  In this article, we review the analytical and numerical approaches for
computing the phase space structures in two degrees-of-freedom Hamiltonian
systems that arise in chemical reactions. In particular, these phase space
structures are the unstable periodic orbit associated with an index-1 saddle,
the periodic orbit dividing surface, and the stable and unstable invariant
manifolds of the unstable periodic orbit. We review the approaches in the
context of a two degrees-of-freedom Hamiltonian with a quartic potential
coupled with a quadratic potential. We derive the analytical form of the phase
space structures for the integrable case and visualize their geometry on the
three dimensional energy surface. We then investigate the bifurcation of the
dividing surface due to the changes in the parameters of the potential energy.
We also review the numerical method of \emph{turning point} and present its new
modification called the \emph{turning point based on configuration difference}
for computing the unstable periodic orbit in two degrees-of-freedom systems.
These methods are implemented in the open-source python package,
UPOsHam~\cite{Lyu2020}.
