Characters of representations for molecular motions

Motion	E	C2	i	σh
Cartesian 3N	21	-1	-3	3
Translation (x,y,z)	3	-1	-3	1
Rotation (Rx,Ry,Rz)	3	-1	3	-1
Vibration	15	1	-3	3

Decomposition to irreducible representations

Motion	Ag	Bg	Au	Bu	Total
Cartesian 3N	5	4	5	7	21
Translation (x,y,z)	0	0	1	2	3
Rotation (Rx,Ry,Rz)	1	2	0	0	3
Vibration	4	2	4	5	15

Molecular parameter

Number of Atoms (N)	7
Number of internal coordinates	15
Number of independant internal coordinates	4
Number of vibrational modes	15

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	Ag	Bg	Au	Bu	Total
Linear (IR)	4	2	4	5	9 / 6
Quadratic (Raman)	4	2	4	5	6 / 9
IR + Raman	- - - -	- - - -	- - - -	- - - -	0* / 0

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2	i	σh
linear	15	1	-3	3
quadratic	120	8	12	12
cubic	680	8	-28	28
quartic	3.060	36	72	72
quintic	11.628	36	-144	144
sextic	38.760	120	300	300

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

Force field	Ag	Bg	Au	Bu
linear	4	2	4	5
quadratic	38	28	26	28
cubic	172	154	172	182
quartic	810	756	738	756
quintic	2.916	2.826	2.916	2.970
sextic	9.870	9.660	9.570	9.660

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 C_2h

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(Ag) ≤ i ≤ pos(Bu)
..10.	AgAg.	..3.	BgBg.	..10.	AuAu.	..15.	BuBu.
Subtotal: 38 / 4 / 4
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
Subtotal: 0 / 0 / 6
Total: 38 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Bu)
..20.	AgAgAg.
Subtotal: 20 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
..12.	AgBgBg.	..40.	AgAuAu.	..60.	AgBuBu.
Subtotal: 112 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Bu)
..40.	BgAuBu.
Subtotal: 40 / 1 / 4
Total: 172 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Bu)
..35.	AgAgAgAg.	..5.	BgBgBgBg.	..35.	AuAuAuAu.	..70.	BuBuBuBu.
Subtotal: 145 / 4 / 4
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
Subtotal: 0 / 0 / 12
Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
..30.	AgAgBgBg.	..100.	AgAgAuAu.	..150.	AgAgBuBu.	..30.	BgBgAuAu.	..45.	BgBgBuBu.	..150.	AuAuBuBu.
Subtotal: 505 / 6 / 6
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Bu)
Subtotal: 0 / 0 / 12
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Bu)
..160.	AgBgAuBu.
Subtotal: 160 / 1 / 1
Total: 810 / 11 / 35

Calculate contributions to

Ag	Bg	Au	Bu

Show only nonzero contributions Show all contributions
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