Characters of representations for molecular motions

Motion	E	C2	i	σh
Cartesian 3N	36	0	0	4
Translation (x,y,z)	3	-1	-3	1
Rotation (Rx,Ry,Rz)	3	-1	3	-1
Vibration	30	2	0	4

Decomposition to irreducible representations

Motion	Ag	Bg	Au	Bu	Total
Cartesian 3N	10	8	8	10	36
Translation (x,y,z)	0	0	1	2	3
Rotation (Rx,Ry,Rz)	1	2	0	0	3
Vibration	9	6	7	8	30

Molecular parameter

Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	9
Number of vibrational modes	30

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	Ag	Bg	Au	Bu	Total
Linear (IR)	9	6	7	8	15 / 15
Quadratic (Raman)	9	6	7	8	15 / 15
IR + Raman	- - - -	- - - -	- - - -	- - - -	0* / 0

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2	i	σh
linear	30	2	0	4
quadratic	465	17	15	23
cubic	4.960	32	0	72
quartic	40.920	152	120	256
quintic	278.256	272	0	680
sextic	1.623.160	952	680	1.904

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

Force field	Ag	Bg	Au	Bu
linear	9	6	7	8
quadratic	130	110	111	114
cubic	1.266	1.214	1.230	1.250
quartic	10.362	10.158	10.174	10.226
quintic	69.802	69.326	69.462	69.666
sextic	406.674	405.246	405.382	405.858

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)
..45.	AgAg.	..21.	BgBg.	..28.	AuAu.	..36.	BuBu.
Subtotal: 130 / 4 / 4
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
Subtotal: 0 / 0 / 6
Total: 130 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Bu)
..165.	AgAgAg.
Subtotal: 165 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
..189.	AgBgBg.	..252.	AgAuAu.	..324.	AgBuBu.
Subtotal: 765 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Bu)
..336.	BgAuBu.
Subtotal: 336 / 1 / 4
Total: 1.266 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Bu)
..495.	AgAgAgAg.	..126.	BgBgBgBg.	..210.	AuAuAuAu.	..330.	BuBuBuBu.
Subtotal: 1.161 / 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)
..945.	AgAgBgBg.	..1.260.	AgAgAuAu.	..1.620.	AgAgBuBu.	..588.	BgBgAuAu.	..756.	BgBgBuBu.	..1.008.	AuAuBuBu.
Subtotal: 6.177 / 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)
..3.024.	AgBgAuBu.
Subtotal: 3.024 / 1 / 1
Total: 10.362 / 11 / 35

Calculate contributions to

Ag	Bg	Au	Bu

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