### Introduction

This project is a comparison of different saddle point finding methods. All the methods we studied only need the force and energy of the system (and not the Hessian matrix of second derivative). Because of this, the methods scale well with system size, and can be used for relatively large systems using interaction potentialas expensive as density functional theory.

### NEB

The Nudged Elastic Band (NEB) method is used to find reaction pathways when both the initial and final states are known. Using this code, the Minimum Energy Path (MEP) for any given chemical process may be calculated, however both the initial and final states must be known. The code works by linearly interpolating a set of images between the known initial and final states (as a "guess" at the MEP), and then minimizes the energy of this string of images. Each "image" corresponds to a specific geometry of the atoms on their way from the initial to the final state, a snapshot along the reaction path. Thus, once the enery of this string of images has been minimized, the true MEP is revealed.

More about the NEB method

### The Dimer Method

The dimer method (or more generally a min-mode method) is used to find saddle points on a potential energy surface. It is complimentary to the nudged elastic band method because it does not require a final state. We have used the dimer method in two ways. The first is to evolve a configuration from an initial guess to a saddle point. The second application, which is much more challenging, is to find all low lying saddle points connecting the initial basin to adjacent basins. This problem is of particular interest in rate theory. If the transitions out of a basin are found, their individual rates can be evaluated, and the system can be evolved over long time scales using kinetic Monte Carlo. This involves randomly picking one of the transitions out of a Boltzmann distribution, and moving the system over that saddle point to an adjacent basin.

### The NEB and Dimer in VASP

The dimer method has been incorperated into the DFT code VASP (Vienna Ab-initio Simulation Package). Our local collection of information, source and scripts is located here.

### BGSD

Similar to the dimer method, the biased gradient squared descent (BGSD) saddle point finding method does not require knowledge of a product state. BGSD transforms a potential energy surface into the square of the gradient which converts all critical points into local minima. A bias term is added to this transformed landscape to stabilize critical points around a specified energy level and destablize all other critical points (See the figure to the right for details of the transformation of the potential energy surface). An advantage of BGSD is its ability to target saddle points at a specified energy, specifically the lowest lying saddle points.

### Benchmarking

See the results of our testing on our benchmarking page.

### Basin constrained κ-Dimer

Dimer searches that are initiated from a reactant state of interest can converge to saddles that are not on the boundary of the reactant state. These disconnected saddles are not directly useful for calculating the escape rate. The reason that the method finds disconnected saddles is a result of the fact that the dimer method tracks local ridges, defined as the set of points where the force is perpendicular to the negative curvature mode, and not the true ridge, defined as the boundary of the set of points which minimize to the reactant. The local ridges tend to deviate from the true ridge away from saddle points. Furthermore, the local ridge can be discontinuous and have holes which allow the dimer to cross the true ridge and escape the initial state. To solve this problem, we employ an alternative definition of a local ridge based upon the minimum directional curvature of the isopotential hyperplane, κ, which provides additional local information to tune the dimer dynamics. We find that hyperplanes of κ=0 pass through all saddle points but rarely intersect with the true ridge elsewhere. By restraining the dimer within the κ<0 region, the probability of converging to disconnected saddles is significantly reduced and the efficiency of finding connected saddles is increased. The left figure shows a 2D example comparing the evolution of the dimer (orange) and κ-dimer (yellow) methods from the same initial position. In 2D, κ is the curvature of the isopotential line.

### References

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