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
28	Ni	58	57.9353	68.0770	1	57.9353

Total					9	169.9150
Isotopomer Natural Abundance (%)					64.5128

Determined Point Group: T_d

Representation Γ_N

**Characters of reducible representation**
E	8C₃	3C₂	6S₄	6σ_d
9	3	1	1	5

**Direct sum of irreducible representation**
A₁	A₂	E	T₁	T₂
3	0	0	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	T_d	24	1	Ni	no	no	⁶⁰Ni ⁶²Ni ⁶¹Ni ⁶⁴Ni	171.9104 173.9080 172.9107 175.9076	24.8501 3.4443 1.0802 0.8770
2	C_3v	6	4	C	no	yes	¹³C	170.9184	2.8701
3	C_3v	6	4	O	no	yes	¹⁸O ¹⁷O	171.9192 170.9192	0.5173 0.098293
Total Number of Atoms:			9	✅ 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	6	0
Double	XY	10	0
Triple	X₃	10	0
Triple	X₂Y	20	2
Triple	XYZ	30	8
Quadruple	X₄	14	1
Quadruple	X₃Y	30	8
Quadruple	X₂Y₂	36	15
Quadruple	X₂YZ	50	38
Quadruple	WXYZ	60	96
Quintuple	X₅	14	1
Quintuple	VWXYZ	60	600
Sextuple	X₆	10	0
Sextuple	UVWXYZ	0	2.520

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
27	0	-1	-1	5

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

Molecular motions and force field analysis

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