Energy Minimization

In computational chemistry, energy minimization (also called energy optimization or geometry optimization) methods are used to compute the equilibrium configuration of molecules and solids.

Stable states of molecular systems correspond to global and local minima on their potential energy surface. Starting from a non-equilbrium molecular geometry, energy minimization employs the mathematical procedure of optimization to move atoms so as to reduce the net forces (the gradients of potential energy) on the atoms until they become negligible.

Like molecular dynamics and Monte-Carlo approaches, periodic boundary conditions have been allowed in energy minimization methods, to make small systems. A well established algorithm of energy minimization can be an efficient tool for molecular structure optimization.

Unlike molecular dynamics simulations, which are based on Newtonian dynamic laws and allow calculation of atomic trajectories with kinetic energy, molecular energy minimization does not include the effect of temperature, and hence the trajectories of atoms during the calculation do not really make any physical sense, i.e. we can only obtain a final state of system that corresponds to a local minimum of potential energy. From a physical point of view, this final state of the system corresponds to the configuration of atoms when the temperature of the system is approximately zero, e.g. as shown in Figure 1, if there is a cantilevered beam vibrating between positions 1 and 2 around an equilibrium position 0 with an initial kinetic motion, whether we start with the state 1, the state 2 or any other state between these two positions, the result of energy minimization for this system will always be the state 0.

Gradient-based algorithms are the most popular methods for energy minimization. The basic idea of gradient methods is to move atoms according to the total net forces acting on them. The force on an atom is calculated as the negative gradient of total potential energy of system, as follows:

where r_i is the position of atom i and Utot is the total potential energy of the system.

An analytical formula of the gradient of potential energy is preferentially required by the gradient methods. If not, one needs to calculate numerically the derivatives of the energy function. In this case, the Powell's direction set method or the downhill simplex method can generally be more efficient than the gradient methods.