The Role of Depth and Flatness of a Potential Energy Surface in Chemical Reaction Dynamics
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The Role of Depth and Flatness of a Potential Energy Surface in Chemical Reaction Dynamics Wenyang Lyu1* , Shibabrat Naik1** , and Stephen Wiggins1*** 1
School of Mathematics, University of Bristol, Fry Building, Woodland Road, BS8 1UG Bristol, United Kingdom Received April 26, 2020; revised September 02, 2020; accepted September 09, 2020
Abstract—In this study, we analyze how changes in the geometry of a potential energy surface in terms of depth and flatness can affect the reaction dynamics. We formulate depth and flatness in the context of one and two degree-of-freedom (DOF) Hamiltonian normal form for the saddlenode bifurcation and quantify their influence on chemical reaction dynamics [1, 2]. In a recent work, Garc´ıa-Garrido et al. [2] illustrated how changing the well-depth of a potential energy surface (PES) can lead to a saddle-node bifurcation. They have shown how the geometry of cylindrical manifolds associated with the rank-1 saddle changes en route to the saddle-node bifurcation. Using the formulation presented here, we show how changes in the parameters of the potential energy control the depth and flatness and show their role in the quantitative measures of a chemical reaction. We quantify this role of the depth and flatness by calculating the ratio of the bottleneck width and well width, reaction probability (also known as transition fraction or population fraction), gap time (or first passage time) distribution, and directional flux through the dividing surface (DS) for small to high values of total energy. The results obtained for these quantitative measures are in agreement with the qualitative understanding of the reaction dynamics. MSC2010 numbers: 37J05,37J15,37J20,34C23,70H05,37G05 DOI: 10.1134/S1560354720050044 Keywords: Hamiltonian dynamics, bifurcation theory, phase space methods
1. INTRODUCTION The topography of a potential energy surface (PES) plays a fundamental role in determining reaction paths and reaction mechanisms [3–5]. For a given chemical reaction, the potential energy surface (PES) describes the variation of the electronic energy with the nuclear coordinates within the Born – Oppenheimer approximation [3]. The electronic structure calculations generate a landscape of mountain ranges with peaks (local maxima) and valleys (local minima) with varying depth and flatness. One approach of crossing the mountain ranges is by going over the lowest point called the index-1 saddle [6] and this mechanism is quite common in chemical reactions. This indicates that the potential energy difference between the saddle and the bottom of the valley, that is, the depth, and the gradient of the landscape, that is, the flatness, dictates the rate and volume of crossings of the saddle. Thus, the role of depth and flatness in crossing the saddle is relevant for understanding reaction dynamics. Furthermore, the asymmetry in the depth of a potential well on either side of an index-1 saddle can lead to difference in forward and backward reaction rates. In addition, it has been noted in ab initio calculations that co
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