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	48	0	-4	-4	0	16	4	4
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	42	2	-2	-2	0	16	4	4

**Decomposition to irreducible representations**
Motion	A_1g	B_1g	B_2g	B_3g	A_1u	B_1u	B_2u	B_3u	Total
Cartesian 3N	8	8	4	4	2	6	8	8	48
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	8	7	3	3	2	5	7	7	42

**Molecular parameter**
Number of Atoms (N)	16
Number of internal coordinates	42
Number of independant internal coordinates	8
Number of vibrational modes	42

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)	8	7	3	3	2	5	7	7	19 / 23
Quadratic (Raman)	8	7	3	3	2	5	7	7	21 / 21
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	42	2	-2	-2	0	16	4	4
quadratic	903	23	23	23	21	149	29	29
cubic	13.244	44	-44	-44	0	1.024	96	96
quartic	148.995	275	275	275	231	5.735	415	415
quintic	1.370.754	506	-506	-506	0	27.568	1.196	1.196
sextic	10.737.573	2.277	2.277	2.277	1.771	117.483	3.979	3.979

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	8	7	3	3	2	5	7	7
quadratic	150	124	94	94	93	96	126	126
cubic	1.802	1.776	1.522	1.522	1.498	1.568	1.778	1.778
quartic	19.577	19.232	17.902	17.902	17.878	17.948	19.278	19.278
quintic	175.026	174.681	167.835	167.835	167.536	168.387	174.727	174.727
sextic	1.358.952	1.355.824	1.327.448	1.327.448	1.327.149	1.328.000	1.356.376	1.356.376

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)
..36.	A_1gA_1g.	..28.	B_1gB_1g.	..6.	B_2gB_2g.	..6.	B_3gB_3g.	..3.	A_1uA_1u.	..15.	B_1uB_1u.	..28.	B_2uB_2u.	..28.	B_3uB_3u.
Subtotal: 150 / 8 / 8
Irrep combinations (i,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(B_3u)
Subtotal: 0 / 0 / 28
Total: 150 / 8 / 36

**Contributions to nonvanishing cubic force field constants**
Irrep combinations (i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(B_3u)
..120.	A_1gA_1gA_1g.
Subtotal: 120 / 1 / 8
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(B_3u)
..224.	A_1gB_1gB_1g.	..48.	A_1gB_2gB_2g.	..48.	A_1gB_3gB_3g.	..24.	A_1gA_1uA_1u.	..120.	A_1gB_1uB_1u.	..224.	A_1gB_2uB_2u.	..224.	A_1gB_3uB_3u.
Subtotal: 912 / 7 / 56
Irrep combinations (i,j,k) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ pos(B_3u)
..63.	B_1gB_2gB_3g.	..70.	B_1gA_1uB_1u.	..343.	B_1gB_2uB_3u.	..42.	B_2gA_1uB_2u.	..105.	B_2gB_1uB_3u.	..42.	B_3gA_1uB_3u.	..105.	B_3gB_1uB_2u.
Subtotal: 770 / 7 / 56
Total: 1.802 / 15 / 120

**Contributions to nonvanishing quartic force field constants**
Irrep combinations (i,i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(B_3u)
..330.	A_1gA_1gA_1gA_1g.	..210.	B_1gB_1gB_1gB_1g.	..15.	B_2gB_2gB_2gB_2g.	..15.	B_3gB_3gB_3gB_3g.	..5.	A_1uA_1uA_1uA_1u.	..70.	B_1uB_1uB_1uB_1u.	..210.	B_2uB_2uB_2uB_2u.	..210.	B_3uB_3uB_3uB_3u.
Subtotal: 1.065 / 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)
..1.008.	A_1gA_1gB_1gB_1g.	..216.	A_1gA_1gB_2gB_2g.	..216.	A_1gA_1gB_3gB_3g.	..108.	A_1gA_1gA_1uA_1u.	..540.	A_1gA_1gB_1uB_1u.	..1.008.	A_1gA_1gB_2uB_2u.	..1.008.	A_1gA_1gB_3uB_3u.	..168.	B_1gB_1gB_2gB_2g.	..168.	B_1gB_1gB_3gB_3g.	..84.	B_1gB_1gA_1uA_1u.
..420.	B_1gB_1gB_1uB_1u.	..784.	B_1gB_1gB_2uB_2u.	..784.	B_1gB_1gB_3uB_3u.	..36.	B_2gB_2gB_3gB_3g.	..18.	B_2gB_2gA_1uA_1u.	..90.	B_2gB_2gB_1uB_1u.	..168.	B_2gB_2gB_2uB_2u.	..168.	B_2gB_2gB_3uB_3u.	..18.	B_3gB_3gA_1uA_1u.	..90.	B_3gB_3gB_1uB_1u.
..168.	B_3gB_3gB_2uB_2u.	..168.	B_3gB_3gB_3uB_3u.	..45.	A_1uA_1uB_1uB_1u.	..84.	A_1uA_1uB_2uB_2u.	..84.	A_1uA_1uB_3uB_3u.	..420.	B_1uB_1uB_2uB_2u.	..420.	B_1uB_1uB_3uB_3u.	..784.	B_2uB_2uB_3uB_3u.
Subtotal: 9.273 / 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)
..504.	A_1gB_1gB_2gB_3g.	..560.	A_1gB_1gA_1uB_1u.	..2.744.	A_1gB_1gB_2uB_3u.	..336.	A_1gB_2gA_1uB_2u.	..840.	A_1gB_2gB_1uB_3u.	..336.	A_1gB_3gA_1uB_3u.	..840.	A_1gB_3gB_1uB_2u.	..294.	B_1gB_2gA_1uB_3u.	..735.	B_1gB_2gB_1uB_2u.	..294.	B_1gB_3gA_1uB_2u.
..735.	B_1gB_3gB_1uB_3u.	..90.	B_2gB_3gA_1uB_1u.	..441.	B_2gB_3gB_2uB_3u.	..490.	A_1uB_1uB_2uB_3u.
Subtotal: 9.239 / 14 / 70
Total: 19.577 / 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