The semi-classical wavepacket method (SWP, also known as the Herman-Kluk propagator or as the initial value representation method) is a viable approach to theoretical description of many molecular processes including collisions, scattering and reaction dynamics. This method accounts for all features of the potential energy surface in a natural dynamic way, incorporates full dimensionality of the problem and avoids using sudden approximation. This is a time-dependent method based on propagating the ensemble of classical trajectories. In addition, every individual trajectory possesses phase and monodromy (stability) matrix, which allows different trajectories to interfere with each other and contribute coherently to the overall semiclassical wave function. All quantum effects appear as the result of this interference. It is well established that this approach describes the quantum zero-point energy, but much less is known about ability of the SWP method to reproduce quantum symmetry and characterize the quantum scattering resonances. These two quantum effects are known to be important for recombination reactions like one that forms ozone:
Thus, it would be instructive to test the validity of the SWP approach using a simplified problem relevant to the reaction dynamics. We have carried out such a methodological study: We applied the SWP method to describe quantum scattering resonances in an ozone-like two-body scattering system. We considered a simple model, where the internal structure of O2 is neglected and the focus is on scattering resonances (metastable states) formed behind the centrifugal barrier due to the angular momentum of relative O2+O motion. We considered all values of angular momentum required to calculate the converged rate of ozone formation and characterized every important resonance by its energy and lifetime. Overall, we looked at more than 80 resonances with lifetimes that evenly cover the range from 0.01 ps to 1000 ps. This certainly represents a thorough test of the SWP approach. To assess its accuracy we compared the SWP results against the data obtained using a fully quantum wavepacket propagation technique. We found that energies of the metastable states obtained using the SWP method are accurate to within 0.1 cm-1 and 2 cm-1 for under-the-barrier and over-the-barrier states, respectively. The SWP lifetimes in the range 0.5 < τ < 100 ps are accurate to within 10%. Within the well region the SWP wave functions are identical to quantum ones. Only in the classically forbidden or in the barrier region the SWP method slightly underestimates tunneling, as can be seen from Fig. 5.
Figure 5: Barrier region of the J = 38 potential function. Initial semi-classical wave packet is placed at r0 = 4.64 a.u. Wave functions of under-the-barrier (v = 15) and over-the-barrier (v = 16) metastable states were calculated by propagating this initial wave packet, calculating the autocorrelation function and obtaining its Fourier transform. Shaded areas show the classically forbidden regions; the barrier top is at rtop = 5.29 a.u.
Recently, we have applied the SWP approach to study scattering resonances in a more realistic two-dimensional problem. The 2D potential energy surface is shown in Fig. 6 which mimics major features of the ozone PES. In this case, besides characterization of the metastable states by their energies and lifetimes, we were able to demonstrate the ΔZPE effect and even obtain the anomalous isotope effect. We hope that this method will allow us to treat the stabilization process of the metastable states, including the quantum symmetry effect.
Figure 6: In order to capture the metastable states in the barrierless potential the initial wave packets should be placed far into the channels.