Energy Minimization - Nonlinear Conjugate Gradient Method

Nonlinear Conjugate Gradient Method

The conjugate gradient algorithm includes two basic steps: adding an orthogonal vector to the current direction of the search, and then move them in another direction nearly perpendicular to this vector. These two steps are also known as: step on the valley floor and then jump down. Figure 2 shows a highly simplified comparison between the conjugate and the simple gradient methods on a 1D energy curve.

In this algorithm, we minimize the energy function by moving the atoms as follows,

where

and gamma is updated using the Fletcher-Reeves formula as:

Here we note that gamma can also be calculated by using the Polak-Ribiere formula, however, it is less efficient than the Fletcher-Reeves one for certain energy functions. At the beginning of calculation (when t = 1), we can make the search direction vector 'h0 = 0.

This algorithm is very efficient. However, it is not quite stable with certain potential functions, i.e. it sometimes can step so far into a very strong repulsive energy range (e.g. when two atoms are too close to each other), where the gradient at this point is almost infinite. It can directly result a typical data-overrun error during the calculation. To resolve this problem, we can combine the conjugate gradient algorithm with the simple one. Figure 3 shows the schematics of the combined algorithm. We note for implementation that steps 2 and 5 can be combined into a single step.