N o t i z e n 3 8 3
Violation of Kruszewski's Rule
I. G u t m a n
Faculty of Science, Kragujevac, Yugoslavia and I. Juranić
Faculty of Science, Belgrade, Yugoslavia Z. Naturforsch. 38 a, 383 - 384 (1983);
received October 11, 1982
Violations of Kruszewski's rule occur in the case of the annelation of (4 m ) - m e m b e r e d rings, provided m is suf- ficiently large. In m a n y cases the violation starts even with m = 2.
Recently Kruszewski [1] proposed an empirical rule for predicting the relative stability of conju- gated isomers obtained by attaching a six- or a four- m e m b e r e d ring to different positions of the parent molecule. Since the effect considered by Kruszewski is obviously closely connected with cyclic conjugation [2], it was reasonable to expect that his observation is a special case of a more general m o d u l o 4 type rule. In [3] this rule has been formulated as follows:
(a) If n = 4 m + 2, then
Prs> PtuO £ ( B i ) > £ ( B2) . (b) If n = 4 m, then
Pr e> Au<=> £ ( B i ) < E ( B2) .
(Here and later we use a symbolism which fully coincides with the notation in [3], where the reader m a y find the necessary details.) In [3] Kruszewski's rule (a) was rigorously proved. A detailed analysis showed, however, that rule (b) can not be generally valid. Qualitative arguments supported the validity of (b) for m = 1, indicating also that violations of rule (b) can occur for larger values of m. In fact, if n = 4 m, then Eq. (5) in [3] can be written as
E ( B \ ) - E(B2) = / | + /2, where
2 Af° Qn-\Rn\XA-vT-vs) .
— log ax , n o Qn-\Rn\(A-vt-vu)
, 2* Qn+ Rn (A-Vr-Vj J
/2 = — log d-Y .
n vo Qn + Rn (A-Vt-Vu) Reprint requests to Prof. Dr. Ivan G u t m a n , Faculty of Science, P.O. Box 60, 34000 Kragujevac, Yugoslavia.
T h e p a r a m e t e r ,v0 is the (unique) real and positive zero of the polynomial Rn, that is, .v0 is the root of the equation
(Pn-2) = 2 . Since
("/2)~' In — 2 — k\
n (n - 2) , „ ,
= 1 + —^ .v2 + . . . + x , a good a p p r o x i m a t i o n for .v0 is
l / ^
X n(n -vo ,/ ,K « - 2 )
It has been demonstrated [3] that the dependence of the first integral, I \ , on molecular topology is in agreement with rule (b). T h e contribution of the second integral, /2, however, is opposite to
to o6 o§'ixJ
8 8 8 8
05 <X5
8 8 8 8
28 36
a ? oo o
2 0 20 28o9
20O OO Ca
8 2U 48 7UQ a Q 32 20
0340-4811 / 83 / 0300-0383 $ 01.3 0 / 0 . - Please o r d e r a reprint r a t h e r t h a n m a k i n g y o u r own copy.
384 Notizen Kruszewski's predictions. W h e t h e r Kruszewski's rule (b) will hold or not, d e p e n d s on t h e relative m a g n i t u d e of the integrals I\ a n d /2. W h e n n is (relatively) small and t h u s .Y0 (relatively) large, t h e effect of 7| will d o m i n a t e . W i t h increasing n, t h e effect of /2 will b e c o m e m o r e a n d m o r e i m p o r t a n t since .Y0 and t h e r e f o r e also I j will tend to zero. T h u s one can expect t h a t if n is a b o v e a certain critical value, «c r i t, the inversion of Kruszewski's rule (b) will occur:
(c) If n = 4 m and n ^ «c r i t, t h e n Au<=> £ ( B i ) > E ( Bz) .
In Fig. 1 various c o n j u g a t e d systems, t o g e t h e r with the c o m p u t e d value for A7crit a r e p r e s e n t e d . T h e following o b s e r v a t i o n s are worth m e n t i o n i n g .
1. An inversion of rule (b) to r u l e (c) was f o u n d in all studied cases. 2. Violations of K r u s z e w s k i ' s rule
(b) h a v e never b e e n observed for n = 4. O n the o t h e r h a n d , in a c o n s i d e r a b l e n u m b e r of cases t h e inversion starts with 8 - m e m b e r e d rings. 3. A re- m a r k a b l e f i n d i n g is t h a t «c r i t has either t h e value 8 or is very large a n d chemically insignificant
(nc r i t ^ 20). I n t e r m e d i a t e ncrit values seem to b e
r a t h e r rare. 4. N o r e a s o n a b l e q u a n t i t a t i v e relation could b e recognized b e t w e e n ncrit and t h e d i f f e r e n c e b e t w e e n the c o r r e s p o n d i n g b o n d orders of t h e p a r - ent c o m p o u n d . 5. O n the o t h e r h a n d , small «c r i t
values usually occur if the two ( n o n e q u i v a l e n t ) b o n d s on which t h e a n n e l a t i o n effect was c o m p a r e d h a v e s i m i l a r topology and consequently h a v e nearly equal b o n d orders.
This latter o b s e r v a t i o n suggests that irrespective of c e r t a i n limitations, Kruszewski's rule can b e used as a s i m p l e q u a l i t a t i v e s c h e m e for guessing t h e relative t h e r m o d y n a m i c stability of isomeric c o n j u - gated molecules.
[1] J. Kruszewski. Pure Appl. Chem. 52, 1525 (1980).
[2] I. G u t m a n and O. E. Polansky, Theor. Chim. Acta 60,203 (1981).
[3] I. G u t m a n . Z. Naturforsch. 35 a, 820 (1980).
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