Suppose that an atom A is hopping from one site to another, during which one bond A-B is broken and the bond A-C is created (see Figure 6.1). Then a constraint is introduced on the relative bond lengths, r_{A-B} and r_{A-C}, and the cluster relaxed maintaining this constraint. The actual constraint used is
c_{1} = r_{A-B}^{2}-r_{A-C}^{2} | (44) |
Now c_{1} is clearly negative in the configuration where the bond A-B is short and positive in the configuration where the bond A-C has been formed.
The saddle point usually, but not always, corresponds to a value of c_{1} around zero where the A-B and A-C bond lengths are equal. For interstitial O motion, a constraint is selected so that A, B and C correspond to the O atom and the two Si atoms which swap bonds with O. However, the imposition of one constraint is insufficient [128]. During the hop, the O atom initially bonded to the Si atoms D and B, becomes bonded to D and C. However at the same time the Si atom D also breaks a bond with C and makes a bond with B. To deal with this a second constraint is required. Here,
c_{2} = r_{D-B}^{2}-r_{D-C}^{2}. | (45) |
In the initial configuration c_{1}, c_{2} are both negative but become positive after the hop. The cluster is relaxed using a conjugate gradient algorithm subject to these constraints for a range of values of c_{i}. The saddle point can then be directly found by interpolation on the resultant 2D energy surface.