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	39	-5	-1	-1	-3	9	5	5
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	33	-3	1	1	-3	9	5	5

**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	4	4	4	2	5	7	7	39
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	3	3	3	2	4	6	6	33

**Molecular parameter**
Number of Atoms (N)	13
Number of internal coordinates	33
Number of independant internal coordinates	6
Number of vibrational modes	33

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	3	3	3	2	4	6	6	16 / 17
Quadratic (Raman)	6	3	3	3	2	4	6	6	15 / 18
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	33	-3	1	1	-3	9	5	5
quadratic	561	21	17	17	21	57	29	29
cubic	6.545	-55	17	17	-55	273	105	105
quartic	58.905	225	153	153	225	1.113	385	385
quintic	435.897	-531	153	153	-531	3.969	1.141	1.141
sextic	2.760.681	1.653	969	969	1.653	12.817	3.325	3.325

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	3	3	3	2	4	6	6
quadratic	94	71	63	63	60	66	72	72
cubic	869	808	784	784	762	806	866	866
quartic	7.693	7.424	7.224	7.224	7.166	7.282	7.446	7.446
quintic	55.174	54.527	53.991	53.991	53.744	54.238	55.116	55.116
sextic	348.174	346.027	343.483	343.483	342.894	344.072	346.274	346.274

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.	..6.	B_1gB_1g.	..6.	B_2gB_2g.	..6.	B_3gB_3g.	..3.	A_1uA_1u.	..10.	B_1uB_1u.	..21.	B_2uB_2u.	..21.	B_3uB_3u.
Subtotal: 94 / 8 / 8
Irrep combinations (i,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(B_3u)
Subtotal: 0 / 0 / 28
Total: 94 / 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)
..36.	A_1gB_1gB_1g.	..36.	A_1gB_2gB_2g.	..36.	A_1gB_3gB_3g.	..18.	A_1gA_1uA_1u.	..60.	A_1gB_1uB_1u.	..126.	A_1gB_2uB_2u.	..126.	A_1gB_3uB_3u.
Subtotal: 438 / 7 / 56
Irrep combinations (i,j,k) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ pos(B_3u)
..27.	B_1gB_2gB_3g.	..24.	B_1gA_1uB_1u.	..108.	B_1gB_2uB_3u.	..36.	B_2gA_1uB_2u.	..72.	B_2gB_1uB_3u.	..36.	B_3gA_1uB_3u.	..72.	B_3gB_1uB_2u.
Subtotal: 375 / 7 / 56
Total: 869 / 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.	..15.	B_1gB_1gB_1gB_1g.	..15.	B_2gB_2gB_2gB_2g.	..15.	B_3gB_3gB_3gB_3g.	..5.	A_1uA_1uA_1uA_1u.	..35.	B_1uB_1uB_1uB_1u.	..126.	B_2uB_2uB_2uB_2u.	..126.	B_3uB_3uB_3uB_3u.
Subtotal: 463 / 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)
..126.	A_1gA_1gB_1gB_1g.	..126.	A_1gA_1gB_2gB_2g.	..126.	A_1gA_1gB_3gB_3g.	..63.	A_1gA_1gA_1uA_1u.	..210.	A_1gA_1gB_1uB_1u.	..441.	A_1gA_1gB_2uB_2u.	..441.	A_1gA_1gB_3uB_3u.	..36.	B_1gB_1gB_2gB_2g.	..36.	B_1gB_1gB_3gB_3g.	..18.	B_1gB_1gA_1uA_1u.
..60.	B_1gB_1gB_1uB_1u.	..126.	B_1gB_1gB_2uB_2u.	..126.	B_1gB_1gB_3uB_3u.	..36.	B_2gB_2gB_3gB_3g.	..18.	B_2gB_2gA_1uA_1u.	..60.	B_2gB_2gB_1uB_1u.	..126.	B_2gB_2gB_2uB_2u.	..126.	B_2gB_2gB_3uB_3u.	..18.	B_3gB_3gA_1uA_1u.	..60.	B_3gB_3gB_1uB_1u.
..126.	B_3gB_3gB_2uB_2u.	..126.	B_3gB_3gB_3uB_3u.	..30.	A_1uA_1uB_1uB_1u.	..63.	A_1uA_1uB_2uB_2u.	..63.	A_1uA_1uB_3uB_3u.	..210.	B_1uB_1uB_2uB_2u.	..210.	B_1uB_1uB_3uB_3u.	..441.	B_2uB_2uB_3uB_3u.
Subtotal: 3.648 / 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)
..162.	A_1gB_1gB_2gB_3g.	..144.	A_1gB_1gA_1uB_1u.	..648.	A_1gB_1gB_2uB_3u.	..216.	A_1gB_2gA_1uB_2u.	..432.	A_1gB_2gB_1uB_3u.	..216.	A_1gB_3gA_1uB_3u.	..432.	A_1gB_3gB_1uB_2u.	..108.	B_1gB_2gA_1uB_3u.	..216.	B_1gB_2gB_1uB_2u.	..108.	B_1gB_3gA_1uB_2u.
..216.	B_1gB_3gB_1uB_3u.	..72.	B_2gB_3gA_1uB_1u.	..324.	B_2gB_3gB_2uB_3u.	..288.	A_1uB_1uB_2uB_3u.
Subtotal: 3.582 / 14 / 70
Total: 7.693 / 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