Note on the asymptotic isomer count of large fullerenes

Abstract

The leading term in the large-$N$ asymptotics of the isomer count of fullerenes with $N$ carbon atoms is extracted from the published enumerations for $N\le 400$ with the help of methods of series analysis. The uncovered simple $N^9$ scaling is distinct from isomer counts of most classes of chemical structures that conform to mixed exponential/power-law asymptotics. The second leading asymptotic term is found to be proportional to $N^{25/3}$. A conjecture concerning isomer counts of the IPR fullerenes is also formulated.

1 Introduction

Recent advances in graph-theoretical algorithms have opened new vistas for enumeration of chemical isomers. In particular, significant progress has been achieved in the case of fullerenes $C_N$, of which all structures with $N \le 400$ have now been generated [1, 2]. The availability of these data has prompted speculations concerning the behavior of the fullerene isomer counts at the $N \rightarrow \infty$ limit, both the $N^9$ [3] and $N^{19/2}$ [4] asymptotics being inferred from crude log–log plots and supported by heuristic arguments. For the reason spelled out in the following, this simple power-law scaling appears unlikely at the first glance.

Complete information about isomer counts for a class of chemical species is encoded in the generating function $F(t)$ given by the for formal series

$$\begin{aligned} F(t) = \sum \limits _{k=1}^{\infty } M(k)\, t^k, \end{aligned}$$

(1)

where $M(k)$ is the number of isomers comprising $k$ units (such as atoms, bonds, rings, etc.). Since, in general, $M(k+1) \ge M(k)$ for all $k > 0$, the series (1) possesses a finite radius of convergence. Consequently, $F(t)$ possesses at least one singular point, at which it behaves like $(t_c-t)^{\zeta _c}$, where $0 < t_c \le 1$. The smallest critical point $t_c$ and the corresponding critical exponent $\zeta _c$ determine the leading term in the large-$k$ asymptotics of $M(k)$, which reads

$$\begin{aligned} M(k) = A\, t_c^{-k}\, k^{-(\zeta _c+1)} + \cdots , \end{aligned}$$

(2)

where $A$ is a constant. Typically, $t_c < 1$ (e.g. ca. $0.35518$ for alkanes [5], ca. $0.20915$ for polyenes [5], and $\frac{1}{5}$ for catafusenes [5–7]), giving rise to the mixed exponential/power-law asymptotics (2). On the other hand, the alleged power-law scaling of the fullerene isomer counts would imply $t_c = 1$.

In order to investigate this matter, in this note we invoke the mathematical formalism of series analysis that is commonly used in lattice statistics [8]. Such a formalism has been previously employed in successful extraction of the asymptotic isomer counts of several classes of chemical structures [5].

2 Series analysis

Let $M(k)$ be the number of isomers of the $C_{2k}$ fullerene. Let $U_0 = 1$, $W_0 = 0$, and $\{U_{k},\,k=1,\ldots m\}$, $\{V_{k},\,k=0,\ldots m\}$, $\{W_{k},\,k=1,\ldots m\}$ be the solution of the system of equations

$$\begin{aligned} \sum \limits _{j=0}^m \,\left[ \, U_j \,(k-j)^2 + V_j \,(k-j) + W_j\right] \,C_{k-j} = 0 ,\quad k = 1, \ldots , 3m-2, \end{aligned}$$

(3)

where

$$\begin{aligned} C_k = \left\{ \begin{array}{lc}M(k+n)&{}\quad \hbox {for}\quad k \ge 0\\ 0&{}\quad \hbox {for}\quad k <0\end{array}\right. . \end{aligned}$$

(4)

Let

$$\begin{aligned} Q(z) = z \sum \limits _{k=0}^m \,U_k\, z^k \end{aligned}$$

(5)

and

$$\begin{aligned} R(z) = \sum \limits _{k=0}^{m} \,(U_k+V_k) \,z^k. \end{aligned}$$

(6)

The smallest positive root $z_c$ of $Q(x)$ and the quantity $\eta _c = 1 - R(z_c)/Q'(z_c)$ yield unbiased estimates for $t_c$ and $\zeta _c$, respectively [5, 8]. In general, the accuracy of these estimates increases with both $n$ and $m$.

3 Results and conclusions

Application of the aforedescribed formalism to the isomer counts reported in Ref. [1] produces estimates that clearly converge to $t_c = 1$ (Fig. 1) and $\zeta _c = -10$ (Fig. 2). Thus, the leading term proportional to $N^9$ in the large-$N$ asymptotics of the isomer count of fullerenes with $N$ carbon atoms is now firmly established (although not rigorously proven).

Fig. 1

The deviations of the estimates $z_c$ from 1 versus $n$ for $1 \le m \le 20$ (gray), $21 \le m \le 40$ (red), and $41\le m \le 66$ (black) (Color figure online)

Fig. 2

The deviations of the estimates $\eta _c$ from $-10$ versus $n$ for $1 \le m \le 20$ (gray), $21 \le m \le 40$ (red), and $41\le m \le 66$ (black) (Color figure online)

The present result imposes the same asymptotics for the isomer count $M_{IPR}(k)$ of the IPR fullerenes with $2k$ carbon atoms as $0 < M_{IPR}(k) < M(k)$ and $\lim _{k\rightarrow \infty } M_{IPR}(k)/M(k) \rightarrow 1$. Curiously, inspection of the published data [1] allows one to formulate the following conjecture (see Fig. 3):

Fig. 3

The ratios $M_{IPR}(k)/M(k-k_0)$ for $k_0 = 24$ (red) and $k_0 = 25$ (black) (Color figure online)

For all $k > 53$, $M(k-24) < M_{IPR}(k) < M(k-25)$, i.e. for all $N > 106$, the number of the IPR fullerene isomers with $N$ carbon atoms is bracketed by the total numbers of isomers of the $C_{N-50}$ and $C_{N-48}$ fullerenes.

The second leading term in the large-$k$ asymptotics of $M(k)$ is also of interest. As revealed by the plot of $M(k)/k^9$ versus $k^{-2/3}$ (Fig. 4), this term scales simply as $k^{25/3}$ and is negative. The combination of the $N^9$ and $N^{25/3}$ asymptotics explains the apparent $N^{19/2}$ scaling deduced from a crude log–log plot [4].

Fig. 4

The reduced isomer count $M(k)/k^9$ versus $k^{-2/3}$