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\title{A Classification Scheme for Toroidal Molecules}
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\author{Jorge Berger and Joseph E.~Avron \\
Department of Physics,
 Technion -  Israel Institute
of Technology \\
 32000 Haifa, Israel\\
 e-mail: PHR76JB@VMSA.TECHNION.AC.IL }

\begin{document}
\addvspace{-25 mm}
\maketitle

\noindent We construct a class of periodic tilings of the plane,
which corresponds to toroidal arrangements of trivalent atoms,  with
pentagonal, hexagonal and heptagonal rings. Each tiling is characterized by a 
set of four integers and gives rise to an 
infinite family of toroidal molecules. These may be chosen to fit a desired 
shape, size or number of atoms in the molecule. The configurational 
energy and the delocalisation energy of several molecules 
obtained in this way were computed 
for Tersoff and H\"uckel models. The results indicate that many of these
molecules would be stable. We studied the influence of size on
the H\"{u}ckel spectrum: it bears both similarities and differences as 
compared with the case of tubules.

\newpage
\section{Introduction}
The last decade witnessed the discovery of diverse \cite{Dov} hollow 
graphitic structures. These may be classified into sphere-like 
\cite{Fullerenes} and cylinder-like \cite{Tubules}. A natural question is 
whether additional topologies, such as tori, also occur. It is
reasonable to expect that molecules that are multiply connected and have
negative curvature (i.e. have regions that approximate a saddle) would have
distinct physical and chemical properties.  For example, 
certain quantum and transport phenomena depend on multiconnectivity 
\cite{Rieman}, and it
is known that negative curvature has important  consequences for the
dynamics on the surface \cite{Chaos}.  Toroidal molecules may have 
applications in host-guest chemistry \cite{hostguest}, where
the ``hole'' in the torus accomodates the ``guest". It
has also been suggested that toroidal molecules would have applications
as components in nanotechnology \cite{torus} and as catalizers 
\cite{Ijima}.

Structures that are actually half tori have been observed 
\cite{Ijima} when two concentric cylinders inter-connect. 
In the present study we shall focus on structures in the shape of a 
doughnut.  Miscellaneous proposals for toroidal molecules have been 
suggested \cite{torus,hexagons,Chern}.

The purpose of the present study is to present a classification 
scheme for a wide class of toroidal molecules . We shall consider  
symmetric (mainly ${\rm D}_{nd}$) molecules.
Our tool is a set of tiling rules \cite{Tiling}, adapted to the
description of  symmetric noncrumpled toroidal molecules. These
tiling rules reflect  the energy penalty for
stretched bonds and deformed bond angles. Our method is 
amenable for computarized search (as the ``orange-peel" method \cite{Peel} 
in the case of spheroids). For simple properties, such as the spectrum 
in H\"{u}ckel
model, it suffices to give the indices which characterize a molecule as 
the input to a short program and the results are automatically obtained. 
The tiling provides also metric information.
An announcement of some of the results in the present study is found in 
\cite{A-B}.

As an application we describe results of a numerical study of several
toroidal carbon molecules. The study involved:
\begin{enumerate}
\item The relaxation of model carbon atoms  represented by tiling to
a stable molecular configuration, through  minimization of the
$\sigma-$bonds geometric potential energy. We have used  a
phenomenological potential due to Tersoff \cite{Tersof}. From here,
average energies and binding energies of the least bound 
atoms were obtained.
\item Homo-lumo  and other properties of the $\pi$-electrons in
H\"uckel model.
\end{enumerate}

 The results (Table I)  support the conjecture that many
toroidal carbon  molecules would be stable.

\section{Tilings}
 A toroidal carbon molecule can be represented by a bi-periodic
planar graph, where  atoms sit at vertices and each vertex is trivalent.
 Since carbon bonds prefer angles near $120^\circ$, graphs
with pentagonal, hexagonal and heptagonal rings are favored and we allow
only these. In hexagons-only tori, as considered in \cite{hexagons},
the hexagons must be deformed in order
to reproduce the curvature of the toroidal surfaces, leading to 
prohibitively large energies.
 We shall represent a molecule by its {\it dual graph}, i.e. a
periodic planar triangular graph with vertices of valences $5,\ 6$ and
$7$. These triangles are called {\it tiles}, and the atoms are 
located at the centers of each tile.

We are interested in smooth tori that approximate the surface of 
revolution with cartesian representation 
\begin{eqnarray}
R\left\{\left(1-\frac{\cos\theta}{\eta}\right)\cos\phi \ ,
\left(1-\frac{\cos\theta}{\eta}\right)\sin\phi \ ,
\frac{\sin\theta}{\eta}\right\}                 %1
\end{eqnarray}
in the toroidal coordinates $\{\theta,\phi\}$.  The parameter
$R>0$ determines the size, and the parameter $\eta > 1$ determines
the shape.  (Large $\eta$ corresponds to thin tori.)  Eq.~(1)
induces the metric and area
\begin{eqnarray}
&&\eta^2 R^{-2}ds^2=d\theta^2+(d\phi)^2  \
(\eta-\cos\theta)^2       \ ; \nonumber \\       %2
&&\eta^2 R^{-2}dA = d\theta d\phi (\eta-\cos\theta) \ .
\end{eqnarray} 

We shall refer to lines of fixed $\theta$ (resp. $\phi$) as latitudes
(resp. longitudes) and draw them horizontal (resp. vertical).  
Our set of tiles includes various sizes,
reflecting the non uniform metric in the $\{\theta,\phi\}$
plane.  (Large tiles are placed where $\cos\theta$ is large and vice
versa.)  For simplicity, we shall take all our tiles as similar
isosceles triangles.  In order to accomodate
vertices with valence 5, 6 and 7, these triangles have to be right-angled.
Due to the privileged role of latitudes and longitudes, we shall (initially)
require that each tile have at least one side along one of these directions. 
More general structures  will be discussed in Section 4.

It follows by Pythagoras that the different tiles are related by 
scaling by powers of $\sqrt{2}$.  The largest tiles will be called the
{\em first generation}; the tiles in the {\em $i$-th generation}
will have an area which is $2^{i-1}$ times smaller.  
In the first generation we shall have the two (big) isosceles right-angled
tiles of area  one in the tiling plane, shown in Fig.~1a.  We
allow only the two orientations (related by a $\pi$ rotation)
with the hypotenuse along a
latitudinal line; this will be required from every odd generation,
giving rise to an array of equidistand horizontal lines. (This
feature agrees with metric (2), which
depends on $\theta$, but not on $\phi$.  In this way the
first generation tiles give the shortest latitudinal lengths.)
 At the second generation we add the four 
 tiles of area $\frac{1}{2}$ in Fig.~1b.  At generation $p+2$, we add
the tiles obtained from  those in generation $p$ by scaling by a factor
of $\frac{1}{2}$. Clearly, tiles that share a common edge 
differ by at most one generation.
Since there is only one way of matching
odd generation tiles, the tiling of an odd-generation region 
is uniquely determined by its border; on the other hand,
since a square can be tiled in two ways (see Fig.~2), the tiling 
of an even-generation region
involves a choice of orientation for each square.

Since the linear dimension of the last generation is smaller than 
that of the first by a factor $2^{(m-1)/2}$, and since the first generation
has the hypotenuse along a latitudinal line (what may introduce
an additional factor $2^{1/2}$ in the metric along the latitudinal 
direction), Eq.~(2) can be used to relate $\eta$ to the number of generations
$m$:
\begin{eqnarray}
\frac{\eta+1}{\eta-1} \sim 2^{[m/2]} \ ,    %3
\end{eqnarray}
where $[m/2]=$ integral part of $m/2$.
Therefore, the
number of generations plays a role analogous to $1/\eta$.
A cylinder can be built with one generation,  skinny tori
 are associated with first and second generation tiles and, the fatter 
the tori, the more generations are needed. 

Unlike the sphere, which is unique up to scaling, the tori of Eq.~(1) 
depend on the shape parameter $\eta$. {\it A priori}, it is not clear 
whether this family has a distinguished member. It turns out that there 
is such a torus and the case $\eta = \sqrt{2}$ is special. In the theory 
of surfaces the torus of revolution with $\eta = \sqrt{2}$ is known as the
Clifford torus and is distinguished by being the minimizer of the 
Willmore functional, i.~e., it minimizes the square of the mean curvature
\cite{Bensimon}.
The fattest torus we shall study has five generations, inspired by 
the Clifford torus. As we shall see, it does have a marked stability. 
For fatter tori,
axially symmetric structures are not expected \cite{Bensimon}.

Valence plays the role of
curvature: valence 5 (resp. 6, 7) vertices carry positive (resp. zero, 
negative) curvature.
We want no 5 and 7 valence vertices embedded in single generation regions.
The pair of tiles of the odd generations have indeed
this behavior built into them:  only sixfold vertices can be made
with them alone.
On the other hand, Fig.~2 shows a tiling by  even
generation tiles with valences 5 and 7.
These pentagon-heptagon pairs do not contribute to the global curvature
required by metric (2), but merely produce crumpling, and should be
avoided by an additional rule; pentagons or heptagons will be allowed 
only at the interfaces between different generations.

The connection between the tiling and the Riemanian curvature can be
made more stringent.  The predominance of pentagons over heptagons
should be proportional to the curvature.  In our case, it should be
a decreasing function of $\cos\theta$ and, also, of the size of the
tile.  However, since there is an energy penalty for every
additional pentagon-heptagon pair, and since we are interested in
molecules with numbers of atoms which are not too large, we shall
compromise the smoothness of the curvature on behalf of a limited
number of pentagons and heptagons.  Accordingly, pentagons (resp. heptagons)
will be located only where the
tiles are smallest (resp. largest).  In summary, we impose the 
following:\\

\noindent {\em 5-6-7 Rule}: For a torus with $m$ generations, all the
heptagons lie at the interfaces between the 1st and the 2nd generation
and all the pentagons lie at the interfaces between the $m$-th and the
$(m-1$)-th generation.  (If there are only two generations,
both the heptagons and the pentagons lie at the interfaces between
them.)  Furthermore, we require that around every heptagon (resp.
pentagon) there be a majority of tiles of generation 1 (resp. $m$).\\

Figs. 3-5 are examples of tilings which obey the 5-6-7 rule.

\noindent {\em Length}:  We define the length $L(\theta)$ of a latitudinal
line as the number of edges it contains. Clearly,
$L(\theta)$ is larger when the line passes through high generation regions.
In Fig.~5, $L(\theta)$ varies from 1 to 3.
Likewise, we define the length of a longitudinal line as the number
of horizontal lines it intersects.  Our tiling rules ensure that the
total length of longitudinal lines is independent of $\phi$, as it
should.  This length will be dubbed the ``girth'' and denoted by
$g$.  We also define $g_i(\phi)$ as the length within the $i$-th
generation regions.  Note that if a longitudinal line crosses
several regions of the same generation $i$, then $g_i(\phi)$
is defined as the {\em total} length across all these regions.
In Fig.~5, $g_1 =g_3 =2$ and $g_2 =g_4 =1$ (independent of $\phi$).
Clearly, $\/ \sum^m_{i=1} g_i(\phi)=g$.\\

\section{Skeletons and Relationships}
Given a set of interfaces between generations, the entire tiling is 
determined uniquely by the 5-6-7 rule (although not for every set of 
interfaces does a legal tiling exist).  The set of interfaces will be called
the {\it skeleton}.  The ($2 p-1,\ 2p)$ interfaces are zigzag lines (e.g.
Fig.~6a) and the $(2p, \ 2p+1$) interfaces are horizontal lines
(e.g. Fig.~6b).  Therefore, regions that contain $2p-1$ and $2p$            
tiles lie within a horizontal stripe, which will be called a
{\em $p$-th stripe}.  (If there are only 2 generations, there is just
one stripe; if the number $m$ of generations is odd, then the
last stripe has tiles of the $m$-th generation only.)
Figs.~3 and 4 have one stipe; Fig.~5 has two.

Since according to metric (2) the length of the latitudinal
lines has only one minimum and one maximum, we require that the
regions of generation $i$, $1\leq i\leq m$, be located in monotonic
and consecutive order, both when going from region 1 to $m$ by
increasing or by decreasing $\theta$.  The 5-6-7 rule imposes a very 
strong constraint on the class of possible skeletons with two or more 
stripes: with the exception of
the vertices at which there is a majority of generation 1, and, if
$m$ is even, the vertices with a majority of generation
$m$, all the vertices of the zigzag interfaces lie at the stripe
borders and all the zigzag interfaces are parallel to each other (see
Fig.~7).  This has
two consequences: First, the tiling obeys the symmetry ${\rm D}_{nd}$, which
is the highest symmetry of a torus in three dimensions
that could be expected from a discrete construction.  The mirrors of the
symmetry are at the planes of constant $\phi$ that contain the
pentagons and the heptagons.  The centers of inversion lie midway
between the heptagons (or pentagons) which are located at different
zigzag lines.  Second, the $g_i$'s are independent of $\phi$, in
agreement with the desired metric (2).
  If the skeleton consists of just one stripe, then the
symmetry ${\rm D}_{nd}$, the constancy of the $g_i$'s and the zigging
between two lines of constant latitude don't follow from the 5-6-7
rule (e. g. Figs.~8-9), but we impose them in order to mimic the 
metric~(2).
The independence of the $g_i$'s on $\phi$ rules out the tori
considered in Ref. \cite{Chern}, which are cylindrical tubes connected by
elbows.

We are thus left with a class of tilings such that the skeleton
(and the entire tiling) is completely determined by four numbers:
$m$ (the number of generations), $g_1$ and $g_m$  (the contributions 
of the first and last generation to the girth), and $z$ (the number of
edges in a zig of the (1,2) interface). If $i \ne 1,m$, then
$g_i=2^{p-1}z$, where $p$ is the number of the stripe.

The girth of the torus is given by
\setcounter{num}{4}
\setcounter{equation}{0}
\def\theequation{\thenum\alph{equation}}
\begin{eqnarray}
g=g_1+g_{\rm mid} + g_{\rm end} + g_m \ ,                %4a
\end{eqnarray}
with
\begin{eqnarray}
g_{\rm mid} = \left\{
        \begin{array}{ll}
        0 & m \leq 2 \\
        z & 2<m<5 \\
        z+(2^k-4)z & m \ge 5
        \end{array} \right. %4b
\end{eqnarray}
where $k=[(m+1)/2]$ is the number of stripes ([ ] denotes integral part), 
and
 \begin{eqnarray}
 g_{\rm end} = \left\{
        \begin{array}{ll}
        2^{k-1}z &  m \mbox{ even and} \ge 4 \\
        0 & \mbox{otherwise}
        \end{array} \right. %4c
\end{eqnarray}       

The number of atoms within a $\phi$-range spanned by a zig and a
zag is given by
\setcounter{equation}{4}
\def\theequation{\arabic{equation}}
\begin{eqnarray}
A = \left\{
       \begin{array}{ll}
       2z\left(g_1+2^{[m/2]}g_m\right) & m<3 \\
 2z\left(g_1+2^{[m/2]}g_m+(2^{m-1}-2)z\right)  & m \ge 3
        \end{array} \right. %5
\end{eqnarray}

The length of the shortest latitudinal line, within a $\phi$-range 
spanned by a zig and a zag, is given by
\begin{eqnarray}
L_{\rm min}=2z-{\rm Min}(g_1,z) \ . %6
\end{eqnarray}
The length of the longest latitudinal line in the same $\phi$-range is 
\begin{eqnarray}
L_{\rm max}= \left\{
       \begin{array}{ll}
       2^{k-1}z & m \mbox{ odd} \\
       2^{k-1}z+{\rm Min}(g_m, 2^{k-1}z) & m \mbox{ even}
       \end{array} \right.   %7
\end{eqnarray}

\section{From Tiles to Tori}
By associating periodic coordinates $\{\theta,\phi\}$ to the tiling
plane, the three-dimensional positions follow by Eq.~(1).
The line $\theta=0$ (resp. $\pi$) is assigned to the horizontal
line through the centers of inversion near the vertices of valence
7 (resp. 5).  We then increase (resp. decrease) $\theta$ by $2\pi/g$
when going up (resp. down) by one horizontal line.  $\phi$ is assigned
by requiring it to span $2\pi$ in the periodic tiling and to be linear
along the horizontal planar distance in the tiling plane.  Finally,
we impose periodicity $\{\theta+2\pi,\phi\} \equiv \{\theta,\phi+2\pi\}
\equiv \{\theta,\phi\}$.

The $\phi$-range spanned by a zig and a zag
will be called a {\em unit cell}, and a torus with $n$ unit 
cells has symmetry ${\rm D}_{nd}$. The number of unit cells
 in the molecule may be estimated by comparing the difference in the
perimeters at $\theta = 0$ and $\theta=\pi$ to that of a geometric
torus.  This gives
\setcounter{equation}{7}
\def\theequation{\arabic{equation}}
\begin{eqnarray}
n \sim n_0 = 2 g/(L_{\rm max}-L_{\rm min}) \ ,                    %8
\end{eqnarray}
where $g$, $L_{\rm max}$ and $L_{\rm min}$ are given by (4), (6) and (7).

To evaluate the parameters $R$ and $\eta$ which
appear in Eq.~(1), $R$ (resp. $R/\eta$) is taken proportional to the 
average latitudinal (resp. longitudinal) length, and the constant of 
proportionality is a ``material parameter". For carbon atoms, 
appropriate constants are
\begin{eqnarray}
 R=0.2n(L_{\rm min}+L_{\rm max}) {\rm \AA} %9
\end{eqnarray}
and
\begin{eqnarray}
R/\eta=0.4g {\rm \AA} \ .          %10 
\end{eqnarray}
The geometric data in Table~I were obtained after 
refinement through minimization of the interatomic potential.

Given a tiling, several additional structures may be generated
from it. This is done by means of Goldberg
inclusion, elongation, chirality and torsion.

{\em Inclusion} consists of drawing $v$ vertices inside each
tile and then connecting both the old
and the new vertices to form a new array of triangles in the dual graph.
Vertices on the edge between two tiles are counted as $\frac{1}{2}$ 
in each. This procedure increases the number of atoms by a factor
$q=1+2v$. Connections are made so that old vertices
retain their valences, new vertices have valence 6, and nearest vertices
are connected. We restrict ourselves to the cases in which this rule for 
the connection of the vertices is unambiguous.
If $q=1+2v$ is of the form
\setcounter{num}{11}
\setcounter{equation}{0}
\def\theequation{\thenum\alph{equation}}
\begin{eqnarray}
q=\ell_1^2 +\ell_2^2 +\ell_1 \ell_2   %11a
\end{eqnarray}
with $\ell_1$ and $\ell_2$ integers, then a distribution of vertices for
which connections are unambiguous is obtained by a variation of
the {\em Goldberg transformation} \cite{Gold}: using local 
coordinates for each tile, such that its vertices are at $(x=0, \ y=0)$, 
$(x=1, \ y=0)$ and $(x=0, \ y=1)$ (with {\bf \^{x}} to the right of 
{\bf \^{y}}), the included vertices are at
\begin{eqnarray}
x &=& (n_1 (\ell_1 +\ell_2 )+ n_2 \ell_2)/q \ , \nonumber 
\\                    y &=& (-n_1 \ell_2 +n_2 \ell_1 )/q  \ ,   %11b
\end{eqnarray}
where $(n_1 ,n_2 )$ are integers such that $(x,y)$ lies inside the tile.
Fig.~10 shows the Goldberg transformation $(\ell_1 ,\ell_2 )=(1,2)$
applied to the tiling $g_1=2, \ g_2=z=1$.

 Inclusion generates a family of
molecules for which the number of pentagons and heptagons and their
relative positions (and therefore the overall shape) remain fixed.
Except for the cases $\ell_1 = \ell_2$ and 
$\ell_1 \mbox{ or } \ell_2 = 0$, which are considered below, the Goldberg 
inclusion gives rise to chiral molecules.

A Goldberg inclusion with indices $(\ell_1 ,\ell_2 )=(\ell,0)$ (or
$(0,\ell)$) will be called {\em inflation}.
 Inflation is the discrete analog of scaling. Inflation
by  a factor $\ell$ replaces  all the tiles by  ``super-tiles"
where each new tile is made of $\ell^2$ scaled copies of the old tile 
(or its
$\pi$ rotated image). 
Inflation  preserves the distinction among generations, increasing by the 
factor $\ell^2$ the number of atoms in each.   
Inflation by a factor $\ell$ gives a tiling within the class we have
considered, leaving $m$ and $n$ unchanged, while leading to
multiplication of $z$ and all the $g_i$'s by $\ell$.  Therefore,
for classification purposes it is convenient to consider primitive
tiling, i.e. sets of numbers $(g_1,g_m,z)$ which have no common
divisor.  (Note that for odd generations $g_i$ must be even.)

A Goldberg inclusion with indices $\ell_1 =\ell_2 =1$ is 
a {\em leapfrog} \cite{Leapfrog}.
In three dimensional space, the leapfrog
operation is a vertex truncation through the mid-points of the edges, so
that every old vertex turns into a hexagon and the number of atoms is
multiplied by 3.  Leapfrogged structures are guaranteed to 
have closed shells in the H\"{u}ckel spectrum \cite{MWF}.
The effect of leapfrogging twice is the same as that of inflation by a
factor 3.

The tori of Ref. \cite{Chern} may be obtained by {\em elongation}. 
Elongation is defined for tilings with two generations. It consists of 
cutting the tiling along an approximately longitudinal line (following 
the edges of the tiles), separating the two pieces by an integer 
latitudinal distance, and filling in with tiles of generation 2, in such 
a way that all additional vertices are hexagonal. Fig.~8 shows an 
elongation of the tiling $g_1=2, \ g_2=z=1$.


For many tilings there exist latitudinal lines  $\theta=\theta^*$ which 
are embedded in a single odd generation (first or last, with 
$L(\theta^*) \equiv L^*$ equal to either $nL_{\rm min}$ or $nL_{\rm 
max}$). If this 
is the case, manifest {\em chirality} \cite{Dress} may be introduced by using 
$\theta$ in the range $\theta^* \leq \theta < \theta^*+2\pi$ and 
associating to the coordinates $\{\theta,\phi\}$ in the tiling the 
coordinates $\{\theta,\phi+j\theta/L^*\}$ in three dimensional space, 
with $j$ integer. (This is the analog of Dehn twist in the theory of 
Riemann surfaces.)
 This amounts to identification of the edge at 
$\theta=\theta^*$ with the edge at $\theta=\theta^*+2\pi$ shifted by $j$. 
The 5-6-7 rule will still be fulfilled. Unless $|j|<<L^* $ and $|j|<<g $ 
, this shift produces considerable strain. 

Shifts in the longitudinal boundaries of the tiling  would lead in general
to a misfit in the skeleton. However, if there
are only two generations, {\em torsion} can be introduced by drawing zigs
($z_1$) longer than the zags ($z_2$), as in Fig.~9. The sinus of the pitch
angle may be estimated as $2(z_1 -z_2 )/(L_{\rm max}-L_{\rm min})$. Helical 
structures have been studied in \cite{helices}.

Let us mention that better approximations to metric~(2)
can be achieved by two-generation tilings in which the skeleton consists
of two parallel jagged lines, so that the number of heptagons minus the 
number of pentagons in the range $[0,\theta ]$ is proportional to 
$|\sin \theta|$. Tilings like this would have a very large number of atoms 
per unit cell and, also, a very large $n_0$ in Eq.~(8).

\section{Implementation}
For each molecule, we calculate two complementary 
phenomenological contributions to the energy. The geometric contribution 
is calculated by means of an interatomic potential, and the 
contribution of delocalized electrons by means of H\"{u}ckel model.

\noindent {\it Interatomic Potential}: We use Tersoff's potential 
\cite{Tersof}. 

The tiling rules of the previous
sections not only describe the connectivity of the molecule, but also
suggest values for  $n$, $R$ and $\eta$ through Eqs.~(8)-(10). The carbon
atoms (located at the centers of each triangle), are assigned
initial positions in space which correspond to their toroidal
coordinates.  The positions of all the atoms are then allowed to vary
in three-dimensional space to minimize the total potential energy of
the bonds. Good tilings flow to a nearby local minimum. Uncontrolled 
guesses for the initial coordinates result in atoms being lost to infinity.

We have used Powell's method \cite{Powel} to flow 
to a local minimum of the molecular
energy. In all cases, our tilings provided initial
guesses for the atomic positions within the basin of convergence of the
minimization algorithm. When the number of coordinates is large, analytic 
expressions for the derivatives of the energy must be supplied in order 
to obtain reliable minima.

For every tiling, we tried several values for the number $n$ of unit cells.
For the tilings considered, the largest binding energies of the entire
molecule were
usually obtained for $n=6$ and the largest binding energy of the most
strained atom, for $n=5$.
The results  are shown in Table I.  Since the differences
between $n=5$ and $n=6$ are small, we present the results for $n=5$ only.
To facilitate
comparison, Tersoff's energies are given relative to that of an ideal
graphitic plane, which is $-7.40$ eV per atom.  It is instructive to
compare these energies with those of a torus with girth 4 and 160
atoms, but with hexagonal rings only: in this
case $N(\bar{E} - E_G)/|E_G|=29$ and $10^3(E_{\rm worst} - E_G)/|E_G|=411$
(and strain increases for larger girth).

\noindent {\em H\"{u}ckel Model}:
H\"uckel model \cite{Graph} associates an $N\times N$  incidence matrix to a
graph of $N$ carbon atoms, whose  eigenvalues, $E_j,\ j\in\{ 1,\dots,N\} $
correspond to the discrete energies of independent delocalized
$\pi$ electrons.
A closed shell, where  half the eigenvalues
are negative and half positive ($N$ is even) implies
stability  (protecting e.g. against Jahn-Teller instability).
The largest negative eigenvalue, $E_{N/2-1}$, is known as {\it homo}-energy
and the smallest positive one, $E_{N/2}$, as {\it lumo}. A large gap
is a criterion for stability. Taking $\frac{1}{N} \sum E_j$ as the reference,
$|E_j|\le 3|\beta|$, where $\beta$ is the nearest-neighbor overlap integral.

A computer program which calculates the H\"{u}ckel 
spectrum for arbitrary $g_1,g_2,z$ and $n$ is available upon request.
 Illustrative 
values  are summarized in Tables I and II.  The average energy
per atom is almost insensitive to the number $n$ of unit cells.

The H\"uckel model also provides an
estimate for the electronic charge distribution in the torus.
 The general trend is that five (resp. seven)-atoms rings
capture (resp. release) electrons from/to their adjacent atoms. The total
capture/release is of the order of a quarter of an electron per ring.

\noindent {\em Choice of Examples}: The method described above allows
to design and analyze infinitely many toroidal molecules.  
One can for instance require the inner radius of the torus
(which is proportional to $nL_{\rm min}$) to fit the size of a 
given ``guest"; given the mass of a molecule,
one could fix $N/n=A$ in Eq.~(5); in order to design an elbow
that fits a given tubule or cap, one would require a given girth.

The case of two connected concentric cylinders \cite{Ijima} may be 
reproduced by a two-generation torus with $g_1 \approx g_2 >> z$. Since 
experimentally found ``turn around" edges have a small difference between 
the external and internal radius (relative to the radii themselves), 
elongation as in Fig.~8 is necessary. The outer cylinder contains tiles 
of generation 2, connections contain the zigzagging boundaries, and 
the inner cylinder contains tiles of generation 1 and separating 
fringes of generation 2. From geometric arguments and 
comparison with Eqs.~(9) and (10), we estimate $g_1$ and $g_2$ as $ 0.4 
\times ($length of the cylinders), $nz$ as $2.5 \times ($difference in 
radii of cylinders) and the total horizontal witdth of the separating 
fringes as $5 \times ($the internal radius) minus $2.5 \times ($the 
external radius), with all lengths in \AA.  Note that in this case
Eq.~(1) is not a good approximation
and that interlayer interaction has not been taken into account.

The primitive tori ${\rm C}_{120}$ and ${\rm C}_{240}$ in \cite{torus}
may be traced to be $g_1 =2, \ g_4 =z=1$ and the leapfrogged of
$g_1 =g_3 =2, \ z=1$, respectively.

For the purpose of illustration, we have focused on small values for
$g_1,g_m$ and $z$. Molecules with longitudinal
perimeters of the order of that of C$_{60}$ imply $g \sim 10$. 
For tori with roughly circular cross section, we may also expect that the 
difference 
in the latitudes of the pentagon and the heptagon lying at the same
longitude be at least about $2\pi/3$.

Fig.~11 shows three dimensional views of the molecule $m=2, \ g_1=g_2=z=4$
after relaxation to the minimal energy, and Fig.~12 shows $m=5,\ g_1 = 
g_5=2, \  z = 1$. This is the five-generations molecule with the smallest 
possible tiling. The difference in $\eta$ is apparent, with Fig.~12 being 
close to a Clifford torus \cite{Bensimon}.

Comparison of different levels of inflation of $m=g_1=g_2=z=2$ shows that
the configurational energy per atom relative to graphite decreases 
roughly as $1/N$. The same result was obtained in \cite{torus}.
  However, the energy of the least bound atoms saturates.
This effect is probably due to the fact that the number of pentagons and
heptagons is kept unchanged, and they have to bear the entire burden of
curvature.  In Fig.~11a it is clearly seen that pentagons stick out as
protuberances.  It is also interesting to note in Fig.~11b how the
pentagons deviate from the latitude $\theta = 0$ which they have in the
tiling.

Fig.~13 shows a chiral molecule, as obtained by shifting by $j=1$ the tiling
of the molecule in Fig.~12 at $\theta^* = \pi $.

The influence of inflation on the H\"{u}ckel spectrum was studied for 
$g_1=g_2=2$ and $1 \leq z \leq 3$. The results are summarized in Figs.~14
and 15. Fig.~14 describes the homo-lumo gap. Since there are $N$ energy
levels, and since they are bound in the interval $[3\beta,-3\beta]$, the gap
might naively be expected to be inversely proportional to $N$, and thus to
$\ell^2$. The trend is, however, that the gap is inversely proportional to
$\ell$. This result resembles the case of tubules, where the gap is 
inversely proportional to the diameter \cite{diameter}. There are some
inflation values for which the regular trend is not obeyed. These are a 
reminiscent of gap closure for tubules whose width is divisible by 3.

Fig. 15 shows how the delocalisation energy per atom approaches that of 
graphite. The difference from graphite is inversely proportional to
$\ell^2$.

\section{Conclusion}
Tori are closed surfaces, like spheres,but unlike spheres  are not
simply connected. They may provide an arena for a variety of physical and
chemical properties.

Table I shows that, from  the energetic point
of view, there are toroidal molecules which are roughly as stable as
C$_{60}$. One could say that the fact that toroidal
carbon molecules have  not been observed is as significant as the fact
that C$_{60}$ molecules were not observed until recently. The energies and
geometries we have found are similar to those in \cite{torus}, in spite
of having used a different potential and minimization technique.

We have developed a classification scheme which can be used to ``design'' 
toroidal molecules with prescribed dimensions. As long as these molecules 
are not found, it may help to suggest candidates; if they are found, it 
may help to analyze them.


{\bf Acknowledgments:}
We are grateful to M.~Kaftory for his help with Figs.~11-13 and 
to M.~Dresselhaus, L.A.~Chernozatonskii and T.~Schlick for correspondence.  The
research is supported by  GIF and DFG through  SFB288, and the Fund for the
Promotion of Research at the Technion.

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\newpage

%\bigskip
\begin{center}
Table I
\end{center}

\begin{tabular}{lcccccccc}
$m$ & 2 & 2 & 2 & 2 & 3 &  4 & 5 & $5^1$ \\
$g_1$ & 2 & 4 & 6 & 2 &4 & 2 & 2 & 2\\
$g_m$ & 2 & 4 & 6 & 2 &2 & 1 & 2 & 2\\
$z$ & 2 & 4 & 6 & 3 &6 & 1 & 1 & 1\\
$N$ & 120 & 480 & 1080 & 180 &1200 & 120 & 240 
& 240 \\
$\epsilon_h(\beta)$ & 0.43 & 0.16 & 0.08 &0.45 &0.07 & 0.04 
& 0.19 & 0.20\\
$\epsilon_\ell(\beta)$ & -0.27~ & -0.22~ & -0.02~ &-0.18~ 
&-0.02~ & 0.04 & 0.01 & -0.04~ \\
$10^3(\bar{\epsilon}-\bar{\epsilon}_G) (|\beta |)$ & 16 & 4 & 2 &11 
&2 & 25 &11 & 9\\
$10^3(\bar{E} - E_G)/|E_G|$ & 123 & 32 & 16 & 118 & 16 &  76
& 39 & 43\\
$10^3(E_{\rm worst} - E_G)/|E_G|$ & 181 & 114 & 108 & 170 & 115 &
122 & 101 & 100\\
$N(\bar{E} - E_G)/|E_G|$ & 14.8 & 15.6 & 17.5 & 21.2 & 19.5 & 9.1
& 9.4 & 10.4\\
inner radius ($\rm \AA$) & 4.05 & 8.33 & 12.57~ & 6.87 & 
14.15~ & 2.02 & 2.09 & 2.12\\
outer radius ($\rm \AA$) & 7.29 & 14.21~ & 21.17~ & 10.14~
&23.42~ & 6.02 & 8.49 & 8.61\\
height ($\rm \AA)$  & 2.93 & 5.68 & 8.00 & 3.37 & 8.13 & 4.57 & 6.89 & 6.96
\end{tabular}
\pagebreak


%\centerline{Table II}
%
%\begin{tabular}{cccc}
%& graphite&C$_{60}$&dodecahedron \\
%\\
%atoms per molecule&$\infty$&60&20 \\
% $\epsilon_h$($\beta$)  & 0  & 0.62  & 0 \\
% $\epsilon_\ell$($\beta$)  & 0  & $-0.14$~ & 0 \\
%$\frac{N(\epsilon_\ell-\epsilon_n)}{6|\beta|}$
%& - & 7.6~ & 0 \\
%$\bar{\epsilon}(\beta)$ & 1.57 & 1.55 & 1.47 \\
%$\bar{E} - E_G$ & 0 & 0.67 & 1.62 \\
%radius ($\AA$) & $\infty$ & 3.69 &  1.93
%\end{tabular}
%\pagebreak

\begin{center} Table II \end{center}

\begin{tabular}{|cc|ccc|ccc|}  \hline
\verb+\+ & $g_2$ & 1 & 2 & 3 & 1 & 2 & 3 \\
$z$ & \verb+\+ & &$g_1=2$ & & & $g_1=4$ & \\      \hline
1 & & 35 & 17 & 22 & 36 & ~0 & ~1 \\
2 & & 61 & 70 & ~8 & 11 & 51 & 13 \\
3 & & 71 & 62 & 22 & 44 & 27 & 21 \\ \hline
\end{tabular}
\pagebreak

\begin{center}
Table Captions
\end{center}

\noindent {\bf Table I}
Energetic and geometric values for several toroidal molecules, according to
H\"uckel's theory and to Tersoff's potential. All the molecules have 5 unit 
cells. The second and third columns are inflations
of the first. $N$ is the number of atoms in the molecule,
 $\epsilon_{h,\ell}$ is the energy of the highest
occupied/lowest unoccupied molecular orbital, $\bar{\epsilon}$ is the
average energy of delocalized electrons, $\bar{E}$ is the average geometric
energy per atom, $E_{\rm worst}$ is the geometric energy 
associated with the least bound atom, and
$\bar{\epsilon}_G=1.575 \beta$ and
$E_G=-7.40{\rm eV}$ are the delocalisation and
the geometric energy per atom of an ideal graphitic plane. 
The molecule in the last column (denoted by $m=5^1$) is
obtained by a chiral shift $j=1$ of $(m,g_1,g_m,z,n)=(5,2,2,1,5)$

\noindent {\bf Table II} $10^2 \times 
(\epsilon_{\ell}-\epsilon_h)/|\beta|$ for molecules with 2 generations 
and 5 unit cells, for various values of $g_1, \ g_2$ and $z$. 
\newpage

\centerline{Figure Captions}

\noindent {\bf Fig. 1} (a) Odd generation tiles. (b) Even generation tiles.

\noindent {\bf Fig. 2} Even generation tiling with unnecessary crumpling.

\noindent {\bf Fig. 3} The tiling $m=g_1=g_2=z=2$.

\noindent {\bf Fig. 4} The tiling $m=g_1=g_2=2, \ z=3$.

\noindent {\bf Fig. 5} The tiling $m=4, \ g_1=2, \ g_4=z=1$.

\noindent {\bf Fig. 6} (a) Interface between even generation higher than
the odd generation. (b) Interface between odd generation higher than
the even generation. 

\noindent {\bf Fig. 7} Skeleton of tiling that obeys the 5-6-7 rule and has 
more than one stripe. (In the figure $m=6$ and there are 3 stripes.)
The numbers denote the generation of the tiles in the
region. $g_1$, $g_m$ and $z$ determine completely the unit cell of the 
tiling.

\noindent {\bf Fig. 8} The tiling $g_1=2, \ g_2=z=1$, after elongation by 1.

\noindent {\bf Fig. 9} Three unit cells of the tiling $g_1=g_2=z_1=2, 
  \ z_2=1$.

\noindent {\bf Fig. 10} Unit cell of the tiling $g_1=2, \ g_2=z=1$, after
a Goldberg inclusion with indices $\ell_1=1, \ \ell_2=2$. The circles signal
the original vertices before the inclusion.

\noindent {\bf Fig. 11} Three-dimensional view of the molecule 
$(m,g_1,g_2,z,n)=(2,4,4,4,5)$, which minimizes Tersoff's potential. For 
clarity, only atoms in the foreground are shown. (a) View from ``above".
(b) Side view.

\noindent {\bf Fig. 12} Same as Fig. 11, for $(m,g_1,g_2,z,n)=(5,2,1,1,5)$.

\noindent {\bf Fig. 13} Same as Fig. 12, with a chiral shift $j=1$.

\noindent {\bf Fig. 14} Gap in the H\"uckel spectrum for molecules 
$(m,g_1,g_2,z,n)=(2,2\ell,2\ell,\ell,5)$, $(2,2\ell,2\ell,\ell,6)$,
$(2,2\ell,2\ell,2\ell,5)$, $(2,2\ell,2\ell,2\ell,6)$,
$(2,2\ell,2\ell,3\ell,5)$, $(2,2\ell,2\ell,3\ell,6)$. For $z/\ell=1$ 
(resp. 2, 3), calculations were carried up to $\ell=9$ (resp. 8, 5).
The numbers in the graph stand for $z/\ell$. Calculations for $n=6$ were 
carried up to $\ell=5$; for the values of $\ell$ for which a number appears
only once, the values of the gap for $n=5$ and $n=6$ are either identical
or indistinguishable within the resolution of the graph. For every $z/\ell$,
$\ell(\epsilon_{\ell}-\epsilon_h)$ exhibits approximate periodicity as a
function of $\ell$, with period 3.

\noindent {\bf Fig. 15} Delocalisation energy for the same molecules studied
in Fig. 14. $\bar{\epsilon}_G=1.575 \beta$ 
is the delocalisation energy per atom of an ideal graphitic plane. 

\end{document}
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