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	1	12.0000
8	O	16	15.9949	99.7620	1	15.9949

Total					4	30.0106
Isotopomer Natural Abundance (%)					98.6338

Determined Point Group: C_2v

Representation Γ_N

**Characters of reducible representation**
E	C₂.(z)	σ_v.(xz)	σ_d.(yz)
4	2	4	2

**Direct sum of irreducible representation**
A₁	A₂	B₁	B₂
3	0	1	0

**Properties of derivatives and isotopomers by single substitution, h(C_2v)=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	H	no	yes	²H	31.0168	0.030779
2	C_2v	4	1	C	no	yes	¹³C	31.0139	1.0970
3	C_2v	4	1	O	no	yes	¹⁸O ¹⁷O	32.0148 31.0148	0.1977 0.037570
Total Number of Atoms:			4	✅ Correct Number of Atoms found

*Atom Orbit
**Subgroup of point group C_2v
***Calculated as h( C_2v)/h(Site Symmetry)
****Natural Abundance of single substituted Isotopomer in %

**Numbers of isomers by substitution**
Replacement	Pattern	Achiral Isomers	Chiral Isomer Pairs
Single	X	3	0
Double	X₂	4	0
Double	XY	7	0
Triple	X₃	3	0
Triple	X₂Y	7	0
Triple	XYZ	12	0
Quadruple	X₄	1	0
Quadruple	X₃Y	3	0
Quadruple	X₂Y₂	4	0
Quadruple	X₂YZ	7	0
Quadruple	WXYZ	12	0
Quintuple	X₅	0	0
Quintuple	VWXYZ	0	0
Sextuple	X₆	0	0
Sextuple	UVWXYZ	0	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.(z)	σ_v.(xz)	σ_d.(yz)
12	-2	4	2

**Direct sum of irreducible representation**
A₁	A₂	B₁	B₂
4	1	4	3

Molecular motions and force field analysis

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