Orthogonality Constraints

  In order to constrain the system in this way it is necessary to firstly obtain the ground state structures of the two end points of the diffusion path, A and B. A vector is then constructed in order to translate A into B, where N is the number of atoms in the system.

$${\bf R_i} = {\bf B_i - A_i}, i = 1,3N$$ (43)

The atoms are then moved a fraction $\alpha$ along this vector, where $\alpha$=0 corresponds to structure A, $\alpha$=1 corresponds to structure B. The atoms are allowed to relax using a conjugate gradient algorithm at each point, subject to the constraint that on each iteration the total system relaxation vector must be orthogonal to ${\bf R_i}$. If only one atom is moving then this is equivalent to allowing the atom to relax in a plane perpendicular to its direction of movement.

This method is simple to implement but can sometimes result in a system that is overconstrained; it tends to be best for systems where the diffusion motion primarily involves a single atom [152]. In particular when examining compound motion such as the oxygen dimer this method will only sample diffusion paths where both atoms are moving simultaneously, which need not be the lowest energy diffusion mechanism.

Therefore although initial investigations were performed with this method, we later switched to a second, bond-length constraint method detailed in the following section.