Point Group D4h

Results for Point Group D_4h

**Characters of representations for molecular motions**
Motion	E	2C₄	C₂	2C'₂	2C''₂	i	2S₄	σ_h	2σ_v	2σ_d
Cartesian 3N	36	4	-4	-4	0	0	0	8	8	4
Translation (x,y,z)	3	1	-1	-1	-1	-3	-1	1	1	1
Rotation (R_x,R_y,R_z)	3	1	-1	-1	-1	3	1	-1	-1	-1
Vibration	30	2	-2	-2	2	0	0	8	8	4

**Decomposition to irreducible representations**
Motion	A_1g	A_2g	B_1g	B_2g	E_g	A_2u	B_2u	E_u	Total
Cartesian 3N	4	2	2	2	4	4	2	6	26
Translation (x,y,z)	0	0	0	0	0	1	0	1	2
Rotation (R_x,R_y,R_z)	0	1	0	0	1	0	0	0	2
Vibration	4	1	2	2	3	3	2	5	22

**Molecular parameter**
Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	4
Number of vibrational modes	22

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A_1g	A_2g	B_1g	B_2g	E_g	A_2u	B_2u	E_u	Total
Linear (IR)	4	1	2	2	3	3	2	5	8 / 14
Quadratic (Raman)	4	1	2	2	3	3	2	5	11 / 11
IR + Raman	- - - -	1	- - - -	- - - -	- - - -	- - - -	2	- - - -	0^* / 3

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)
Force field	E	2C₄	C₂	2C'₂	2C''₂	i	2S₄	σ_h	2σ_v	2σ_d
linear	30	2	-2	-2	2	0	0	8	8	4
quadratic	465	1	17	17	17	15	-1	47	47	23
cubic	4.960	0	-32	-32	32	0	0	208	208	72
quartic	40.920	8	152	152	152	120	8	792	792	256
quintic	278.256	16	-272	-272	272	0	0	2.640	2.640	680
sextic	1.623.160	8	952	952	952	680	-8	8.008	8.008	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_g	A_1u	A_2u	B_1u	B_2u	E_u
linear	4	1	2	2	3	0	3	0	2	5
quadratic	47	21	37	31	52	22	31	23	29	60
cubic	356	286	330	312	598	260	330	270	320	650
quartic	2.795	2.457	2.689	2.555	5.012	2.417	2.603	2.443	2.577	5.180
quintic	17.956	17.126	17.714	17.360	34.486	16.796	17.626	16.894	17.520	35.146
sextic	103.527	100.573	102.813	101.287	201.860	99.965	101.967	100.199	101.725	203.692

Contributions to nonvanishing force field constants

pos(X) : Position of irreducible representation (irrep) X in character table of D_4h

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_u)
..10.	A_1gA_1g.	..1.	A_2gA_2g.	..3.	B_1gB_1g.	..3.	B_2gB_2g.	..6.	E_gE_g.	..6.	A_2uA_2u.	..3.	B_2uB_2u.	..15.	E_uE_u.
Subtotal: 47 / 8 / 10
Irrep combinations (i,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_u)
Subtotal: 0 / 0 / 45
Total: 47 / 8 / 55

**Contributions to nonvanishing cubic force field constants**
Irrep combinations (i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(E_u)
..20.	A_1gA_1gA_1g.
Subtotal: 20 / 1 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_u)
..4.	A_1gA_2gA_2g.	..12.	A_1gB_1gB_1g.	..12.	A_1gB_2gB_2g.	..24.	A_1gE_gE_g.	..24.	A_1gA_2uA_2u.	..12.	A_1gB_2uB_2u.	..60.	A_1gE_uE_u.	..3.	A_2gE_gE_g.	..10.	A_2gE_uE_u.	..12.	B_1gE_gE_g.
..30.	B_1gE_uE_u.	..12.	B_2gE_gE_g.	..30.	B_2gE_uE_u.
Subtotal: 245 / 13 / 90
Irrep combinations (i,j,k) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ pos(E_u)
..4.	A_2gB_1gB_2g.	..12.	B_1gA_2uB_2u.	..45.	E_gA_2uE_u.	..30.	E_gB_2uE_u.
Subtotal: 91 / 4 / 120
Total: 356 / 18 / 220

**Contributions to nonvanishing quartic force field constants**
Irrep combinations (i,i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(E_u)
..35.	A_1gA_1gA_1gA_1g.	..1.	A_2gA_2gA_2gA_2g.	..5.	B_1gB_1gB_1gB_1g.	..5.	B_2gB_2gB_2gB_2g.	..36.	E_gE_gE_gE_g.	..15.	A_2uA_2uA_2uA_2u.	..5.	B_2uB_2uB_2uB_2u.	..190.	E_uE_uE_uE_u.
Subtotal: 292 / 8 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_u)
Subtotal: 0 / 0 / 90
Irrep combinations (i,i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_u)
..10.	A_1gA_1gA_2gA_2g.	..30.	A_1gA_1gB_1gB_1g.	..30.	A_1gA_1gB_2gB_2g.	..60.	A_1gA_1gE_gE_g.	..60.	A_1gA_1gA_2uA_2u.	..30.	A_1gA_1gB_2uB_2u.	..150.	A_1gA_1gE_uE_u.	..3.	A_2gA_2gB_1gB_1g.	..3.	A_2gA_2gB_2gB_2g.	..6.	A_2gA_2gE_gE_g.
..6.	A_2gA_2gA_2uA_2u.	..3.	A_2gA_2gB_2uB_2u.	..15.	A_2gA_2gE_uE_u.	..9.	B_1gB_1gB_2gB_2g.	..18.	B_1gB_1gE_gE_g.	..18.	B_1gB_1gA_2uA_2u.	..9.	B_1gB_1gB_2uB_2u.	..45.	B_1gB_1gE_uE_u.	..18.	B_2gB_2gE_gE_g.	..18.	B_2gB_2gA_2uA_2u.
..9.	B_2gB_2gB_2uB_2u.	..45.	B_2gB_2gE_uE_u.	..36.	E_gE_gA_2uA_2u.	..18.	E_gE_gB_2uB_2u.	..300.	E_gE_gE_uE_u.	..18.	A_2uA_2uB_2uB_2u.	..90.	A_2uA_2uE_uE_u.	..45.	B_2uB_2uE_uE_u.
Subtotal: 1.102 / 28 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ pos(E_u)
..36.	E_gE_gA_2uB_2u.	..12.	A_1gA_2gE_gE_g.	..40.	A_1gA_2gE_uE_u.	..48.	A_1gB_1gE_gE_g.	..120.	A_1gB_1gE_uE_u.	..48.	A_1gB_2gE_gE_g.	..120.	A_1gB_2gE_uE_u.	..12.	A_2gB_1gE_gE_g.	..30.	A_2gB_1gE_uE_u.	..12.	A_2gB_2gE_gE_g.
..30.	A_2gB_2gE_uE_u.	..12.	B_1gB_2gE_gE_g.	..40.	B_1gB_2gE_uE_u.	..90.	A_2uB_2uE_uE_u.
Subtotal: 650 / 14 / 360
Irrep combinations (i,j,k,l) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(E_u)
..16.	A_1gA_2gB_1gB_2g.	..48.	A_1gB_1gA_2uB_2u.	..180.	A_1gE_gA_2uE_u.	..120.	A_1gE_gB_2uE_u.	..12.	A_2gB_2gA_2uB_2u.	..45.	A_2gE_gA_2uE_u.	..30.	A_2gE_gB_2uE_u.	..90.	B_1gE_gA_2uE_u.	..60.	B_1gE_gB_2uE_u.	..90.	B_2gE_gA_2uE_u.
..60.	B_2gE_gB_2uE_u.
Subtotal: 751 / 11 / 210
Total: 2.795 / 61 / 715

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Last update November, 13^th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement

Results for Point Group D4h

Force field analysis

Further Reading

Contributions to nonvanishing force field constants

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Results for Point Group D_4h