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	2	24.0000

Total					8	30.0470
Isotopomer Natural Abundance (%)					97.7206

Determined Point Group: D_3d

Representation Γ_N

**Characters of reducible representation**
E	2C₃	3C'₂	i	2S₆	3σ_d
8	2	0	0	0	4

**Direct sum of irreducible representation**
A_1g	A_2g	E_g	A_1u	A_2u	E_u
2	0	1	0	2	1

**Properties of derivatives and isotopomers by single substitution, h(D_3d)=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	C_3v	6	2	C	no	yes	¹³C	31.0503	2.1738
2	C_s	2	6	H	no	yes	²H	31.0532	0.091481
Total Number of Atoms:			8	✅ Correct Number of Atoms found

*Atom Orbit
**Subgroup of point group D_3d
***Calculated as h( D_3d)/h(Site Symmetry)
****Natural Abundance of single substituted Isotopomer in %

**Numbers of isomers by substitution**
Replacement	Pattern	Achiral Isomers	Chiral Isomer Pairs
Single	X	2	0
Double	X₂	5	1
Double	XY	6	2
Triple	X₃	6	2
Triple	X₂Y	10	9
Triple	XYZ	12	22
Quadruple	X₄	8	4
Quadruple	X₃Y	14	17
Quadruple	X₂Y₂	18	29
Quadruple	X₂YZ	18	61
Quadruple	WXYZ	12	134
Quintuple	X₅	6	2
Quintuple	VWXYZ	0	560
Sextuple	X₆	5	1
Sextuple	UVWXYZ	0	1.680

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	3C'₂	i	2S₆	3σ_d
24	0	0	0	0	4

**Direct sum of irreducible representation**
A_1g	A_2g	E_g	A_1u	A_2u	E_u
3	1	4	1	3	4

Molecular motions and force field analysis

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