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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