We would like to solve the second-order differential equation
(8.8)
where
is related to the forces in equation (8.2) by
.
The first step of this algorithm consists in
evaluating the atomic positions and velocities at time
from the positions and the velocities at time
,
where
i = 0,...,k-2, k being the order of the predictor part. The
extrapolation is given by
(8.9)
for the atomic positions, and
(8.10)
for the velocities. The coefficients
and
satisfy
the equation
(8.11)
These predicted values are then corrected from the value of at a time
(calculated from the predicted values
themselves), using the expressions
(8.12)
for the atomic positions, and
(8.13)
for the velocities. The coefficient
and
satisfy
similar equations as
and .
The coefficient used in
the present work are for k = 4, and are given in table 8.1.
Table 8.1:
Coefficient of the Predictor-Corrector algorithm for k = 4 for second-order differential equation.
k = 4 (
1/24)
1
2
3
19
-10
3
27
-22
7
3
10
-1
7
6
-1
The Predictor-Corrector algorithm gives more accurate positions and
velocities than the leapfrog algorithm, and is therefore suitable in
very ``delicate'' calculations. However, it is computationally
expensive and needs significant storage (the forces at the last two
stages, and the coordinated and velocities at the last step).