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The conjugate gradient is an algorithm to find the local minimum of a
function, that can be of many variables. In the present case, the
potential energy of the sample has to be minimized, with a number 3N
of variables, where N is the number of atoms in the system. The Taylor
expansion of a function
about a point
of 3Ndimension is given by

(8.20) 
where
,
and
.
From G one can
define a set of {} conjugate directions according to

(8.21) 
for
and
i,j = 1,...,N. A line minimization is then
performed along these directions, with an initial direction defined by

(8.22) 
that is, by the forces applied on each atom. The i+1 step is derived
from the i step by

(8.23) 
where

(8.24) 
and

(8.25) 
The vectors
and
satisfy the orthogonality
conditions

(8.26) 
It can be shown that the directions
and
fulfill the condition in equation 8.21, and that the function is
minimized along the conjugate directions. At each step the position of
the atoms are calculated according to

(8.27) 
where
is determined so that
minimizes
in the direction
from .
Next: Results for damaged diamond
Up: The numerical techniques
Previous: Molecular dynamics at constant
David Saada
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