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	2	2.0157
6	C	12	12.0000	98.9000	2	24.0000
8	O	16	15.9949	99.7620	4	63.9797

Total					8	89.9953
Isotopomer Natural Abundance (%)					96.8540

Determined Point Group: C_2h

Representation Γ_N

**Characters of reducible representation**
E	C₂	i	σ_h
8	0	0	8

**Direct sum of irreducible representation**
A_g	B_g	A_u	B_u
4	0	0	4

**Properties of derivatives and isotopomers by single substitution, h(C_2h)=4**
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	C_s	2	2	C	no	yes	¹³C	90.9987	2.1545
2	C_s	2	2	O	no	yes	¹⁸O ¹⁷O	91.9996 90.9995	0.3883 0.073785
3	C_s	2	2	O	no	yes	¹⁸O ¹⁷O	91.9996 90.9995	0.3883 0.073785
4	C_s	2	2	H	no	yes	²H	91.0016	0.030223
Total Number of Atoms:			8	✅ Correct Number of Atoms found

*Atom Orbit
**Subgroup of point group C_2h
***Calculated as h( C_2h)/h(Site Symmetry)
****Natural Abundance of single substituted Isotopomer in %

**Numbers of isomers by substitution**
Replacement	Pattern	Achiral Isomers	Chiral Isomer Pairs
Single	X	4	0
Double	X₂	16	0
Double	XY	28	0
Triple	X₃	28	0
Triple	X₂Y	84	0
Triple	XYZ	168	0
Quadruple	X₄	38	0
Quadruple	X₃Y	140	0
Quadruple	X₂Y₂	216	0
Quadruple	X₂YZ	420	0
Quadruple	WXYZ	840	0
Quintuple	X₅	28	0
Quintuple	VWXYZ	3.360	0
Sextuple	X₆	16	0
Sextuple	UVWXYZ	10.080	0

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	C₂	i	σ_h
24	0	0	8

**Direct sum of irreducible representation**
A_g	B_g	A_u	B_u
8	4	4	8

Molecular motions and force field analysis

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