Distortive properties of σ- and π-electrons and aromaticity: a semiempirical localized molecular orbital approach

THEO CHEM

ELSEVIER Journal of Molecular Structure (Theochem) 369 (1996) 39-52

Distortive properties of O- and welectrons and aromaticity: a semiempirical localized molecular orbital approach

Albert MoyanoaY*, Juan Carlos Paniaguab

‘Departament de Quimica Orghica, Facultat de Quimica, Universitat a’e Barcelona, Marti i Franquks l-11, E-08028 Barcelona, Spain bDepartament de Quimica Fisica, Facultat de Quimica, Vniversitat de Barcelona, Marti i Franquks I-11, E-08028 Barcelona, Spain

Received 13 February 1996; accepted 12 April 1996

Abstract

This paper analyzes the changes experienced by the valence localized molecular orbitals (LMOs) of benzene, singlet cyclobutadiene and (E)-1,3,Shexatriene - with respect to both energy and degree of delocalization - under the effect of several geometrical distortions of the carbon-carbon frame. The analysis shows that, at the AM1 level of theory, there is a unique symmetrizing tendency of the x-LMOs of benzene, which are at the same time maximally delocalized and energetically most stable for a totally symmetric geometry. The x-LMOs of both cyclobutadiene and hexatriene show opposite behavior, increasing their energies when subjected to bond length equalizing distortions and decreasing them when the carbon-carbon bond length alternation of the system is enhanced. The distortive properties of the x-electrons, when analyzed in terms of the corresponding AMl-LMOs, thus appear to correlate well with the aromatic character of the molecule.

Keywords: Aromaticity; Benzene; Cyclobutadiene; Electronic delocalization; Hexatriene; Localized molecular orbital

1. Introduction

Since the pioneering work of Pauling [l] and Hiickel [2], the concept of aromaticity has become associated with both the enhanced energetic stability and the bond length regularity exhibited by planar conjugated molecules having cyclic arrays of [4n + 21 x-electrons. These properties stand in sharp contrast with the thermodynamic instability and alternating geometry shown by acyclic conjugated polyenes [3]. In the early 1960s these ideas were given a sound theoretical basis by Dewar [4], who showed, within the framework of the ?r SCF-MO approximation, that the energies and geometries of s-conjugated

molecules could be calculated with chemical accuracy. In addition, the fact that for acyclic polyenes there is a linear relationship between the total energy and the number of carbon atoms allowed the definition of a delocalization energy index, the Dewar resonance energy (DRE), as the difference between the total x-energy of a cyclic conjugated molecule and the x-energy of a localized, acyclic polyene-like reference structure [4]c,[5]. The DRE index classifies in general x-systems as being either aromatic, non- aromatic, or antiaromatic, depending on the relative stability of the x-electrons, and for monocyclic systems is in perfect agreement with the Htickel aromaticity rule [2],[3]c.

There therefore appeared to be a firmly grounded, * Corresponding author. well-established relationship between the stability and

0166-1280/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved PII SO166-1280(96)04599-X

40 A. Moyano, J.C. PaniaguaJJournal of Molecular Structure (Theochem) 369 (1996) 39-52

geometric shape of a conjugated system on the one the distortive properties of the energy components hand, and the delocalization degree of its r-electrons depend strongly on both the energy partition scheme on the other. and the reaction path for the localizing distortion

This situation began to change some ten years ago, however, when this connection between aromatic

properties and n-electrons was questioned by several

authors, heralding an opposing view in which the

delocalization of n-electrons was not the cause but

the result of the energetic and geometric requirements

of the a-framework [6,7]. In the past few years, a

number of theoretical studies have addressed the

question of the a-electron delocalization in planar

conjugated systems [8-121, some of which reached

contradictory conclusions. At present, the best that

can be said is that the once highly relied upon and interrelated concepts of aromaticity, resonance

stabilization and electronic delocalization are being perceived as confusing by organic chemists (e.g.

‘ ‘aromaticity . . . remains an ill-defined and unquanti-

fied concept with an unsatisfactory theoretical support” [13]). Nevertheless, it is also true that

aromaticity continues to be a most important unifying

theme in several areas of chemistry [14], and in recent years there have been numerous and brilliant efforts to

produce quantitative criteria of aromaticity [ 151.

WlW61.

One of the major arguments in support of the idea

that r-electron delocalization is a mere by-product of the geometric constraints imposed by u-electrons

relies on the analysis of the (T- and n-components of

the total energy of the system in front of geometric

distortions [&lo-121, generally at the one- configuration, restricted Hartree-Fock (RHF) level.

The results of these studies, with one exception [lO]b, show that in effect the a-energy component

of planar conjugated n-systems decreases when the

molecular geometry is distorted so as to enhance the C-C bond alternation, irrespective of the “aromatic”

character of the system. The immediate conclusion is

that s-electrons always tend to a geometry with alternating bond lengths, and that their apparent delocali-

zation in aromatic molecules like benzene is

determined by the fact that the energy of the u- electrons is minimal at a geometry with equal bond lengths. However, it is important to realize that, contrary to what happens with the total energy, its CJ- or ?r- components are not observable quantities, and therefore cannot be unambiguously defined. In fact, it is now recognized that the results obtained in studies of

On the other hand, a very convenient way to analyze the relationship between aromatic stabiliza-

tion and electronic delocalization relies on the use of

localized molecular orbitals (LMOs)’ [ 171. More than 20 years ago, England and Ruedenberg [22] showed

that the R-LMOs of aromatic systems were consider-

ably more delocalized than the ethylene n-orbital, and

that therefore the resonance stabilization of conju-

gated systems could be traced out to the local delocalization of the r-electrons, since in general (other

things being equal) the energy of an orbital decreases with increasing orbital size [23]. In 1979, Haddon [24]

analyzed the degree of localization of the a-LMOs of 1,3,.5hexatriene, cyclobutadiene, benzene, and cyclo- octatetraene, and found that it could be correlated with

the Dewar resonance energies (DREs) [3]b,[4]c,[5] of

the compounds. More recently, and as a result of extensive studies of the rr-LMOs of conjugated

acyclic polyenes [25], and of mono- and polycyclic conjugated hydrocarbons [26], we found that, in

general, any conjugated alternant [27] hydrocarbon with a positive DRE exhibits a higher degree of local electronic delocalization than its corresponding acyclic polyene-like reference structure, and that the reverse is true for antiaromatic alternant hydrocarbons

[28]. Therefore, as first hypothesized by England and Ruedenberg [22], there appears to be a close relation-

ship between aromaticity, n-electron energy stabilization and local electronic delocalization, at least in the

case of alternant conjugated hydrocarbons [29]. This finding is, however, in conflict with the current

interpretation of the results of computational studies of the behavior of the u- and a-components of the

energy in front of geometric distortions [8],[9]e-g, [lO]a,[ll]. It states that while the u-electrons find an

energetic minimum at a geometry with equal bond

‘Itshould be emphasized that the localized molecular orbitals (LMOs) referred to here are those obtained by a unitary (orthogonal) transformation of the canonical molecular orbitals (CMOS, i.e. those obtained by diagonalization of the Fock matrix), as defined originally by Lennard-Jones [18], Boys [19] and Ruedenberg [ZO]. These should not be confused with the truncated localized orbitals used by other authors [lO]b,[ll]b,[21] for the evaluation of delocalization (resonance) energies.

A. Moyano, J.C. PaniagualJournal of Molecular Structure (Theochem) 369 (1996) 39-52 41

lengths, the rr-electrons, irrespective of the aromatic

or antiaromatic character of the system, favor a

geometry with alternating bond lengths. In effect,

this would imply one of the following: (1) The n-LMOs of an aromatic system (e.g.

benzene) become more delocalized when the C-C

distances of the system are distorted towards an acyclic polyene-like geometry, contrary to the fact

that ?r-LMOs of acyclic polyenes are always more localized than those of aromatic hydrocarbons

[24-261. (2) Even if the a-LMOs of the distorted system

become more localized, their electronic energies

decrease, so that the well-proven relationship (at the

Hiickel level of theory) between the energy and delocalization degree of a LMO would not hold in

this case. In order to throw some light on this important issue,

we have investigated (with regard to both energy and degree of delocalization) the behavior of G- and n-

LMOs of three representative systems (benzene,

1,3,5hexatriene and singlet cyclobutadiene) towards geometric distortions of the molecules, at the AM1 level of theory. As we will see presently, when

described in terms of AMl-LMOs aromatic X- electrons become in fact more localized and ener-

getically destabilized when subjected to a localizing distortion; at the same time, both non-aromatic and

antiaromatic n-electrons show an opposite behavior, increasing their energies in more symmetric geo-

metries and decreasing them when the C-C bond length alternation of the system is enhanced.

2. Partitioning of the Hartree-Fock energy and unitary transformations of molecular orbitals

In the framework of the Hartree-Fock self-

consistent field theory (HF SCF), the 2n-electron wave function for a closed-shell molecule in its

ground state is given by the Slater determinant:

\k=(2n)!-‘i2detl~,(l)a(l)~P,(2)/3(2)...(o,(2n)P(2n)l

(1)

where CY and @ are spin functions and vi is a molecular orbital (MO), obtained by resolution of the Fock equations. Usually, the MOs in 9 are expressed as a linear

combination of atomic orbitals {xr} (LCAO),

pi = T ci,x, (i= 1, 2, . . . . n) (2)

The total energy (Em) of the system is the sum of the electronic energy (Eel) and the nuclear (or core-core,

in the case of semiempirical Hamiltonians like the

AM1 one) repulsion energy (EN),

EHF=Ee’+E N (3)

At this level of theory, the electronic energy is given

by [301

Eel= ~ [hii +&i] i=l

where hi” the so-called “core Hamiltonian” term, contains the kinetic energy and the potential energy due to the interaction with the nuclei (or with the

cores) of an electron described by the MO pi, and ei is the “orbital energy” term, which in fact is the ith diagonal term of the Fock matrix in the MO basis and

equals the core Hamiltonian energy plus the electronic repulsion terms, R ij,

(5)

In this last expression, J, and Kij are the Coulomb and exchange integrals, respectively.

Let us now suppose that the symmetry of the molecule allows us to divide the MO basis set {pi} into two

disjoint subsets {ur,u2,. ..,u,J (“sigma” MOs) and

{~lJZ>..., ?T,+,,J (“pi” MOs); in this case, we can

divide the electronic energy E”’ given by Eq. (4) in two terms, Ez, and E”,]:

Ee’ =Ef +E$ (6)

with

Pa)

42 A. Moyano, J.C. PaniaguaiJournaf of Moiecular Structure (Theochem) 369 (1996) 39-52

where the meaning of the individual terms hz, hi, Rr,

R r and R r should be clear from the definitions given in Eqs. (4) and (5). This constitutes a logical way to partition the electronic energies into two components, and is the approach used by Jug and Koster [ 1 l]a,[31] and, more recently, by Ichikawa and Kagawa [32]. It should be kept in mind, however, that due to the existence of the 5--x repulsion term C z RT, any partition is intrinsically arbitrary, and that other authors [8,10]b,[12]a have proposed alternative approaches.

On the other hand, it has long been recognized that, due to the mathematical properties of determinants, the effect of a unitary transformation U of the occupied MOs, as defined by the following set of equations:

$‘i = ,il pkuki, i = 1, 2, . ..) n (8a)

i u&u; = kil uLukiXsii, i,j= 1, 2, . . . . n (8b) k-l _

on the 2n-electron wavefunction 9 (Eq. (1)) is simply its multiplication by an immaterial phase factor. Therefore, the electronic energy E”’ (Eq. (4)) of the system does not depend on the MO basis selected to construct the determinant wavefunction. While it is evident that the partition of the electronic energy into (T- and x-components is no longer feasible after a unitary transformation that mixes 5 and ?r CMOS, it is important to note that, in spite of the presence of the u-u repulsion terms in the expressions (7a and 7b, respectively) for Et and E”,‘, these components will remain invariant if the unitary transformation does not mix MOs belonging to different subsets, as we will presently see.

Let us apply in effect a unitary transformation U”‘” to the MO basis such that it only mixes MOs in the same subset, as exemplified by the following equations:

5i’ = ji UjU]T, i= 1, 2, . . . . m (94 *

n-m rkr= !I1 TIU& k=l, 2,..., n-m Pb)

Now, if we introduce the set of equations (7) into Eq. (6), we see that the total electronic energy of the

system can be expressed as

E”‘=2 ,tl h;+ iFl j$l Rr+2 1s; h;

(10)

It is clear that 1 hz and z 2 R7 are not changed by the partial unitary transformation defined in Eq. (9a); by the same token, both 1 hz and 1 1 Ry are invariant under the partial transformation (9b). On the other hand, Ee’ will not be altered by application of the global transformation U’*, so that we have the interesting result that the u-x electron repulsion term 12 R 7 is also invariant under any unitary transformation that preserves P-K separation. As a conse- quence, the 5- and x-components of the electronic energy, as defined by Eqs. (7) (or, for that matter, by any other combination of their constituents) are independent of the particular MO basis chosen within the 5 and a subsets to construct the global wavefunction of the system. In particular, this justifies the use of 5- and rr-LMOs to describe the energetic changes undergone by a conjugated system under the effect of a geometric distortion.

Although in general the localization process leads to the mixing of u- and n-orbitals [33], we have previously demonstrated [34] that if one uses a localization criterion based on the maximization of the quantity

(11)

(where A are the atoms embraced by the ith MO, and the coefficients Ci, are those corresponding to the LCAO expansion defined by Eq. (2)) the subsets of CT- and ?r-LMOs will be localized separately. This has been therefore the localization procedure used in this work.

According to Eqs. (4) and (5), the part of the electronic energy contributed by an MO (either localized or canonical) is given by

(12)

In this work, we use these four quantities (Er’, hii, Ei

and C Rv) to analyze the energetic changes experienced by 5- and a-LMOs when subjected to distortions of the core geometry.

A. Moyano, J.C. PaniaguaiJournal of Molecular Structure (Theochem) 369 (1996) 39-52 43

On the other hand, in order to evaluate the degree of

delocalization of an LMO we will use the number of

centers (not), defined as [35]

(noc)i = I/( F ( r& CfJ2) (13)

and which is a measure of the effective number of

centers spanned by the orbital. It is easy to see that the not value for a strictly bicentric LMO is exactly

2.0, and that (not), increases with increasing deloca-

lization of the orbital. The localization process has been effected on

CMOS obtained with the semiempirical AM1 method

[36], in the standard RHF version, so that the level of

analysis of the resulting LMOs is simple (one needs to consider only valence electrons and the LCAO expan-

sion of the MOs uses a minimal A0 basis), yet

physically meaningful (since the geometric and electronic description of conjugated hydrocarbons by AM1

is very accurate). It should be noted, however, that the present analysis can be effected on any monodetermi-

nant wave function, provided that a suitable localiza-

tion method, with preservation of u-x separability, is

used.

3. Results and discussion

As the subject of our study, we chose three repre-

sentative molecules: benzene (l), singlet cyclobutadiene (2) - the prototypical aromatic and

antiaromatic systems, respectively - and (E)-1,3,5- hexatriene (3) a nonaromatic conjugated species.

The LMOs corresponding to these molecules were obtained, first for the equilibrium AM1 geometries,

and subsequently for structures having distorted

C-C bond lengths, in which both the C-H distances

H H 1110.0pm 1 1 OS.8 pm

and the remaining angular coordinates were optimized

with AMl. This allowed us to construct correlation

diagrams in which we analyze the effects of geometric

distortions on the electronic energy contributions (Eq. (12)) and degree of delocalization (Eq. (13)) of both (T- and ?r-LMOs. We now examine in some detail the

results obtained for each of the representative systems

l-3, and the conclusions that can be derived from

them.

4. Distortive properties of AMl-LMOs in benzene

We considered four different geometries (la-ld) of the benzene molecule, with increasingly distorted

(along the BZu mode) C-C frameworks (Fig. 1). The

first species (la) corresponds to the AM1 equilibrium

geometry of benzene, which is in perfect agreement with the most reliable experimental and theoretical

values [37]. The second structure (lb) is a distorted one in which the short (“double”) and long

(“single”) C-C distances correspond to the values

proposed by Baird [9]a for the most adequate alter- nated structure of benzene. The third geometry (lc), having shorter “C=C” bonds, is the one originally

used by Shaik, Hiberty and coworkers [6]d,[6]e,

[8]b,[8]d, and has subsequently been employed in other theoretical studies on the distortive propensity

of a-electrons in benzene [lO]b,[12]a. Finally, the cyclohexatriene structure Id is the one usually

employed for the evaluation of the theoretical reso-

nance energy (TRE) of benzene [38]. The effects of

the localizing distortions exemplified by lb-ld on

both the degree of delocalization and the energies of

the three different types of valence electron LMOs of benzene (i.e. (Jc-u, uc- and nc=c) are summarized in

Fig. 2. A more detailed analysis of the different terms

l39.5~@: l46.3iq.8 pm 146.3;vi.O pm 15!2.0;~;.0 Pm

H H H H

la lb lc

Fig. 1. Equilibrium (la) and distorted (lb-ld) geometries of the benzene molecule.

Id

44 A. Moyano, J.C. Paniagua/Joumal of Molecular Structure (Theochem) 369 (1996) 39-52

Orbital electronic energy (eV)

-200 -

-205 - w=c

(2.401)__..---x _-- (2.8p?!___----- _203.53 ---------__ w=c -205.00 -204.85

(1.985) (1.985) (1.988) (1.988)

W-H ____------ ------_______ _____------

-1gg.8g (2.473) - GC-H

-200.50 -200,25 -199.50 (2.250)

-210 1

I

-230 -, (2.050)

-235 -

-240 -, ccc

.-- (2.043) (2.04+---=-

-232.50

(2.037)_/--*

,._235.14---------____* -235.93

,_ : _ _ _ _ - _ _ - - - (2.034) (2.032)

(2.030)

-239.21 -239.22 ---------_ _ _._-------- -241.28 -241.04

w-c

‘s&C

la lb IC Id

Fig. 2. Effect of localizing geometric distortions on the degree of delocalization (not index, in parentheses) and on the electronic energy

contributions (Ef’, eV) of the AM1 cr- and s-LMOs in benzene.

contributing to the ?r-LMO energies can be found in Table 1.

Several interesting points emerge from the data shown in Fig. 2. With respect to the delocalization degree (as measured by the not index, Eq. (13)), it is evident that the (T- and ?r-LMOs exhibit totally different behavior: whereas the (noc)i value of the former is rather insensitive to the geometric distortions (the maximum observed variation is a mere 0.64% for a (I~_~ LMO), the degree of delocalization of the ?r-LMOs clearly diminishes when the bond length of the “double” bond is shortened, being minimal for the “cyclohexatriene” structure Id. In other words, the a-electrons become more localized when the acyclic polyene-like character of the carbon- carbon framework is enhanced. This result is in

perfect agreement with the known properties of T- LMOs of conjugated systems [22,24-26,281. It is worth noting, however, that the degree of delocalization of the x-LMOs of Id is even greater than that of the “central” x-LMO of 1,3,5_hexatriene 3a (2.163; see below), so that this appreciably distorted structure still retains a certain degree of aromaticity (according to the definition given by Ruedenberg [22]) in its ?r- electrons.

If we analyze now the behavior of the orbital electronic energy contributions, we can see that in general the ?r-electrons of benzene are destabilized when subjected to the localizing distortions exemplified by lb-ld, although there is no clear correlation between EF’ (or its components Ei, hii and 1 R;j; see Table 1) and the degree of delocalization of the corresponding

Table 1

Energy component analysisa of the AM1 K-LMOs of benzene (la) and of distorted species (lb-ld)

la lb LC Id

E:’ (A) -205.005 (-) -203.533 (1.472) -204.652 (0.353) -203.433 (1.572)

~1 (A) -10.895 (-) -10.786 (0.109) -10.942 (-0.047) -10.929 (-0.034)

hii (A) -194.110 (-) -192.747 (1.363) -193.710 (0.400) -192.504 (1.606)

C R, (A) 183.214 (-) 181.961 (-1.253) 182.767 (-0.447) 181.575 (-1.639)

a Electronic energy terms, according to Eq. (12), in eV. The quantities in parentheses (A) are the differences from the la value.


Table 2

Energy partition analysis of benzene (la) and of distorted species (lb-ld) (values in kcal mol-‘)

Ml

A Wf) EN

A (EN)

Eocc

A (Eocc)

E@CH

A @CH)

ErcC

A (Escc)

la

+ 22.02

(-) +55412.5

(-) -33097.3

(-) -27741.6

c-1

-14182.1

(-)

lb

+28.09

+6.07

+54923.4

-489.1

-32816.0

+281.3

-27629.6

+112.0

-14080.4

+101.7

lc

+29.94

+7.92

+55276.0

-136.5

-33012.2

+85.1

-27706.8

+34.8

-14157.9

+24.2

Id

+43.79

+21.77

+54848.7

-563.8

-32759.7

+337.6

-27602.4

+139.2

-14073.5

+108.6

R-LMO. In fact, the energetic destabilization of the ?r- electron system clearly grows with the net amount of carbon-carbon bond lengthening (i.e. the lengthening of the C-C bonds minus the shortening of the C=C ones) in the distorted structure, which increases in the order lc < lb < Id. The behavior of the @c-H LMOs is similar, although the energetic differences are smaller. The most evident effect of the geometric distortions (which in every case reduce the D6,, symmetry of la to a D3h one), however, is on the six equivalent ucc LMOs, which are split into two subsets: first, the three a-components of the C=C bonds (a&, which are slightly stabilized when the length of the “double” bonds is shortened (their energy being minimal for structure lc, with 134.0 pm); and second, the three u-components of the C-C bonds (a&, which are strongly destabilized when the length of the “single” carbon-carbon bonds is increased. This destabilization is maximal for the “cyclohexatriene” species Id, which has the longest “single” CC bonds, and largely outweighs the energetic stabilization of the ucZc counterparts. A global balance of the partitioned energies of the benzene- related structures la-ld can be found in Table 2.

With respect to the global energy values, the AM1

calculated differences of the enthalpy of formation are in very good agreement with previously reported values [6]d[6]e[8]b[8]d[l2]a. Although the core- core repulsion energy term (EN) favors the distorted structures lb-ld (even in the case of lc, for which the relative decrease is very small), the accompanying larger increase in the electronic energies shifts the equilibrium geometry towards la. The largest contri- bution to the electronic destabilization comes from the reluctance of both ucc and CJ~ electrons to depart from the symmetric Deb structure, but it is worth noting that according to the present analysis the ?r- electrons of the “archaromatic’ ’ benzene molecule can also be strongly destabilized (up to 18.1 kcal mol-’ per r-electron for ld) by effect of the considered localizing distortions.

5. Distortive properties of AMl-LMOs in singlet cyclobutadiene

Starting from the AM1 equilibrium geometry of singlet cyclobutadiene (2a, Fig. 3), which is reason- ably close to the best theoretical estimates [39], we have effected two different geometric distortions:

,07.7 iJzc.4 pm ,07.8 pi]E(.O pm ,07.6 pmIJflc.O Pm

2a 2b 2c

Fig. 3. Equilibrium (2a) and distorted (2b, 2c) geometries of singlet cyclobutadiene.

46 A. Moyano, J.C. PaniagualJournal of Molecular Structure (Theochem) 369 (1996) 39-52


-155 -’ (2.000)

(2.000) (2.000)

w=c _158.39-------__ -- -158.82 __------_157.91 Jk=c

-160 -!

I

%-H __----- ----______

-160.36 -160.24 -160.36 %-H

-165- (1.952) (1.954) (1.951)

I s

(2.057)

CYsc (2.051)___------182.47=._

-- . . -185- ‘.

I

“long” -184.62 .\ (2.047) OC-C *-

-186.15 “long”

-190-J

w=c (2.037) _-(2.040) cTC,C

-- -1 90.89 “short” -195

-, “short”-192.87-----._ -I (2.y----

-195.08

2a 2b 2c Fig. 4. Effect of localizing (2b) and of delocalizing (2~) geometric distortions on the degree of delocalization (not index, in parentheses) and on

the electronic energy contributions (E:‘, eV) of the AM1 e- and r-LMOs in singlet cyclobutadiene.

first, elongation of the “single” C-C bonds and shortening of the “double” C-C bonds leads to a distorted

structure (2b) in which the alternating character of the system is enhanced; and second, we have chosen a

more symmetric structure (2~) with diminished differ-

ences in C-C bond lengths, which corresponds to a “delocalizing’ ’ distortion of the more stable geometry.

As in the case of benzene, we have analyzed the effect

of these distortions on the LMOs of the system. With the help of Fig. 4, we can follow diagramma-

tically the changes experienced by the LMOs corre-

sponding to the three types of bonds present in singlet

cyclobutadiene. A detailed energy component analy-

sis of the ?r-LMOs is presented in Table 3. We consider first the localizing distortion leading to

structure 2b. Inspection of Fig. 4 shows that while the

UC._u LMOs are only slightly destabilized with respect

to structure 2a, the other LMOs experience more important changes: both the ?r- and a-LMOs asso-

ciated with the short (C=C) bonds become more

stable, while the a-LMOs describing the long (C-C) bonds and the C-H bonds are destabilized. The global

balance of the partitioned energies (see Table 4) reveals that the net effect of the distortion on the

ucc framework is somewhat stabilizing, so that in

this case, contrary to what happens with the benzene

molecule (where both u- and x-electrons are appreciably destabilized by localizing distortions) increasing

the bond length alternation of the ring leads to a

decrease in the a-electronic energy of the system. This energetic stabilization cannot be related here to

an increase in the localized character of the orbital,

Table 3

Energy component analysisa of the AM1 s-LMOs of singlet cyclobutadiene (2a) and of distorted species (2b,2c)

2a 2b 2c

E;’ (A) -158.393 (-) -158.825 (-0.432) -157.911 (0.482) si (A) -10.543 (-) -10.855 (-0.312) -10.290 (0.253) h,, (A) -147.850 (-) -147.970 (-0.120) -147.621 (0.229) X R,(A) 137.307 (-) 137.115 (-0.192) 137.331 (0.024)

’ Electronic energy terms, according to Fq. (12) in eV. The quantities in parentheses (A) are the differences from the 2a value.


Table 4

Energy partition analysis of singlet cyclobutadiene (2a) and of dis-

torted species (2b,2c) (values in kcal molK’)

2a 2b 2c

Afff A wff) EN

A (EN) Eocc A (Eocc) ECCH A (EUCH)

Ercc

A @*cc)

+111.26

C-1 +26531.0

(-) -17410.1

(-) -14791.6

(-) -7304.9

(-1

+117.46

+6.20

+26548.3

+17.3

-17412.8

-2.7

-14780.3

+11.3

-7325.0

-20.1

+115.27

+4.01

+26492.0

-39.0

-17389.3

+20.8

-14791.8

-0.2

-7282.8

+22.1

since the two r-LMOs of singlet cyclobutadiene are

already maximally localized (i.e. they are strictly

bicentric) at the equilibrium geometry 2a, and remain

so in the distorted structures 2b and 2c, due to the

symmetry properties of the DZh point group [40]. The energetic changes experienced by these orbitals are therefore only due to the variations of the inter-

molecular distances. In particular, examination of Table 3 reveals that the energetic stabilization of the

n-LMOs in 2b is due to significant decreases in both

133.6 pm 134.5 pm

“I/H H

144.8 pm

3a 139.5 pm

/\ i HpH H H H

139.5 pm 3c

the core Hamiltonian and the electronic repulsion

terms. One should compare this result with the trends

observed in the case of benzene, where the overall

energetic destabilization experienced by a ?r-LMO under the effect of a localizing distortion is due to

an increase in the core Hamiltonian (hii) term, which

clearly outweighs the accompanying diminution of

the electronic repulsions (see Table 1). Finally,

another differential feature of singlet cyclobutadiene is the small increase in the core-core repulsion energy

caused by the localizing distortion, which stands in

sharp contrast to the large stabilizations exhibited by

structures lb and Id (cf. the corresponding entries in Table 2 and Table 4).

We now turn our attention to the distortion leading

to structure 2c, in which the differences between C-C

bond distances have been reduced in comparison with

2a. Not unexpectedly, the effects of such a distortion are the opposite to those of the “localizing” one just

discussed: in effect, both the ‘lr- and the a-LMOs

corresponding to the “short” (C=C) bonds are

strongly destabilized, while the a-LMOs describing

the “long” (C-C) bonds become stabilized relative to those of 2a. On the other hand, the energy of the (Tc-H LMOs remains virtually unaffected (see Fig. 4).

130.0 pm

H H +$$A / / H

H H

150.0 pm

3b

Fig. 5. Equilibrium (3a) and distorted (3b, 3c) geometries of (E)-1,3,5-hexatriene (s-rrans, s-trans conformer). The C-H distances (not shown)

fall in the range of 109.3-110.4 pm for each of the three structures and are rather insensitive to geometric distortions.


The overall effect is that both the u- and the x-CC is very close to both the experimental and theoretical electrons are destabilized by roughly the same amount ab initio values [37]. From there, we considered a through the distortion, at the same time the core-core “localizing” distortion leading to a structure with interaction energy decreases (see last column of Table enhanced C-C bond length alternation, 3b, as well 4). The energetic destabilization of the r-LMOs in 2c as a “delocalizing” one, exemplified by structure is due mainly to the decrease (in absolute value) in the 3c, in which all of the carbon-carbon distances have core Hamiltonian term. as shown in Table 3. been equalized to an intermediate value (Fig. 5).

6. Distortive properties of AMl-LMOs in Q-1,3,5- hexatriene

We have also studied the LMOs of (I?)-1,3,5hexa- triene, as a representative example of linear polyenes. The equilibrium AM1 geometry (3a) has been determined for a completely planar, s-tram, s-tram conformer (belonging to the Cz, point group) and as usual

-180


-190

-210

-23C

-240

(2.088) (2.133)

7kkC2 ----_____ P.uao) --_ ___--- _----- _,87.5, ttcl=C2

-188.22 -188.37

The energetic changes experienced by the LMOs of hexatriene by going from 3a to 3b are similar to those observed for the localizing distortion of singlet cyclobutadiene (see Fig. 6): both the rr- and the a-LMOs corresponding to the short (C=C) bonds are stabilized, while the two equivalent (T~_~ LMOs describing the “single” C-C bonds increase their electronic energy. The net effect of the (r electrons is only slightly stabilizing, as happens with the ec_u LMOs; the overall electronic stabilization partially compensates for

(2.163) (2.248)

(2.116) __------ -207.16 @3=c4

Q3=c4 -208.11 -------____ __--

-206.63

(2.030) (2.026)

%l=C2 _2227, ----a___ --_ (2.(X4)----= --- -219.78 %l=C2

-224.24

%2X3 (2.042) ______--

(2.049) -___

___--- -231.86 ----___

-233.94 -.

(2.036)

-236.08 *C2-C3

(2.030) (2.034)

mJ=c4 %3=c4 --. (2.027) _[------- - -241.42

-243.85 -------__ __--

-245.76

3a 3b 3c

Fig. 6. Effect of localizing (3b) and of delocalizing (3~) geometric distortions on the degree of delocalization (not index, in parenthesis) and on

the electronic energy contributions (Er’, eV) of the AM1 IY cc- and r-LMOs in (I?)-1,3,Shexatriene. For clarity, the ocH LMOs have been

omitted from the analysis.


Table 5

Energy component analysis” of the AM1 *-LMOs of (E)-1,3,Shexatriene (3a) and of distorted species (3b,3c)

3a 3b 3c

1. Terminal bonds

EP’ (4 -188.222 (-) -188.369(-0.147) -187.512 (0.710)

e: (A) -10.718 (-) -10.908 (-0.190) -10.405 (0.313)

hz, (A) -177.504 (-) -177.461 (0.043) -177.107 (0.397)

C Rij (A) 166.786 (-) 166.553 (-0.233) 166.702 (-0.084)

II. Central bond

E:’ (A)

8, (A)

h, (A)

C R,,(A)

-208.110 (-) -208.634 (-0.524) -207.162 (0.948)

-10.812 (-) -11.023 (-0.211) -10.603 (0.209)

-197.298 (-) -197.611 (-0.313) -196.559 (0.739)

186.486 (-) 186.588 (0.102) 185.956 (-0.530)

a Electronic energy terms, according to Eq. (12), in eV. The quantities in parentheses (A) are the differences from the 3a value

the increase in the core-core repulsion energy caused by the distortion (see the second column of Table 6). In this case, and contrary to the behavior of K- electrons in benzene, the energetic stabilization of the ?r-LMOs is accompanied by an increase in the localized character of the orbital (as measured by the corresponding diminution of the (noc)i value). The energy component analysis of the ?r-LMOs (Table 5) is more complicated than that of cyclobutadiene, since the changes in both the core Hamiltonian and electron repulsion terms have different signs for the terminal ?T,-~=cz and for the central uc3=c4 LMOs.

This parallel behavior between singlet cyclobutadiene and (E)-1,3,5_hexatriene is also observed when we subject the starting structure 3a to the bond length equalizing change exemplified by geometry 3c. In effect, this produces a destabilization of both the

Table 6

Energy partition analysis of (E)-1,3,5-hexatriene (3a) and of dis-

torted species (3b,3c) (values in kcal mol-‘)

3a 3b 3c

AHf +42.91 +48.52 +52.06

A (AH3 (-) +5.61 +9.15

EN +55836.1 +55878.8 +55634.3

A (EN) (-) +42.7 -201.8

E%C -26684.1 -26693.8 -26590.5

A (&c) (-) -9.7 +93.6

EDCH -35889.9 -35898.4 -35827.2

A WCH) (-) -8.5 +62.7

Ercc -13479.7 -13498.6 -13425.3

A (Ercc) (-) -18.9 +54.4

u- and 7r-components of the C=C electrons, and the only LMOs which are stabilized are the (T~_~ ones (see Fig. 6). In the case of the ?r-LMOs, this energetic destabilization is accompanied by an important increase in the degree of delocalization of the orbital. Here, the energy component analysis (last column of Table 5) of both terminal and central r-bonds shows that the increase in electronic energy is due mainly to the core Hamiltonian energy term, which becomes more positive. The overall energetic balance (Table 6) shows that the response of the system to a “delocalizing” distortion is an increase in the ucH, ucc and rcc components of the electronic energy, which overrides the important decrease experienced by the core-core energy repulsion term.

7. Conclusions

Having established that a separate localization of the u and r-subsets of a planar conjugated system is compatible with a u-u partitioning of the RHF electronic energy, we have analyzed with some detail the changes experienced by the AMl-LMOs of benzene, singlet cyclobutadiene and 1,3,5hexatriene under the effect of several distortions of the carbon-carbon framework.

Although in every case the overall energetic balance of the distortion appears to be the result of contributions of opposite sign coming from the core- core repulsion term, on the one hand, and from the overall (uc~ + uc_c + uc=c + A& electronic energies,


on the other, we can classify the LMOs of the systems under study in two different types:

(1) Localization-prone LMOs: These are orbitals that simultaneously become more localized - i.e. having a smaller (noc)i value - and more stable when the length of the bond upon which they are centered experiences a decrease from the equilibrium value, and which show the opposite behavior upon an increase in the bond length. All of the (Tc_n, (J~_~ and ucEc LMOs of the systems we have been considering, as well as the *c=c LMOs of both the “antiaromatic” singlet cyclobutadiene and of the “non-aromatic” 1,3,5-hexatriene, fall into this category. The electrons described by these LMOs will favor alternating (as happens with cyclobutadiene and hexatriene) or symmetric geometries (benzene) depending on the balance between the energetic changes caused by the shortening and elongation of the bond distances. In particular, the fact that the ecEc LMOs of benzene are only slightly stabilized when the molecule is subjected to a BZu distortion, while at the same time the u- LMOs centered at the long bonds sharply increase their energies, is responsible for the overall tendency of the carbon-carbon u-frame in benzene to oppose any localizing distortion [41,42].

(2) Deloculization-prone LMOs: These LMOs simultaneously become less stable and more localized when the bond length alternation of the system is enhanced. Only the T-LMOs of the prototypical “aromatic”

molecule, benzene, exhibit this behavior among the

systems studied. The electrons described by these LMOs can be said to prefer a symmetric geometry.

We can conclude therefore that, at least at the AM1 level of theory, the description in terms of LMOs emphasizes once again the uniqueness of the r-electron system of benzene, in accordance with the classical concept of aromaticity. The present findings have paved the way for computationally more advanced calculations at the RHF ab initio level, which are currently under study in our laboratories. The results will be reported in due course.

Acknowledgements

We wish to thank the DGICYT (PB93-0806 and PB92-0766-C02-01) for financial support, and the former Centre de Calcul de la Universitat de Barcelona for computational facilities.

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Documents

Distortive properties of σ- and π-electrons and aromaticity: a semiempirical localized molecular orbital approach