Equations of motion, trajectories, and excitation functions

In order to keep the expressions transparent we, once again, restrict the discussion to the dissociation of the triatomic molecule ABC into products A and BC. Furthermore, the total angular momentum is limited to J = 0. In this chapter we consider the vibration and the rotation of the fragment molecule simultaneously. The corresponding Hamilton function, i.e., the total energy as a function of all coordinates and momenta, using action-angle variables (McCurdy and Miller 1977 Smith 1986), reads 

At each instant t the classical system is uniquely specified by the vector r(t) = R(t), r(t),7(t), P(t),p(t), j(t) in the six-dimensional phase-space. The total of all coordinates and momenta as a function of time is called a classical trajectory. The evolution of the trajectory r(t) is determined by the Hamilton equations (Goldstein 1951 ch.7 Arnold 1978) 

Hamilton s equations form a set of coupled first-order differential equar tions which under normal conditions can be numerically integrated without any problems. The forces —dVi/dR and —dVi/dr and the torque —dVj/d7, which reflect the coordinate dependence of the interaction potential, control the coupling between the translational (R,P), the vibrational (r,p), and the rotational (7,j) degrees of freedom. Due to this coupling energy can flow between the various modes. The translational mode becomes decoupled from the internal motion of the diatomic fragment (i.e., dP/dt = 0 and dR/dt =constant) when the interaction potential diminishes in the limit R — 00. As a consequence, the translational energy 

In order to calculate final rotational state distributions it is useful to define the so-called rotational excitation function (McCurdy and Miller 1977 Schinke and Bowman 1983 Schinke 1986a,c, 1988a,b) 

J(ro) is the classical counterpart of the rotational quantum number j of the fragment molecule. (Note that J(to) is defined as a dimensionless quantity.) It represents the final angular momentum of the fragment molecule as a function of all initial variables to- In the same way, we define the vibrational excitation function (Miller 1974, 1975, 1985) 

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