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	36	36.2817
6	C	12	12.0000	98.9000	36	432.0000
7	N	14	14.0031	99.6340	24	336.0738
8	O	16	15.9949	99.7620	12	191.9390

Total					108	996.2945
Isotopomer Natural Abundance (%)					59.4279

Determined Point Group: D_6h

Representation Γ_N

**Characters of reducible representation**
E	2C₆	2C₃	C₂	3C'₂	3C''₂	i	2S₃	2S₆	σ_h	3σ_d	3σ_v
108	0	0	0	0	0	0	0	0	24	12	8

**Direct sum of irreducible representation**
A_1g	A_2g	B_1g	B_2g	E_1g	E_2g	A_1u	A_2u	B_1u	B_2u	E_1u	E_2u
8	3	4	3	7	11	1	6	5	6	11	7

**Properties of derivatives and isotopomers by single substitution, h(D_6h)=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_s	2	12	C	no	yes	¹³C	997.2978	7.9317
2	C_s	2	12	H	no	yes	²H	997.3007	0.1113
3	C₁	1	24	N	yes	yes	¹⁵N	997.2915	5.2393
4	C_s	2	12	C	no	yes	¹³C	997.2978	7.9317
5	C_s	2	12	O	no	yes	¹⁸O ¹⁷O	998.2987 997.2987	1.4297 0.2716
6	C_s	2	12	C	no	yes	¹³C	997.2978	7.9317
7	C_s	2	12	H	no	yes	²H	997.3007	0.1113
8	C_s	2	12	H	no	yes	²H	997.3007	0.1113
Total Number of Atoms:			108	✅ Correct Number of Atoms found

*Atom Orbit
**Subgroup of point group D_6h
***Calculated as h( D_6h)/h(Site Symmetry)
****Natural Abundance of single substituted Isotopomer in %

**Numbers of isomers by substitution**
Replacement	Pattern	Achiral Isomers	Chiral Isomer Pairs
Single	X	7	1
Double	X₂	79	217
Double	XY	93	435
Triple	X₃	567	8.226
Triple	X₂Y	1.041	24.999
Triple	XYZ	1.426	50.326
Quadruple	X₄	3.914	221.756
Quadruple	X₃Y	8.323	889.021
Quadruple	X₂Y₂	11.935	1.334.641
Quadruple	X₂YZ	16.537	2.671.279
Quadruple	WXYZ	24.642	5.346.774
Quintuple	X₅	21.734	4.633.682
Quintuple	VWXYZ	450.480	557.120.640
Sextuple	X₆	114.708	79.681.359
Sextuple	UVWXYZ	8.247.120	57.402.502.080

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	2C₆	2C₃	C₂	3C'₂	3C''₂	i	2S₃	2S₆	σ_h	3σ_d	3σ_v
324	0	0	0	0	0	0	0	0	24	12	8

**Direct sum of irreducible representation**
A_1g	A_2g	B_1g	B_2g	E_1g	E_2g	A_1u	A_2u	B_1u	B_2u	E_1u	E_2u
17	12	13	12	25	29	10	15	14	15	29	25

Molecular motions and force field analysis

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