_1gE_1g.	..10.	E_2gE_2g.	..1.	A_2uA_2u.	..3.	B_1uB_1u.	..3.	B_2uB_2u.	..6.	E_1uE_1u.	..3.	E_2uE_2u.
Subtotal: 34 / 10 / 12
Irrep combinations (i,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
Subtotal: 0 / 0 / 66
Total: 34 / 10 / 78

**Contributions to nonvanishing cubic force field constants**
Irrep combinations (i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(E_2u)
..4.	A_1gA_1gA_1g.	..20.	E_2gE_2gE_2g.
Subtotal: 24 / 2 / 12
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
..4.	E_1gE_1gE_2g.	..2.	A_1gA_2gA_2g.	..6.	A_1gB_2gB_2g.	..2.	A_1gE_1gE_1g.	..20.	A_1gE_2gE_2g.	..2.	A_1gA_2uA_2u.	..6.	A_1gB_1uB_1u.	..6.	A_1gB_2uB_2u.	..12.	A_1gE_1uE_1u.	..6.	A_1gE_2uE_2u.
..6.	A_2gE_2gE_2g.	..3.	A_2gE_1uE_1u.	..1.	A_2gE_2uE_2u.	..24.	E_2gE_1uE_1u.	..12.	E_2gE_2uE_2u.
Subtotal: 112 / 15 / 132
Irrep combinations (i,j,k) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ pos(E_2u)
..4.	A_2gB_1uB_2u.	..8.	B_2gE_1gE_2g.	..4.	B_2gA_2uB_1u.	..12.	B_2gE_1uE_2u.	..3.	E_1gA_2uE_1u.	..4.	E_1gB_1uE_2u.	..4.	E_1gB_2uE_2u.	..6.	E_1gE_1uE_2u.	..8.	E_2gA_2uE_2u.	..24.	E_2gB_1uE_1u.
..24.	E_2gB_2uE_1u.
Subtotal: 101 / 11 / 220
Total: 237 / 28 / 364

**Contributions to nonvanishing quartic force field constants**
Irrep combinations (i,i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(E_2u)
..5.	A_1gA_1gA_1gA_1g.	..1.	A_2gA_2gA_2gA_2g.	..5.	B_2gB_2gB_2gB_2g.	..1.	E_1gE_1gE_1gE_1g.	..55.	E_2gE_2gE_2gE_2g.	..1.	A_2uA_2uA_2uA_2u.	..5.	B_1uB_1uB_1uB_1u.	..5.	B_2uB_2uB_2uB_2u.	..21.	E_1uE_1uE_1uE_1u.	..6.	E_2uE_2uE_2uE_2u.
Subtotal: 105 / 10 / 12
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
..40.	A_1gE_2gE_2gE_2g.	..20.	A_2gE_2gE_2gE_2g.	..2.	B_2gE_1gE_1gE_1g.	..4.	A_2uE_2uE_2uE_2u.	..20.	B_1uE_1uE_1uE_1u.	..20.	B_2uE_1uE_1uE_1u.
Subtotal: 106 / 6 / 132
Irrep combinations (i,i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
..3.	A_1gA_1gA_2gA_2g.	..9.	A_1gA_1gB_2gB_2g.	..3.	A_1gA_1gE_1gE_1g.	..30.	A_1gA_1gE_2gE_2g.	..3.	A_1gA_1gA_2uA_2u.	..9.	A_1gA_1gB_1uB_1u.	..9.	A_1gA_1gB_2uB_2u.	..18.	A_1gA_1gE_1uE_1u.	..9.	A_1gA_1gE_2uE_2u.	..3.	A_2gA_2gB_2gB_2g.
..1.	A_2gA_2gE_1gE_1g.	..10.	A_2gA_2gE_2gE_2g.	..1.	A_2gA_2gA_2uA_2u.	..3.	A_2gA_2gB_1uB_1u.	..3.	A_2gA_2gB_2uB_2u.	..6.	A_2gA_2gE_1uE_1u.	..3.	A_2gA_2gE_2uE_2u.	..3.	B_2gB_2gE_1gE_1g.	..30.	B_2gB_2gE_2gE_2g.	..3.	B_2gB_2gA_2uA_2u.
..9.	B_2gB_2gB_1uB_1u.	..9.	B_2gB_2gB_2uB_2u.	..18.	B_2gB_2gE_1uE_1u.	..9.	B_2gB_2gE_2uE_2u.	..20.	E_1gE_1gE_2gE_2g.	..1.	E_1gE_1gA_2uA_2u.	..3.	E_1gE_1gB_1uB_1u.	..3.	E_1gE_1gB_2uB_2u.	..12.	E_1gE_1gE_1uE_1u.	..6.	E_1gE_1gE_2uE_2u.
..10.	E_2gE_2gA_2uA_2u.	..30.	E_2gE_2gB_1uB_1u.	..30.	E_2gE_2gB_2uB_2u.	..138.	E_2gE_2gE_1uE_1u.	..66.	E_2gE_2gE_2uE_2u.	..3.	A_2uA_2uB_1uB_1u.	..3.	A_2uA_2uB_2uB_2u.	..6.	A_2uA_2uE_1uE_1u.	..3.	A_2uA_2uE_2uE_2u.	..9.	B_1uB_1uB_2uB_2u.
..18.	B_1uB_1uE_1uE_1u.	..9.	B_1uB_1uE_2uE_2u.	..18.	B_2uB_2uE_1uE_1u.	..9.	B_2uB_2uE_2uE_2u.	..39.	E_1uE_1uE_2uE_2u.
Subtotal: 640 / 45 / 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)
..2.	E_1gE_1gA_2uE_2u.	..6.	E_1gE_1gB_1uE_1u.	..6.	E_1gE_1gB_2uE_1u.	..20.	E_2gE_2gA_2uE_2u.	..24.	E_2gE_2gB_1uB_2u.	..60.	E_2gE_2gB_1uE_1u.	..60.	E_2gE_2gB_2uE_1u.	..8.	A_1gE_1gE_1gE_2g.	..4.	A_2gE_1gE_1gE_2g.	..12.	A_2uE_1uE_1uE_2u.
..12.	A_1gA_2gE_2gE_2g.	..6.	A_1gA_2gE_1uE_1u.	..2.	A_1gA_2gE_2uE_2u.	..48.	A_1gE_2gE_1uE_1u.	..24.	A_1gE_2gE_2uE_2u.	..24.	A_2gE_2gE_1uE_1u.	..12.	A_2gE_2gE_2uE_2u.	..20.	B_2gE_1gE_2gE_2g.	..12.	B_2gE_1gE_1uE_1u.	..6.	B_2gE_1gE_2uE_2u.
..12.	B_1uB_2uE_1uE_1u.	..4.	B_1uB_2uE_2uE_2u.	..18.	B_1uE_1uE_2uE_2u.	..18.	B_2uE_1uE_2uE_2u.
Subtotal: 420 / 24 / 660
Irrep combinations (i,j,k,l) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(E_2u)
..8.	A_1gA_2gB_1uB_2u.	..16.	A_1gB_2gE_1gE_2g.	..8.	A_1gB_2gA_2uB_1u.	..24.	A_1gB_2gE_1uE_2u.	..6.	A_1gE_1gA_2uE_1u.	..8.	A_1gE_1gB_1uE_2u.	..8.	A_1gE_1gB_2uE_2u.	..12.	A_1gE_1gE_1uE_2u.	..16.	A_1gE_2gA_2uE_2u.	..48.	A_1gE_2gB_1uE_1u.
..48.	A_1gE_2gB_2uE_1u.	..8.	A_2gB_2gE_1gE_2g.	..4.	A_2gB_2gA_2uB_2u.	..12.	A_2gB_2gE_1uE_2u.	..3.	A_2gE_1gA_2uE_1u.	..4.	A_2gE_1gB_1uE_2u.	..4.	A_2gE_1gB_2uE_2u.	..6.	A_2gE_1gE_1uE_2u.	..8.	A_2gE_2gA_2uE_2u.	..24.	A_2gE_2gB_1uE_1u.
..24.	A_2gE_2gB_2uE_1u.	..4.	B_2gE_1gA_2uE_2u.	..12.	B_2gE_1gB_1uE_1u.	..12.	B_2gE_1gB_2uE_1u.	..24.	B_2gE_2gA_2uE_1u.	..32.	B_2gE_2gB_1uE_2u.	..32.	B_2gE_2gB_2uE_2u.	..48.	B_2gE_2gE_1uE_2u.	..8.	E_1gE_2gA_2uB_1u.	..8.	E_1gE_2gA_2uB_2u.
..12.	E_1gE_2gA_2uE_1u.	..16.	E_1gE_2gB_1uE_2u.	..16.	E_1gE_2gB_2uE_2u.	..72.	E_1gE_2gE_1uE_2u.	..12.	A_2uB_1uE_1uE_2u.	..12.	A_2uB_2uE_1uE_2u.
Subtotal: 619 / 36 / 495
Total: 1.890 / 121 / 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