2. Potential energy surface (PES) and its characteristic points

The value E of the potential energy surface depends on the position (x, y, z) of the N _at atoms that make up the system. If we place ourselves in the framework of a series expansion, a scalar function can be characterized at order 2 by its gradients and second derivatives (which then correspond to its local curvature). When this function is energy, gradients and curvatures are called "forces" and "Hessian" or "Hessian matrix", respectively.

The gradient vector, denoted $\nabla E,$ corresponds to the vector in 3N _at dimensions each component of which is the derivative of E along the direction x _i . The force...