Characters of representations for molecular motions

Motion	E	2C4	C2	2C'2	2C''2	i	2S4	σh	2σv	2σd
Cartesian 3N	15	1	-1	-3	-1	-3	-1	5	3	1
Translation (x,y,z)	3	1	-1	-1	-1	-3	-1	1	1	1
Rotation (Rx,Ry,Rz)	3	1	-1	-1	-1	3	1	-1	-1	-1
Vibration	9	-1	1	-1	1	-3	-1	5	3	1

Decomposition to irreducible representations

Motion	A1g	A2g	B1g	B2g	Eg	A1u	A2u	B1u	B2u	Eu	Total
Cartesian 3N	1	1	1	1	1	0	2	0	1	3	11
Translation (x,y,z)	0	0	0	0	0	0	1	0	0	1	2
Rotation (Rx,Ry,Rz)	0	1	0	0	1	0	0	0	0	0	2
Vibration	1	0	1	1	0	0	1	0	1	2	7

Molecular parameter

Number of Atoms (N)	5
Number of internal coordinates	9
Number of independant internal coordinates	1
Number of vibrational modes	7

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1g	A2g	B1g	B2g	Eg	A1u	A2u	B1u	B2u	Eu	Total
Linear (IR)	1	0	1	1	0	0	1	0	1	2	3 / 4
Quadratic (Raman)	1	0	1	1	0	0	1	0	1	2	3 / 4
IR + Raman	- - - -	0	- - - -	- - - -	- - - -	0	- - - -	0	1	- - - -	0* / 1

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C4	C2	2C'2	2C''2	i	2S4	σh	2σv	2σd
linear	9	-1	1	-1	1	-3	-1	5	3	1
quadratic	45	1	5	5	5	9	1	17	9	5
cubic	165	-1	5	-5	5	-19	-1	45	19	5
quartic	495	3	15	15	15	39	3	103	39	15
quintic	1.287	-3	15	-15	15	-69	-3	211	69	15
sextic	3.003	3	35	35	35	119	3	399	119	35

Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted

Force field	A1g	A2g	B1g	B2g	Eg	A1u	A2u	B1u	B2u	Eu
linear	1	0	1	1	0	0	1	0	1	2
quadratic	8	2	5	4	4	1	2	1	2	6
cubic	15	9	13	12	12	6	12	6	12	28
quartic	52	31	43	37	52	20	26	20	26	68
quintic	100	79	94	88	124	62	83	62	83	194
sextic	251	195	232	211	336	147	168	147	168	406

Further Reading

• J.K.G. Watson, J. Mol. Spec. 41 229 (1972)
  The Numbers of Structural Parameters and Potential Constants of Molecules
  
  
• X.F. Zhou, P. Pulay. J. Comp. Chem. 10 No. 7, 935-938 (1989)
  Characters for Symmetric and Antisymmetric Higher Powers of Representations:
  Application to the Number of Anharmonic Force Constants in Symmetrical Molecules
  
  
• F. Varga, L. Nemes, J.K.G. Watson. J. Phys. B: At. Mol. Opt. Phys. 10 No. 7, 5043-5048 (1996)
  The number of anharmonic potential constants of the fullerenes C₆₀ and C₇₀

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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(A1g) ≤ i ≤ pos(Eu)
..1.	A1gA1g.	..1.	B1gB1g.	..1.	B2gB2g.	..1.	A2uA2u.	..1.	B2uB2u.	..3.	EuEu.
Subtotal: 8 / 6 / 10
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 45
Total: 8 / 6 / 55

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..1.	A1gA1gA1g.
Subtotal: 1 / 1 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..1.	A1gB1gB1g.	..1.	A1gB2gB2g.	..1.	A1gA2uA2u.	..1.	A1gB2uB2u.	..3.	A1gEuEu.	..3.	B1gEuEu.	..3.	B2gEuEu.
Subtotal: 13 / 7 / 90
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..1.	B1gA2uB2u.
Subtotal: 1 / 1 / 120
Total: 15 / 9 / 220

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..1.	A1gA1gA1gA1g.	..1.	B1gB1gB1gB1g.	..1.	B2gB2gB2gB2g.	..1.	A2uA2uA2uA2u.	..1.	B2uB2uB2uB2u.	..11.	EuEuEuEu.
Subtotal: 16 / 6 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 90
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..1.	A1gA1gB1gB1g.	..1.	A1gA1gB2gB2g.	..1.	A1gA1gA2uA2u.	..1.	A1gA1gB2uB2u.	..3.	A1gA1gEuEu.	..1.	B1gB1gB2gB2g.	..1.	B1gB1gA2uA2u.	..1.	B1gB1gB2uB2u.	..3.	B1gB1gEuEu.	..1.	B2gB2gA2uA2u.
..1.	B2gB2gB2uB2u.	..3.	B2gB2gEuEu.	..1.	A2uA2uB2uB2u.	..3.	A2uA2uEuEu.	..3.	B2uB2uEuEu.
Subtotal: 25 / 15 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..3.	A1gB1gEuEu.	..3.	A1gB2gEuEu.	..1.	B1gB2gEuEu.	..3.	A2uB2uEuEu.
Subtotal: 10 / 4 / 360
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu)
..1.	A1gB1gA2uB2u.
Subtotal: 1 / 1 / 210
Total: 52 / 26 / 715

Calculate contributions to

A1g	A2g	B1g	B2g	Eg	A1u	A2u	B1u	B2u	Eu

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