Constrained shrinking dimer dynamics for saddle point search with constraints

In this paper, we study the constrained shrinking dimer dynamics (CSDD) which leads to numerical procedures for locating saddle points (transition states) associated with an energy functional defined on a constrained manifold. We focus on the most generic case corresponding to a constrained stationary point where the projected Hessian of the energy onto the tangent hyperplane of the constrained manifold has only one unstable direction and demonstrate, in this case, the local stability of the CSDD. We examine various numerical implementation issues and consider some interesting applications of CSDD including the computation of periodic centroidal Voronoi tessellations, generalized Thomson problems about particle/charge distribution on the unit sphere, and the critical nuclei morphology in binary phase transformations.