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	10	159.9491
15	P	31	30.9738	100.0000	4	123.8950

Total					14	283.8442
Isotopomer Natural Abundance (%)					97.6453

Determined Point Group: T_d

Representation Γ_N

**Characters of reducible representation**
E	8C₃	3C₂	6S₄	6σ_d
14	2	2	0	6

**Direct sum of irreducible representation**
A₁	A₂	E	T₁	T₂
3	0	1	0	3

**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	285.8484 284.8484	1.1745 0.2232
2	C_3v	6	4	P	no	yes	-	-	-
3	C_3v	6	4	O	no	yes	¹⁸O ¹⁷O	285.8484 284.8484	0.7830 0.1488
Total Number of Atoms:			14	✅ 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	3	0
Double	X₂	10	0
Double	XY	15	1
Triple	X₃	22	7
Triple	X₂Y	42	26
Triple	XYZ	60	61
Quadruple	X₄	42	26
Quadruple	X₃Y	90	126
Quadruple	X₂Y₂	111	200
Quadruple	X₂YZ	150	427
Quadruple	WXYZ	180	911
Quintuple	X₅	61	58
Quintuple	VWXYZ	360	9.830
Sextuple	X₆	79	92
Sextuple	UVWXYZ	360	89.910

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
42	0	-2	0	6

**Direct sum of irreducible representation**
A₁	A₂	E	T₁	T₂
3	0	3	4	7

Molecular motions and force field analysis

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