Efficient Procedure to Compute the Microcanonical Volume of Initial Conditions that Lead to Escape Trajectories from a Multidimensional Potential Well

A procedure is presented for computing the phase space volume of initial conditions for trajectories that escape or “react” from a multidimensional potential well. The procedure combines a phase space transition state theory, which allows one to construct dividing surfaces that are free of local recrossing and that minimize the directional flux, and a classical spectral theorem. The procedure gives the volume of reactive initial conditions in terms of a sum over each entrance channel of the well of the product of the phase space flux across the dividing surface associated with the channel and the mean residence time in the well of trajectories which enter through the channel. This approach is illustrated for HCN isomerization in three dimensions, for which the method is several orders of magnitude more efficient than standard Monte Carlo sampling.

Figure 1

Isopotential surfaces of the HCN potential energy surface of 15 in polar representation of the Jacobi coordinates $r$, $R$, and $\gamma$.

Figure 2

Magnification of the neighborhood of the saddle at $\gamma ={\gamma }^{*}$ in Fig. 1 and the configuration space projection of cell complexes (meshes) constructed for visualization purposes on (a) the NHIM, (b) its stable and unstable manifolds, and (c) the dividing surface.

Figure 3

(a) Monte Carlo computation of mean passage time $⟨t{⟩}_j$ as a function of the number of sample points $j$. (b) Survival probability of trajectories with initial conditions in the HCN component (the units along the time axis are picoseconds). (c) Convergence of a Monte Carlo computation of energy surface volume $N_{\mathrm{HCN};j}$ as a function of the number of sample points $j$.

Figure 4

Configuration space projection of the volume of reactive initial conditions [(a); pink (gray)] in the HCN energy surface component, and subvolumes that are filled out by trajectories that enter the HCN component, perform one [(c); brown (shaded subvolume)], two [(d); green (shaded subvolume)], or three [(e); blue (shaded subvolume)] oscillation(s) in the HCN well before escaping. Panels (c)–(e) also show a representative trajectory of the relevant type. (b) Distribution of passage times $P_{\mathrm{pt}}(t)$ for initial conditions on the DS hemisphere controlling access to the HCN well.