Heilbronner modes

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 7r-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... 

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 /3 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(8), the number of independent Heilbronner modes, to depend on a difference function of edges and vertices. The next section deals with this connection. 

Fig. 2. Construction of Heilbronner modes (a) in [4]polyacene, attachment of parameters to the 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 — —b and c = 0 (c) in butadiene, specification of a single parameter on any one edge of the graph determines the whole mode.

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 3n(v)/2 — n(v) = n v)/2 modes, as do the fullerenes. 

Case (b) tells us that even monocycles have one Heilbronner mode and indeed, since n e) — n(v) + 1 is the number of rings in a polycyclic molecule, that any alleven 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 n(v ) — 1 Heilbronner... 

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. 

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 (ID) irreducible representation, 7. which has character +1 under those operations that permute vertices only within their starred and unstarred sets, and character - 1 under all the other operations, those that permute starred with unstarred vertices. The symmetry T is that of the inactive vertex constraint. With it, the scalar relation n(S) = n(e) — n(v) + 1 becomes... 

Equations (4)-(6) are, respectively, the characters (/i) of equations (9)—(11) under the identity operation E. In the trivial group C, 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 /3 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 cr-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 [w]-polyene chain, all form orbits. Each orbit has an associated permutation representation that contains the totally symmetric representation Jo 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(vi) by the respective numbers of orbits. The +1 entry in equation (5) is retained when I = T0... 

Heilbronner mode symmetries have been tabulated for various series of n systems [13]. Some specific results are in unbranched polyenes, the unique Heilbronner mode is either totally symmetric (2/ )-polyene] or has the symmetry of a dipole moment along the chain [(2n + l)-polyene] in 2n -linear acenes the Heilbronner modes span nAg + nB u of D2h, and in [2n + l]-linear acenes have an extra BXu component the Heilbronner modes of the tetrahedron, cube and dodecahedron span E(Td), Eg + T2u(Oh), and //, + Hu(Ih), 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 . 

The symmetry of the Heilbronner modes of Ceo 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 a framework. The potentially distortive Heilbronner modes of Ceo span... 

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

Fig. 4. Demonstration that a pattern of edge parameters obeying XL = — 2 is also a Heilbronner 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 central edge unless —a — b — x = — a — b + 8x, x = 0.

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 7r framework. A formal theory of the distortive tendencies of tt systems was proposed in an early paper by Binsch et al. [12] and gives a point of reference for the simpler model. 

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... 

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