On Topology-Property Relations of Polycyclic Aromatic
Hydrocarbons
M. Z a n d e r
Rütgerswerke AG, Castrop-Rauxel
Z. Naturforsch. 40a, 636-638 (1985); received April 18, 1985
The hydrogen-depleted graphs of polycyclic aromatic hydrocarbons contain two types of
vertices (with regard to their degree) and correspondingly three types of edges. The respective
sums of these edges reflect the molecular topology of the hydrocarbons and were used for constructing a new topological index for kata-annellated aromatic hydrocarbons that correlates well
with their topological resonance energies per electron. Dependent on the degrees of their first
neighbours, the vertices of degree 2 of polycyclic hydrocarbons can be distinguished as to high,
medium and low chemical reactivity of the corresponding carbon atoms, in agreement with the
results from MO theory and experiments.
T h e electronic and o t h e r types of molecular properties of polycyclic a r o m a t i c hydrocarbons (PAH)
d e p e n d strongly on the molecular topology of the
systems [1]. T h e topology of PAH can be conveniently f o r m u l a t e d in m a t h e m a t i c a l terms by
m e a n s of so-called "topological indices" [2]. If such
indices fulfill certain presuppositions [3] they correlate well with theoretically derived and experimental data and thus enable the chemist to obtain
i n f o r m a t i o n relevant for his practical work, e.g.
i n f o r m a t i o n on stability, reactivity etc. of as yet
u n k n o w n h y d r o c a r b o n s . In this note a new topological a p p r o a c h to kata-annellated
PAH is presented.
T h e h y d r o g e n - d e p l e t e d g r a p h s ("Hückel g r a p h s "
[4]) of PAH contain two types of vertices, i.e.
vertices of degree 2 and 3 (denoted here by v(2) and
r ( 3 ) ). Accordingly they contain three types of edges:
<?0'(2), ' " ( 2 ) ) , e ( r ( 2 ) , r l 3 ) ) and e ( r ( 3 ) , r ( 3 ) ) [5]. The
corresponding sums of these edges
Z
n
Z
n
Z
n
''(2)) = £2,2,
e
(l'(2), r
( 3
) )
=
<?0-(3), ' (3)) =
£ 2 . 3 ,
£3.3
reflect the topology of the individual PAH. By way
of example, with increasing " b r a n c h i n g " [6, 7] (e.g.
tetracene, triphenylene) EI 3 decreases while £2,2
and E33 increases. T h e ratio E2J/E3<3 proves to be
Reprint requests to Professor Dr. M. Zander, Rüteerswerke AG. D-4620 Castrop-Rauxel, FRG.
particularly suitable for designing new topological
indices.
For the p u r p o s e of correlation analyses "topological resonance energies per electron" (TREPE) [8]
were used because T R E P E d e p e n d strongly on the
topology of P A H and, f u r t h e r m o r e , are linearly
connected with o t h e r i m p o r t a n t (theoretical and
experimental)
p a r a m e t e r s , e.g. first ionisation
potentials of P A H [9]. F o r isomeric kata-annellated
P A H linear correlations were f o u n d to exist between
T R E P E and the ratio
E23/E3^3:
T R E P E {ß) = Ax
j + Ao •
(1)
C - n u m b e r s (C), A0 and A\ values, s a m p l e sizes (n)
and correlation coefficients (/•) are given in Table 1.
Although the ratio E23/E33
is not solely a function of b r a n c h i n g but reflects to s o m e extent also the
size ( n u m b e r of c a r b o n atoms) of the systems, an
additional term is necessary to c o p e with PAH of
different size in a single regression equation. T h e
sum £ t o t of all edges present in the g r a p h s was taken
as a m e a s u r e for size, and m u l t i p l e regression
analysis did yield
T R E P E ( / ? ) = - 0.00379
+ 0.05£tot
+ 0.04878.
I
(2)
T h e correlation coefficient is 0.971 for a sample size
n = 34. T h e s a m p l e included systems with 14. 18, 22,
26 and 30 c a r b o n a t o m s (i.e. PAH with 3 to 7 rings)
and did not include linear annellated PAH (acenes:
anthracene, tetracene etc.).
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M. Zander • Polycyclic Aromatic Hydrocarbons
637
Table 1. Parameters related to (1).
c
A0(x 102)
At (x 102)
n
r
18
22
26
30
4.436
4.288
4.265
4.349
- 0.365
-0.341
- 0.377
- 0.437
4
9
10
10
0.986
0.977
0.980
0.980
deviation between a p p r o x i m a t e and exact T R E P E
values is in the range f r o m 0 to 5% and the averaged
deviation a m o u n t s to 1.6%. T h u s the p r o p o s e d new
topological index
£3.3
Table 2. Topological
(ß units).
PAH
resonance
energies
TREPE (Eq. 2 )
&
COCO?
per
electron
T R E P E (Ref. [ 8 ] )
0.0384
0.0378
0.0315
0.0308
0.0343
0.0349
0.0379
0.0374
0.0325
0.0325
0.0344
0.0341
0.0359
0.0361
0.0371
0.0374
0.0309
0.0303
0.0363
0.0382
>yN
c c a
n
006^
0
<
c g
^
M
c
c
c
f f ^ c o
To test f u r t h e r the quality of the relationship,
T R E P E were calculated according to (2) for 10 randomly chosen k a t a - a n n e l l a t e d PAH that were not
contained in the original s a m p l e and c o m p a r e d with
analytically derived T R E P E [8] (Table 2). T h e
+ 0.05 £ t ,
(3)
seems to work quite satisfactorily. T h e design of
related topological indices a p p l i c a b l e to both kataannellated and peri-condensed PAH should be
possible.
COO o$9
1
A n o t h e r f e a t u r e of this topological a p p r o a c h
related to chemical reactivity shall be m e n t i o n e d
briefly. T h e hydrogen-depleted g r a p h s of P A H contain three types of vertices of d e g r e e 2 (i.e. positions
that can react in substitution reactions), viz. r ( 2 )
a d j a c e n t to 2 vertices r ( 2 ) (e.g. the 2-position in
a n t h r a c e n e (I)), v 2 adjacent to 1 vertex r ( 2 ) and to
a n o t h e r vertex r ( 3 ) (e.g. the 1-position in I) and r ( 2 )
adjacent to 2 vertices r ( 3 ) (e.g. the 9-position in I).
Generally the reactivity of PAH increases in this
order, as the c o m p a r i s o n with reactivity indices
derived f r o m M O theory [10] and experimental data
shows. Even in highly unsymmetrical PAH with
m a n y different reactive positions the most reactive
position can in many cases thus be clearly identified
by inspection of the graphs. By way of example, the
hydrogen-depleted graph of b e n z o [ a ] p y r e n e (II) contains only 1 vertex r ( 2 ) a d j a c e n t to 2 vertices r ( 3 ) ,
and it is well established that this 5-position (see
f o r m u l a II) is the most reactive one.
1 thank Professor O. E. Polansky, Max-PlanckInstitut f ü r Strahlenchemie, M ü l h e i m a. d. R u h r , for
valuable discussions and Dr. J. Palm, R ü t g e r s w e r k e
A G . C a s t r o p - R a u x e l , for p e r f o r m i n g the c o m p u t e r assisted regression analyses.
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638
[1] M. Zander and O. E. Polansky, Naturwiss. 71, 623
(1984).
[2] see: A. T. Balaban, in: Steric Fit in Quantitative
Structure-Activity Relations (A. T. Balaban, A.
Chiriac, I. Motoc, and Z. Simon), p. 22, Lecture Notes
in Chemistry, No. 15, Springer-Verlag, Berlin-Heidelberg-New York 1980; D. H. Rouvray, in: Chemical
Applications of Topology and Graph Theory (R. B.
King, ed.). p. 159, Elsevier. Amsterdam 1983.
[3] K.-D. Gundermann. C. Lohberger, and M. Zander,
Croat. Chem. Acta 56(3), 357 (1983).
[4] A. Graovac and N. Trinajstic. J. Mol. Struct. 30, 416
(1976).
[5] For an introduction to graph theory see: F. Harary.
Graphentheorie. R. Oldenboura-Verlag. München
1974.
M. Zander • Polycyclic Aromatic Hydrocarbons 638
[6] A. Graovac. 1. G u t m a n , and N. Trinajstic, Topological Approach to the Chemistry of Conjugated Molecules, Lecture Notes in Chemistry, No. 4, SprineerVerlag, Berlin 1977.
[7] M. Zander, Naturwiss. 69,436 (1982).
[8] I. G u t m a n , Coll. Sei. Papers, Faculty of Science,
Kragujevac, Yugoslavia 3, 43 (1982) (this paper contains a full list of references on the concept and
applications of TREPE).
[9] I. G u t m a n . Z. Naturforsch. 37 a, 248 (1982).
[10] see: A. Streitwieser. Molecular Orbital Theory for
Organic Chemists, p. 328. John Wiley & Sons, Inc.,
New York 1961.
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