Properties of potential energy surfaces are integral to understanding the dynamics of unimolecular reactions. As discussed in chapter 2, the concept of a potential energy surface arises from the Born-Oppenheimer approximation, which separates electronic motion from vibrational/rotational motion. Potential energy surfaces are calculated by solving Eq. (2.3) in chapter 2 at fixed values for the nuclear coordinates R. Solving this equation gives electronic energies Eie(R) at the configuration R for the different electronic states of the molecule. Combining Eie(R) with the nuclear repulsive potential energy VNN(R) gives the potential energy surface Vi(R) for electronic state i (Hirst, 1985). Each state is identified by its spin angular momentum and orbital symmetry. Since the electronic density between nuclei is different for each electronic state, each state has its own equilibrium geometry, sets of vibrational frequencies, and bond dissociation energies. To illustrate this effect, vibrational frequencies for the ground singlet state (S0) and first excited singlet state (S1) of H2CO are compared in table 3.1. For a diatomic molecule, potential energy surfaces only depend on the internuclear separation, so that a potential energy curve results instead of a surface. Possible potential energy curves for a diatomic molecule are depicted in figure 3.1. Of particular interest in this figure are the different equilibrium bond lengths and dissociation energies for the different electronic states. The lowest potential curve is referred to as the ground electronic state potential. The primary focus of this chapter is the ground electronic state potential energy surface. In the last section potential energy surfaces are considered for excited electronic states. A unimolecular reactant molecule consisting of N atoms has a multidimensional potential energy surface which depends on 3N-6 independent coordinates. For the smallest nondiatomic reactant, a triatomic molecule, the potential energy surface is four-dimensional (three independent coordinates plus the energy). Since it is difficult, if not impossible, to visualize surfaces with more than three dimensions, methods are used to reduce the dimensionality of the problem in portraying surfaces. In a graphical representation of a surface the potential energy is depicted as a function of two coordinates with constraints placed on the remaining 3N-8 coordinates.
Ultrafast X-Ray Scattering Measurements of Coherent Structural Dynamics on the Ground-State Potential Energy Surface of a Diplatinum Molecule