Grouptheoretical Analysis

Grouptheoretical Analysis

Composition Input Structure

Atomic Number	Atomic Symbol	Isotope	Isotope Mass	Isotope Abundance	Number Atoms	Mass
1	H	1	1.0078	99.9844	12	12.0939
6	C	12	12.0000	98.9000	12	144.0000
24	Cr	52	51.9405	83.7890	1	51.9405

Total					25	208.0344
Isotopomer Natural Abundance (%)					73.2368

Determined Point Group: D₆

Representation Γ_N

**Characters of reducible representation**
E	2C₆	2C₃	C₂	3C'₂	3C''₂
25	1	1	1	1	1

**Direct sum of irreducible representation**
A₁	A₂	B₁	B₂	E₁	E₂
3	2	2	2	4	4

**Properties of derivatives and isotopomers by single substitution, h(D₆)=12**
Atom Set^*	Site Symmetry^**	h(Site Symmetry)	Identical Atoms^***	Element	Chrial	Polar	Isotopomer
Atom Set^*	Site Symmetry^**	h(Site Symmetry)	Identical Atoms^***	Element	Chrial	Polar	Isotope	Mass	Abundance^****
1	D₆	12	1	Cr	yes	no	⁵³Cr ⁵⁰Cr ⁵⁴Cr	209.0345 206.0399 210.0328	8.3045 3.7978 2.0672
2	C₁	1	12	C	yes	yes	¹³C	209.0378	9.7748
3	C₁	1	12	H	yes	yes	²H	209.0407	0.1371
Total Number of Atoms:			25	✅ Correct Number of Atoms found

*Atom Orbit
**Subgroup of point group D₆
***Calculated as h( D₆)/h(Site Symmetry)
****Natural Abundance of single substituted Isotopomer in %

**Numbers of isomers by substitution**
Replacement	Pattern	Chiral Isomers
Single	X	3
Double	X₂	32
Double	XY	50
Triple	X₃	200
Triple	X₂Y	582
Triple	XYZ	1.150
Quadruple	X₄	1.094
Quadruple	X₃Y	4.218
Quadruple	X₂Y₂	6.402
Quadruple	X₂YZ	12.650
Quadruple	WXYZ	25.300
Quintuple	X₅	4.466
Quintuple	VWXYZ	531.300
Sextuple	X₆	14.892
Sextuple	UVWXYZ	10.626.000

Further Reading

P.W. Fowler, J. Chem. Soc. Faraday Trans. 91(15) 2241 (1995)
Isomer Counting using Point Group Symmetry

Representation Γ_3N

**Characters of reducible representation**
E	2C₆	2C₃	C₂	3C'₂	3C''₂
75	2	0	-1	-1	-1

**Direct sum of irreducible representation**
A₁	A₂	B₁	B₂	E₁	E₂
6	7	6	6	13	12

Molecular motions and force field analysis

Last update November, 13^th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement