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European Journal of Applied Sciences – Vol. 10, No. 6

Publication Date: December 25, 2022

DOI:10.14738/aivp.106.13277. Tsassi, L. M. D., Emadak, A., Ndassa, I. M., Patouossa, I., & Ndongue, J. C. E. (2022). Algorithm for Combinatorial Bipartite

Enumeration of Chiral and Achiral Stereoisomers of Homo-and Heteropolysubstituted Derivatives of the Cubane Molecule.

European Journal of Applied Sciences, 10(6). 371-391.

Services for Science and Education – United Kingdom

Algorithm for Combinatorial Bipartite Enumeration of Chiral and

Achiral Stereoisomers of Homo- and Heteropolysubstituted

Derivatives of the Cubane Molecule

Leonel Marcelin Djoumessi Tsassi

Department of Inorganic Chemistry, Faculty of Science

University of Yaounde I, Yaounde, Cameroon

Alphonse Emadak

Department of Inorganic Chemistry, Faculty of Science

University of Yaounde I, Yaounde, Cameroon

Ibrahim Mbouombouo Ndassa

Department of Chemistry, Higher Teacher Training College

University of Yaounde I, Yaounde, Cameroon

Issofa Patouossa

Department of Inorganic Chemistry, Faculty of Science

University of Yaounde I, Yaounde, Cameroon

Jules César Epée Ndongue

Department of Inorganic Chemistry, Faculty of Science

University of Yaounde I, Yaounde, Cameroon

Abstract

This paper is devoted to the application of an algorithm for the enumeration of

chiral and achiral stereoisomers of homo and heteropolysubstituted derivatives of

the cubane molecule. The sequence of this algorithm is in three steps, the first of

which is the determination of the permutations of 8 hydrogen atoms of the cuban

molecule ( ) subjected to the action of the Oh group. The second step will be

the composition of these permutations later into permutational representations

and controlling respectively the chirality and achirality of this

molecular system. The transformation of these 2 operators into a pair of generic

formulae used for bipartite combinatorial enumeration giving chiral and

achiral isomer numbers of the homo- and hetero-polysubstituted

derivatives of the cubane molecule respectively symbolised by the empirical

formulae and being the final step.

Keywords: Enumeration, stereoisomer, chirality, substitution, diastereomers, cuban

C H8 8

RcH8 RacH8

(8, ) s N q c

(8, ) s N q ac

CH X 8 8-q q 8 0 1 ... ... qq q q i n CH X Y Z

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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 6, December-2022

Services for Science and Education – United Kingdom

NOTATIONS

: Permutations generated by symmetry operations of the point group ; and

average permutations generated by rotations and rotoreflections respectively; and

representations of permutations controlling chirality and achirality respectively; and

are the numbers of chiral and achiral stereoisomers respectively; is the total number of

stereoisomers

INTRODUCTION

Graph theory applied to the study of molecular structures is an interdisciplinary science, called

chemical graph theory or molecular topology. Interdisciplinary science, called chemical graph

theory or molecular topology. By using the tools taken from graph theory, set theory and

statistics one tries to identify the

try to identify the structural feature involved in the structure-property activity relationship[1] .

To do a computational chemistry study one has to represent the different molecules as

molecular graphs. Where each atom is represented as a ball and linked together by rods.

Enumeration in chemistry is therefore an important step since Humbolt[2] which states the

possibility of different chemical substances with the same elemental composition. Later

Berzelius referred to these types of chemical compounds as isomers[3]. Structures with atypical

geometries have always been of interest to researchers. Even more so when these structures

do not occur in nature. Organic structures in the form of polyhedra[4] are the ones that fascinate

researchers the most; especially those with a geometric structure in the form of the Platonic

solids. The tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron are

the five Platonic solids; of these, three are possible in chemistry: the tetrahedron, the cubane

and the dodecahedron, which derive respectively from the tetrahedron, the cube and the

dodecahedron. The cubane molecule[5] or pentacyclo octane is the one that is

the subject of our study. This molecule has several derivatives that can be useful industrially,

militarily and medicinally[6-10] . Given all the interests that derivatives of the cubane molecule

can have, we have undertaken to count the stereoisomers of the homo and hetero

polysubstituted derivatives of the cubane molecule using an enumeration algorithm. The

polysubstitutions are performed with non-isomerisable substituents representing atoms or

groups of atoms. This work is presented according to the following plan: a first part where we

present the mathematical formulation of our algorithm; a second part in which we apply this

formulation and at the end we conclude this work.

Mathematical formulations and applications

Point group symmetry of the cuban molecule

Let G be the stereograph of the cubane molecule . From the literature we know that the

point group of this stereograph is [11]. The 48 operations of this group are listed below:

= { , , , , , , , , , , , , , , , ,

, , , , , , , , , , , , , , , , ,

( )

8 R Oh H Oh Rro Rrr

Rc Rac

s Nc

s Nac

T N

2,5 3,8 4,7 é ù 4.2.0.0 .0 .0 ë û

C H8 8

Oh

Oh E C2(1) C2(2) C2(3) C3(1) C3(3) C3(2) C3(4)

2 C3(1)

2 C3(4)

2 C3(3)

2 C3(2)

' C2(6)

' C2(1)

' C2(4)

' C2(2)

' C2(5)

' C2(3)

3 C4(3) C4(3)

3 C4(1) C4(1) C4(2)

3 C4(2) i sh(3) sh(2) sh(1)

5

6(1) S 5

6(3) S 5

6(2) S 5

6(4) S S6 (1)

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373

Tsassi, L. M. D., Emadak, A., Ndassa, I. M., Patouossa, I., & Ndongue, J. C. E. (2022). Algorithm for Combinatorial Bipartite Enumeration of Chiral and

Achiral Stereoisomers of Homo-and Heteropolysubstituted Derivatives of the Cubane Molecule. European Journal of Applied Sciences, 10(6). 371-

391.

URL: http://dx.doi.org/10.14738/aivp.106.13277

, , , , , , , , , , , , , , }.

(1)

These symmetry operations contain 24 proper operations ( , , , , , ,

) and 24 improper operations ( , , , , , , ).

The representation of the different symmetry operations of on the cubane molecule is

shown in Figure 1:

Figure 1: Representation of the different symmetry operations of on the cubane molecule.

S6(4) S6(3) S6(2) sd (1) sd (6) sd (2) sd (4) sd (3) sd (5) S4(3)

3

4(3) S S4 (1)

3

4(1) S 3

4(2) S S4(2)

E 3C2

' 6C2 4C3

2 4C3 3C4

3 3C4

i 3sh d 6s 4 3S 3 3S4 4S6

5 4S6

Oh

d (1) s

d (6) s

h (3) s

h (2) s

h (1) s

d (2) s d (4) s

d (3) s

d (5) s

6(4) S

6(3) S

4(3) S

2 (1) C

2(2) C

2(3) C

3 (1) C

3(3) C

3(2) C

3(4) C

'

2(6) C

'

2 (1) C

'

2(4) C

'

2(2) C

'

2(5) C '

2(3) C

4(3) C

4(2) C

Oh