\section*{Problems}

\begin{enumerate}
\renewcommand{\labelenumi}{ [\arabic{chapter}-\arabic{enumi}] }

\item Consider a CH$_4$ molecule which has T$_d$
symmetry.  Obtain from some textbook the character table for T$_d$
symmetry. Which irreducible representations correspond to the
vector ($x$, $y$ and $z$)? Show the matrix for a $C_3$ rotation
of this molecule.

\item Which irreducible representations of the $T_d$ group
correspond to rotations around the $x$, $y$ and
$z$ axes? Explain that the results do not depend on
how you select the axes.

\item Obtain the atom site representation for a CH$_4$ molecule and
decompose your atom site representations into a set of irreducible
representations.

\item Calculate the reducible representation for the molecular
vibrations for the CH$_4$ molecule and decompose the reducible
representation into irreducible representations.  How many vibrational
modes are there for a CH$_4$ molecule?


\item Obtain the symmetries of the Raman and IR active phonon modes of
a CH$_4$ molecule. In the case of a CH$_4$ molecule, show that there
is no inversion symmetry both by plotting a model of the CH$_4$ molecule
and by showing its atomic coordinates.

\item Now let us consider the single atomic layer of graphite which we
call graphene. Consider the character table of the point group for the
hexagonal unit cell of monolayer graphene. Obtain the atom site representation of the
two carbon atoms of graphene.

\item Obtain the symmetry of the vibrations of graphene at the zone
center of the Brillouin zone for graphene.

\item Obtain the symmetry of the tight binding
orbitals of graphene at the zone center of the Brillouin zone.

\item Show that an optical transition from one $2p$ to another
$2p$ carbon atomic orbital within the same carbon atom is
not allowed.  On the other hand, the optical
transition from $\pi$ to $\pi^*$ energy bands is
allowed. Explain what kind of matrix element
contributes to  such an optically allowed transition? Expand
the tight binding wavefunctions for the optical transition matrix.

\item In the case of graphene, optical
transitions occur around the hexagonal corners of
the Brillouin zone ($K$ and $K'$ points). When we consider a
large unit cell, the K point can be folded into
the $\Gamma$ point (zone center). Plot the extended unit
cell and the folded Brillouin zone. What is the point
group for the extended unit cell?

\item We can consider a $C_2$ rotation around the
axis at the center of the C-C atomic bond in the
direction perpendicular to the bond. However, this
rotation is not included in the operations for the
hexagonal unit cell of graphene. On the other
hand, when we consider the rhombic unit cell of
graphene, we can include the $C_2$ rotation.
Discuss the difference in the results of a symmetry
discussion for the two different shapes of the unit
cell of graphene.

\item Consider the point group of double layer
graphene, in which the two layers have AB
stacking. Obtain the
irreducible representations  for
the atom sites and vibrations of bilayer graphene.

\item In the case of double layer graphene with AB stacking, the
interlayer interaction can be treated as a
perturbation to the unperturbed two graphene layerd.  What
is the irreducible representation for the interlayer
interaction?

\item For double layer graphene, we expect four
energy bands derived from the $\pi$ and $\pi^*$ bands. Obtain the
irreducible representations for the four electronic energy bands
at the zone center. Here and in the next graphene related
problems, we always consider the AB Bernal stacking.


\item For double layer graphene, show by group theory that the four
$\pi$ bands at the K point consist of one doubly
degenerate energy state and two non-degenerate
energy states. Discuss the optical selection rules
of monolayer graphene and bilayer graphene near the K point.

\item Discuss the Raman selection rules for
the vibrations occurring in monolayer  graphene and in bilayer
graphene.

\item Discuss the symmetry for trilayer
graphene and obtain the
symmetries of its Raman-active modes. Obtain the
optical selection rules for triple layer graphene.

\item Discuss the symmetry for four layer graphene and obtain the
symmetries of its Raman-active modes. Obtain the optical selection
rules for four layer graphene.

\item When we consider three dimensional graphite with AB layer
stacking, show the point group of the graphite unit cell and discuss
the IR and Raman-active modes for graphite.

\item Discuss the symmetry of a unit cell for a graphene ribbon with
zigzag edges or one with armchair edges. What is the difference in the
symmetry of these unit cells relative to the symmetry of a unit cell
for monolayer graphene?

\item The C$_{60}$ molecule has $I_h$ symmetry. Find the atom site
irreducible representations and symmetries of the vibrational modes
for the C$_{60}$ molecule. How many Raman-active modes
are there for the C$_{60}$ molecule and what are their symmetries?

\item Obtain Eqs.\,(\ref{tuv}), (\ref{eq_u}) and (\ref{eq_v}).

\item Demonstrate Eqs.\,(\ref{eq:phon_sym}) and (\ref{elmzigzag}).

\item Consider $(n,n)$ armchair single wall carbon
nanotubes. Obtain $\Gamma^{\rm vector}$, $\Gamma^{\rm a.s.}$,
$\Gamma_{\rm lat.mode}$ and discuss their Raman-active modes.

\item Consider $(n,m)$ chiral single wall carbon
nanotubes ($n \ne m$). Obtain $\Gamma^{\rm vector}$,
$\Gamma^{\rm a.s.}$, $\Gamma_{\rm lat.mode}$ and discuss
their Raman-active modes.


\end{enumerate}
