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Amongst the different versions of integrators,
the predictorcorrector ( PC ) [5456], method was chosen for our simulations.
The goal is to solve the second order ordinary differential equations:

(8.1) 
where

(8.2) 
and

(8.3) 
This is Newton's equations of motion for the component of the th atom, and the same equation could
be written for and . We omit the index now for simplicity.
The PC method is composed of three steps: prediction, evaluation, and correction.
In particular, from the initial position
and velocity at time t, the steps are as follows:
Prediction
1.) Predict the position , where is the time step of integration:

(8.4) 
In our simulation we use k=4, and therefore we can write explicitly the formula,
using the
from [54] :

(8.5) 
2.) Predict the velocities:

(8.6) 
and therefore:

(8.7) 
Evaluation
3.) Evaluate the force using the predicted values:

(8.8) 
Correction
4.) The final step is to correct the predictions by using some combinations
of the predicted and previous values of position and velocity:

(8.9) 
or

(8.10) 
and for velocity:

(8.11) 
and therefore:

(8.12) 
The set of parameters
promotes
numerical stability of the algorithm. Gear [56] determined their
values by applying the predictorcorrector algorithm to linear differential equations
and analyzing the stability of the method.
The values of
were chosen specially
to make the local truncation error of order of ,
and the global error for the second order differential equations is order of .
It worth to mention, that interactions are evaluated using the results of the predictor step,
but they are not reevaluated again following the corrector step.
It was shown that mean error, induced by the absence of force reevaluation, is insignificant [55].
The main ingredient of the predictor  corrector method is the corrector step,
which accounts for a feedback mechanism. The feedback can damp the instabilities that
might be introduced by the predictor step.
This method provides both the positions and velocities of the atoms at the same time
and it can be used to calculate forces,
which depend explicitly on the velocity,
this is in turn usually needed in algorithms which control temperature and pressure.
Next: NoseHoover algorithm
Up: POINT DEFECTS, LATTICE STRUCTURE
Previous: Summary and conclusions
20030115