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	54	0	0	-2	0	18	2	0
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	48	2	2	0	0	18	2	0

**Decomposition to irreducible representations**
Motion	A_1g	B_1g	B_2g	B_3g	A_1u	B_1u	B_2u	B_3u	Total
Cartesian 3N	9	9	5	4	4	5	9	9	54
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	9	8	4	3	4	4	8	8	48

**Molecular parameter**
Number of Atoms (N)	18
Number of internal coordinates	48
Number of independant internal coordinates	9
Number of vibrational modes	48

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)	9	8	4	3	4	4	8	8	20 / 28
Quadratic (Raman)	9	8	4	3	4	4	8	8	24 / 24
IR + Raman	- - - -	- - - -	- - - -	- - - -	4	- - - -	- - - -	- - - -	0^* / 4

* 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	48	2	2	0	0	18	2	0
quadratic	1.176	26	26	24	24	186	26	24
cubic	19.600	50	50	0	0	1.410	50	0
quartic	249.900	350	350	300	300	8.670	350	300
quintic	2.598.960	650	650	0	0	45.594	650	0
sextic	22.957.480	3.250	3.250	2.600	2.600	211.922	3.250	2.600

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	9	8	4	3	4	4	8	8
quadratic	189	164	124	123	124	124	164	164
cubic	2.645	2.620	2.280	2.255	2.280	2.280	2.620	2.620
quartic	32.565	32.240	30.160	30.135	30.160	30.160	32.240	32.240
quintic	330.813	330.488	319.252	318.927	319.252	319.252	330.488	330.488
sextic	2.898.369	2.895.444	2.843.276	2.842.951	2.843.276	2.843.276	2.895.444	2.895.444

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)
..45.	A_1gA_1g.	..36.	B_1gB_1g.	..10.	B_2gB_2g.	..6.	B_3gB_3g.	..10.	A_1uA_1u.	..10.	B_1uB_1u.	..36.	B_2uB_2u.	..36.	B_3uB_3u.
Subtotal: 189 / 8 / 8
Irrep combinations (i,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(B_3u)
Subtotal: 0 / 0 / 28
Total: 189 / 8 / 36

**Contributions to nonvanishing cubic force field constants**
Irrep combinations (i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(B_3u)
..165.	A_1gA_1gA_1g.
Subtotal: 165 / 1 / 8
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(B_3u)
..324.	A_1gB_1gB_1g.	..90.	A_1gB_2gB_2g.	..54.	A_1gB_3gB_3g.	..90.	A_1gA_1uA_1u.	..90.	A_1gB_1uB_1u.	..324.	A_1gB_2uB_2u.	..324.	A_1gB_3uB_3u.
Subtotal: 1.296 / 7 / 56
Irrep combinations (i,j,k) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ pos(B_3u)
..96.	B_1gB_2gB_3g.	..128.	B_1gA_1uB_1u.	..512.	B_1gB_2uB_3u.	..128.	B_2gA_1uB_2u.	..128.	B_2gB_1uB_3u.	..96.	B_3gA_1uB_3u.	..96.	B_3gB_1uB_2u.
Subtotal: 1.184 / 7 / 56
Total: 2.645 / 15 / 120

**Contributions to nonvanishing quartic force field constants**
Irrep combinations (i,i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(B_3u)
..495.	A_1gA_1gA_1gA_1g.	..330.	B_1gB_1gB_1gB_1g.	..35.	B_2gB_2gB_2gB_2g.	..15.	B_3gB_3gB_3gB_3g.	..35.	A_1uA_1uA_1uA_1u.	..35.	B_1uB_1uB_1uB_1u.	..330.	B_2uB_2uB_2uB_2u.	..330.	B_3uB_3uB_3uB_3u.
Subtotal: 1.605 / 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)
..1.620.	A_1gA_1gB_1gB_1g.	..450.	A_1gA_1gB_2gB_2g.	..270.	A_1gA_1gB_3gB_3g.	..450.	A_1gA_1gA_1uA_1u.	..450.	A_1gA_1gB_1uB_1u.	..1.620.	A_1gA_1gB_2uB_2u.	..1.620.	A_1gA_1gB_3uB_3u.	..360.	B_1gB_1gB_2gB_2g.	..216.	B_1gB_1gB_3gB_3g.	..360.	B_1gB_1gA_1uA_1u.
..360.	B_1gB_1gB_1uB_1u.	..1.296.	B_1gB_1gB_2uB_2u.	..1.296.	B_1gB_1gB_3uB_3u.	..60.	B_2gB_2gB_3gB_3g.	..100.	B_2gB_2gA_1uA_1u.	..100.	B_2gB_2gB_1uB_1u.	..360.	B_2gB_2gB_2uB_2u.	..360.	B_2gB_2gB_3uB_3u.	..60.	B_3gB_3gA_1uA_1u.	..60.	B_3gB_3gB_1uB_1u.
..216.	B_3gB_3gB_2uB_2u.	..216.	B_3gB_3gB_3uB_3u.	..100.	A_1uA_1uB_1uB_1u.	..360.	A_1uA_1uB_2uB_2u.	..360.	A_1uA_1uB_3uB_3u.	..360.	B_1uB_1uB_2uB_2u.	..360.	B_1uB_1uB_3uB_3u.	..1.296.	B_2uB_2uB_3uB_3u.
Subtotal: 14.736 / 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)
..864.	A_1gB_1gB_2gB_3g.	..1.152.	A_1gB_1gA_1uB_1u.	..4.608.	A_1gB_1gB_2uB_3u.	..1.152.	A_1gB_2gA_1uB_2u.	..1.152.	A_1gB_2gB_1uB_3u.	..864.	A_1gB_3gA_1uB_3u.	..864.	A_1gB_3gB_1uB_2u.	..1.024.	B_1gB_2gA_1uB_3u.	..1.024.	B_1gB_2gB_1uB_2u.	..768.	B_1gB_3gA_1uB_2u.
..768.	B_1gB_3gB_1uB_3u.	..192.	B_2gB_3gA_1uB_1u.	..768.	B_2gB_3gB_2uB_3u.	..1.024.	A_1uB_1uB_2uB_3u.
Subtotal: 16.224 / 14 / 70
Total: 32.565 / 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