Applications of the Perturbational Molecular Orbital Method
Fillmore Freeman University of California Irvine. California 92717
The perturbational molecular orbital (PMO) approach (1-1 1 ) involves simnle calculations which give .~~~.~~ .. ..ecod oualitative . predictions of the n.acti\.ities. and staldities ol'oryani~molecules and unif\f much chemical ta.hwior. This rnt,thod cnlculates the diffeiences between systems rather than analyzing each system independently and then evaluating the differences. The approximate PMO method, which is developed in terms of the Nonbonded Molecular Orbital Approximation and the pairing theorem (alternant hydrocarbons, AH'S) (12), is useful for the treatment of aromaticity of conjugated hydrocarbons, effects of heteroatoms on conjugated systems, electrophilic and nucleophilic aromatic substitution reactions, elimination-addition reactions, nucleophilic aliphatic substitution reactions, pericyclic reactions, s complex stabilities, and other reactions. It will be shown below that the simple properties of AHaMO's, with the PMO formulas, make qualitative calculations particularly facile. Before applying the PMO method to various reaction systems, it is of interest to define and discuss alternant (AH) and nonalternant hydrocarbons and the Nonbonded Molecular Orbital Approximation. An alternant hydrocarbon (AH) is defined as a planar, conjugated hydrocarbon which contains no odd-membered rings and in which it is passihle to divide the carbon atoms into starred and unstarred sets so that no two atoms of the same set are joined by a bond. Examples of even AH's are ~
-
1
~
'
Energy
Antibonding N----a Nonbonding (NBMO)
u u u u
Orbital energies in (a) in an even AH: (b) in an odd AH carbocation:(c)in an 0%. neutral AH radical: and (dl in an AH carbanion. = a. According to the pairing theorem, in a neutral, odd AH, the qj is unity at each position and t h e p i j between two starred atoms or between two unstarred atoms is zero. Moreover, Longuet-Higgins (5) has shown that the coefficients (c,) of the NBMO are easilv calculated without solvine secular determinants (7, 12). The coefficients (cn)of NBMO's at each unstarred atom in odd AH'S are zero (zero sum rule). For the benzyl system (I), where cl = cs = cs = 0, the NBMO is
*m*
*and examples of nonalternant hydrocarbons are
An odd AH has more starred than unstarred atoms and bas a nonbonding molecular orbital (NBMO). Odd AH's are generally reaction intermediates such as anions, carbocations, and radicals. Examples of odd AH's are
*J ;-*
*B\'
* .
The pairing theorem (12)shows that the 2n a MO's of even AH's are symmetrically distrihuted about the energy a (Coulomb integral) of a carbon 2p atomic orbital (AO) (fig.). These MO's appear in pairs with energies E = a + k B (honding MO) and E = a - kB (antibonding MO), where 0 is the resonance integral. The bonding MO differs from the antibonding MO by a change in the sign of the coefficient ( c ) a t each starred or unstarred carbon atom. Further, for neutral AH's, the a electron density (q,) a t each atom is unity and the bond order ( 0 ;;) between atoms of the same set (starred or un&&e4 ';s'zero (12). For odd AH's the numher of MO's is odd since the number of contributing AO's is odd. Application of the pairing theorem to an odd AH reveals that one unpaired MO is left over (fig.). This orbital must be paired with itself and have energy a. The electrons in this odd MO are neither bonding nor antibonding. Thus, the odd MO which is confined to starred atoms, is described as a nonhonding molecular orbital (NBMO) with E 26 1 Journal of Chemical Education
= $0 = C Z +~~ 4 J . 4+ ~ 6 # 6+ ~ 7 1 1 7 $NAMO (1) Another characteristic of the NBMO is that the sum of the coefficients adjacent to an unstarred atom is zero. Thus, for (11) ez + er = 0 (about c d q + es = 0 (about cs) cz + cg + c7 = 0 (about e l ) The normalization condition (eqn. (2)) is characteristic of all MO's.
Zei2 = 1
(2)
Application of the normalization condition to (11) gives
Therefore, for the henzyl system J.o = -0.38J.2
+ 0.38J.a - 0.38J.s + 0.76$q
(4)
The henzyl radical is electrically neutral with the unpaired electron in the NBMO, and the charge densities (zi = N - qi, where N is the number of electrons (=1 for carbon) donated by the atom in question) are zero at all positions. The charge densities (2;) in the carbocation (IV) or carbanion (V) are simply the coefficients of the NBMO squared.
*@,,
-4%
,-, -.,.,
-8
.ma,
-wa
(N)
(V)
The NBMO coefficients for the propenyl (allyl, VI), pentadienyl (VII), and a-naphthylmethyl (VIII) systems are
c = h
c - 1G
c-+
(VII
Thus, as expected, benzene is more stable than 1,3,5-hexatriene because the new bonding MO's of benzene are more stable by one 6 unit. The above procedure is easily applied to cyclic systems by using an odd AH radical and the methyl radical ( 0 ) . For example, the union of the pentadienate radical (VII) and methyl radical could afford 1,3,5-hexatriene or benzene.
IVII)
WnI)
The simplest odd AH is the hypothetical methyl unit (CHz) which has an NBMO coefficient (co) of unity. The methyl unit is often symbolized by a heavy dot ( 0 ) . The overall difference between two conjugated systems can be treated as a sum of three kinds of perturbations: intermolecular, intramolecular, and monocentric. Monocentric perturbations (eqn. (5)) involve alteration or replacement of a given atom
an intermolecular perturbation occurs when two smaller conjugated systems unite to form a larger system (eqn. (6))
As with the propenyl system (VI), the a energy for the double union to give benzene is larger than that for the single union to give the polyene. Similar calculations for the union of propenyl (VI) and methyl to give 1,s-butadiene or cyclobutadiene show that the latter should he less stable than the former. Cyclobutadiene is antiaromatic (destabilized by its cyclic structure) since 6E, = 0.
It is an unstable and highly reactive compound which is difficult to isolate owing to its tendency to dimerize. In the above PMO analysis, a cyclic conjugated system will be less stable than its straight chain analog if the terminal coefficients (c) on the odd AH have opposite signs (cf. eqns. (12) and (13)).Thus, if the odd AH contains 4n + 1 atoms, the monocyclic system will contain 4n 2 atoms and (4n 2)a electrons (n = an integer) and will be aromatic (Huckel's rule). In contrast, if the odd AH contains 4n - 1 atoms, the resulting polyene will contain 4n a electrons and will be antiaromatic (6E, = 0, vide infra). Alternant and nonalternant bicyclic polyenes may also be treated by the union of a straight chain odd AH and methyl (eqn. (14)). Thus, cyclooctatetraene (1x1 and pentalene (X) are classified as antiaromatic since 6E, = 0.
+
and an intramolecular perturbation leads to an alteration of the connectivity of a conjugated system (eqn. (I)).
A first-order perturbation, which is additive, is obtained by assuming the electron distribution in a system remains unchanged hy the perturbation (13).That is, the a energy of the union of two odd AH'S a t two or more points is the sum of the individual union. The energy gained from filling the new honding MO which is generated by the intermolecular union of degenerate orbitals in atoms i and j is given by bE, = 2ZeoieojP
(8) where coi and coj are the NBMO coefficients on atoms i and j . The larger the value of 6E,, the more stable is the new honding MO which results from the intermolecular union. The hypothetical union of two propenyl radicals (VI) can lead to benzene (eqn. (6)) or to 1,3,5-hexatriene (eqn. (9))
t
U
-
---.
.A
ax 1 6E, = 0 PMO also predicts that benzenoid hydrocarbons are especially stable (eqn. (15)).
(9)
+v.
The energy change for the formation of 1,3,5-hexatriene is
and the energy change for the formation of benzene is
+
SE, = 8 c P The PMO method is also applicable to individual rings in benzenoid hydrocarbons. Thus, it is seen that the central ring in phenanthrene is more aromatic than the central ring in anthracene by 0.366 units (eqns. (16) and (17)).
Volume 55, Number 1, January 1978 1 27
-iC
p-; -& \
2"
c=-
1
6E,
(17)
/
=
1.80 0
m
The concepts from intermolecular perturbations are easily applied to the electrophilic aromatic substitution reactions of even AH'S which involve arenonium ions (a complexes) and localization of one n bond in the rate determining step.
This change in localization energy a t atom i is given by dEl‘,, = -2@
1:eo,
(19)
The intermolecular union of an odd AH with an even AH permits the calculation of the delocalization energy for the nnimolecular reaction of arylmethyl chlorides. 6+
b-
. . Cl]t
-
(B
Ar-CH2
+ CIe
(26)
The following expression is used to calculate delocalization energy dEdeluc = 28(1 -coil (27) where co, is the NBMO coefficient of the positively charged carbon atom (3).The smaller eoi, the faster the reaction. Thus, 1-naphthylmethyl chloride is predicted to react faster than propenyl or henzyl chloride.
-x
where the sum is over all atams ( r ) directly linked to atom i. Since Dewar's reactivity number (N,) is defined as
eqn. (19) may he expressed as dEl,,, = -@N, (21) Although N, values are generally lower than Hiickel molecular orbital (HMO) localization energies, there is an excellent correlation between the two (15). Construction of a henzenoid hydrocarbon from an appropriate odd AH and methyl results in aromatization.
810~.
+[Ar-CH2..
Ar-CH-CI
I.
c , = 0.76
c , = 0.67
c.
=
0.71
The n energy of the union of any two even AH'S (eqn. (28)), which is a second-order perturbation, is small and has approximately the same value (3).
The a energy change for the intramolecular union of alternant hydrocarbons is given by (291 where pi;is the a bond order between atoms i and j . The following a bond orders have been demonstrated. 6E,
= 2pijP
The resulting delocalization energy is given by eqn. (8) or (23) 6Edp~,,c = 2 ( ~ 0+ i cojM The smaller the value of (iEdel,,e,the lower the energy of activation and the faster the reaction. Calculation of 6Edeloc or N, shows that the 1 position of naphthalene and the 9 position of anthracene are favored in electrophilic addition reactions. These simple calculations are in excellent agreement with the more time consuming Hiickel and semi-empirical molecular orhital calculations.
Nucleophilic and radical aromatic substitution reactions are analogous to electrophilic substitution reactions.
between two starred atoms or between two unstarred atoms in even AH. between two starred atoms or between two unstarred pij = 0 atoms in an odd AH radical. in an odd AH anion. pjj = eojcoj pa = -eo,caj in an odd AH cation. The intramolecular cyclization of (1x1 to (XI (eqn. (7)) shows that (X) is no more aromatic (stable) than (IX) since 6 E , = 2(0)0 = 0. Similarly, fulvene appears to be no more stable than 1,3,5-hexatriene, (XII) and (XIII) are less stable than the triene, and benzene is more stable (aromatic) than the triene. pij = 0
P , =~ 0.30 6E, = 0.60p
(xm)
p,, = -0.39 6 E , = -0.78
The intramolecular union of an AH via two starred or two unitarred atoms t,, give an odcl-mi!mhered ring can he treated hs the a bmil c,nlr:r deiinitions d~>cussed abow. In this 1'\10 analysis, if opening the ring leads to no change in a energy 28 / Journal of Chemical Education
(8E, = 01, the ring is called nonaromatic. Examples include the ring closure of 1,3,5-hexatriene to fulvene (eqn. (30)) and the formation of azulene from cyclodecapentaene. If the ring opening leads to a decrease in a energy, the ring is called antiaromatic, and if the opening leads to an increase in a energy, the rine is described as aromatic.
is less stable than the linear form.
where EN-H is the nitrogen-hydrogen bond strength, q~ is the a electron density on nitrogen (16, 17), a N is the Coulomb integral fnr nitrogen, PC-N is the total hond order for the carhon-nitroeen bond. and 8"-w . ., .. is the resonance inteeral. It is assumed t i a t the a h o n d s formed by heteroatoms 20 not differ sienificantlv in strength from those formed bv carbon atoms. Thus, the pyridinium ion is isoelectronic with benzene and the change in T energy on passing from pyridine to the isoconjugate system is attrihuted to the changes in the Coulomb integral or
a ~ =, 2 q,a,,,
(34)
Since qj = I for an even AH nE, =
Similar PMO analysis of the [I] annulenium ions shows that the anion is antinromatic and the (rycloheptatrienyl) carhocation is aromatic.
x a,,,
(35)
Consequently, the rules discussed ahuve for even AH's also aoolv containing heteroatoms. . . . to isoconiueate svstems . The a h w e e s ~ ~ n p l ilwn e c t h r a i m p l i ~ ~.and t \ us~.t'ulnrisoi ~l. I < >I rher reucticm *\.ctwnsha,,. tht 1'\10 n ~ ~ . t h:\~wlinrions been discussed ( 3 ,d, i0). Literature Cited 1 1 1 Anderim.d. M.."lntr~,ductiir ihQmantum l'hernirlry." Ycnk, 196%
6 E , = -2c' 0 6 E , = 2c2 0 Odd AH's will have coefficients a t the terminal carhon atoms of opposite sign if the number of carhon atoms is (4n 3)(3, 7, 11, 15, etc.) which leads to negative resonance energies for cyclic compounds of 4n carbon atoms (4,8, 12, etc.). Odd AH'S with (4n 1)(5,9,13,17, etc.) carhon atoms have coefficients a t the terminal carhon atoms of the same sign which leads to positive resonance energies for cyclic com. pounds with (4n 2) carbon atoms (6, 10, 14, e t ~ . )Thus, anions with (4n 1) carhon atoms and carhocations with (4n + 3) carhon atoms are aromatic and anions with (4n 3 carhon atoms and cations with.(4n 1) carhon atoms are antiaromatic. Cyclic conjugated systems Of this type are nonaromaticif 6E, = 0 (uide suura).
+
+
+
+
+
I?) Dewrr. M. .!. 5.. "The M&cdar Orl!itnl
W. A. Henjamin. l n c . New
'l'hei~ryi,f'OqsnicChemirtry," Me(:mw~Hill,
New Twk. IYR9.,,. li. 111 Dewar. M. .I. S.. and I l e t ~ ~ h e r t y11. . c.. '"The PMO l'he,,ry dor~anic Chemirirv." l'lenum I'resr. New Yurk. 1975. I41 l ' f ~ ~ ~ l ~ ~ ~ n , C . A . , ~ ~ ? d l . ~ ~ ~ ~ ~ ~ ~ ~ ~ - H11,~~nd~~n1,A19!,:!91~9741; i~~~ns.H.l'..f'r~~~.K~~~.S~~ A192, 1 6 1 i 9 4 7 l : A 1 9 1 . 4 l i . I l ( i 1191X):A195, I I W . I l U X ! . 151 i.c,e~aet-Higrin*. H. C . . I U w m . I'L)r.. lR.16%%5. 283 119601. I61 Dewav, M . 4 S.,.) Am,w ('lwtn. .Sc,?., 71,,
