CROATICA
CHEMICA
ACTA
42 (1970)
433
541.57:539.19
Original Scientific Paper
CCA-616
Some Remarks on the Use of Experimental Bond
Lengths in the Maximum Overlap Method
Z. B. M aksić and M. E c k e rt-M a k sić
Institute »Ruđer Bošković«, Zagreb, Croatia, Yugoslavia
B e c e iv e d A p r i l 2, 1970
The hybridization in norbornane, ethane, acetylene, cyclopropane, cyclobutane and cyclopentane has been reexamined by
the maximum overlap calculations based on experimental bond
lengths. The irregularity in the variations of the C—C bond lengths
in small ring hydrocarbons is discussed and a qualitative agreement
with the experimental bond lengths has been found. The results
also indicate that the use of experimental molecular geometry is
advantageous for the maximum overlap method.
The m axim um overlap m ethod w as recently applied to a large num ber
of cyclic and polycyclic hydrocarbons1-4. It has been found th a t the hybrids so
obtained are v ery useful in discussing the bonding in these m olecules and
several em pirical correlations betw een the hybridization param eters and
various physical and Chemical bond properties w ere established. However,
m ost of the calculations so fa r reported w ere based on stan d ard bond lengths:
d (C —C) = 1.535 A and d (C—H) = 1.07 A for the CC and CH bonds respectively. Some m ost recent results on correlations of CC and CH bond
lengths w ith corresponding bond overlaps indicate th a t b e tte r q uantitative
agreem ent w ith experim ental data is possible if experim ental bond lengths
are involved in th e m axim um overlap procedure5. Therefore, it seems w o rth w hile to reevaluate th e hybridization in some interesting molecules by adopting experim ental m olecular geometries.
In this p aper w e consider in p articu lar the hybridization obtained by
the m axim um overlap m ethod for norbornane, ethane, acetylene, cyclopropane, cyclobutane and cyclopentane for w hich fairly accurate bond lengths
are available. The details of the m axim um overlap procedure have already
been rep o rted 1’2. B riefly, coefficients of th e hybrids ipjj = a^ (2 s) + (1 —
— a 2 )l/2 (2 p) are v aried in order to obtain th e m axim um of th e sum of
the suitably w eighteđ bond overlap integrals
(1)
w here th e w eighting factors kCH and k cc tak e into accoount the difference
in energies of th e CC and CH bonds. The w eighting factors depend on the
n a tu re of th e atomic functions em ployed in the calculations. For the radial
p arts of the orbitals we adopted Clem enti double zeta atomic functions6.
Then the w eighting factors are: k cn = 135.9 kcal m o le'1 and k cc = 121.2 kcal
m o le'1. The basic overlap integrals w ere taken from th e available tables7.
The hybrids of th e sam e carbon atom are m utually orthogonal since the
434
Z. B. M A K S IC A N D M. E C K E R T -M A K S IĆ
interaction betw een the adjacent bonds is neglected in our scheme. This
leads to th e following relationship betw een the coefficients of m ixing a^
and aik and th e in terh y b rid angle {)jk:
a ij
~ al
)v* d -
a ?k > * c o s *ik =
(
1 lf J “ k
I
0 if j + k
(2)
The in terb o n d angles in cyclic system s are frequently different from the
in terh y b rid angles, i. e. the hybrids are inclined to the internuclear line by
the bending angles 6^. The actual search for the optim um param eters is a
num erical one assum ing the initial set of param eters and varying them until
the m axim um of expression (1) is achieved.
RESU LTS
The results of the application of the m axim um overlap m ethod on
norbornane, ethane, acetylene, cyclopropane, cyclobutane and cyclopentane
are sum m arized in th e Table. The hybrids are presented in the shorthand
sp” notation w here n is a noninteger defined by the relation n = (1 — a2)/a2.
E arlier obtained results are given in parentheses for comparison.
D IS C U S S IO N
Norbornane
The geom etry of norbornane and stru ctu rally related compounds was
thoroughly investigated by d ifferent experim ental m ethods8. However, the
bond lengths in norbornane are still the subject of controversy since there
is a long CC bond [d (C—C) = 1.57 A] which is not uniquely assigned in this
molecule. Thus, Morino et al.9 ascribed the long CC bond to the bridge bond
w hile D allinga and Tonem an8 attrib u ted it to the wing p a rt of the molecule.
E arlier m axim um overlap calculations based on the assum ption th a t ali CC
bonds are equal (1.54 A) have shown th at the C,—CT bridge bond has the
sm allest overlap*. Since th e sm aller overlap m eans the larger bond length5
we accepted in this paper the geom etry of norbornane skeleton as proposed
by Morino et al.9. The resulting hybridization param eters show some interesting
changes as com pared w ith earlier results. The p characters of the hybrid
orbitals form ing the CC bonds of the wings are decreased being still larger
th an in th e sp* hybridization State. This is in accordance w ith the unusual
reactivity evidence of the norbornylbenzenes10. The s character of the ty12 =
= tyio hydrids is tran sferre d to the neighbouring C, — H bond which is
described by th e ijj14 = sp2-48 h y b rid being thus interm ediate betw een the sp*
and sp2 States. This hybrid exhibited the largest change illustrating at the
sam e tim e th a t th e noninteger hybriđs sometimes deviate considerably from
th e ideal hybridization States. The bonds of the norbornane bridge increased
th e p character th u s contributing to the w eakening of these bonds. The
hybrids ij>17 and i[i71 and the corresponding bending angles are sim ilar to
those found in the cyclobutane ring. This is not suprising since (a) the C1C7C4
angle is v ery close to the rig h t angle and (b) the hybrids form ing the bond
tend to have the same hybridization param eter. If we neglect, as a first
approxim ation, the relatively sm all deviations of the hybrids participating
in th e CC bonds, we can apply the form ulae which correlate the bond lengths
w ith th e bond overlaps
H Y B R ID IZ A T IO N A N D BO N D L E N G T H S
435
d (C — C) = — 1.166 Scc + 2.298
(3)
d (C — H) = — 0.869 SCH + 1.726
(4)
for the CC and CH bonds respectively5. The results obtained are cited in
the last column of the Table. A very close agreem ent betw een the calculated
and the experim ental bond lengths provided by electron diffraction m easurem ents of Morino et aZ.9 supports th eir claim th a t the C,—CT bond is
particularly long.
E than e and A c e t y le n e
The hybridization in ethane rem ains u naltered since the new CC and CH
bond lengths are sim ilarly decreased resulting in alm ost p arallel change
of the basic overlap integrals. This confirms at the sam e tim e an earlier
conclusion th at the hybridization in the m ethyl group is practically constant
and unaffected by its environm ent11.
Acetylene is quoted in m ost textbooks as a typical exam ple of the sp
hybridization. "cv /ev er, the CC and CH bonds are of different kind and
consequently th ere is no reason for the sp hybridization. We found th a t the
CC and CH bonds are described by i|>cc = sp9-80 and i|)HC = sp1-25 hybrids.
There is only a slight difference betw een the obtained hybrids and those
calculated e a rlier12. If the M uller-P ritchard13 relationship J Cu—h = 5 (s ° / o)
is applied to acetylene assum ing the ideal sp hybridization, the calculated
cm—h value of 250 cps is in surprisingly good agreem ent w ith the e x p e ri-J
m ental value of 248.7 cps14. The sp hybridization is only a crude approxim ation
and if the m ore reliable hybridization sp1-25 obtained by the m axim um overlap
m ethod is used th e M uller-P ritchard form ula fails to give a satisfactory
result: J Ci3_ H calculated is only 222.6 cps. However, the M uller-P ritchard
correlation can be modified and im proved by th e inclusion of th e bond
overlap integral arising from the norm alization constant of the localized
orbital describing the C—H fragm ent16. Then the calculated J Ci3_ H spin-spin
coupling constant is 247.8 cps w hat is in excellent agreem ent w ith the
experim ental value.
C y c lo a lk a n e s
The m ain featu re of the present calculation in the series cyclopropane,
cyclobutane and cyclopentane is the increase in the s character of the
carbon-hyđrogen bonds; it is most pronounced in cyclopropane. The CH bond
lengths calculated by m eans of eqn. (4) are in excellent agreem ent w ith the
experim ental ones. The carbon-carbon bond lengths in cyclopropane and
cyclobutane deserve some m ore comments. It is evident from the Table th a t
the CC bond overlaps in strained rings are sm aller than, for instance, in
ethane. This is the resu lt of the hybriđ deviations which necessarily appear
in sm all rings. It follows from eqns. (3) and (4) th a t the sm aller overlap
m eans the longer CC bond. Therefore we would expect th a t the CC bond
in cyclopropane is considerably longer th an in ethane. This, how ever, is not
the case since the cyclopropane CC bond of 1.510 A is unusually short.
Coulson and M offitt16 tried to explain this shortening by the ji-character
of th e CC bond, b u t they did not m ake any calculations. In order to settle
this point let us consider two neighbouring hybrides inclined by the angle 8
436
TABLE
Maxim um Overlap Hybrids, Corresponding Deviation Angles, Bond Overlaps
and Experimental and Calculated Bond Lengths
H y b r id iz a tio n
M o le c u le
O v e r la p
B e n d in g
812 = 2.8° (5°)
= SP3-15 (sp 3-26)
1|>28 = lp21
82x = 2.5° (1.5°)
823 = 821
3(7X7 = SP3'36 (Sp3-33)
817 = 6.5° (8°)
1p71 = SP3'42 (sp 3-36)
3pXl£ = SP2'48 (SP2-29)
3(I2H = SP2'89 (SP2'77)
871 = 7.5° (4.5°)
S 12 = 0.650 (0.643)
,
7
k
A<
/ /
6
*
b
0
d
*
7
S23 = 0.650 (0.645)
S i7 = 0.631 (0.639)
Sin = 0.725 (0.746)
S2h = 0.718 (0.737)
1.539 ± 0.0158
(1.54)
1.539 ± 0.025
(1.54)
1.568 ± 0.016
(1.54)
1.114 ± 0.012 av
(1.07)
Bond
L e n g th s
(calcd .)
1.540
1.540
1.562
1.096
1.102
1|’7II = SP2'65 (Sp2-69)
S 7h = 0.722 (0.738)
E th a n e
tpCH = Sp2'94 (Sp2-94)
ijjcc = sp 3-20 (sp3-20)
SCH = 0.722 (0.720)
Sce = 0.650 (0.648)
1.096“ (1.102)
1.534 (1.543)
1.099
1.540
A c e ty le n e
ipCH = sp 1-25 (sp 1-29)
1(CC = sp 0-80 (sp 0-77)
S ch = 0.771 (0.769)
Sce = 0.858 (0.861)
1.059 (1.063)
1.205” (1.201)
1.056
C y c lo p r o p a n e
ijicn = sp 2-50 (sp 2-40)
Tpcc = sp 3-67 (sp 3-86)
8 = 22.9° (22.5°)
S ch = 0.733 (0.744)
Sce = 0.613 (0.599)
1.089° (1.07)
1.510 (1.535)
1.089
1.491*
Cyclobutane
3(ICH = sp 2-65 (sp 2-62)
ipcc = sp 3-42 (sp 3-47)
8 = 8.5° (8.4°)
S ch = 0.729 (0.739)
Sce = 0.635 (0.636)
1.092d (1.07)
1.548 (1.535)
1.092
1.528*
C y c lo p e n ta n e
IpCH = sp 2-82 (sp 2-75)
IpOC = sp 3-20 (sp3-28)
8 = 0.1° (— 0.1°)
S ch = 0.724 (0.736)
Sce = 0.649 (0.646)
1.095d (1.07)
1.538 (1.535)
1.097
1.534*
K . K u c h its u , J . C h e m . P h y s . 49 (1968) 4456.
C .C . C o s ta in a n d B . P . S to lc h e f f , J . C h e m . P h y s . 30 (1959) 777.
O . B a s tia n s e n , F . N . F r i t s c h , a n d K . H e đ b e r g , A c ta C r y s t. 17 (1964) 538.
A . A lm e n in g e n , O . B a s tia n s e n , a n d P . N . S k a n c k e , A c ta C h e m . S c a n d . 15 (1961) 711.
C a l c u la te d b y m e a n s o f e q n . (6).
1.099
Z. B. MAKSIC AND M. ECKERT-MAKSIĆ
lp 12 = sp 3-14 (sp 3-29)
B o n d L e n g th s
(exp.)
H Y B R 1D IZ A T 10N A N D BO N D L E N G T H S
to the in tern u clear line
electronic density curve
assum e th a t a parabola,
its tangents, describes
437
(Fig. 1). Since they form a b ent bond the m axim um
is displaced from the in tern u clear vector. We shall
w hich approaches points A and B the hybrids being
approxim ately this line. A calculation shows th at
d ( c _ c I bent
F ig . 1. S c h e m a tic P r e s e n t a t i o n o f t h e B e n t B o n đ .
the distance betw een th e vertex of this parabola in cyclopropane and the
intern u clear line is 0.319 A; it is in full agreem ent w ith th e displacem ent
of 0.32 A of th e m axim um electronic density point found by X -ray m easurem ents in cis-l,2,3-tricyanocyclopropane17. It can be easily found th at
d(C — C) = K ( 5 ) d ( C — C)bent
(5)
w here d (C—C)bent is th e length of the parabola betw een th e points A and B
w hile K (8) = 2 tg 8/[tg 8 (1 + tg2 8)1/2 + ln (tg 8 + (1 + tg2 8)1/2)]. In order to
get some insight into th e shortening of the CC bonds in strain ed rings we
have to replace d (C—C)bent by a reference straig h t CC bond length. We
adopted th e value of 1.534 A, appearing in ethane, as the stan d ard CC bond
length. Then by th e following eqn.
d (C — C) = 1.534 K (8) A
(6)
q uantitativ e estim ates of the CC bond lengths in strained hydrocarbons can
be obtained. The results of the applications of eqn. (6) on the series cyclopropane, cyclobutane and cyclopentane are presented in th e Table. The
calculated bond lengths are in good accordance w ith the experim ental ones
but th eir m agnitudes are consistently low. However, it has to be recalleđ
th at our reference bond length corresponds to th e sp3-2—sp3-2 hybriđization
and th a t th e hybrids describing the CC bonds in cyclopropane and cyclobutan e are sp3-67 and sp3-42 respectively. It is likely th a t the increased
p-character in th e la tte r m olecules w ould lengthen the bonds, thus im proving
the agreem ent w ith th e experim ental bond lengths.
Finally, we can say th a t the use of experim ental bond lengths in m axim um
overlap calculations does not alter the qualitative conclusions obtained w ith
standard bond lengths. However, qu an titativ e agreem ent w ith th e experim ental
data is generally b e tte r w hen experim ental bond lengths are used in the
calculations. Thus we can recom m end th eir use in fu rth e r applications of
the m axim um overlap method.
Acknowledgment. We thank Professor M. Randić for his interest in this work
and useful comments.
438
Z. B. M A K SIĆ A N D M. E C K E R T -M A K S IC
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IZVOD
Primjedbe na primjenu eksperimentalnih dužina veza u metodi maksimalnog
prekrivanja
Z. B. Maksić i M. Eckert-Maksić
Ponovo je primjenom m etode maksimalnog prekrivanja, te uzimanjem u obzir
eksperim entalnih dužina veza ispitana hibridizacija u norbornanu, etanu, acetilenu,
ciklopropanu, ciklobutanu i ciklopentanu. Rezultati idu u prilog daljnje upotrebe
eksperim entalne molekularne geom etrije u računima maksimalnog prekrivanja.
Diskutirane su dužine C—C veza u malim prstenastim ugljikovodicima, i postignuto
je dobro kvalitativno slaganje s eksperimentalnim podacima.
IN S T IT U T
» R U Đ E R B O S K O V IC «
ZA GREB
P rim lje n o
2.
tra v n ja
1970.