during optimization, that are tangential to the reaction path. For example, if for convenience, it is defined that , then the energy gradient at each bead minus the component of the energy gradient that is tangential to the reaction pathway is given by where is the identity matrix and is a unit vector representing the reaction path tangent at . By projecting out components of the energy gradient or the optimization step that are parallel to the reaction path, an optimization algorithm significantly reduces the tendency of each of the beads to be optimized directly to a minimum. Synchronous transit The simplest chain-of-state method is the linear synchronous transit (LST) method. It operates by taking interpolated points between the reactant and product geometries and choosing the one with the highest energy for subsequent refinement via a local search. The quadratic synchronous transit (QST) method extends LST by allowing a parabolic reaction path, with optimization of the highest energy point orthogonally to the parabola. Nudged elastic band In Nudged elastic band (NEB) method, the beads along the reaction pathway have simulated spring forces in addition to the chemical forces, , to cause the optimizer to maintain the spacing constraint. Specifically, the force on each point i is given by where is the spring force parallel to the pathway at each point (k is a spring constant and , as before, is a unit vector representing the reaction path tangent at ). In a traditional implementation, the point with the highest energy is used for subsequent refinement in a local search. There are many variations on the NEB method, such including the climbing image NEB, in which the point with the highest energy is pushed upwards during the optimization procedure so as to (hopefully) give a geometry which is even closer to that of the transition state. There have also been extensions to include Gaussian process regression for reducing the number of evaluations. For systems with non-Euclidean (R^2) geometry, like magnetic systems, the method is modified to the geodesic nudged elastic band approach. String method The string method uses splines connecting the points, , to measure and enforce distance constraints between the points and to calculate the tangent at each point. In each step of an optimization procedure, the points might be moved according to the force acting on them perpendicular to the path, and then, if the equidistance constraint between the