First derivatives methods

The methods that require up to first derivatives of the energy with respect to the nuclear coordinates are mainly the steepest descent and the conjugate gradient family methods [12].

Since the magnitude of the gradient indicates the steepness of the local slope, the energy of the system can be lowered by moving each atom in response to the force acting on it. This is the basis of the steepest descent, where the displacement of the geometry at iteration k $\Delta {\bf q}$ may be obtained from the gradient ${\bf g}_k$ at the current geometry

$$\Delta {\bf q}_k = -\alpha_k \frac{{\bf g}_k}{\vert{\bf g}_k\vert}$$ (2.71)

Where $\alpha_k$ is the step length determined by trust radius or line search.

In conjugate gradient method the displacement is computed from the gradient at the current point plus the scaled previous displacement

$$\Delta q_k = \alpha_k \bigl[-\frac{{\bf g}_k}{\vert{\bf g}_k\vert} + \gamma_k\Delta q_{k-1} \bigr]$$ (2.72)

where the scaling factor $\gamma_k$ is computed using the previous gradient vectors. There are several expressions for this factor, the easiest form is the Fletcher-Reeves

$$\gamma_k = \frac{{\bf g}_k\cdot{\bf g}_k}{{\bf g}_{k-1}\cdot{\bf g}_{k-1}}$$ (2.73)