Relaxing an Atomistic Configuration

Position vector that comprises 3V components specifying the spacial positions 

Since convergence by steepest descents can be rather slow in the final stages of relaxation, the search for the minimum at this point can be switched to the Newton-Raphson method. In this procedure the second derivative at x is also calculated and compounded into a Hessian matrix H, whose elements are 

y = 3 V /dxjdxj (i and j run from 1 to 3A/). The Hjt is the curvature of the potential energy surface at the point in configuration space, which with the slope vector allows the energy V at a slightly displaced point to be expressed as a Taylor expansion 

Note that using Eq. (10) does not eliminate the gradient vector which provides the direction of the required displacement along the energy surface toward the minimum. An improved estimate of the configuration of the system is then 

The procedure is repeated iteratively until gj is smaller than a preset discriminant, i.e., until an energy minimum is located. 

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