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Journal of Mathematical Chemistry

, Volume 56, Issue 8, pp 2512–2524 | Cite as

Partition of \(\pi \)-electrons among the faces of polyhedral carbon clusters

  • Tomislav Došlić
  • Ivana Zubac
Original Paper
  • 55 Downloads

Abstract

We apply the concepts of importance and redundancy to compute and analyze the partition of \(\pi \)-electrons among faces of actual and potential polyhedral carbon clusters. In particular, we present explicit formulas and investigate asymptotic behavior of total and average \(\pi \)-electron content of all faces of prisms and n-barrels. We also discuss the observed deviations from the uniform distribution and show that the patterns of net migration of \(\pi \)-electrons differ from those computed for narrow nanotubical fullerenes. Some possible directions of future work are also indicated.

Keywords

\(\pi \)-electron partition Polyhedral carbon cluster Prism graph Perfect matching 

1 Introduction

This paper extends the line of research of Ref. [3] that was concerned with patterns of distribution of \(\pi \)-electrons between faces of several actual and potential polyhedral carbon clusters. In particular, we complete their program for all cubic Archimedean solids by obtaining explicit formulas for the \(\pi \)-electron content in all faces of prisms. Besides prisms, we also consider the so-called n-barrels, that is, an infinite series of polyhedra consisting of two n-gonal bases and 2n lateral pentagonal faces. Two members of the series represent the unique fullerenes on 20 and on 24 atoms, respectively. Again, explicit formulas have been obtained. In both cases we have performed a comparison with the uniform distribution and computed the net electron excess and deficiencies of particular faces.

There are several ways in literature to define the average \(\pi \)-electron content of a ring of carbon atoms. Here we adopt the edge-based approach, as used in several recent papers [1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18] dealing with planar benzenoid and coronoid species as well as with three-dimensional polyhedra. Our way of computing the \(\pi \)-electron content relies on the concept of edge importance, closely related to the Pauling bond order. It requires counting perfect matchings in certain subgraphs of considered structures. The resulting expressions are then divided by the total number of prefect matchings. In that way we obtain explicit formulas which are then compared with distribution patterns of uniform distribution, in which a vertex (i.e., an atom) shared by k faces contributes 1 / k electrons to each face. We obtain that there is a net migration of electrons among various regions of considered structures and we determine the patterns of such migrations.

2 Preliminaries

The carbon clusters are modeled by their graphs, in which vertices represent carbon atoms, and edges represent the bonds between them. All structures considered here are represented by finite, simple and connected graphs. All of them are also 3-regular, i.e., cubic, 3-connected and 3-edge-connected. We refer the reader to any textbook on graph theory (such as, e.g., [20]) for definitions of those and related concepts not explained here.

2.1 Importance and Pauling bond order

A matching M in a graph G is a collection of edges of G such that no two edges from M share a vertex. A matching M is called perfect if each vertex of G is incident with an edge of M. The number of perfect matchings of G is usually denoted by \(\Phi (G)\) in mathematical literature. In chemical literature, however, this quantity is usually denoted by K(G), reflecting the fact that in chemical contexts perfect matchings are usually called Kekulé structures. We refer the reader to [14] for a comprehensive survey of all aspects of (not only perfect) matchings.

A classical result of Petersen states that every cubic bridgeless graph has at least one perfect matching (Here bridgeless means that it contains no edges whose removal results in a disconnected graph. The condition can be weakened by admitting at most two bridges, but we do not need it here). Hence all graphs considered here have perfect matchings. Even more, from Theorem 3.4.2 on p. 113 of [14] and its corollary two pages after, it follows that every edge of a cubic and 2-edge-connected graph G is contained in some perfect matching of G. Since the graphs representing polyhedral carbon clusters satisfy this condition, it follows that each edge of such a cluster is contained in at least one perfect matching. In order to quantitatively refine participation of edges in perfect matchings, we introduce the concept of their importance.

The importance \(\iota (e)\) of an edge e of G is defined as the number of perfect matchings in G that contain e [6]. A similar concept, known as the Pauling bond order, has been long known in chemistry. The Pauling bond order p(e) of an edge e is defined as the fraction of all Kekulé structures in G that contain e. It is computed by dividing the number of perfect matchings containing e, i.e., the importance of e, by the total number of perfect matchings in G, \(p(e) = \iota (e)/K(G)\). A closely related concept of redundancy of an edge is defined as the number of perfect matchings in G that do not contain e. We denote it by \(\rho (e)\). Since all perfect matchings in G can be partitioned to those matchings that contain e and to those that do not contain e, the relation \(\iota (e) + \rho (e) = K (G)\) must be valid for any edge e of G. If the edge e connects vertices u and v in G, we can write [6]
$$\begin{aligned} \iota (e) = K (G - u - v), \qquad \rho (e) = K (G - e). \end{aligned}$$
Here by \(G - u - v\) we denote the graph obtained from G by deleting vertices u and v and all edges incident with them, while \(G-e\) denotes the graph obtained from G by deleting edge e and keeping both of its end-vertices. Sometimes we also write \(\iota (e) = K (G\backslash e)\), where by \(G\backslash e\) we denote the graph obtained from G be deleting e and both its end-vertices.
For any vertex \(u \in V(G)\) we have
$$\begin{aligned} K(G) = \sum _{e = uv} \iota (e), \end{aligned}$$
where we sum over all edges of G incident with u. This follows from the fact that every perfect matching must saturate every vertex u and it can do it by using exactly one of the edges incident with u.

2.2 \(\pi \)-electron distribution, content and excess

It is clear that in our clusters no carbon atom can participate in more than one double bond. This means that any pattern of double bonds must form a Kekulé structure, represented by a perfect matching. Since each carbon atom is adjacent to three others, it is also clear that exactly one \(\pi \)-electron per carbon atom remains for participation in double bonds. Hence the total number of \(\pi \)-electrons that participate in Kekulé structures must be equal to the number of atoms in the cluster. What we are interested here is the exact pattern of distribution of those electrons among the faces of the cluster. We start from the pattern of their distribution among the bonds that is, in turn, given by the distribution of Pauling bond orders [10]. We construct the face distribution as follows.

Let G be a planar graph with a perfect matching, and H one of its faces. The boundary of H is defined as the set of all edges incident with H. We denote it by \(\partial H\). Let M be any perfect matching of G. Each edge from \(\partial H\) participating in M represents a double bond carrying two electrons. As the edge is shared between two faces, it is natural to assume that each of them gets one of its electrons. Then the total number of electrons given to H by M is obtained by simply counting edges from \(\partial H\) in M. The total \(\pi \)-electron content of H, \(\pi (H)\), is then obtained by adding contributions of all perfect matchings of G, while the average \(\pi \)-electron content of H, \(\overline{\pi }(H)\), is obtained by dividing its total \(\pi \)-electron content by the number of perfect matchings in G. Hence, \(\overline{\pi }(H) = \frac{\pi (H)}{K (G)}\).

By double counting of edges, one obtains a simple expression for the total \(\pi \)-electron content of a face in terms of importances of edges from its boundary.

Theorem I

$$\begin{aligned} \pi (H) = \sum _{e \in \partial H} \iota (e) . \end{aligned}$$

If we divide the above expression by K(G), we will obtain a formula for \(\overline{\pi }(H)\) that is equivalent to formula (1) of reference [10]. However, the formula from Theorem I is much better suited for computations that follow.

A simple alternative way to define the \(\pi \)-electron content of a face of G is to assume that an atom (i.e., a vertex) shared by k faces gives 1 / k of its \(\pi \)-electron to each face. In cubic, and generally in all regular clusters, this gives the same number of electrons to all faces of the same size. Nevertheless, this uniform distribution, although quite coarse, is still useful as it can serve as a baseline for comparisons with other distributions. We denote the so constructed \(\pi \)-electron content of a face H by \(\pi _a (H)\), where index a reminds us that the distribution is atom-based. The difference \(\overline{\pi }(H) - \pi _a (H)\) we call the \(\pi \)-electron excess of a face H of G and denote by \(\varepsilon _{\pi } (H)\). Hence, \(\varepsilon _{\pi } (H) = \overline{\pi }(H) - \pi _a (H)\).

In the next sections we compute \(\overline{\pi }(H)\) and \(\varepsilon _{\pi } (H)\) for all faces of prisms and m-barrels. As mentioned before, a similar problem was considered in ref. [3], where the authors considered all trivalent (i.e., cubic) regular and semi-regular polyhedra, hence three Platonic and seven Archimedean solids. Their results were obtained by first constructing all perfect matchings and then computing their contributions to particular faces. We compute the corresponding quantities here by employing a much faster approach following from Theorem I.

We start our considerations with prisms, thus completing the program of ref. [3] for all cubic semi-regular polyhedra.

3 Prisms

Prisms form one of the two infinite families of Archimedean or semi-regular polyhedra; the other one, the antiprisms, is made of 4-regular graphs and hence does not fit the scope of this paper. An n-sided prism has 2 regular n-gons as its bases and n squares as it lateral sides. We denote its graph by \(Z_n\). Hence, \(Z_n\) has 2n vertices, 3n edges and \(n+2\) faces. The Schlegel diagram of an 8-sided prism is shown in Fig. 1.
Fig. 1

A prism with n lateral faces

Clearly, there are only 2 types of symmetry-nonequivalent edges in any prism on \(n \ne 4\) sides: those separating a base and a lateral face, and those separating two lateral faces. We denote them by f and e, respectively, as shown in Fig. 1. (For \(n=4\) we get a cube in which all edges are equivalent.) It is an easy exercise to verify that the importance of e is equal to the number of perfect matchings in a ladder graph on \(n-1\) rungs, for a long time known to equal the n-th Fibonacci number \(F_{n}\). Hence, \(\iota (e) = F_n\). A simple parity argument implies that, for odd n, any perfect matching that contains f must also contain the other edge of type f from the same lateral side. The remaining graph is again a ladder, this time on \(n-2\) rungs, and its number of perfect matchings is equal to \(F_{n-1}\). For even n one must also take into account perfect matchings that do not contain any edge of type e. There are two such perfect matchings containing f, but one of them has been counted by \(F_{n-1}\). Hence, \(\iota (f) = F_{n-1} + \frac{1 + (-1)^n}{2}\). Since each vertex of \(Z_n\) is incident with one edge of type e and two edges of type f, we obtain
$$\begin{aligned} K(Z_ n) = \iota (e) + 2 \iota (f) = F_{ n+1} + F_{ n-1} + 1 + (-1)^ n = L_ n + 1 + (-1)^ n, \end{aligned}$$
where \(L_n\) denotes the n-th Lucas number. The sequence counting perfect matchings in \(Z_n\) appears as sequence A102081 in The On-Line Encyclopedia of Integer Sequences [19].
Now we can compute the Pauling bond orders and also the \(\pi \)-electrons contents in lateral faces and in the bases of the prism. We denote with \(\overline{\pi } _{m}(Z_n)\) average number of electrons that belong to an m-sided face of \((Z_n)\). For a lateral side we obtain
$$\begin{aligned} \overline{\pi } _{4}(Z_n)&=\frac{2\iota \left( e\right) +2\iota \left( f\right) }{K \left( Z_{n}\right) }= \frac{2F_{n+1}+1+\left( -1\right) ^{n}}{F_{n+1}+F_{n-1}+1+\left( -1\right) ^{n}} =2-\frac{2F_{n-1}+1+\left( -1\right) ^{n}}{F_{n+1}+F_{n-1}+1+\left( -1\right) ^{n}}. \end{aligned}$$
Since the ratio of two successive Fibonacci numbers tends to the Golden Section ratio \(\varphi = \frac{1+\sqrt{5}}{2} \approx 1.618\) for large values of n, the right-hand side behaves as \(2 - \frac{n}{\varphi ^{2}+1} \approx 1.447\) for large values of n. This quantity is greater than 4 / 3, the number of electrons assigned to a lateral side by the uniform distribution. Hence each lateral side has a net excess of approximately 0.114 electrons.
For a base we obtain
$$\begin{aligned} \overline{\pi } _{n}(Z_n)=\frac{n\iota \left( f\right) }{K \left( Z_{n}\right) }=\frac{ nF_{n-1}+n+n\left( -1\right) ^{n}}{F_{n+1}+F_{n-1}+1+\left( -1\right) ^{n}}. \end{aligned}$$
For large values of n this behaves as \(\overline{\pi } _{n}(Z_n) \sim \frac{n}{\varphi ^{2}+1}\), a quantity smaller than \(\frac{n}{3}\) electrons assigned to each base by the uniform distribution. Hence, there is a net migration of \(\pi \)-electrons from bases to lateral sides.

4 Barrels with n -gonal base

Next we look at polyhedra known ad n-gonal barrels. They are special cases of m-generalized fullerene graphs [5] with no hexagonal faces. The simplest way to construct an n-barrel is to take 2 concentric regular n-gons, insert a concentric 2n-gon between them, and then connect vertices of 2n-gon alternating between internal and external n-gon in the way shown in Fig. 2. The obtained graph is a Schlegel diagram of a polyhedron having two n-gonal bases and 2n pentagonal lateral sides. An n barrel has 4n vertices, 6n edges and \(2n+2\) faces. We denote an n-barrel by \(B_n\). Obviously, n-barrels exist for all \(n \ge 3\). A 5-barrel is the icosahedron, and a 6-barrel is the only fullerene graph on 24 vertices.
Fig. 2

An 8-barrel

From Fig. 2 we can see that there are three different classes of symmetry-nonequivalent edges in \(B_n\) for \(n \ne 5\). We denote them by r (for radial), b (for base) and g (for girth), respectively. We know that their importances are equal to the number of perfect matchings in the subgraph obtained from \(B_n\) by deleting the considered edge together with its end-vertices. Hence, we introduce notation \(\iota (r)=K \left( B_{n}^{\prime }\right) ,\iota (g)=K \left( B_{n}^{\prime \prime }\right) ,\iota (b)=K \left( B_{n}^{\prime \prime \prime }\right) .\)
Fig. 3

\(B_{n}^{\prime }\) case for radial edge

We look first at the importance of a radial edge. The graph obtained by its removal is shown in Fig. 3. Each perfect matching in this graph either contains or does not contain the edge marked by the arrow. By decomposing this graph with respect to the marked edge, we obtain auxiliary graphs \(R_n\) and \(S_n\) shown in Fig. 4. The same figure shows another two auxiliary graphs, \(R_n'\) and \(S_n'\). The number of perfect matchings in \(R_n, R_n', S_n, S_n'\) we denote by \(r_n, r_n', s_n, s_n'\), respectively. From the decomposition we get \(\iota (r)=r_{n-1}+s_{n-3}\), and our task now reduces to finding \(r_n\) and \(s_n\). We look at the case of \(r_n\) first.
Fig. 4

Four families of auxiliary graphs for a radial edge

Fig. 5

Relationship between auxiliary graphs \(R_{n}\) i \(R_{n}^{\prime }\)

From Fig. 5 we obtain a system of coupled recurrence relations for \(r_n\) and \(r_n'\):
$$\begin{aligned} r_{n}= & {} r_{n-1}^{\prime }+r_{n-2} \\ r_{n-1}^{\prime }= & {} r_{n-1}+r_{n-3}^{\prime }. \end{aligned}$$
By expressing \(r_{n-1}^{\prime }\) from the first formula and plugging it into second, we obtain a recurrence relation for \(r_n\),
$$\begin{aligned} r_{n}=r_{n-1}+2r_{n-2}-r_{n-4}. \end{aligned}$$
The recurrence is of the fourth order and it needs four initial conditions that are readily obtained by explicit enumeration of perfect matchings in the corresponding graphs. The initial conditions are \(r_{1}=1,r_{2}=3,r_{3}=5\), and \(r_{4}=10\).
The situation is similar for \(s_n\). As in the previous case, but this time from Fig. 6, we obtain a system of recurrence relations
$$\begin{aligned} s_{n}= & {} s_{n-2}+s_{n-3}^{\prime } \\ s_{n-3}^{\prime }= & {} s_{n-3}+s_{n-5}^{\prime }, \end{aligned}$$
Fig. 6

Relationship between auxiliary graphs \(S_{n}\) i \(S_{n}^{\prime }\)

which reduces to a single fourth-order recurrence for \(s_n\),
$$\begin{aligned} s_{n}=2s_{n-2}+s_{n-3}-s_{n-4}, \end{aligned}$$
with initial conditions \(s_{3}=1,s_{4}=3,s_{5}=4,s_{6}=5\).

From the above results we could, in principle, derive explicit formula for \(\iota (r)\). However, such formula would be quite complicated and hence not very informative. Instead, we are going to look at the asymptotic behavior of \(\iota (r)\) for large values of n. Before doing so, we determine the importances of the other two types of edges.

For the base edge, b, we have \(\iota (b) = K(B_n''')\). Edges of this type are, in a sense, the most interesting, since once we know their importance, we immediately know the \(\pi \)-electron content of bases, and then, by symmetry, also the \(\pi \)-electron content of lateral pentagons. The auxiliary graphs for computing \(\iota (b) = K(B_n''')\) are shown in Figs. 7 and 8.
Fig. 7

\(K \left( B_{n}^{\prime \prime \prime }\right) \) case for a base edge

Proposition 1

$$\begin{aligned} i(b)=2r_{n-2}-2r_{n-4}+s_{n}-s_{n-4}. \end{aligned}$$

Proof

Look at Fig. 7. We have \(K \left( B_{n}\backslash b\right) =K \left( B_{n}\backslash b-g\right) +K \left( B_{n}\backslash b\backslash g\right) \). In Figure 7a, we look at the case \(B_{n}\backslash b-g\). In this case, the edge marked by the arrow is forced, and it must participate in any perfect matching containing b. The remaining graph is shown in Fig. 7b. Decomposing this graph with respect to the edge marked with dotted line and arrow, we obtain two cases, \(R_{n-3}^{\prime }\) when this edge is included in a perfect matching, and \(S_{n-3}^{\prime }\) when this edge is not included. It follows \(K \left( B_{n}\backslash b-g\right) =r_{n-3}^{\prime }+s_{n-3}^{\prime }\).
Fig. 8

\(K \left( B_{n}\backslash b\backslash g\right) \) case for a girth edge

Now we consider the other case, \(K \left( B_{n}\backslash b\backslash g\right) \). The participation of g in a perfect matching necessarily includes the edge on the central circle, thus leaving the graph on Fig. 8. The key is the edge whose two ends are level 2. When it participates in perfect matching, the remaining graph is \(R_{n-3}^{\prime }\). When it does not participate, then both adjacent edges must be included, and this leaves the graph \( S_{n-5}^{\prime }\). According to this, \(K \left( B_{n}\backslash b\backslash g\right) =r_{n-3}^{\prime }+s_{n-5}^{\prime }\), and hence \( i(b)=2r_{n-3}^{\prime }+s_{n-3}^{\prime }+s_{n-5}^{\prime }.\) The result can be more neatly expressed as
$$\begin{aligned} i(b)=2r_{n-2}-2r_{n-4}+s_{n}-s_{n-4}. \end{aligned}$$
Since every vertex of \(B_n\) is incident either with a radial edge and two base edges, or with a radial edge and two girth edges, it follows immediately that the importances of radial and girth edges must be equal:
$$\begin{aligned} K(B_n) = \iota (r) + 2 \iota (b) = \iota (r) + 2 \iota (g) \Longrightarrow \iota (b) = \iota (g). \end{aligned}$$
Now we can express \(K(B_n)\) in terms of \(r_n\) and \(s_n\).

Theorem 1

$$\begin{aligned} K \left( B_{n}\right) =r_{n-1}+4\left( r_{n-2}-r_{n-4}\right) +s_{n-3}+2\left( s_{n}-s_{n-4}\right) . \end{aligned}$$
Now we can determine the \(\pi \)-electron contents of bases and lateral faces. Again, we denote the average number of electrons that belong to an m-gonal side of \(B_n\) by \(\overline{\pi }_m(B_n)\). Obviously,
$$\begin{aligned} \overline{\pi }_5(B_n)= & {} 2-\frac{\iota (b)}{K \left( B_{n}\right) }, \\ \overline{\pi }_n(B_n)= & {} \frac{n\cdot \iota (b)}{K \left( B_{n}\right) }. \end{aligned}$$
We first check the formula for the 24-atom fullerene \(B_{6}\). We obtain \(\iota (b)=2r_{4}-2r_{2}+s_{6}-s_{2}=17\), \(\iota (r)=r_{5}+s_{3}=20\) and hence \(K \left( B_{6}\right) =54\). From there it follows \(p_{6}=\frac{6\cdot \iota (b)}{K \left( B_{6}\right) }=\frac{17}{9} \approx 1.88888\), and \(p_{5}=2-\frac{17}{54}=\frac{91}{54}\approx 1.685185\). As in the prismatic case, there seem to be an excess of electrons in lateral faces and electron deficiency in bases, compared with the uniform distribution. Let us see whether the pattern persists also for large n.

We first compute the generating functions for sequences \(r_n\) and \(s_n\) and then use the variant of Darboux theorem to extract the information about the asymptotic behavior of those sequences. By a routine computation we obtain \(R(x)= \mathop {\sum }\nolimits _{n=0}^{\infty }r_{n}x^{n}\) and \(S(x) =\mathop {\sum }\nolimits _{n=0}^{\infty }s_{n}x^{n}\).

Lemma 1

$$\begin{aligned} R(x)=\frac{1}{1-x-2x^{2}+x^{4}}, \qquad S(x)=\frac{1}{1-2x^{2}-x^3+x^{4}}. \end{aligned}$$

It follows from Darboux theorem that \(r_{n}\) asymptotically behaves as \(\alpha ^{-n},\) where \(\alpha =0.524889\) is the smallest positive solution of the characteristic equation \(x^{4}-2x^{2}-x+1=0\). Similarly, \(s_{n}\) asymptotically behaves as \(\beta ^{-n},\) where \(\beta =0.671044\) is the smallest positive solution of the characteristic equation \(x^{4}-x^3-2x^{2}+1=0\). Since \(\beta ^{-1} < \alpha ^{-1}\), the asymptotic behavior is dominated by \(\alpha ^{-n}\) and we have the following result.

Lemma 2

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\frac{\iota (b)}{K \left( B_{n}\right) }=\frac{2\alpha ^{-2}-2}{\alpha ^{-3}+4\alpha ^{-2}-4}\approx 0.30167=\tau _{\infty }. \end{aligned}$$

Corollary 1

$$\begin{aligned} p_{5}\rightarrow 2-\tau _{\infty }, \qquad p_{n} \sim n\tau _{\infty }. \end{aligned}$$

Since \(\tau _{\infty } < \frac{1}{3}\), the bases are poorer with electrons and lateral sides richer than for the uniform distribution. As in the prismatic case, we have a net migration of \(\pi \)-electrons from bases toward lateral faces.

It is interesting to mention here that the pattern of \(\pi \)-electron migration toward lateral faces is not present in fullerene nanotubes, at least not in the narrowest ones. According to findings of ref. [8], where nanotubes with two hemi-dodecahedral caps were considered, all their hexagonal faces receive exactly two electrons, as is the case for the atom-based distribution. The polar pentagons, however, exhibit a slight excess of \(\pi \)-electrons, achieved at the expense of adjacent rings of 5 non-polar pentagons. The net migration is very small, amounting to two thirds of one electrons for long nanotube, one third of an electron on each end. We refer the reader to [8] for more details and for discussion of possible relevance for fullerene stability.

5 Concluding remarks

In this paper we have investigated the patterns of partition of \(\pi \)-electrons among faces of several classes of polyhedral carbon clusters. In particular, we have obtained explicit formulas for prisms and n-barrels.

There are also others classes of polyhedral compounds that could be treated by methods exposed here. Promising candidates are open narrow nanotubes and m-generalized fullerenes from ref. [5]. Also, there are still many classes of catacondensed benzenoid and coronoid compounds for which explicit formulas for \(\pi \)-electron contents could be obtained in a similar way. For example, in our work under preparation, we have been able to obtain explicit formulas and asymptotic behavior for several classes of conjugated polycyclic compounds considered in ref. [4].

Notes

Acknowledgements

Partial support of the Croatian Science Foundation via research project LightMol (Grant no. IP-2016-06-1142) is gratefully acknowledged by T. Došlić.

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Authors and Affiliations

  1. 1.Faculty of Civil EngineeringUniversity of ZagrebZagrebCroatia
  2. 2.Faculty of Mechanical Engineering and ComputingUniversity of MostarMostarBosnia and Herzegovina

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