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A quantum-chemical approach to the chemical definition of aromaticity
Tetrahedron Letters No.4, ppo 519-524,
1970,
Pergamn
Press,
Printed in Great Britain.
A QUANTUM-CHEA'iICAL APPROACH TO THE CHEMICAL DECINITION OF AROtiiATICITY J.Kruseewskl a/ and T.I%.Krygowsklb' a/ Dept. of Physical Chemlstry,Univ. of Ebdk, Poland b/ Inst.of Fundamental Problems of Chemlstry,Warsaw 22,Pasteura 1 ,POlaUd
(Received in UK 15th Decefiber
1969;accepted for publication 23 December 1969)
Recently, the problem of aromaticlty In general as well as some attempts at a quantitative approach to it have been extensively discussed in numerous publlcatlons(i-9). The earllest chemical definition of the aroImtlclty was based on the susceptibility of the aromatic systems to substitution reactions such as halogenatlcn,sulphcnatlon or nitratlon. In these reactions, the oleflnlc analogues yield addition products (10,ll). As the basis for our considerations, we adopted a somewhat Idealized reaction between a pi-electron compound (i.e. one with conjugated double bonds) and some reagent XY whose properties, we assume, are analogous to that of Cl2 and Br2. Furthermore, we assume that the reaction may be reversible, and that It may run from the beginning state (I) through the sigmacomplex (II) to yield either the products of addltlon (III) or substitution (IV): x
No,4
Thus,
the
(cf.
final
Fig.
~1~.
would
be determined
by a change
of
the
reaction
1 A~_ exam+
The addltlon
: the mdecu\e
reaction
-dFS
nFA
effect
free
energy
1):
<
1s determined
0,
or
dHA-
path
C[ benzene by satisfying
T4S,-
<4HS
the
- TASS)
<
inequality: (2a)
0
i.e. d HA where
-
AHS + T(LI Ss -
S,H and T stand
OSA)
for
<
the
0
(2b)
entropy,
enthalpy
and
(12,13),
that
the
absolute
temperature,
respectively. Further
we assume,
inequality the
(2b)
same kind and
XII),
if
In terms
Is (a
the of
as
constant
(IV)
under
reaction the
in scheme energy
done the
compound runs
under
delocallzatlon the (I)
consideration
pi-electron
often */ , if
pi-electron
we may postulate
(S), and
Is
for
to
be
which
the
under
reacting
model
in sigma
the
molecules
ccnstant.
At such
differentiates
last
term
consideration with
the
the
for
chemical
electron
i . e . the molecules,
term e.g.
T(n
Ss -AS,)
for
benzene
is
the
the
are
of
same reagent
of
the
pi-electron
approximation, molecules
reactivity
energy
same for
and anthracene
every
between 1s only
‘involved
in the
compared
or benzene
(III)
compounds
it
reaction (1) and determlnes its flnal effect. Hence, It Is only _-__________________~~~~~~~~-~~~~-~~~~~~~-~~~~~~~~~~~-~~-~~~-~~_~--~-~~~~-*I
of
same conditions.
energy
differences all
compounds always
the
the
the palr
and fulvene
of etc.
321
No,4
pi-energy
term
flcence
in the
substltuents that
X is
in the
enthalpy,
formula
seleotlon
of
dlrectlon
weakly
the
interacting
a substituent
of
that
(2b),
which
of
the
with
the’pl-eleotron
kind
-
It
is
has
essential
reaction.
Thus,
system
possible
slgnl-
to
-
for
we assume
write
the
following
expression: = Egt
Eil stands
where Eg’ reaction.
for
is
Included
and the
for
addition,
In the
state
say be
of
take
the
place
the
This
the
On the
the
electrophylllc
reagent
localization deflnea.
the
of
that
approximation
(12)
- as -
the
of
the
to yield
inequality
constant
from
(4)
(3),
sigma-electron
the
formation
of
the
(12,14) en
and then
is the
posit
the
on the
energy
is
atoms
term the
t\e
greatly
adaltlon
r and (r,s) first
sigma-complex that
mlnlmum or,
most
often
In units
of
the chosen
- EE1(
has
attack
of
position
(II)
thus
the
Wheland
more precisely,
calculated i3’
depends
s are
the
In
1s most
loss
beforehand,
condition
in
two carbon
smallest
of
its
1s measured
established pi-elec-
\Ei’
Ion
after
the
positions
in the
be at
of
but
determined
determined
system
of
this
expression
the
be
that
there
words,
the
delocalization
new bonds,
where
of
C-X and C-Y are
that
the C-X bona
energy
CONST. of
energy
the
hand,
is
energy
term
1s obvious
X should of
the
compound
a subtraction
of
of
other
before
enthalpy.
It
modulus
The position localization
In tne
of
In other
centre
pi-electron
pi-electron
pi-electron
in posltlons
that
the
means
posltlon
system
we obtain:
when two new bonds
delocallzatlon.
the
term T(A SS -LISA), result
is
of
0
XY.
In the
(4)
minimum value.
Siickel
the
r and s.
of
condition
modulus
reagent
(IV)
the
being
for
sp3.
to
the
condition
hybridization
likely
of
<
the
case
and (3)
as a result
selection
on the
in
<2b),
decreased
on the
pi-electron
of
and
term
two positions, will
term
(III)
as
of
energy
+ CCNST.
the
constancy
The first
the
with
constant
energies
trons
from
- EEL
products
(3)
pi-electron
an indispensable
addition are
the
Consequently, Eii
which
+ const.
(
in the 0.
This
energy
approximately
I --9
II,
to
thus
be formed
sult
In the
EIi
Lr a stands , a result of
the
reaction.
tion
= Eil
part
the
formation the
absolute
result
in mind
of
unlformlty
with
Table
It
I presents
energy
calculated
can be
seen
from
values
were
calculated
sidered
to
be aromatic,
it lity
can Fe assumed (4)
Is
of
the
that range
term
is
the the
as r and
The
the
addltlon
has
a strictly
energy
gain of
first
which
the
forma-
term,
the
rcactlon.
and assumptions, index
posltlons
energy
1s the
of
loses
reaction.
molecular
direction
from
for
which
is justlf
which
It
indices, values
one
bond
Table
use
we postulate
we define
value 3/i’,
ori;inctes,
the
as
follows:
not
Only
by the
but
also
by a
(6)) the
of
uf.
Zahradnlic‘
and Julg’s
see EK (3
constant dotted
for
(12;) to
which
delocali-
s
be
in
Fig.
index ?K>i
i;enerally
In the :.
A(3).
J (ZK-
are
non-romntic.
term CL!:.‘?‘. iine
I.
aromatlclty
compounds
where the
in Table
specific
HL.C-method,
those
ied
as presented (t:,e
that
the
whereas
of
the
CCUST.,
constant
aromaticity
and 3Esp
the
re-
1 of
tg
the
pi-electron
probcble
the
and
aromaticlty
the
where
a molecule
in
less
the
objections
aromaticity 3
which
i4),
addition,
cheo:lstry
other
likely
(5)
the
represents
above
criterion
organic
term,
t:ie
bi-centric
to addition
inequality
determine
sbtalned
and C-Y
C-j:
a molecule
this
= mln 1 Lr,s
numerical
tradltion
zatlon
it
KK as a numerical KK
The
the
Is most
value. Is
energy
new bonds
thls
process
as:
pi-electron
of
the
- 2oc
EGql
of
energy
of
C-X bond
a
called
-
of
the
so
(r,s)
of
activation
an expression
1s defined
Since,
wlll
lowest
the
the XP reagent
one,
expression
(4)
of
molecule
which
value
two new bonds.
Keeping
the
merely
meaning:
of
pi-electron
of
the
term
phi;sical
of
of
energy
reacting
resistance
The last
opposite 1s the
(15)
for
and defines
greater
is
- Eg’
r9s
of
lnequallty
energy II
II
the
position
the
(r,s)
localization
s,
in state
from
Hence,
estimates
conThus, inequa-
No.4
323
Table
I
1)
be neene
3.53
I
.oox/:
0.333
2)
pyrldine
3.53
1
.oo
0.350
3)
annulene-1
3.36
0.95
4)
aeulene
3.31
0.91-l
5)
anthracene
3.20
O.9l-O.96x/
0.332 0.342
a
0.279
.oox/ 0.306
6)
pyrene
3.06
0.81
7) 8) 9) 10)
fulvene
2.99
0 . 62x’-O
2.67
0.44
annulene-8
(cyclooctatetraene)
0.244 0.207
hexatrlene-1,3,5
2.52
0.71
0.198
p-be nzo qu in0 ne
2.51
0.39
0.164
1.53
0.00
0.000
Kekule‘s
11)
. 72
x/
Ref.
structure
of
benzene
(3)
EXAMPLES For
benzene
only
one
value
one-centric
whereas
three
1s possible,
lizatlon
energy
are
= 4.47
-
8.00
=
- 3.53
(butadlene
= 2.83
-
8.00
=
- 5.17
/(ally1
= 4.00
-
8.00
=
-
(2
(6)
it
Ll ,3 % ,4
possible
From condltlon is
KK =I L,
2l
r=l
so
that
calculate
Ll ,2 Ll ,3 L1,4
‘. the
L,
(in
units
that
the
for
Hence
butadlene
we select
formation
of
the
the
glve
-
lowest
the
benzene) index
following
value
sigma-complex
for
benzene
of
values:L,=-1.64/3’
the
looallzatlon
II
1s easier.
-
butadiene)
energy
Now we can
s:
9
- 2.47
(ethylene
= 0.00
-
4.47
=
-
/(two
= 2.00
- 4.47
=
- 2.47
As can be
computing
looa-
+C-radIoal)-benzene/
aromatlclty
=
for
bl-centric
- benzene)
ethylenes
- 4.47
one L,
of
radical
= 2.00
only
values
locallzatlon
ofp’):
4.00
follows
Wheland‘s
=3.53.
Slmllar’calculatlons and L2 =-2.47/j
‘,
the
energy
L1,2
L,=-2.54p
of
seen
from
s value
9
EX for
the
4.47
above
Is possible butadlene:
C-radloals)
(ethylene calculations, in order that
is
to
-
- butadlene)
butadlene
within satisfy
KK = 2.4’7.
the
)
HiXO-approximation, condition
necessary
No,4
324
The aromatlolty Index ILKmakes a convenient basis for the classlfloatlon of the pi-electron compounds (16), as - it refers to the chemical properties of the moleoule and also makes possible a comparison with other numerical criteria which are adopted and accepted as a measure of aromatlolty. Acknowledgements: One of the authors (T.N.K.) 1s deeply indebted to Dr Leoh Stolaroayk and Dr Plotr Tomasik for their valuable remarks, as well as to Prof. Wlktor Kemula for his friendly supervision. We are also very grateful to Prof. Stanlshaw Turskl end the staff of the Computing Department of the Warsaw University for their help and for making available to us the GIER computer. REFERENCES (1) K.Fu~l,A.Imamura,T.Joneeawa,C.Nagata,Bull.Chem.Soc.Ja~an~~~,l591,(l960) (2) (3) (4) (5) (6) (7) (8)
R.Breslow, Chem.Eno.News, ;s$a, 90 A.Julg, Ph. Franoois, Theor. Chlm. Acta, 2, 249, (1967) L.J.S.Dewar, Tetrahedron SUPD~., 2, 75, (1966) F.Sondhelmer, P_ure and Apnl.Chem., 'z, 363, (1963) R.Zahradnlk, J.Mlohl, J.Pancir,T&rahedron, $i?,1355, (1966) W.Kemula, T.M.Krygowskl, Tetrahedron Letters, 126_8_, ---_ 3135 M.J.S.Dewarw ChemlcQ Reactivity in: Chlmioa Teorlca, VIII Corso dl --I-estavo di Chimloa, Roma (pp. 181-237), 1965 (9) G.M.Badger, AJomatlo Character and Arormtlcltv, Cambridge, Unlv.Press, (10) (11) (12) (13) (14) (15) (16)
1969 C.Nenltescu, Organic Chemistry, (in polish), Warsaw, PWN,l968 H.Duewell, J.Chem.Ed., $2, 138,(1966) A. Streltwieser Jr., Molecular Orbltal Theory, J.Wiley,l961 O.E. Polansky, P.Schuster, -In~,Quant.Chem.,;,l82,(1967) G.W.Wheland, J&.Chem.Soc., 62,900, (1942) R.D.Brown, Ouart.Revs., 6,63, (1952) T.M.Krygowskl, to be published in this Journal
1970,
Pergamn
Press,
Printed in Great Britain.
A QUANTUM-CHEA'iICAL APPROACH TO THE CHEMICAL DECINITION OF AROtiiATICITY J.Kruseewskl a/ and T.I%.Krygowsklb' a/ Dept. of Physical Chemlstry,Univ. of Ebdk, Poland b/ Inst.of Fundamental Problems of Chemlstry,Warsaw 22,Pasteura 1 ,POlaUd
(Received in UK 15th Decefiber
1969;accepted for publication 23 December 1969)
Recently, the problem of aromaticlty In general as well as some attempts at a quantitative approach to it have been extensively discussed in numerous publlcatlons(i-9). The earllest chemical definition of the aroImtlclty was based on the susceptibility of the aromatic systems to substitution reactions such as halogenatlcn,sulphcnatlon or nitratlon. In these reactions, the oleflnlc analogues yield addition products (10,ll). As the basis for our considerations, we adopted a somewhat Idealized reaction between a pi-electron compound (i.e. one with conjugated double bonds) and some reagent XY whose properties, we assume, are analogous to that of Cl2 and Br2. Furthermore, we assume that the reaction may be reversible, and that It may run from the beginning state (I) through the sigmacomplex (II) to yield either the products of addltlon (III) or substitution (IV): x
No,4
Thus,
the
(cf.
final
Fig.
~1~.
would
be determined
by a change
of
the
reaction
1 A~_ exam+
The addltlon
: the mdecu\e
reaction
-dFS
nFA
effect
free
energy
1):
<
1s determined
0,
or
dHA-
path
C[ benzene by satisfying
T4S,-
<4HS
the
- TASS)
<
inequality: (2a)
0
i.e. d HA where
-
AHS + T(LI Ss -
S,H and T stand
OSA)
for
<
the
0
(2b)
entropy,
enthalpy
and
(12,13),
that
the
absolute
temperature,
respectively. Further
we assume,
inequality the
(2b)
same kind and
XII),
if
In terms
Is (a
the of
as
constant
(IV)
under
reaction the
in scheme energy
done the
compound runs
under
delocallzatlon the (I)
consideration
pi-electron
often */ , if
pi-electron
we may postulate
(S), and
Is
for
to
be
which
the
under
reacting
model
in sigma
the
molecules
ccnstant.
At such
differentiates
last
term
consideration with
the
the
for
chemical
electron
i . e . the molecules,
term e.g.
T(n
Ss -AS,)
for
benzene
is
the
the
are
of
same reagent
of
the
pi-electron
approximation, molecules
reactivity
energy
same for
and anthracene
every
between 1s only
‘involved
in the
compared
or benzene
(III)
compounds
it
reaction (1) and determlnes its flnal effect. Hence, It Is only _-__________________~~~~~~~~-~~~~-~~~~~~~-~~~~~~~~~~~-~~-~~~-~~_~--~-~~~~-*I
of
same conditions.
energy
differences all
compounds always
the
the
the palr
and fulvene
of etc.
321
No,4
pi-energy
term
flcence
in the
substltuents that
X is
in the
enthalpy,
formula
seleotlon
of
dlrectlon
weakly
the
interacting
a substituent
of
that
(2b),
which
of
the
with
the’pl-eleotron
kind
-
It
is
has
essential
reaction.
Thus,
system
possible
slgnl-
to
-
for
we assume
write
the
following
expression: = Egt
Eil stands
where Eg’ reaction.
for
is
Included
and the
for
addition,
In the
state
say be
of
take
the
place
the
This
the
On the
the
electrophylllc
reagent
localization deflnea.
the
of
that
approximation
(12)
- as -
the
of
the
to yield
inequality
constant
from
(4)
(3),
sigma-electron
the
formation
of
the
(12,14) en
and then
is the
posit
the
on the
energy
is
atoms
term the
t\e
greatly
adaltlon
r and (r,s) first
sigma-complex that
mlnlmum or,
most
often
In units
of
the chosen
- EE1(
has
attack
of
position
(II)
thus
the
Wheland
more precisely,
calculated i3’
depends
s are
the
In
1s most
loss
beforehand,
condition
in
two carbon
smallest
of
its
1s measured
established pi-elec-
\Ei’
Ion
after
the
positions
in the
be at
of
but
determined
determined
system
of
this
expression
the
be
that
there
words,
the
delocalization
new bonds,
where
of
C-X and C-Y are
that
the C-X bona
energy
CONST. of
energy
the
hand,
is
energy
term
1s obvious
X should of
the
compound
a subtraction
of
of
other
before
enthalpy.
It
modulus
The position localization
In tne
of
In other
centre
pi-electron
pi-electron
pi-electron
in posltlons
that
the
means
posltlon
system
we obtain:
when two new bonds
delocallzatlon.
the
term T(A SS -LISA), result
is
of
0
XY.
In the
(4)
minimum value.
Siickel
the
r and s.
of
condition
modulus
reagent
(IV)
the
being
for
sp3.
to
the
condition
hybridization
likely
of
<
the
case
and (3)
as a result
selection
on the
in
<2b),
decreased
on the
pi-electron
of
and
term
two positions, will
term
(III)
as
of
energy
+ CCNST.
the
constancy
The first
the
with
constant
energies
trons
from
- EEL
products
(3)
pi-electron
an indispensable
addition are
the
Consequently, Eii
which
+ const.
(
in the 0.
This
energy
approximately
I --9
II,
to
thus
be formed
sult
In the
EIi
Lr a stands , a result of
the
reaction.
tion
= Eil
part
the
formation the
absolute
result
in mind
of
unlformlty
with
Table
It
I presents
energy
calculated
can be
seen
from
values
were
calculated
sidered
to
be aromatic,
it lity
can Fe assumed (4)
Is
of
the
that range
term
is
the the
as r and
The
the
addltlon
has
a strictly
energy
gain of
first
which
the
forma-
term,
the
rcactlon.
and assumptions, index
posltlons
energy
1s the
of
loses
reaction.
molecular
direction
from
for
which
is justlf
which
It
indices, values
one
bond
Table
use
we postulate
we define
value 3/i’,
ori;inctes,
the
as
follows:
not
Only
by the
but
also
by a
(6)) the
of
uf.
Zahradnlic‘
and Julg’s
see EK (3
constant dotted
for
(12;) to
which
delocali-
s
be
in
Fig.
index ?K>i
i;enerally
In the :.
A(3).
J (ZK-
are
non-romntic.
term CL!:.‘?‘. iine
I.
aromatlclty
compounds
where the
in Table
specific
HL.C-method,
those
ied
as presented (t:,e
that
the
whereas
of
the
CCUST.,
constant
aromaticity
and 3Esp
the
re-
1 of
tg
the
pi-electron
probcble
the
and
aromaticlty
the
where
a molecule
in
less
the
objections
aromaticity 3
which
i4),
addition,
cheo:lstry
other
likely
(5)
the
represents
above
criterion
organic
term,
t:ie
bi-centric
to addition
inequality
determine
sbtalned
and C-Y
C-j:
a molecule
this
= mln 1 Lr,s
numerical
tradltion
zatlon
it
KK as a numerical KK
The
the
Is most
value. Is
energy
new bonds
thls
process
as:
pi-electron
of
the
- 2oc
EGql
of
energy
of
C-X bond
a
called
-
of
the
so
(r,s)
of
activation
an expression
1s defined
Since,
wlll
lowest
the
the XP reagent
one,
expression
(4)
of
molecule
which
value
two new bonds.
Keeping
the
merely
meaning:
of
pi-electron
of
the
term
phi;sical
of
of
energy
reacting
resistance
The last
opposite 1s the
(15)
for
and defines
greater
is
- Eg’
r9s
of
lnequallty
energy II
II
the
position
the
(r,s)
localization
s,
in state
from
Hence,
estimates
conThus, inequa-
No.4
323
Table
I
1)
be neene
3.53
I
.oox/:
0.333
2)
pyrldine
3.53
1
.oo
0.350
3)
annulene-1
3.36
0.95
4)
aeulene
3.31
0.91-l
5)
anthracene
3.20
O.9l-O.96x/
0.332 0.342
a
0.279
.oox/ 0.306
6)
pyrene
3.06
0.81
7) 8) 9) 10)
fulvene
2.99
0 . 62x’-O
2.67
0.44
annulene-8
(cyclooctatetraene)
0.244 0.207
hexatrlene-1,3,5
2.52
0.71
0.198
p-be nzo qu in0 ne
2.51
0.39
0.164
1.53
0.00
0.000
Kekule‘s
11)
. 72
x/
Ref.
structure
of
benzene
(3)
EXAMPLES For
benzene
only
one
value
one-centric
whereas
three
1s possible,
lizatlon
energy
are
= 4.47
-
8.00
=
- 3.53
(butadlene
= 2.83
-
8.00
=
- 5.17
/(ally1
= 4.00
-
8.00
=
-
(2
(6)
it
Ll ,3 % ,4
possible
From condltlon is
KK =I L,
2l
r=l
so
that
calculate
Ll ,2 Ll ,3 L1,4
‘. the
L,
(in
units
that
the
for
Hence
butadlene
we select
formation
of
the
the
glve
-
lowest
the
benzene) index
following
value
sigma-complex
for
benzene
of
values:L,=-1.64/3’
the
looallzatlon
II
1s easier.
-
butadiene)
energy
Now we can
s:
9
- 2.47
(ethylene
= 0.00
-
4.47
=
-
/(two
= 2.00
- 4.47
=
- 2.47
As can be
computing
looa-
+C-radIoal)-benzene/
aromatlclty
=
for
bl-centric
- benzene)
ethylenes
- 4.47
one L,
of
radical
= 2.00
only
values
locallzatlon
ofp’):
4.00
follows
Wheland‘s
=3.53.
Slmllar’calculatlons and L2 =-2.47/j
‘,
the
energy
L1,2
L,=-2.54p
of
seen
from
s value
9
EX for
the
4.47
above
Is possible butadlene:
C-radloals)
(ethylene calculations, in order that
is
to
-
- butadlene)
butadlene
within satisfy
KK = 2.4’7.
the
)
HiXO-approximation, condition
necessary
No,4
324
The aromatlolty Index ILKmakes a convenient basis for the classlfloatlon of the pi-electron compounds (16), as - it refers to the chemical properties of the moleoule and also makes possible a comparison with other numerical criteria which are adopted and accepted as a measure of aromatlolty. Acknowledgements: One of the authors (T.N.K.) 1s deeply indebted to Dr Leoh Stolaroayk and Dr Plotr Tomasik for their valuable remarks, as well as to Prof. Wlktor Kemula for his friendly supervision. We are also very grateful to Prof. Stanlshaw Turskl end the staff of the Computing Department of the Warsaw University for their help and for making available to us the GIER computer. REFERENCES (1) K.Fu~l,A.Imamura,T.Joneeawa,C.Nagata,Bull.Chem.Soc.Ja~an~~~,l591,(l960) (2) (3) (4) (5) (6) (7) (8)
R.Breslow, Chem.Eno.News, ;s$a, 90 A.Julg, Ph. Franoois, Theor. Chlm. Acta, 2, 249, (1967) L.J.S.Dewar, Tetrahedron SUPD~., 2, 75, (1966) F.Sondhelmer, P_ure and Apnl.Chem., 'z, 363, (1963) R.Zahradnlk, J.Mlohl, J.Pancir,T&rahedron, $i?,1355, (1966) W.Kemula, T.M.Krygowskl, Tetrahedron Letters, 126_8_, ---_ 3135 M.J.S.Dewarw ChemlcQ Reactivity in: Chlmioa Teorlca, VIII Corso dl --I-estavo di Chimloa, Roma (pp. 181-237), 1965 (9) G.M.Badger, AJomatlo Character and Arormtlcltv, Cambridge, Unlv.Press, (10) (11) (12) (13) (14) (15) (16)
1969 C.Nenltescu, Organic Chemistry, (in polish), Warsaw, PWN,l968 H.Duewell, J.Chem.Ed., $2, 138,(1966) A. Streltwieser Jr., Molecular Orbltal Theory, J.Wiley,l961 O.E. Polansky, P.Schuster, -In~,Quant.Chem.,;,l82,(1967) G.W.Wheland, J&.Chem.Soc., 62,900, (1942) R.D.Brown, Ouart.Revs., 6,63, (1952) T.M.Krygowskl, to be published in this Journal