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	24	0	0	-4	0	0	0	8	4	0
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	18	-2	2	-2	2	0	0	8	4	0

**Decomposition to irreducible representations**
Motion	A_1g	A_2g	B_1g	B_2g	E_g	A_2u	B_2u	E_u	Total
Cartesian 3N	2	2	2	2	2	2	2	4	18
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	2	1	2	2	1	1	2	3	14

**Molecular parameter**
Number of Atoms (N)	8
Number of internal coordinates	18
Number of independant internal coordinates	2
Number of vibrational modes	14

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)	2	1	2	2	1	1	2	3	4 / 10
Quadratic (Raman)	2	1	2	2	1	1	2	3	7 / 7
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	18	-2	2	-2	2	0	0	8	4	0
quadratic	171	3	11	11	11	9	1	41	17	9
cubic	1.140	-4	20	-20	20	0	0	160	48	0
quartic	5.985	9	65	65	65	45	5	525	133	45
quintic	26.334	-14	110	-110	110	0	0	1.512	308	0
sextic	100.947	19	275	275	275	165	5	3.941	693	165

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	2	1	2	2	1	0	1	0	2	3
quadratic	21	9	15	13	16	8	9	7	9	24
cubic	88	76	84	82	120	56	68	52	74	160
quartic	454	377	423	401	680	337	349	331	353	800
quintic	1.784	1.707	1.760	1.738	3.089	1.518	1.595	1.494	1.626	3.467
sextic	6.762	6.410	6.646	6.514	12.112	6.033	6.110	6.002	6.134	13.056

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)
..3.	A_1gA_1g.	..1.	A_2gA_2g.	..3.	B_1gB_1g.	..3.	B_2gB_2g.	..1.	E_gE_g.	..1.	A_2uA_2u.	..3.	B_2uB_2u.	..6.	E_uE_u.
Subtotal: 21 / 8 / 10
Irrep combinations (i,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_u)
Subtotal: 0 / 0 / 45
Total: 21 / 8 / 55

**Contributions to nonvanishing cubic force field constants**
Irrep combinations (i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(E_u)
..4.	A_1gA_1gA_1g.
Subtotal: 4 / 1 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_u)
..2.	A_1gA_2gA_2g.	..6.	A_1gB_1gB_1g.	..6.	A_1gB_2gB_2g.	..2.	A_1gE_gE_g.	..2.	A_1gA_2uA_2u.	..6.	A_1gB_2uB_2u.	..12.	A_1gE_uE_u.	..3.	A_2gE_uE_u.	..2.	B_1gE_gE_g.	..12.	B_1gE_uE_u.
..2.	B_2gE_gE_g.	..12.	B_2gE_uE_u.
Subtotal: 67 / 12 / 90
Irrep combinations (i,j,k) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ pos(E_u)
..4.	A_2gB_1gB_2g.	..4.	B_1gA_2uB_2u.	..3.	E_gA_2uE_u.	..6.	E_gB_2uE_u.
Subtotal: 17 / 4 / 120
Total: 88 / 17 / 220

**Contributions to nonvanishing quartic force field constants**
Irrep combinations (i,i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(E_u)
..5.	A_1gA_1gA_1gA_1g.	..1.	A_2gA_2gA_2gA_2g.	..5.	B_1gB_1gB_1gB_1g.	..5.	B_2gB_2gB_2gB_2g.	..2.	E_gE_gE_gE_g.	..1.	A_2uA_2uA_2uA_2u.	..5.	B_2uB_2uB_2uB_2u.	..36.	E_uE_uE_uE_u.
Subtotal: 60 / 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)
..3.	A_1gA_1gA_2gA_2g.	..9.	A_1gA_1gB_1gB_1g.	..9.	A_1gA_1gB_2gB_2g.	..3.	A_1gA_1gE_gE_g.	..3.	A_1gA_1gA_2uA_2u.	..9.	A_1gA_1gB_2uB_2u.	..18.	A_1gA_1gE_uE_u.	..3.	A_2gA_2gB_1gB_1g.	..3.	A_2gA_2gB_2gB_2g.	..1.	A_2gA_2gE_gE_g.
..1.	A_2gA_2gA_2uA_2u.	..3.	A_2gA_2gB_2uB_2u.	..6.	A_2gA_2gE_uE_u.	..9.	B_1gB_1gB_2gB_2g.	..3.	B_1gB_1gE_gE_g.	..3.	B_1gB_1gA_2uA_2u.	..9.	B_1gB_1gB_2uB_2u.	..18.	B_1gB_1gE_uE_u.	..3.	B_2gB_2gE_gE_g.	..3.	B_2gB_2gA_2uA_2u.
..9.	B_2gB_2gB_2uB_2u.	..18.	B_2gB_2gE_uE_u.	..1.	E_gE_gA_2uA_2u.	..3.	E_gE_gB_2uB_2u.	..18.	E_gE_gE_uE_u.	..3.	A_2uA_2uB_2uB_2u.	..6.	A_2uA_2uE_uE_u.	..18.	B_2uB_2uE_uE_u.
Subtotal: 195 / 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)
..2.	E_gE_gA_2uB_2u.	..6.	A_1gA_2gE_uE_u.	..4.	A_1gB_1gE_gE_g.	..24.	A_1gB_1gE_uE_u.	..4.	A_1gB_2gE_gE_g.	..24.	A_1gB_2gE_uE_u.	..2.	A_2gB_1gE_gE_g.	..12.	A_2gB_1gE_uE_u.	..2.	A_2gB_2gE_gE_g.	..12.	A_2gB_2gE_uE_u.
..12.	B_1gB_2gE_uE_u.	..12.	A_2uB_2uE_uE_u.
Subtotal: 116 / 12 / 360
Irrep combinations (i,j,k,l) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(E_u)
..8.	A_1gA_2gB_1gB_2g.	..8.	A_1gB_1gA_2uB_2u.	..6.	A_1gE_gA_2uE_u.	..12.	A_1gE_gB_2uE_u.	..4.	A_2gB_2gA_2uB_2u.	..3.	A_2gE_gA_2uE_u.	..6.	A_2gE_gB_2uE_u.	..6.	B_1gE_gA_2uE_u.	..12.	B_1gE_gB_2uE_u.	..6.	B_2gE_gA_2uE_u.
..12.	B_2gE_gB_2uE_u.
Subtotal: 83 / 11 / 210
Total: 454 / 59 / 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