Grouptheoretical Analysis

Grouptheoretical Analysis

Composition Input Structure

Atomic Number	Atomic Symbol	Isotope	Isotope Mass	Isotope Abundance	Number Atoms	Mass
6	C	12	12.0000	98.9000	4	48.0000
8	O	16	15.9949	99.7620	4	63.9797

Total					8	111.9797
Isotopomer Natural Abundance (%)					94.7645

Determined Point Group: D_4h

Representation Γ_N

**Characters of reducible representation**
E	2C₄	C₂	2C'₂	2C''₂	i	2S₄	σ_h	2σ_v	2σ_d
8	0	0	4	0	0	0	8	4	0

**Direct sum of irreducible representation**
A_1g	A_2g	B_1g	B_2g	E_g	A_1u	A_2u	B_1u	B_2u	E_u
2	0	2	0	0	0	0	0	0	2

**Properties of derivatives and isotopomers by single substitution, h(D_4h)=16**
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_2v	4	4	C	no	yes	¹³C	112.9830	4.2160
2	C_2v	4	4	O	no	yes	¹⁸O ¹⁷O	113.9839 112.9839	0.7599 0.1444
Total Number of Atoms:			8	✅ Correct Number of Atoms found

*Atom Orbit
**Subgroup of point group D_4h
***Calculated as h( D_4h)/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	7	0
Double	XY	10	0
Triple	X₃	10	0
Triple	X₂Y	26	0
Triple	XYZ	48	0
Quadruple	X₄	15	0
Quadruple	X₃Y	42	0
Quadruple	X₂Y₂	65	0
Quadruple	X₂YZ	114	0
Quadruple	WXYZ	216	0
Quintuple	X₅	10	0
Quintuple	VWXYZ	840	0
Sextuple	X₆	7	0
Sextuple	UVWXYZ	2.520	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	2C₄	C₂	2C'₂	2C''₂	i	2S₄	σ_h	2σ_v	2σ_d
24	0	0	-4	0	0	0	8	4	0

**Direct sum of irreducible representation**
A_1g	A_2g	B_1g	B_2g	E_g	A_1u	A_2u	B_1u	B_2u	E_u
2	2	2	2	2	0	2	0	2	4

Molecular motions and force field analysis

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