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	6	6.0470
6	C	12	12.0000	98.9000	3	36.0000
8	O	16	15.9949	99.7620	3	47.9847

Total					12	90.0317
Isotopomer Natural Abundance (%)					95.9572

Determined Point Group: C_3v

Representation Γ_N

**Characters of reducible representation**
E	2C₃	3σ_v
12	0	4

**Direct sum of irreducible representation**
A₁	A₂	E
4	0	4

**Properties of derivatives and isotopomers by single substitution, h(C_3v)=6**
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	3	C	no	yes	¹³C	91.0350	3.2018
2	C_s	2	3	O	no	yes	¹⁸O ¹⁷O	92.0359 91.0359	0.5771 0.1097
3	C_s	2	3	H	no	yes	²H	91.0380	0.044915
4	C_s	2	3	H	no	yes	²H	91.0380	0.044915
Total Number of Atoms:			12	✅ Correct Number of Atoms found

*Atom Orbit
**Subgroup of point group C_3v
***Calculated as h( C_3v)/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	10	6
Double	XY	12	16
Triple	X₃	20	28
Triple	X₂Y	28	96
Triple	XYZ	24	208
Quadruple	X₄	31	67
Quadruple	X₃Y	52	304
Quadruple	X₂Y₂	66	462
Quadruple	X₂YZ	60	960
Quadruple	WXYZ	24	1.968
Quintuple	X₅	40	112
Quintuple	VWXYZ	0	15.840
Sextuple	X₆	44	134
Sextuple	UVWXYZ	0	110.880

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	2C₃	3σ_v
36	0	4

**Direct sum of irreducible representation**
A₁	A₂	E
8	4	12

Molecular motions and force field analysis

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