Point Group D2h

Results for Point Group D_2h

**Characters of representations for molecular motions**
Motion	E	C₂.(z)	C₂.(y)	C₂.(x)	i	σ.(xy)	σ.(xy)	σ.(xy)
Cartesian 3N	36	0	-2	-2	0	12	2	2
Translation (x,y,z)	3	-1	-1	-1	-3	1	1	1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1	3	-1	-1	-1
Vibration	30	2	0	0	0	12	2	2

**Decomposition to irreducible representations**
Motion	A_1g	B_1g	B_2g	B_3g	A_1u	B_1u	B_2u	B_3u	Total
Cartesian 3N	6	6	3	3	2	4	6	6	36
Translation (x,y,z)	0	0	0	0	0	1	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	0	0	0	0	3
Vibration	6	5	2	2	2	3	5	5	30

**Molecular parameter**
Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	6
Number of vibrational modes	30

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A_1g	B_1g	B_2g	B_3g	A_1u	B_1u	B_2u	B_3u	Total
Linear (IR)	6	5	2	2	2	3	5	5	13 / 17
Quadratic (Raman)	6	5	2	2	2	3	5	5	15 / 15
IR + Raman	- - - -	- - - -	- - - -	- - - -	2	- - - -	- - - -	- - - -	0^* / 2

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)
Force field	E	C₂.(z)	C₂.(y)	C₂.(x)	i	σ.(xy)	σ.(xy)	σ.(xy)
linear	30	2	0	0	0	12	2	2
quadratic	465	17	15	15	15	87	17	17
cubic	4.960	32	0	0	0	472	32	32
quartic	40.920	152	120	120	120	2.112	152	152
quintic	278.256	272	0	0	0	8.184	272	272
sextic	1.623.160	952	680	680	680	28.336	952	952

Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field	A_1g	B_1g	B_2g	B_3g	A_1u	B_1u	B_2u	B_3u
linear	6	5	2	2	2	3	5	5
quadratic	81	65	47	47	47	48	65	65
cubic	691	675	557	557	557	573	675	675
quartic	5.481	5.345	4.847	4.847	4.847	4.863	5.345	5.345
quintic	35.907	35.771	33.725	33.725	33.725	33.861	35.771	35.771
sextic	207.049	206.233	199.319	199.319	199.319	199.455	206.233	206.233

Contributions to nonvanishing force field constants

pos(X) : Position of irreducible representation (irrep) X in character table of D_2h

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(B_3u)
..21.	A_1gA_1g.	..15.	B_1gB_1g.	..3.	B_2gB_2g.	..3.	B_3gB_3g.	..3.	A_1uA_1u.	..6.	B_1uB_1u.	..15.	B_2uB_2u.	..15.	B_3uB_3u.
Subtotal: 81 / 8 / 8
Irrep combinations (i,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(B_3u)
Subtotal: 0 / 0 / 28
Total: 81 / 8 / 36

**Contributions to nonvanishing cubic force field constants**
Irrep combinations (i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(B_3u)
..56.	A_1gA_1gA_1g.
Subtotal: 56 / 1 / 8
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(B_3u)
..90.	A_1gB_1gB_1g.	..18.	A_1gB_2gB_2g.	..18.	A_1gB_3gB_3g.	..18.	A_1gA_1uA_1u.	..36.	A_1gB_1uB_1u.	..90.	A_1gB_2uB_2u.	..90.	A_1gB_3uB_3u.
Subtotal: 360 / 7 / 56
Irrep combinations (i,j,k) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ pos(B_3u)
..20.	B_1gB_2gB_3g.	..30.	B_1gA_1uB_1u.	..125.	B_1gB_2uB_3u.	..20.	B_2gA_1uB_2u.	..30.	B_2gB_1uB_3u.	..20.	B_3gA_1uB_3u.	..30.	B_3gB_1uB_2u.
Subtotal: 275 / 7 / 56
Total: 691 / 15 / 120

**Contributions to nonvanishing quartic force field constants**
Irrep combinations (i,i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(B_3u)
..126.	A_1gA_1gA_1gA_1g.	..70.	B_1gB_1gB_1gB_1g.	..5.	B_2gB_2gB_2gB_2g.	..5.	B_3gB_3gB_3gB_3g.	..5.	A_1uA_1uA_1uA_1u.	..15.	B_1uB_1uB_1uB_1u.	..70.	B_2uB_2uB_2uB_2u.	..70.	B_3uB_3uB_3uB_3u.
Subtotal: 366 / 8 / 8
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(B_3u)
Subtotal: 0 / 0 / 56
Irrep combinations (i,i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(B_3u)
..315.	A_1gA_1gB_1gB_1g.	..63.	A_1gA_1gB_2gB_2g.	..63.	A_1gA_1gB_3gB_3g.	..63.	A_1gA_1gA_1uA_1u.	..126.	A_1gA_1gB_1uB_1u.	..315.	A_1gA_1gB_2uB_2u.	..315.	A_1gA_1gB_3uB_3u.	..45.	B_1gB_1gB_2gB_2g.	..45.	B_1gB_1gB_3gB_3g.	..45.	B_1gB_1gA_1uA_1u.
..90.	B_1gB_1gB_1uB_1u.	..225.	B_1gB_1gB_2uB_2u.	..225.	B_1gB_1gB_3uB_3u.	..9.	B_2gB_2gB_3gB_3g.	..9.	B_2gB_2gA_1uA_1u.	..18.	B_2gB_2gB_1uB_1u.	..45.	B_2gB_2gB_2uB_2u.	..45.	B_2gB_2gB_3uB_3u.	..9.	B_3gB_3gA_1uA_1u.	..18.	B_3gB_3gB_1uB_1u.
..45.	B_3gB_3gB_2uB_2u.	..45.	B_3gB_3gB_3uB_3u.	..18.	A_1uA_1uB_1uB_1u.	..45.	A_1uA_1uB_2uB_2u.	..45.	A_1uA_1uB_3uB_3u.	..90.	B_1uB_1uB_2uB_2u.	..90.	B_1uB_1uB_3uB_3u.	..225.	B_2uB_2uB_3uB_3u.
Subtotal: 2.691 / 28 / 28
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ pos(B_3u)
Subtotal: 0 / 0 / 168
Irrep combinations (i,j,k,l) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(B_3u)
..120.	A_1gB_1gB_2gB_3g.	..180.	A_1gB_1gA_1uB_1u.	..750.	A_1gB_1gB_2uB_3u.	..120.	A_1gB_2gA_1uB_2u.	..180.	A_1gB_2gB_1uB_3u.	..120.	A_1gB_3gA_1uB_3u.	..180.	A_1gB_3gB_1uB_2u.	..100.	B_1gB_2gA_1uB_3u.	..150.	B_1gB_2gB_1uB_2u.	..100.	B_1gB_3gA_1uB_2u.
..150.	B_1gB_3gB_1uB_3u.	..24.	B_2gB_3gA_1uB_1u.	..100.	B_2gB_3gB_2uB_3u.	..150.	A_1uB_1uB_2uB_3u.
Subtotal: 2.424 / 14 / 70
Total: 5.481 / 50 / 330

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Results for Point Group D2h

Force field analysis

Further Reading

Contributions to nonvanishing force field constants

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Results for Point Group D_2h