Symmetry Aspects of Distortivity in p Systems

P. W. Fowler

Department of Chemistry, University of Exeter, Stocker Road, Exeter EX4 4QD, UK

AbstractThe distortive effect of p electrons in conjugated systems can be modelled, following

Heilbronner, by varying the Huckel bond integrals subject to retention of a constant sum at

each (non-pendant) unsaturated atom. In this simplified graph theoretical model, symmetry

theorems give the numbers and types of constrained distortion modes leading away from the

s-optimal geometry. For the class of systems without pendant edges, these Heilbronner

modes correspond to specific eigenvectors of the edge-incidence matrix. They also model

directly the most efficient distortive modes as predicted from the eigenvectors of the bond-

polarisability matrix, and add specificity to the symmetry rules for non-alternants that are

based on the second-order Jahn–Teller effect. An orbital model for induced ring currents in p

systems shows that p currents are resilient to s framework changes, indicating a survival of p

delocalisation and mobility, in systems such as pentalene even after distortion has taken place,

and under significant bond alternation in (s) ‘clamped’ benzenes.

Contents

1. Introduction 219

2. The Heilbronner model 221

3. Counting Heilbronner modes 222

4. Symmetry and Heilbronner modes 225

5. Heilbronner modes in polyhedra 227

6. Heilbronner modes and line graphs 229

7. When does distortion occur? 232

8. Distortion and aromaticity 234

References 236

1. INTRODUCTION

It is a truism that many of the most useful concepts in chemistry draw their strength

from a certain amount of fuzziness of definition. Aromaticity is perhaps the most

celebrated example. Used to predict and rationalise reactivity and properties of

cyclic p systems, this concept is associated with a loose cluster of geometric,

energetic, magnetic and reactivity-based criteria, and debate still rumbles on about

the extent to which these logically distinct criteria can or do give consistent

measures of a single, identifiable characteristic [1]. Probably all chemists would

ADVANCES IN QUANTUM CHEMISTRY, VOLUME 44 q 2003 Elsevier Inc.ISSN: 0065-3276 DOI 10.1016/S0065-3276(03)44014-8 All rights reserved

agree that benzene is the archetypal aromatic p system, though, if a much-quoted

exchange at the 1970 Jerusalem conference [2] is to be taken seriously, some would

also say that it is the only such system.

Benzene has a number of properties that might be taken as the basis of a definition

of aromaticity. For one, to revive a term from a paper presented at the same

conference, the p system of benzene has the ability to sustain a ring-current

circulation—the property of strobilism (from jnlina aloga strobilizomena;Greek for a children’s merry-go-round, according to Ref. [3]). In recent years

attention has focused on this particular magnetic response property as the defining

characteristic of an aromatic molecule: an aromatic molecule is taken to be one that

sustains a diamagnetic (diatropic) ring current [4a,b]. From the points of view of

physical detection, through the effect of the ring current on 1H NMR shifts [5], and

of quantum mechanical calculation, either indirectly through chemical shifts at

nuclei and at ring centres [6], or directly through visualisation of the induced current

density [7a,b], this concentration on induced mobility of the p electrons has the

definite advantage of providing a simple yes/no answer for any given system.

Another striking aspect of benzene is its regular hexagonal structure, which raises

the question of the connection between geometry, electron delocalisation, ring

current and aromaticity. After vigorous debate, it now seems to be generally

accepted that the geometric structures of many conjugated p systems result from a

compromise between the tendency of a s framework to produce symmetrical

arrangements with equal or near-equal bondlengths, and the opposing tendency of p

electrons to favour bond alternation, bond fixation and, typically, broken symmetry.

The history, defence, computational exploration and experimental verification of

this concept is detailed in recent comprehensive review articles by Shaik and co-

workers [8a,b].

Simple considerations suggest that all p systems are inherently distortive, whether

or not this tendency is realised in observable geometric changes. Again the paradigm

is benzene, wherep stabilisation would be greater in a D3h geometry that resembles a

single Kekule structure with three fully realized double bonds, but distortion in this

direction is prevented by the stiffness of the s framework which prefers the regular

equilateral hexagon with longer bonds and smaller b integrals [9]. The opposite case

is exemplified by pentalene, where the p-favoured C2h localised Kekule structure is

lower in total energy than the s-favoured D2h delocalised transition-state [10], and

the structure distorts. Various criteria for predicting whether it is the s or the p

electrons that win the energy–geometry competition in any particular case have been

developed [11,12].

Once the possibility of distortion is raised, an immediate next question is the

direction of this distortion, considered as a vibrational mode of the undistorted

system. A simple model for the directional aspect of p-distortivity was proposed in a

tutorial context for benzene itself by Heilbronner [9]. It has an easy generalisation,

which turns out to predict the most likely pattern of distortion of many systems by

pencil-and-paper construction of the Heilbronner vectors or Heilbronner modes

[13]. The present paper reviews some symmetry- and graph-theoretical aspects

P. W. Fowler220

of this model, their connection with more formal treatments and with the pseudo-

Jahn–Teller effect, and in view of recent calculations of ring-current maps [14],

discusses the connection between p-distortivity and p ring currents.

2. THE HEILBRONNER MODEL

In his classic paper in the Journal of Chemical Education entitled ‘Why do some

molecules have symmetry different from that expected?’ [9], Heilbronner pointed

out that the notion of p-distortivity is already implicit in the Huckel treatment of

benzene.

The standard treatment employs equal Coulomb parameters a for all six carbon

atoms, and equal resonance parameters b for all six CC bonds. In effect, a is the

origin and b the unit in a scaled representation of the Huckel hamiltonian matrix as

H ¼ a1 þ bA ð1Þ

where, for a conjugated system, 1 is the nðvÞ £ nðvÞ unit matrix and A is the

adjacency matrix (Aij ¼ 1 for vertices i, j on s-bonded neighbouring atoms, and

Aij ¼ 0 otherwise, i; j ¼ 1;… ; nðvÞ). The unsaturated carbon atoms are the n(v)

vertices and the s-bond CC links are the n(e) edges of the molecular graph of the p

system. The weights of pp atomic orbitals in the p molecular orbitals ci are defined

by eigenvectors of A. The p orbital energies e i are defined by the eigenvalues of A,

{liA}, i ¼ 1;… ; nðvÞ; with e i ¼ aþ blA

i :In Heilbronner’s model, the p-Hamiltonian for benzene is modified by replacing

the constant entries of the scaled adjacency matrix with entries ð1 ^ dÞb which

alternate around the edges of the ring (Fig. 1). The sum of integrals around each

vertex thus remains constant at 2b. The modified eigenvalues {lAi } are

lA1 ¼ þ2; lA

2 ¼ lA3 ¼ þ

ffiffiffiffiffiffiffiffiffiffi1 þ 3d2

p; lA

4 ¼ lA5 ¼ 2

ffiffiffiffiffiffiffiffiffiffi1 þ 3d2

p; lA

6 ¼ 22 ð2Þ

giving a total, distortion-dependent, energy for the p configuration c 21c

22c

23 of

Ep ¼ 6aþ 4bð1 þffiffiffiffiffiffiffiffiffiffi1 þ 3d2

pÞ ð3Þ

Fig. 1. Heilbronner model for the distortive mode of benzene.

Symmetry Aspects of Distortivity in p Systems 221

In this simplified model, the p stabilisation energy is at its lowest in the regular

hexagonal structure is at a minimum, with curvature ð›2E=›½db�2Þ ¼ 12b21: Hence

the prediction is that distortion along the b2u Kekule mode, in which three disjoint

bonds strengthen at the expense of the other three, will produce a quadratic

stabilisation of the p system.

As the energy depends on d, the implication is that it will also depend on bond

length, and so the prediction is that, if p electronic effects are given free rein, the

equilibrium structure of the molecular framework will be one in which there has

been distortion towards a short–long bond-alternated Kekule structure. That

benzene does not in fact distort in this way is then explained by the relative strengths

and force constants of s and p bonds – the s framework of benzene is simply too

stiff to be distorted by the relatively weak p forces [9,12].

The precise form of the model has clearly a limited range in that the constancy of

the sum of bond integrals around a given vertex will not persist indefinitely: the limit

d!1 would give bond integrals 0b and 2b, which probably exaggerates the increase

in magnitude of the integrals in the localized form of the molecule, but does give a

basis for prediction of the initial direction of any distortive behaviour.

The generalisation to other p systems is [13] that the d CC bonds around a given

d-valent vertex of the carbon framework graph should be allowed to change by

dib ði ¼ 1;…; dÞ subject to the constraint thatP

i di ¼ 0:In his tutorial paper [9], Heilbronner considers two examples: benzene, in which

d ¼ 2 for every vertex, and the allyl radical, in which d ¼ 1 for end vertices, and

d ¼ 2 for the central vertex. In the latter case, he allows the two bonds to change by

^d, preserving the sum at the central vertex but not at the terminal vertices. For

systems that include some vertices of degree 1, the most natural generalisation

seems to be that the constant-sum constraint should be applied only at vertices of

degree d greater than 1. We can define any such set of changes to the b integrals,

with the implied changes in the bond lengths, as a Heilbronner mode, and term it

more precisely a Heilbronner distortion if these changes would lead to loss of

symmetry.

An obvious first question for a given carbon skeleton is: how many distinct

Heilbronner modes and distortions are possible? Given the form of the model, where

the freedoms involve the values of b on the edges of the graph of the carbon

framework, and the constraints are imposed at the (multivalent) vertices, it is

reasonable to expect n(d), the number of independent Heilbronner modes, to depend

on a difference function of edges and vertices. The next section deals with this

connection.

3. COUNTING HEILBRONNER MODES

Given a particular graph representing a p system, construction of sets of distortions

that obey the Heilbronner condition is not difficult. A solution can be propagated

from an initial vertex, bringing in new parameters for each edge as needed, subject

P. W. Fowler222

to the constant-sum constraint, and making a final sweep to remove any

redundancies introduced by cycles in the graph. The procedure is reminiscent of

the well known construction for non-bonding eigenvectors of a Huckel framework

[15], and it can be used in any individual case to obtain the number of independent

Heilbronner distortions. The number of free parameters left at the end of the

procedure is equal to n(d).

An example of the construction is given in Fig. 2 for the [n ]-polyacene.

Parameters a1;… are used to describe the departure of a bond b integral from its

original value: b! ð1 þ aiÞb: Beginning at one end of the chain, a parameter a1 is

attached to the starting ring. Propagation along the top half of the perimeter leads to

introduction of one new parameter for each hexagon encountered, a2; a3;… until

the final ring is reached, taking parameter ^an21 on the five of its edges that lie in

the boundary. Consideration of the sum rule at the vertices on internal edges, starting

from the ends of the molecule and working into the centre, then shows that the

assignment of parameters to the lower perimeter follows that of the top as an exact

mirror image, requiring no new independent parameters.

In this case there are exactly as many independent Heilbronner vectors as there

are rings in the molecule, although as Fig. 2 also shows, the number of parameters

can be less than the number of rings, as in pentacene, or more, as in butadiene.

Fig. 2. Construction of Heilbronner modes: (a) in [4]polyacene, attachment of parameters tothe perimeter clockwise from a to d determines all remaining parameters; (b) in pentalene,propagation outwards from one vertex of the central bond leads to a mode with a ¼ 2b andc ¼ 0; (c) in butadiene, specification of a single parameter on any one edge of the graphdetermines the whole mode.

Symmetry Aspects of Distortivity in p Systems 223

These three examples cover all the possible cases. The relation between n(d), n(e)

and n(v) depends on two factors: the presence/absence of pendant vertices, and the

presence/absence of odd cycles in the graph. The set of independent parameters is

determined by solution of a set of linear equations, one for each vertex at which the

sum constraint is to be applied, and with every edge appearing in two such

equations. The distinct cases are as follows [13]. If the graph has no pendant

vertices, then either

(a) the graph contains at least one odd cycle, and each edge introduces a freedom

and each vertex contributes an independent constraint, i.e.,

nðdÞ ¼ nðeÞ2 nðvÞ ð4Þ

or

(b) all cycles in the graph are even, and

nðdÞ ¼ nðeÞ2 nðvÞ þ 1 ð5Þ

where the term þ1 on the LHS arises from the fact that in such a graph it is

possible to find (exactly) one combination of vertex constraints such that each

edge parameter appears with a weight of zero, and hence to find one vertex

constraint that is no constraint at all (see below).

If the graph has pendant vertices, then

(c) the number of constraints is now equal to the number of non-pendant vertices

and the number of Heilbronner modes is

nðdÞ ¼ nðeÞ2 nðvÞ þ nðv1Þ ð6Þ

where n(v1) is the number of pendant vertices. This formula is independent of

the parity of any cycles that may be present, as for nðv1Þ – 0 it is no longer

possible, even for bipartite graphs, to construct the special combination of

vertex constraints in which every edge has zero weight: a pendant edge is

constrained only at one end, so that its parameter appears in only one linear

equation.

Thus, case (a) tells us, for example, that odd monocycles have no Heilbronner

modes, odd–odd bicycles have 1 mode, and all-odd trivalent polyhedra on n(v)

vertices, such as the tetrahedon and the dodecahedron, have 3nðvÞ=2 2 nðvÞ ¼

nðvÞ=2 modes, as do the fullerenes.

Case (b) tells us that even monocycles have one Heilbronner mode and indeed,

since nðeÞ2 nðvÞ þ 1 is the number of rings in a polycyclic molecule, that any all-

even polycyclic has as many Heilbronner modes as it has rings. Any all-even

trivalent polyhedron such as the cube or 2n-gonal prism has as many Heilbronner

modes as it has rings in its Schlegel diagram, i.e., one fewer than the number of faces

of the polyhedron itself.

Case (c) tells us that all unbranched polyene chains have one Heilbronner mode,

and more generally, all trees (acyclic connected graphs) have nðv1Þ2 1 Heilbronner

P. W. Fowler224

modes, since nðeÞ ¼ nðvÞ þ 1 for any tree. (The unbranched tree is the path, with

nðv1Þ ¼ 2 terminal vertices.)

The three relations connect the raw counts of structural components with the

numbers of Heilbronner modes, without considering the equivalences introduced by

symmetry, which reduce the number of independent distortions that need to be

considered. Section 4 introduces this aspect.

4. SYMMETRY AND HEILBRONNER MODES

Scalar counting relations for sets of structural components can seen as expressions

for characters under the identity operation of more general relations between

representations of those sets. For example, the Euler relation in topology can be

generalised to connect not only the numbers of edges, vertices and faces of a

polyhedron, but also various symmetries associated with the structural features. The

well-known Euler theorem

nðvÞ þ nð f Þ ¼ nðeÞ þ 2 ð7Þ

for a spherical polyhedron with n(v) vertices, n(e) edges and n( f ) faces, is converted

in this way to an equation in reducible representations [16]:

GsðvÞ þ Gsð f Þ £ Ge ¼ GkðeÞ þ G0 þ Ge ð8Þ

where the subscripts s and k denote a permutation representation and a

representation of vectors along edges, respectively. G0 is the totally symmetric

representation, and Ge is the determinantally antisymmetric representation in the

point group of the polyhedron.

Similarly, in mechanics, the extended Maxwell condition for rigidity of bar and

joint assemblies can be used to generate a relation between the permutation

representations of the bars and joints and those of the states of self-stress and

mechanisms of the assembled framework [17a,b].

In the present case, the extension of the scalar counting rules for n(d) to symmetry

theorems is straightforwardly achieved by replacing n(d), n(v), n(e) n(v1) by the

permutation representations GsðdÞ; GsðvÞ; GsðeÞ and Gsðv1Þ [13]. A permutation

representation GsðxÞ; of a set of objects x has character xðRÞ under operation ðRÞ of

the symmetry group of the undistorted framework, where xðRÞ is equal to the

number of objects unshifted under operation R. The subscript s is often dropped if

there is no danger of confusion. With these replacements, equation (4) becomes

GðdÞ ¼ GðeÞ2 GðvÞ ð9Þ

The translation of equation (5) into symmetry-extended form requires more

discussion. A bipartite graph is one in which the vertices can be partitioned into two

sets, starred and unstarred, say, such that every starred vertex has only unstarred

neighbours and vice versa. The origin of the þ1 in the scalar equation (5) is that it is

possible to define a vector of coefficients on the vertices of a bipartite graph such that

Symmetry Aspects of Distortivity in p Systems 225

each starred vertex carries coefficient þ1 and each unstarred vertex 21; in the

absence of pendant vertices, the two sets are of equal size, and the bipartite property

then ensures that every edge of the graph links a coefficient þ1 to a coefficient 21.

In this case of a bipartite graph without pendant vertices, if the coefficients in the

vector are used to define a weighted combination of vertex sum constraints, each

edge parameter will appear with equal and opposite signs and the weighted

‘constraint’ condition will be identically satisfied by any set of edge parameters. As

noted earlier, this restores one edge freedom to the system.

The symmetry corresponding to the null constraint on the Heilbronner modes is

the representation of the vector of ^1 coefficients. This is a one-dimensional (1D)

irreducible representation, Gw; which has character þ1 under those operations that

permute vertices only within their starred and unstarred sets, and character 21 under

all the other operations, those that permute starred with unstarred vertices. The

symmetry Gw is that of the inactive vertex constraint. With it, the scalar relation

nðdÞ ¼ nðeÞ2 nðvÞ þ 1 becomes

GðdÞ ¼ GðeÞ2 GðvÞ þ Gw ð10Þ

For case (c), the scalar relation (6) for graphs with pendant vertices becomes

GðdÞ ¼ GðeÞ2 GðvÞ þ Gðv1Þ ð11Þ

Equations (4)–(6) are, respectively, the characters xðEÞ of equations (9)–(11)

under the identity operation E. In the trivial group C1, the scalar and representation

forms of the equations are the same, but whenever an undistorted molecular

framework has some non-trivial symmetry, the symmetry-extended forms provide

potentially useful extra information in that they reduce the set of Heilbronner modes

to the minimum set of distinct distortions, and give their symmetry characteristics,

which in many cases serve to define completely the distortions allowed by the model.

The significance of the symmetry treatment is that it shows explicitly which of the

Heilbronner modes are inherently distortive. Any totally symmetric Heilbronner

mode corresponds to readjustment of b parameters and hence of bondlengths, but

without loss of symmetry. Since the point of departure of the model is that the

molecule is already in its s-optimal geometry, such modes can be ignored, and only

the non-totally symmetric distorting modes retained.

The number of totally symmetric Heilbronner modes follows from counting of

orbits. An orbit is a set of equivalent (structureless) objects, which are permuted

amongst themselves by symmetry operations of the group: every operation of the

group either leaves a given member of the orbit in place and unchanged, or moves it

to another location. Thus the six edges of benzene, the two face centres of pentalene

and the pair of terminal vertices of an [n ]-polyene chain, all form orbits. Each orbit

has an associated permutation representation that contains the totally symmetric

representation G0 exactly once.

Totally symmetric Heilbronner modes are therefore counted in cases (a)–(c) by

replacing the total numbers of components n(e), n(v), n(v1) by the respective

numbers of orbits. The þ1 entry in equation (5) is retained when Gw ¼ G0

P. W. Fowler226

(i.e., when the starred and unstarred sets are each composed of whole orbits), and

deleted otherwise. As an example, D2h naphthalene has four orbits of edges (sets of

4, 4, 2, 1), and three orbits of vertices (4, 4, 2). As its starred and unstarred vertex

sets contain partial orbits, Gw – G0 is a dipole symmetry for this molecular graph.

Hence, naphthalene has 4 2 3 ¼ 1 totally symmetric Heilbronner modes, and one

distortive mode consisting of alternating enhancement and depletion of b around

the perimeter, with no contribution from the central bond.

Heilbronner mode symmetries have been tabulated for various series of p

systems [13]. Some specific results are: in unbranched polyenes, the unique

Heilbronner mode is either totally symmetric [(2n)-polyene] or has the symmetry

of a dipole moment along the chain [(2n þ 1)-polyene]; in [2n ]-linear acenes the

Heilbronner modes span nAg þ nB1u of D2h, and in [2n þ 1]-linear acenes have an

extra B1u component; the Heilbronner modes of the tetrahedron, cube and

dodecahedron span EðTdÞ; Eg þ T2uðOhÞ; and Hg þ HuðIhÞ; respectively, reducing

the sets of modes to be considered from 2, 5 and 10 to just 1, 2, and 2 independent

distortive modes which can be constructed easily ‘by hand’.

Symmetry also gives insight into how the scalar counts n(d) are achieved. In

benzene, for example, the edges and vertices are in different orbits of size 6,

distinguished by their behaviour under C02 and C00

2 operations. Standard

spherical-harmonic analysis shows that both edge and vertex permutation

representations contain s, p and d components corresponding to functions with

components of angular momentum along the principal axis, L, equal to 0, ^1,

^2. The remaining sixth combination differs between the two permutation

representations and is one of the orthogonal halves of the lLl ¼ 3 pair of

f-functions, intersected by three nodal lines in the molecular plane. Of this pair,

the vertex combination is cancelled by Gw; leaving as the only contribution to

GðdÞ the fully alternating edge combination. This, of course, is the Kekule mode

derived by the pencil-and-paper method.

The power of the spherical-harmonic argument is that it shows that the same

cancellation process will occur for all even monocycles, always leaving GðdÞ as the

maximally sign-alternated combination of edges with the highest lLl quantum

number. Likewise for the odd monocycle, since edge and vertex orbits coincide

exactly in the D2nþ1h groups of these systems, where all C02 axes form a single class,

Heilbronner modes of every conceivable symmetry must vanish identically, and the

result nðdÞ ¼ 0 is seen to have roots deeper than the simple equality of numbers of

edges and vertices.

5. HEILBRONNER MODES IN POLYHEDRA

Symmetry also adds useful detail to the picture for polyhedral 3D p systems. A

trivalent polyhedron obeys

GsðvÞ £ GT ¼ GsðeÞ þ GkðeÞ ð12Þ

Symmetry Aspects of Distortivity in p Systems 227

where GT is the representation of the three translations, as is easily verified by noting

that if a trivalent polyhedron is marked with a pair of points at one-third and two-

thirds of the way along each edge, the markings can be grouped in two ways: every

edge carries a distinct pair of marks, by construction, and every vertex is surrounded

by a distinct triangle of marks. Considered as edge pairings, the in- and out-of-phase

combinations transform as a scalar on and a vector along the edge, but considered as

vertex triples, the three independent combinations at a vertex transform as a radial

vector (s) and a pair of tangential vectors (p), matching the representation of a set of

local xyz-axes on all vertices, hence

GsðvÞ þ GpðvÞ ¼ GsðvÞ £ GT ¼ GsðeÞ þ GkðeÞ ð13Þ

The edge-vertex difference function of equations (9) and (10) then becomes

GsðeÞ2 GsðvÞ ¼ GpðvÞ2 GkðeÞ ð14Þ

and so the Heilbronner symmetry GðdÞ can be seen either as a difference of scalar

functions on edges and vertices or of vector functions tangential to vertices and

parallel to edges. Thus for trivalent polyhedra, equations (9) and (10) may be re-

expressed as

GðdÞ ¼ GpðvÞ2 GsðvÞ2 GkðeÞ þ ðGwÞ: ð15Þ

The extended Euler theorem can also be used to give a version of equation (15) in

which vertex and face, but not edge, terms appear.

A special subclass of trivalent polyhedra, of special interest in the theory of the

electronic structure of the fullerenes, is that of the leapfrogs [18a,b]. A leapfrog

polyhedron is obtained from a general trivalent polyhedron by a transformation

process which can be described combinatorially as first capping all faces and then

taking the dual of the result, or as an operation on the graph in which every edge of

the parent is first crossed by a new tangential edge, the ends of the new edges are

joined to form faces inset in those of the parent, and then all vertices and edges of the

parent are deleted. The leapfrog graph obtained from a trivalent polyhedron is itself

the skeleton of a trivalent polyhedron, containing all the original faces of the parent

in place but rotated, and with a new hexagonal face centred on every original

trivalent vertex.

Leapfrogging a fullerene polyhedron produces another fullerene, of the same

symmetry but with three times as many vertices and with a properly closed p shell

(in this case, with a bonding HOMO and antibonding LUMO). The leapfrog

construction gives rise to nearly all of the fullerenes that have this p-ideal electronic

configuration. The set of crossing edges constitutes one-third of the edges of the

enlarged fullerene. A Fries-type Kekule structure achieving the maximum possible

number of simultaneous benzenoid hexagons can be formed by treating the crossing

edges as formally double and all others as formally single. In consequence, the

symmetry of the occupied orbitals in a leapfrog fullerene is exactly that of the edges

of the parent, Gsðe;PÞ:

P. W. Fowler228

Consideration of the description of leapfrogging in terms of edge crossings

suggests formulas for the permutational representations of the leapfrog polyhedron

Lf in terms of structural representations of the parent, P. From

Gsðe; Lf Þ ¼ Gsðe;PÞ þ Gsðv;PÞ £ GT ¼ 2Gsðe;PÞ þ Gkðe;PÞ ð16Þ

and

Gsðv; LÞ ¼ Gsðe;PÞ þ G’ðe;PÞ ð17Þ

the symmetry of the Heilbronner modes of the leapfrog can be expressed in terms of

parent properties, involving the difference in symmetry of tangential vectors along

and across edges:

Gðd; Lf Þ ¼ Gsðe;PÞ þ Gkðe;PÞ2 G’ðeÞ þ ðGwÞ ð18Þ

where Gw is to be included only for bipartite polyhedra. (If P is a bipartite spherical

polyhedron, then so is Lf.)

In general, G’ðeÞ and Gkðe;PÞ are not equal, being related by a global twist

Gkðe;PÞ ¼ G’ðeÞ £ Ge ð19Þ

but the two are equal in some circumstances, as when P is chiral (implying Ge ¼ G0)

or when all faces of P are of odd size, as in the tetrahedron or the dodecahedron.

Thus, for the icosahedral C60 fullerene, which is the leapfrog of the dodecahedron,

Gðd;C60Þ ¼ Gsðe;C20Þ ð20Þ

and the Heilbronner modes span exactly the same 30-orbit of the Ih group as the

edges of the dodecahedron, the formal double bonds of the dominant Kekule

structure of C60, and the set of 30 occupied p orbitals of this, the smallest leapfrog

fullerene.

The symmetry of the Heilbronner modes of C60 thus includes just one totally

symmetric mode, already implicitly taken into account by the difference in length of

the 60 long (pentagon–hexagon) and 30 short (hexagon–hexagon) edges of the s

framework. The potentially distortive Heilbronner modes of C60 span

Gðd;C60Þ ¼ Gg þ 2Hg þ T1u þ T2u þ Gu þ Hu ð21Þ

6. HEILBRONNER MODES AND LINE GRAPHS

A Heilbronner mode is defined by a set of scalar quantities attached to the edges of a

graph, which suggests a possible connection with the eigenvectors of the edge-

incidence matrix, i.e., the adjacency matrix of the line graph. The line graph L(G)

has one vertex for every edge of G, and two vertices of the line graph are adjacent if

and only if the corresponding edges of G have a vertex of G in common.

We restrict attention to graphs G that have no pendant edges. For concreteness,

take a graph G corresponding to a trivalent polyhedron P, where each edge has four

Symmetry Aspects of Distortivity in p Systems 229

neighbouring edges. Let coefficients be attached to a central edge and its neighbours

as in Fig. 3.

Now, if the local pattern of coefficients is part of a Heilbronner mode, we have

a0 þ a1 þ a2 ¼ 0 ð22Þ

a0 þ a3 þ a4 ¼ 0 ð23Þ

and similar equations centred in turn on every edge of P. On the other hand,

if the local pattern of coefficients forms part of an eigenvector of the

adjacency matrix of the line graph derived from G, then for each edge considered

as a centre

2lLi a0 þ a1 þ a2 þ a3 þ a4 ¼ 0 ð24Þ

for some eigenvalue lLi : Comparison of equations (22) and (23) with equation (24)

shows that any Heilbronner mode of a trivalent polyhedron must also be an

eigenvector of the adjacency matrix of L(G), with lLi ¼ 22:

Furthermore, any Heilbronner mode of any graph without pendant edges

corresponds to such an eigenvector, as can be verified by deleting one but not both

neighbouring edges at each vertex of the central edge, equivalent to setting some of

the ai to zero.

The reverse implication also holds. Suppose there is an eigenvector with lL ¼ 22

that is not a Heilbronner mode. This would mean that a set a1;…; a4 satisfied

equation (24) without satisfying either of equations (22) or (23). The central edge of

that set must then have an excess at one vertex, a0 þ a1 þ a2 ¼ x . 0; say, balanced

by a deficiency at the other, a0 þ a3 þ a4 ¼ 2x: Condition (24) demands that the

excess/deficiency propagate in a consistent pattern of ^signs on vertices such that

every edge has a positive and a negative end with the same absolute value lxl: This is

impossible if the graph is non-bipartite. Even if the graph is bipartite, it leads to a

contradiction for x – 0 once edge parameters are assigned: on a walk along edges

away from a given central edges, the assigned parameters have contributions of x or

2x multiplied by ever larger integers as the number of steps increases; since every

Fig. 3. Local pattern of coefficients on the edges of a trivalent graph.

P. W. Fowler230

edge occurs in a cycle, this leads sooner or later to equality of x with a multiple of x

and hence x ¼ 0 (see Fig. 4 for an example).

In the case of regular graphs (i.e., those where all vertices of G are of equal

degree), the one-to-one correspondence between Heilbronner modes and edge-

adjacency eigenvectors can also be easily be proved from a mathematical result for

the spectra of their line graphs. The eigenvalues of an adjacency matrix are the roots

of the characteristic polynomial of the graph. The characteristic polynomial of the

line graph L(G) is related to that of G by [19]

PLðGÞðlÞ ¼ ðlþ 2ÞnðeÞ2nðvÞPGðl2 d þ 2Þ ð25Þ

where d is the degree of G (assumed regular), and n(e) and n(v) are the numbers of

edges and vertices of G, as before. For d ¼ 3;

PLðGÞðlÞ ¼ ðlþ 2ÞnðvÞ=2PGðl2 1Þ ð26Þ

and so the eigenvalue spectrum of L(G) contains a shifted copy of the spectrum of G,

plus nðeÞ2 nðvÞ ¼ nðvÞ=2 roots 22. Every root 23 of PG generates an additional

22 root of PL(G). Thus we have nðeÞ2 nðvÞ eigenvalues 22 for non-bipartite G and

nðeÞ2 nðvÞ þ 1 for bipartite G, as a connected trivalent bipartite graph has exactly

one adjacency eigenvalue lA ¼ 23: As these are exactly the values of n(d) for the

two cases (a) and (b), the one-to-one nature of the correspondence between

Heilbronner modes and lL ¼ 22 eigenvectors is demonstrated.

Monocycles offer an easy example of this match; a monocycle G is its own line

graph and has d ¼ 2; an even monocycle has one eigenvalue lA ¼ lL ¼ 22 and

nðdÞ ¼ 1; an odd monocycle has no 22 eigenvalue, and indeed nðdÞ ¼ 0:The link with line graphs could be used to generate Heilbronner modes

(inefficiently) by diagonalisation of the adjacency matrix of the line graph, but more

interestingly, it gives symmetry information about the eigenspaces of line graphs

themselves. The structure of the spectrum of the line graphs of fullerenes and other

trivalent polyhedra has already been noted in passing in the study of ‘decorated’

fullerenes [20], where various transformations of the polyhedra lead to spectra that

Fig. 4. Demonstration that a pattern of edge parameters obeying lL ¼ 22 is also aHeilbronner mode. The signs indicate the presumed excess/defect ^x in each vertex sum.Propagation clockwise from b and anticlockwise from a leads to a contradiction on the centraledge unless 2a 2 b 2 x ¼ 2a 2 b þ 8x; x ¼ 0:

Symmetry Aspects of Distortivity in p Systems 231

consist of heavily degenerate special eigenvalues plus one or more distorted copies

of the spectrum of the original graph. The copies arise where the eigenvalues of the

decorated structure are roots of polynomial equations whose coefficients are

functions of the original eigenvalues lA:In the case of the line graph of a fullerene Cn, the spectrum consists of the entire

adjacency spectrum of the fullerene itself (shifted by a constant),

lLi ¼ lA

i þ 1; i ¼ 1;…; n ð27Þ

with lLn . 22; together with an n/2-fold degenerate eigenvalue

lLi ¼ 22 i ¼ n þ 1;…; 3n=2: ð28Þ

The eigenvectors of L(G) span Gðe;GÞ; as the basis is a set of scalar quantities on the

edges of the graph G, and the correspondence with Heilbronner modes then shows

that the eigenvectors of L(Cn) span G(v) for i ¼ 1;…; n and GðdÞ ¼ GðeÞ2 GðvÞ for

i ¼ n þ 1;…; 3n=2: Given an eigenvector of the fullerene graph, with coefficients ci

on vertices, an eigenvector of L(G) in the G(v) subspace follows by assigning to each

edge the sum of the coefficients on its end vertices; given an eigenvector on L(G) in

this space, the process can be reversed, using the implied eigenvalue, to find an

eigenvector of G.

7. WHEN DOES DISTORTION OCCUR?

Interesting though the Heilbronner modes may be as graph theoretical objects, their

chemical interest lies in their utility or otherwise in predicting distortion of a p

framework. A formal theory of the distortive tendencies of p systems was proposed

in an early paper by Binsch et al. [12] and gives a point of reference for the simpler

model.

The bond-polarisability matrix is a second-order perturbation property whose

elements pij,kl define the derivative of the p bond order of ij with respect to the b

integral of bond kl. Bond order is defined from the product of the orbital coefficients

for the two ends of the bond, summed over all occupied spin orbitals, and elements

pij,kl are functions of the energies and coefficients of occupied and unoccupied

orbitals, and hence follow directly from the adjacency matrix. With additional

approximations, which are more severe for non-bipartite systems, the eigenvectors

that result from diagonalisation of the nðeÞ £ nðeÞ bond-polarisability matrix

correspond to the principal directions of distortion by which the p system can lower

its energy. The eigenvectors of this matrix span the full permutation representation

of the edges, G(e); its eigenvalues give the curvature of the p energy with respect to

distortion of b integrals along the eigenvector and hence can be used to predict

whether distortion will actually occur. Estimation of the variation of bond length

with b, and typical force constants, leads to a rule of thumb that a p system will

distort if the bond-polarisability matrix has an eigenvalue larger than ,1.7b 21 [12].

P. W. Fowler232

The classic case for application of the rule is pentalene. The 9 £ 9 bond-

polarisability matrix of this system has one large eigenvalue (2.357b 21),

corresponding to a b2g vector that leads away from the maximum D2h symmetry

of the graph to the twisted C2h structure that is found in full ab initio optimisations.

This is clearly a success for the bond-polarisability model. However, application of

the simple Heilbronner model gives directly, without calculation, a single mode of

the correct symmetry, almost indistinguishable from the eigenvector of the bond-

polarisability matrix [10]; the large second derivative of Ep is easily confirmed by a

single-point energy calculation.

More generally, the Heilbronner vectors have a strong tendency to pick out the

most distortive modes of the p system [13]. As noted earlier, the symmetry rule (9)

gives the Heilbronner modes for dodecahedral C2þ20 as the pair of 5-fold degenerate

sets Hg þ Hu; of these Hu shadows exactly the most distortive bond-polarisability

eigenvector [13]. Symmetry gives an immediate pictorial explanation, showing

the cylindrically symmetric component of the distortion mode as an alternating

expansion and contraction of bonds along the equator of the dodecahedron.

Likewise for neutral C60, the Heilbronner modes that were noted earlier to span the

permutation representation of the formal double bonds of the fully symmetric

Kekule structure, give 7 distinct Ep profiles, all clustering around the most distortive

of the eigenvectors of the 90 £ 90 bond-polarisability matrix [13]. In neither

fullerene is the distortion threshold reached, but the model gives a clear indication of

the modes that are closest to achieving distortion. Distortion has been predicted for

some fullerenes [21] and in scanning for the very large isomer sets for further

possibilities, the advantages of the simple model, either by itself or as a method of

rationalising bond-polarisability results, are evident.

An alternative approach to the prediction of distortive behaviour in closed-shell

systems uses the well-established language of the second-order Jahn–Teller effect.

Nakajima et al. [11] derived a rule of thumb for the type of symmetry loss to be

expected in non-bipartite polycyclic systems: the irreducible representation of the

symmetry breaking mode should be one that is present in the product of the HOMO

and LUMO representations in the higher group. Prediction of whether, by how

much, and with what detailed geometric changes the molecule actually distorts

along a pathway of the favoured symmetry then requires an explicit calculation.

Within the limits of the information that it can provide, the approach is generally

successful for the non-bipartite (non-alternant) molecules for which it was designed

[11]. It has nothing to say about possible distortions of bipartite (alternant) systems,

where the HOMO–LUMO product symmetry contains Gw; which in many cases is

not a possible Heilbronner-mode symmetry [13]. The rule is also less specific for

non-alternants of high symmetry, where degeneracies may imply that the HOMO–

LUMO product spans several symmetries, with no guide to which will be the

symmetry of the softest mode.

The crucial advantage of the Heilbronner-mode approach is that both symmetry

and detailed form of potential distortions are predicted, without requiring

knowledge of the electronic structure. Its success in picking out the most distortive

Symmetry Aspects of Distortivity in p Systems 233

modes from amongst the eigenvectors of the bond-polarisability matrix stems from

its recognition of the oscillatory nature of the response of conjugated systems to

perturbation: modification of one bond integral tends to produce alternating

strengthening and weakening of bonds along chains away from the perturbed site.

The bond-polarisability matrix elements reflect this trend [12], and Heilbronner’s

intuitive constraint is therefore an ideal device for concentrating attention on the

eigenvectors of maximum eigenvalue.

8. DISTORTION AND AROMATICITY

The picture of a p system that emerges from the work of Heilbronner, Shaik and

others is one where delocalisation is a consequence rather than a cause of the regular

geometry in systems such as benzene. On this view, an important characteristic of p

electrons is the tolerance of p bonding to geometric constraints. Recent progress in

calculation and interpretation of the magnetic properties of p systems sheds further

light on this aspect in that it shows the geometric resilience of characteristic ring

currents.

Ring currents are inferred in experiment from 1H NMR chemical shifts at sites

attached to the carbon skeleton [5], and by analogy, are often inferred in calculation

from calculated ring-centre shifts [6]. However, it is now possible to calculate the

pattern of current density induced in a molecule by an external magnetic field

directly, and to assign ring-current aromaticity on the basis of the current-density

map itself. A key advance in ab initio methods was the proposal [22a–d] to use a

distributed-gauge method for computation of current density in which the origin of

vector potential is a function of the point at which the current density is to be

calculated. In the simplest version, current density at each point is calculated with

the point itself taken as the origin of vector potential. This ipsocentric [7a,b] choice

yields maps of excellent quality, even with basis sets of only moderate size, and the

maps can be used routinely to discuss the sense, intensity and physical location of

ring currents.

Orbital contributions to the perturbed wavefunction are sum-over-states

expressions with several special features: (i) each term is a transition integral

divided by an orbital energy difference; (ii) the integrals define virtual excitations

from occupied to unoccupied molecular orbitals – there is no remixing of occupied

orbitals; (iii) the terms leading to diamagnetic (paramagnetic) circulations in planar

molecules in the presence of a perpendicular magnetic field obey in-plane

translational (rotational) selection rules. As has been discussed extensively

elsewhere, these features lead to an interpretation of the current map as governed,

in delocalised systems, by transitions from a few occupied frontier orbitals. The

sense of a current can often be deduced from the symmetry characteristics of the

HOMO, LUMO and nearby orbitals, and even simple Huckel calculations may

be sufficient to give these symmetries. Orbital contributions defined in this way also

have a unique physical status, in allowing first-order prediction of the changes in

P. W. Fowler234

maps caused by electron gain or loss, or chemically modification of substituents on

the ring, in terms of blocking and opening up of channels for occupied-virtual

excitations.

For example, a pure symmetry argument based on the angular-momentum

characteristics ofp molecular orbitals implies diamagnetic ring currents for 4n þ 2p

monocycles, and paramagnetic ring currents for 4n p monocycles, and furthermore

ascribes essentially all the ring current in each case to the activity of 4 HOMO

electrons for the aromatic (diatropic) and 2 HOMO electrons for the antiaromatic

(paratropic) systems [7a,b].

The few-electron character of ring currents uncovered by the ipsocentric approach

prompts many questions about the links between aromaticity/strobilism and

distortivity. Just two examples from recent work [14] will be discussed briefly here.

The classic case of pentalene is one where the sense of the ring current and the

direction of the distortion are linked. Bond-polarisability, Heilbronner-mode

analysis and the HOMO–LUMO rule all agree in this case. The unique Heilbronner

distortion has the symmetry of the HOMO–LUMO product which is also the

symmetry of a rotation about the out-of-plane C2 axis. The HOMO–LUMO gap is

small. Thus, the D2h molecule is predicted to be unstable to distortion and to exhibit

a strong paratropic ring current, both predictions being confirmed by ab initio

calculation of the geometry and the current map [14c]. One and the same transition

supplies a large bond-polarisability eigenvalue, drives the distortion of the

framework, and gives a ring current with the paratropic sense expected for an 8p

antiaromatic perimeter. Significantly, the paratropic current persists, albeit with

reduced intensity, even after the distortion has taken place (Fig. 5); this is to be

expected on a perturbation argument as the topologies of HOMO and LUMO remain

essentially unchanged by the symmetry breaking.

A spectacular example of resilience of p currents in the presence of s constraints

is given by the current densities of various clamped benzenes. Production of a

Fig. 5. Computed map of the p current density calculated in the ipsocentric approach forpentalene optimised at the RHF/6-31G**level [14c]. Arrows show the projection of thecurrent density induced by unit perpendicular magnetic field in the plane 1 bohr above themolecule. The clockwise sense indicates paramagnetic (anti-Lenz) circulation.

Symmetry Aspects of Distortivity in p Systems 235

localised 1,3,5-cycohexatriene form of the benzene ring has been a long-standing

target of the art and sport of synthetic organic chemistry. Significant bond

alternation in the benzene ring can be induced by annelation with rigid clamping

groups, and p localisation has been claimed for several molecules synthesised on

this pattern [23a,b].

Mapping of the current densities in these and a wide range of hypothetical

clamped systems gives a surprising result [14a,b]. Substantial bond alternation can

coexist with the characteristic 4-electron diatropic ring current of benzene, but

systems with similar alternation can also be found where the current is quenched.

The orbital picture based on the ipsocentric approach gives a systematic rationale

for these findings. Essentially, current is retained where the topology of the HOMO

and LUMO orbitals is retained, and quenched where it is not. In systems where the

geometric change is produced by s clamping groups, the frontier orbitals remain

benzene-like, and the current associated with the benzene HOMO ! LUMO

transition is present. In those systems where the clamping involves groups that retain

a p system, new p and p w orbitals are introduced into the frontier region, pushing

the benzene-based orbitals further apart, and the strong transitions responsible for

the specific ring current of the central ring are lost by attenuation and mixing. An

entirely similar dichotomy explains the retention and loss of paratropic ring current

in various s and p-clamped planarised cyclooctatetraene molecules.

The message from both pentalene and benzene/cyclohexatriene examples is of the

persistence of the mobility of p electrons under constraints imposed by a s

framework. This resilience and p-distortivity are opposite sides of the same coin.

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Symmetry Aspects of Distortivity in p Systems 237