In order to maintain constant values of D2 and D3 during the relaxation of the remaining molecular structure constraints must be applied to the motion of several of the ions in the molecule. The angles between planes X and Y and planes Y and Z in Figure 4.3 must remain constant. These conditions will be met if the following constraints are applied to the ionic motions.
Figure 4.3: A schematic diagram defining the planes relating to the
constraints on the dihedral angles D2 and D3.
This removes 8 degrees of freedom from the motion of the ions as it also prohibits bulk rotational and translational modes. These constraints will fix the angle D3, however a change in D2 remains possible as the ion may move out of plane X, thereby changing this plane and hence D2. In order to ensure that this may not occur a correction must be applied to the position of , , and associated hydrogen ions. As the movements of the ions in one iteration will be small, a linear shift may be applied to correct for any such deviation. The shift necessary may be calculated in the following way.
=4mm =4mm Figure 4.4 shows a view along the bond indicated by unit vector . Let be the unit normal vector to plane Y and be the (non unit) vector between and . After a relaxation iteration will remain the same due to the constraints applied however and will change to and respectively. Finally define
Figure 4.4: A schematic diagram defining the vectors relating to the
correction of dihedral angle D2.
In any permitted motion must remain constant, or equivalently
must be constant.
After a relaxation, in general we will have , and such that
Therefore a shift must be applied such that
Let , and which gives
from Equation 4.4. Furthermore we wish to conserve the bond length -- which gives
Solving Equations 4.5 and 4.6 simultaneously gives
which in turn gives
The smallest shift obeying these relations is chosen and applied to
correct the deviation.