 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 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

formula_3

where is the identity matrix and is a unit vector representing the reaction path tangent at .