Accelerated Relaxation Engines for Optimizing to Minimum Energy Path

Abstract

In the last few decades, several novel algorithms have been designed for finding critical points on PES and the minimum energy paths connecting them. This has led to considerably improve our understanding of reaction mechanisms and kinetics of the underlying processes. These methods implicitly rely on computation of energy and forces on the PES, which are usually obtained by computationally demanding wave-function or density-function based ab initio methods. To mitigate the computational cost, efficient optimization algorithms are needed. Herein, we present two new optimization algorithms: adaptively accelerated relaxation engine (AARE), an enhanced molecular dynamics (MD) scheme, and accelerated conjugate-gradient method (Acc-CG), an improved version of the traditional conjugate gradient (CG) algorithm. We show the efficacy of these algorithms for unconstrained optimization on 2D and 4D test functions. Additionally, we also show the efficacy of these algorithms for optimizing an elastic band of images to the minimum energy path on two analytical potentials (LEPS-I and LEPS-II) and for HCN/CNH isomerization reaction. In all cases, we find that the new algorithms outperforms the standard and popular fast inertial relaxation engine (FIRE).

Figures (6)

• Figure 1: Decomposition of $\hat{\mathbf{v}}$ into components parallel ($v_{\parallel}\mathbf{\hat{e}}_{\parallel}$) and perpendicular ($v_{\perp}\mathbf{\hat{e}}_{\perp}$) to the $\hat{\mathbf{F}}$. Here, $\hat{\mathbf{F}}$ is aligned along the $y$-axis and $|v_{\parallel}|^2 + |v_{\perp}|^2 = 1$. For this analysis, we assume a $two$-dimensional system and the power factor greater than zero.
• Figure 2: Trajectories of FIRE and AARE on 2D test functions. The mathematical formulas for these functions are provided in Supplementary Information (Table S1). (a) and (b) demonstrates the problems with the FIRE algorithm. In (a) and (c), we use the Himmelblau function: starting point $= \left(0,0\right)$; minima $= \left(3, 2\right)$; and convergence criteria taken as $\|\mathbf{F}\| < 0.01$. In (b) and (d), we use the SCEQ function: starting point $= \left(-0.4,0\right)$; minima $= \left(-0.83, -0.52\right)$; and convergence criteria taken as $\|\mathbf{F}\| < 0.001$.
• Figure 3: Trajectories of FIRE, ACC-CG, AARE-PR and AARE-FR on selected 2D test functions. The mathematical formulas for these functions are provided in Supplementary Information (Table S1). We used a convergence criteria of $\|\mathbf{F}\| < 0.01$ for these runs.
• Figure 4: Euclidean norm of forces as a function of number of force evaluations for FIRE, Acc-CG, AARE-PR and AARE-FR. Mathematical formulas for the selected 2D test functions are provided in Supplementary Information (Table S1). The algorithm was terminated when $\|\mathbf{F}\| < 0.01$.
• Figure 5: (a) Illustrates NEB pathways for the LEPS-I potential using AARE-FR as an optimizer. The black curve represents the initial discretized pathway. The numbers on each intermediate pathway represents the number of force evaluations required to reach at that point. The final MEP, achieved after 982 force evaluations, is shown in red. (b) Euclidean norm of forces as function of the number of force evaluations for FIRE, Acc-CG, AARE-PR, and AARE-FR during an optimization run on LEPS-I potential. The mathematical formula for the LEPS-I is provided in Supplementary Information (Section S2). We use the convergence criterion as $\|\mathbf{F}\| < 0.01$ and a $k_{\text{spring}} = 1$ for these runs.