Grouptheoretical Analysis

Grouptheoretical Analysis

Composition Input Structure

Atomic Number	Atomic Symbol	Isotope	Isotope Mass	Isotope Abundance	Number Atoms	Mass
8	O	16	15.9949	99.7620	6	95.9695
15	P	31	30.9738	100.0000	4	123.8950

Total					10	219.8645
Isotopomer Natural Abundance (%)					98.5805

Determined Point Group: T_d

Representation Γ_N

**Characters of reducible representation**
E	8C₃	3C₂	6S₄	6σ_d
10	1	2	0	4

**Direct sum of irreducible representation**
A₁	A₂	E	T₁	T₂
2	0	1	0	2

**Properties of derivatives and isotopomers by single substitution, h(T_d)=24**
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	6	O	no	yes	¹⁸O ¹⁷O	221.8688 220.8688	1.1858 0.2253
2	C_3v	6	4	P	no	yes	-	-	-
Total Number of Atoms:			10	✅ Correct Number of Atoms found

*Atom Orbit
**Subgroup of point group T_d
***Calculated as h( T_d)/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	0
Double	XY	6	1
Triple	X₃	8	3
Triple	X₂Y	12	10
Triple	XYZ	12	24
Quadruple	X₄	12	5
Quadruple	X₃Y	20	27
Quadruple	X₂Y₂	24	43
Quadruple	X₂YZ	24	94
Quadruple	WXYZ	12	204
Quintuple	X₅	12	6
Quintuple	VWXYZ	0	1.260
Sextuple	X₆	12	5
Sextuple	UVWXYZ	0	6.300

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	8C₃	3C₂	6S₄	6σ_d
30	0	-2	0	4

**Direct sum of irreducible representation**
A₁	A₂	E	T₁	T₂
2	0	2	3	5

Molecular motions and force field analysis

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