Energy Minimization - Simple Gradient Method

Simple Gradient Method

Here we have a single function of the potential energy to minimize with 3N independent variables, which are the 3 components of the coordinates of N atoms in our system. We calculate the net force on each atom F at each iteration step t, and we move the atoms in the direction of F with a multiple factor k. k can be smaller at the beginning of calculation if we begin with a very high potential energy. Note that similar strategy can be used in molecular dynamics for reducing the probability of divergence problems at the beginning of simulations.

We repeat this step in the above equation t = 1,2,... until F reaches to zero for every atom. The potential energy of system goes down in a long narrow valley of energy in this procedure.

Though it is also called “steepest descent”, the simple gradient algorithm is in fact very time-consuming if we compare it to the nonlinear conjugate gradient approach, it is therefore known as a not very good algorithm. However, its advantage is its numerical stability, i.e., the potential energy can never increase if we take a reasonable k. Thus, it can be combined with a conjugated gradient algorithm for solving the numerical divergence problem when two atoms are too close to each other.