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In this article, we review the analytical and numerical approaches forcomputing the phase space structures in two degrees-of-freedom Hamiltoniansystems that arise in chemical reactions. In particular, these phase spacestructures are the unstable periodic orbit associated with an index-1 saddle,the periodic orbit dividing surface, and the stable and unstable invariantmanifolds of the unstable periodic orbit. We review the approaches in thecontext of a two degrees-of-freedom Hamiltonian with a quartic potentialcoupled with a quadratic potential. We derive the analytical form of the phasespace structures for the integrable case and visualize their geometry on thethree dimensional energy surface. We then investigate the bifurcation of thedividing 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 newmodification 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}.

Elementary exposition of realizing phase space structures relevant to chemical reaction dynamics