Characters of representations for molecular motions

Motion	E	2C5	2(C5)2	5C'2	σh	2S5	2(S5)3	5σv
Cartesian 3N	30	0.000	-0.000	-2	10	-0.000	-0.000	2
Translation (x,y,z)	3	1.618	-0.618	-1	1	-0.382	-2.618	1
Rotation (Rx,Ry,Rz)	3	1.618	-0.618	-1	-1	0.382	2.618	-1
Vibration	24	-3.236	1.236	0	10	0.000	-0.000	2

Decomposition to irreducible representations

Motion	A'1	A'2	E'1	E'2	A''1	A''2	E''1	E''2	Total
Cartesian 3N	2	2	4	4	0	2	2	2	18
Translation (x,y,z)	0	0	1	0	0	1	0	0	2
Rotation (Rx,Ry,Rz)	0	1	0	0	0	0	1	0	2
Vibration	2	1	3	4	0	1	1	2	14

Molecular parameter

Number of Atoms (N)	10
Number of internal coordinates	24
Number of independant internal coordinates	2
Number of vibrational modes	14

---

Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'1	A'2	E'1	E'2	A''1	A''2	E''1	E''2	Total
Linear (IR)	2	1	3	4	0	1	1	2	4 / 10
Quadratic (Raman)	2	1	3	4	0	1	1	2	7 / 7
IR + Raman	- - - -	1	- - - -	- - - -	0	- - - -	- - - -	2	0 / 3

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C5	2(C5)2	5C'2	σh	2S5	2(S5)3	5σv
linear	24	-3.236	1.236	0	10	0.000	-0.000	2
quadratic	300	5.854	-0.854	12	62	0.618	-1.618	14
cubic	2.600	-7.236	-2.764	0	290	0.000	0.000	26
quartic	17.550	5.854	-0.854	78	1.128	-0.618	1.618	104
quintic	98.280	2.764	7.236	0	3.822	2.000	2.000	182
sextic	475.020	-18.416	8.416	364	11.634	-1.000	-1.000	546

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

Force field	A'1	A'2	E'1	E'2	A''1	A''2	E''1	E''2
linear	2	1	3	4	0	1	1	2
quadratic	25	12	37	35	12	13	24	23
cubic	150	137	289	290	108	121	231	232
quartic	980	889	1.868	1.867	815	828	1.643	1.641
quintic	5.152	5.061	10.209	10.210	4.678	4.769	9.445	9.446
sextic	24.559	24.104	48.663	48.669	23.123	23.214	46.336	46.342

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

---

Contributions to nonvanishing force field constants

pos(X) : Position of irreducible representation (irrep) X in character table of D_5h

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'1) ≤ i ≤ pos(E''2)
..3.	A'1A'1.	..1.	A'2A'2.	..6.	E'1E'1.	..10.	E'2E'2.	..1.	A''2A''2.	..1.	E''1E''1.	..3.	E''2E''2.
Subtotal: 25 / 7 / 8
Irrep combinations (i,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
Subtotal: 0 / 0 / 28
Total: 25 / 7 / 36

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E''2)
..4.	A'1A'1A'1.
Subtotal: 4 / 1 / 8
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
..24.	E'1E'1E'2.	..2.	A'1A'2A'2.	..12.	A'1E'1E'1.	..20.	A'1E'2E'2.	..2.	A'1A''2A''2.	..2.	A'1E''1E''1.	..6.	A'1E''2E''2.	..3.	A'2E'1E'1.	..6.	A'2E'2E'2.	..1.	A'2E''2E''2.
..30.	E'1E'2E'2.	..9.	E'1E''2E''2.	..4.	E'2E''1E''1.
Subtotal: 121 / 13 / 56
Irrep combinations (i,j,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E''2)
..3.	E'1A''2E''1.	..6.	E'1E''1E''2.	..8.	E'2A''2E''2.	..8.	E'2E''1E''2.
Subtotal: 25 / 4 / 56
Total: 150 / 18 / 120

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E''2)
..5.	A'1A'1A'1A'1.	..1.	A'2A'2A'2A'2.	..21.	E'1E'1E'1E'1.	..55.	E'2E'2E'2E'2.	..1.	A''2A''2A''2A''2.	..1.	E''1E''1E''1E''1.	..6.	E''2E''2E''2E''2.
Subtotal: 90 / 7 / 8
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
..40.	E'1E'1E'1E'2.	..2.	E''1E''1E''1E''2.	..60.	E'1E'2E'2E'2.	..4.	E''1E''2E''2E''2.
Subtotal: 106 / 4 / 56
Irrep combinations (i,i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
..3.	A'1A'1A'2A'2.	..18.	A'1A'1E'1E'1.	..30.	A'1A'1E'2E'2.	..3.	A'1A'1A''2A''2.	..3.	A'1A'1E''1E''1.	..9.	A'1A'1E''2E''2.	..6.	A'2A'2E'1E'1.	..10.	A'2A'2E'2E'2.	..1.	A'2A'2A''2A''2.	..1.	A'2A'2E''1E''1.
..3.	A'2A'2E''2E''2.	..78.	E'1E'1E'2E'2.	..6.	E'1E'1A''2A''2.	..12.	E'1E'1E''1E''1.	..21.	E'1E'1E''2E''2.	..10.	E'2E'2A''2A''2.	..10.	E'2E'2E''1E''1.	..66.	E'2E'2E''2E''2.	..1.	A''2A''2E''1E''1.	..3.	A''2A''2E''2E''2.
..3.	E''1E''1E''2E''2.
Subtotal: 297 / 21 / 28
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E''2)
..12.	E'1E'1A''2E''2.	..12.	E'1E'1E''1E''2.	..10.	E'2E'2A''2E''1.	..20.	E'2E'2E''1E''2.	..48.	A'1E'1E'1E'2.	..24.	A'2E'1E'1E'2.	..2.	A''2E''1E''1E''2.	..6.	A'1A'2E'1E'1.	..12.	A'1A'2E'2E'2.	..2.	A'1A'2E''2E''2.
..60.	A'1E'1E'2E'2.	..18.	A'1E'1E''2E''2.	..8.	A'1E'2E''1E''1.	..30.	A'2E'1E'2E'2.	..9.	A'2E'1E''2E''2.	..4.	A'2E'2E''1E''1.	..12.	E'1E'2E''1E''1.	..36.	E'1E'2E''2E''2.	..3.	A''2E''1E''2E''2.
Subtotal: 328 / 19 / 168
Irrep combinations (i,j,k,l) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E''2)
..6.	A'1E'1A''2E''1.	..12.	A'1E'1E''1E''2.	..16.	A'1E'2A''2E''2.	..16.	A'1E'2E''1E''2.	..3.	A'2E'1A''2E''1.	..6.	A'2E'1E''1E''2.	..8.	A'2E'2A''2E''2.	..8.	A'2E'2E''1E''2.	..12.	E'1E'2A''2E''1.	..24.	E'1E'2A''2E''2.
..48.	E'1E'2E''1E''2.
Subtotal: 159 / 11 / 70
Total: 980 / 62 / 330

Calculate contributions to

A'1	A'2	E'1	E'2	A''1	A''2	E''1	E''2

Show only nonzero contributions Show all contributions
Up to quartic force fieldUp to quintic force fieldUp to sextic force field