to the reaction coordinate, that is, it must represent the geometric transformation relating to the process whose transition state is sought. Given the above pre-requisites, a local optimization algorithm can then move "uphill" along the eigenvector with the most negative eigenvalue and "downhill" along all other degrees of freedom, using something similar to a quasi-Newton method. Dimer method The dimer method can be used to find possible transition states without knowledge of the final structure or to refine a good guess of a transition structure. The “dimer” is formed by two images very close to each other on the PES. The method works by moving the dimer uphill from the starting position whilst rotating the dimer to find the direction of lowest curvature (ultimately negative). Activation Relaxation Technique (ART) The Activation Relaxation Technique (ART) is also an open-ended method to find new transition states or to refine known saddle points on the PES. The method follows the direction of lowest negative curvature (computed using the Lanczos algorithm) on the PES to reach the saddle point, relaxing in the perpendicular hyperplane between each "jump" (activation) in this direction. Chain-of-state methods Chain-of-state methods can be used to find the approximate geometry of the transition state based on the geometries of the reactant and product. The generated approximate geometry can then serve as a starting point for refinement via a local search, which was described above. Chain-of-state methods use a series of vectors, that is points on the PES, connecting the reactant and product of the reaction of interest, and , thus discretizing the reaction pathway. Very commonly, these points are referred to as beads due to an analogy of a set of beads connected by strings or springs, which connect the reactant and products. The series of beads is often initially created by interpolating between and , for example, for a series of beads, bead might be given by where . Each of the beads has an energy, , and forces, and these are treated with a constrained optimization process that seeks to get an as accurate as possible representation of the reaction pathway. For this to be achieved, spacing constraints must be applied so that each bead does not simply get optimized to the reactant and product geometry. Often this constraint is achieved by projecting out components of the force on each bead , or alternatively the movement of each bead