Saddle point sampling using scaled normal coordinates

Abstract

The sampling of saddle points (SPs) on a potential energy surface (PES) is critical for describing the dynamics and transport properties of materials. Here, we propose a saddle point search (SPS) method that enables efficient sampling of practically meaningful SPs around a local minimum by converting the PES to the scaled normal coordinates (SNCs), together with the dimer method. We find that the pathway leading to a SP using SNCs is highly efficient and is independent of the system size. This results in the significantly increased SPS efficiency and the enhanced possibility of a complete catalog of the SPs. In addition, we perform SPSs using the SNCs for the diffusion of a vacancy and a dumbbell in body-centered cubic Fe and demonstrate (1) the required number of system force evaluations for a SPS decreases by at least an order of magnitude, (2) the effects of the system size on the number of force evaluations are greatly decreased, and (3) the effects of the number of atoms involved in the SPS on the probability of finding practically meaningful SPs is mostly eliminated. We also discuss the computational cost of introducing the SNCs. It should be highlighted that the SNCs could also be applied to other minimum-mode following methods, demonstrating the general versatility of the proposed method.

Introduction

The exploration of a potential energy surface (PES) of a system provides critical information on both the stability and dynamics of the system. For instance, each local minimum (LM) on a PES corresponds to a different static state of the system, and we can understand the relative stabilities of different system configurations by searching for the LMs. In addition, neighboring LMs on a PES are connected by at least one saddle point (SP), and the SPs dictate the detailed processes of the system transitions between different states and the energy barriers for the system to overcome. Therefore, obtaining a complete energy landscape, including the essential LMs and SPs, provides powerful means to understand the system properties and behaviors, e.g. atomic diffusion, chemical reaction, and protein folding [1], [2], [3].

The most straightforward approach to find a LM would be energy minimization, e.g. following the steepest descent path from an arbitrary point on the PES, though it is daunting to locate all the LMs in most practical applications. Meanwhile, saddle point search (SPS) methods can be categorized into two types: double- and single-ended methods [1]. In the double-ended methods, both the neighboring LMs that are connected by a SP of interest need to be known beforehand. For instance, the nudged elastic band method [4], [5] constructs a chain of system images between the two LMs and makes the chain converge to the minimum energy path (MEP), along which the SP is located. If all the essential LMs are known, the double-ended methods can be utilized as powerful tools for energy landscape samplings. In contrast, when one does not have the information on the neighboring LMs around a particular LM, the single-ended methods need to be employed to find potential SPs around it. In addition, multiple SPSs are usually performed until SPs are sampled sufficiently. However, SPSs by single-ended methods could be more difficult because they need to be conducted without reliable clues to the locations of SPs.

One of the commonly-used single-ended methods is the minimum-mode following approach [6], [7], [8], [9], [10], [11], where the system climbs up the PES from a particular LM based on the direction of the lowest curvature to identify a SP. Particularly, the dimer method [6], [7] has been employed as a powerful single-ended method for a wide range of problems, such as point defect diffusion in Fe [12] and on W surfaces [13], interstitial cluster formation process in Fe [14], change in a grain boundary structure in Cu under annealing [15], H atom diffusion at grain boundaries in Al [16], Pd cluster formation on the MgO(1 0 0) surface [17], ${\text{H}}_{\text{2}}\text{O}$ admolecule diffusion on ice surfaces [18], and methanol molecule decomposition on the Cu(1 0 0) surface [19]. While SPSs are successfully performed in these studies, the computational effort required for sufficiently sampling SPs by the dimer method drastically increases as the system size becomes larger, in general. This can significantly limit its applicability to processes involving a large number of atoms and is caused by the following two facts. First, the cost of force evaluations increases dramatically as the system gets larger, and force evaluation is the bottleneck of a SPS. Second, the total number of SPSs required for sufficiently sampling essential SPs increases because the chance to find a practically meaningful SP decreases [20]. For instance, a SPS by the dimer method sometimes converges to a SP that is not directly connected to the starting LM. Such a SP is not useful for estimating the frequency of the system to escape from the current LM to either of the neighboring LMs [21]. The drastic increase in the computational cost associated with increasing system size, arising from these facts, could be a serious drawback for sufficiently sampling SPs that are critical for many practical applications.

In this study, we propose a SPS method for efficiently sampling meaningful SPs around a LM on a PES. Specifically, we perform a SPS by the dimer method [6] in the scaled normal coordinates (SNCs) [22], [23], termed the SNC-dimer. As discussed later, the SNCs help the system go through a very efficient pathway toward a SP, regardless of the system size. We perform benchmark calculations for the SNC-dimer using point defect diffusion in body-centered cubic (bcc) Fe. It should be highlighted that the SNCs might be useful not only for the dimer method but also for other minimum-mode following methods, which share the same spirit of utilizing the lowest curvature information to navigate the system toward a SP.