Projection operator method for geometry optimization with constraints

A new approach is presented for performing geometry optimization for stationary points on potential energy hypersurfaces with equality constraints on the internal coordinates of a polyatomic system. The working equations are the same as for unconstrained Newton–Raphson optimization in Cartesian coordinates except that projection operators are applied to the gradient and Hessian to enforce the constraints. Two reactive systems with different kinds of constraints are treated as examples: OH + H₂ → OH 3≠ → H₂O + H with one constrained OH bond distance and CH₃ + H₂ → CH 5≠ → CH₄ + H with one constrained HCH bond angle in the CH₃ group or with one constrained bond distance and one simultaneously constrained bond angle. In each case we optimized all reactants and products as well as the saddle point, all subject to the constraints.