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.
constrained to move along 
bond.
constrained to move in plane Y.
constrained to move in plane Z.
constrained to move in plane X.
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.
