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	3	3.0235
5	B	11	11.0093	80.1000	1	11.0093
8	O	16	15.9949	99.7620	3	47.9847

Total					7	62.0175
Isotopomer Natural Abundance (%)					79.4922

Determined Point Group: C_3h

Representation Γ_N

**Characters of reducible representation**
E	C₃	(C₃)²	σ_h	S₃	(S₃)⁵
7	1	1	7	1	1

**Direct sum of irreducible representation**
A'	E'*	A''	E''*
3	2	0	0

**Properties of derivatives and isotopomers by single substitution, h(C_3h)=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_3h	6	1	B	no	no	¹⁰B	61.0212	19.7490
2	C_s	2	3	O	no	yes	¹⁸O ¹⁷O	64.0218 63.0217	0.4781 0.090837
3	C_s	2	3	H	no	yes	²H	63.0238	0.037208
Total Number of Atoms:			7	✅ Correct Number of Atoms found

*Atom Orbit
**Subgroup of point group C_3h
***Calculated as h( C_3h)/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	7	0
Double	XY	14	0
Triple	X₃	13	0
Triple	X₂Y	35	0
Triple	XYZ	70	0
Quadruple	X₄	13	0
Quadruple	X₃Y	48	0
Quadruple	X₂Y₂	70	0
Quadruple	X₂YZ	140	0
Quadruple	WXYZ	280	0
Quintuple	X₅	7	0
Quintuple	VWXYZ	840	0
Sextuple	X₆	3	0
Sextuple	UVWXYZ	1.680	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	C₃	(C₃)²	σ_h	S₃	(S₃)⁵
21	0	0	7	-2	-2

**Direct sum of irreducible representation**
A'	E'*	A''	E''*
4	5	3	2

Molecular motions and force field analysis

Last update April, 29^th 2022 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement