Characters of representations for molecular motions

Motion	E	2C5	2(C5)2	5C'2	σh	2S5	2(S5)3	5σv
Cartesian 3N	60	0.000	-0.000	0	0	-0.000	-0.000	4
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	54	-3.236	1.236	2	0	0.000	0.000	4

Decomposition to irreducible representations

Motion	A'1	A'2	E'1	E'2	A''1	A''2	E''1	E''2	Total
Cartesian 3N	4	2	6	6	2	4	6	6	36
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	4	1	5	6	2	3	5	6	32

Molecular parameter

Number of Atoms (N)	20
Number of internal coordinates	54
Number of independant internal coordinates	4
Number of vibrational modes	32

---

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)	4	1	5	6	2	3	5	6	8 / 24
Quadratic (Raman)	4	1	5	6	2	3	5	6	15 / 17
IR + Raman	- - - -	1	- - - -	- - - -	2	- - - -	- - - -	6	0 / 9

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	54	-3.236	1.236	2	0	0.000	0.000	4
quadratic	1.485	5.854	-0.854	29	27	0.618	-1.618	35
cubic	27.720	-7.236	-2.764	56	0	0.000	0.000	120
quartic	395.010	5.854	-0.854	434	378	-0.618	1.618	610
quintic	4.582.116	8.764	13.236	812	0	0.000	0.000	1.856
sextic	45.057.474	-37.833	15.833	4.466	3.654	-1.000	-1.000	7.134

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	4	1	5	6	2	3	5	6
quadratic	92	60	152	150	72	75	146	145
cubic	1.429	1.341	2.772	2.773	1.369	1.401	2.772	2.773
quartic	20.031	19.509	39.539	39.538	19.688	19.776	39.464	39.462
quintic	229.775	228.441	458.210	458.211	228.847	229.369	458.210	458.211
sextic	2.255.954	2.250.154	4.506.108	4.506.120	2.252.022	2.253.356	4.505.377	4.505.389

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)
..10.	A'1A'1.	..1.	A'2A'2.	..15.	E'1E'1.	..21.	E'2E'2.	..3.	A''1A''1.	..6.	A''2A''2.	..15.	E''1E''1.	..21.	E''2E''2.
Subtotal: 92 / 8 / 8
Irrep combinations (i,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
Subtotal: 0 / 0 / 28
Total: 92 / 8 / 36

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E''2)
..20.	A'1A'1A'1.
Subtotal: 20 / 1 / 8
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
..90.	E'1E'1E'2.	..4.	A'1A'2A'2.	..60.	A'1E'1E'1.	..84.	A'1E'2E'2.	..12.	A'1A''1A''1.	..24.	A'1A''2A''2.	..60.	A'1E''1E''1.	..84.	A'1E''2E''2.	..10.	A'2E'1E'1.	..15.	A'2E'2E'2.
..10.	A'2E''1E''1.	..15.	A'2E''2E''2.	..105.	E'1E'2E'2.	..105.	E'1E''2E''2.	..90.	E'2E''1E''1.
Subtotal: 768 / 15 / 56
Irrep combinations (i,j,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E''2)
..6.	A'2A''1A''2.	..50.	E'1A''1E''1.	..75.	E'1A''2E''1.	..150.	E'1E''1E''2.	..72.	E'2A''1E''2.	..108.	E'2A''2E''2.	..180.	E'2E''1E''2.
Subtotal: 641 / 7 / 56
Total: 1.429 / 23 / 120

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E''2)
..35.	A'1A'1A'1A'1.	..1.	A'2A'2A'2A'2.	..120.	E'1E'1E'1E'1.	..231.	E'2E'2E'2E'2.	..5.	A''1A''1A''1A''1.	..15.	A''2A''2A''2A''2.	..120.	E''1E''1E''1E''1.	..231.	E''2E''2E''2E''2.
Subtotal: 758 / 8 / 8
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
..210.	E'1E'1E'1E'2.	..210.	E''1E''1E''1E''2.	..280.	E'1E'2E'2E'2.	..280.	E''1E''2E''2E''2.
Subtotal: 980 / 4 / 56
Irrep combinations (i,i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
..10.	A'1A'1A'2A'2.	..150.	A'1A'1E'1E'1.	..210.	A'1A'1E'2E'2.	..30.	A'1A'1A''1A''1.	..60.	A'1A'1A''2A''2.	..150.	A'1A'1E''1E''1.	..210.	A'1A'1E''2E''2.	..15.	A'2A'2E'1E'1.	..21.	A'2A'2E'2E'2.	..3.	A'2A'2A''1A''1.
..6.	A'2A'2A''2A''2.	..15.	A'2A'2E''1E''1.	..21.	A'2A'2E''2E''2.	..465.	E'1E'1E'2E'2.	..45.	E'1E'1A''1A''1.	..90.	E'1E'1A''2A''2.	..550.	E'1E'1E''1E''1.	..465.	E'1E'1E''2E''2.	..63.	E'2E'2A''1A''1.	..126.	E'2E'2A''2A''2.
..465.	E'2E'2E''1E''1.	..1.107.	E'2E'2E''2E''2.	..18.	A''1A''1A''2A''2.	..45.	A''1A''1E''1E''1.	..63.	A''1A''1E''2E''2.	..90.	A''2A''2E''1E''1.	..126.	A''2A''2E''2E''2.	..465.	E''1E''1E''2E''2.
Subtotal: 5.084 / 28 / 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)
..60.	E'1E'1A''1A''2.	..180.	E'1E'1A''1E''2.	..270.	E'1E'1A''2E''2.	..450.	E'1E'1E''1E''2.	..90.	E'2E'2A''1A''2.	..210.	E'2E'2A''1E''1.	..315.	E'2E'2A''2E''1.	..630.	E'2E'2E''1E''2.	..360.	A'1E'1E'1E'2.	..90.	A'2E'1E'1E'2.
..180.	A''1E''1E''1E''2.	..270.	A''2E''1E''1E''2.	..40.	A'1A'2E'1E'1.	..60.	A'1A'2E'2E'2.	..40.	A'1A'2E''1E''1.	..60.	A'1A'2E''2E''2.	..420.	A'1E'1E'2E'2.	..420.	A'1E'1E''2E''2.	..360.	A'1E'2E''1E''1.	..105.	A'2E'1E'2E'2.
..105.	A'2E'1E''2E''2.	..90.	A'2E'2E''1E''1.	..450.	E'1E'2E''1E''1.	..630.	E'1E'2E''2E''2.	..60.	A''1A''2E''1E''1.	..90.	A''1A''2E''2E''2.	..210.	A''1E''1E''2E''2.	..315.	A''2E''1E''2E''2.
Subtotal: 6.560 / 28 / 168
Irrep combinations (i,j,k,l) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E''2)
..24.	A'1A'2A''1A''2.	..200.	A'1E'1A''1E''1.	..300.	A'1E'1A''2E''1.	..600.	A'1E'1E''1E''2.	..288.	A'1E'2A''1E''2.	..432.	A'1E'2A''2E''2.	..720.	A'1E'2E''1E''2.	..50.	A'2E'1A''1E''1.	..75.	A'2E'1A''2E''1.	..150.	A'2E'1E''1E''2.
..72.	A'2E'2A''1E''2.	..108.	A'2E'2A''2E''2.	..180.	A'2E'2E''1E''2.	..300.	E'1E'2A''1E''1.	..360.	E'1E'2A''1E''2.	..450.	E'1E'2A''2E''1.	..540.	E'1E'2A''2E''2.	..1.800.	E'1E'2E''1E''2.
Subtotal: 6.649 / 18 / 70
Total: 20.031 / 86 / 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