Lagrangian Descriptors of Thermalized Transition States on Time-Varying Energy Surfaces

Thermalized chemical reactions driven under dynamical load are characteristic of activated dynamics for arbitrary nonautonomous systems. Recent generalizations of transition state theory to obtain formally exact rates have required the construction of a time-dependent transition state trajectory. Here, we show that Lagrangian descriptors can be used to obtain this structure directly. By developing a phase space separatrix that is void of recrossings, these constructs allow for the principal criterion in the implementation of modern rate theories to be satisfied. Thus, the reactive flux over a time-varying barrier can be determined without ambiguity in chemical reactions. The generality of the formalism suggests that this approach is applicable to any activated system subjected to arbitrary driving and thermal fluctuations.

Figure 1

Phase space contour plots of $L({\mathbf{q}}_0,t_0)$ for varying values of $t_0$ are shown in (a)–(d) for a symmetrical barrier ($\kappa =1,\gamma =0$) and in (g)–(j) for a thermalized asymmetrical barrier ($\kappa =0.5,\gamma =25m_u/\mathrm{ps}$). In all panels, the TS trajectory $\mathcal{T}(t)$ is shown as a striped curve (white-orange) and the stable manifold is shown as a dashed curve (white). In the athermal case, $\mathcal{T}$ is shown over an entire period of oscillation, while in the thermal case, it is shown over the interval $[0,t_0]$. The time-varying potential surface is shown above in units of $k_{B}T$ at 298 K, which is the temperature of the thermal bath. The corresponding values of $t_0$ are marked in (e) and (f) for thermal and athermal cases, respectively, with trajectories of $\mathcal{T}$, $\mathcal{F}$, as well as $\dot{\mathcal{T}}$ (units shown at right). The thermal driving (gray) given by ${\int }_t^{t+\mathrm{\Delta }t}{\xi }_{\alpha }(t^{\prime })d{t}^{\prime }$ is shown in (f) for each integration time step $\mathrm{\Delta }t=0.001\mathrm{ps}$ [36]. Parameters in all panels are $\tau =0.5\mathrm{ps}$, $m=10\hspace{0.17em}{m}_u$, $c_1=c_2=0.75\AA$, ${\mathrm{\Omega }}_1=15{\mathrm{ps}}^{-1}$, $a=0.85{\AA }^{-1}$, and ${\tau }_{\mathrm{w}}=0.2\mathrm{ps}$.

Figure 2

Surfaces formed by $L({\mathbf{q}}_0,t_0)$ with $\tau =0.5\mathrm{ps}$ are shown above for athermal symmetrical (a)–(c) and thermalized asymmetrical (d)–(f) barriers with the corresponding contour plot shown below in each panel. The initial times are (from top to bottom) $t_0\in \{0,0.129,0.356\}$ (left panel) and $t_0\in \{0,0.209,0.4\}$ (right panel). All units and parameters are as in Fig. 1. (g) Calculation of $L_{f,b}$ with $q_0$ held constant at the marked values and $\tau =0.75\mathrm{ps}$.

Figure 3

Phase portraits of (a) forward-time and (b) backward-time integration of an ensemble of trajectories in a thermal environment. The initial position for every trajectory is $q_0=-1\AA$. Pieces of the stable ${\mathcal{W}}^s$ and unstable ${\mathcal{W}}^u$ manifolds at $t_0=0$ are shown in white and marked accordingly. Contour plots of the forward-time and backward-time surface $L_{f,b}$ with $\tau =1.0\mathrm{ps}$ are shown in (c) and (d), respectively. Parameters in all panels are as in the asymmetrical thermalized case in Fig. 1.

Figure 4

Time evolution of a swarm of trajectories in (a) athermal and (b) thermal environments. Each trajectory is colored according to the difference in initial velocity with respect to the stable manifold at $q_0=\mathcal{T}(0)-1\AA$ and $t_0=0$. The trajectories of two dividing surfaces: the TS trajectory (orange) and the instantaneous BT (striped black) are also shown. The respective reactant populations $P$ for each choice of dividing surface and the additional static surface $q=0$ are shown below. The system parameters are as in Fig. 1.