Finding a minimum of a function is frequent problem in computational chemistry. One reason for this is that spontaneous processes drive the system toward lowest free energy state. At zero Kelvin, where the entropy and thermal energy contributions to free energy are negligible, the potential energy determines the structure and behavior of molecules and materials. Some examples where minimization is used are:
The richness of examples above hints that functions that need to be minimized can be quite different. For example, minimum energy distance between two covalently bound atoms can be found by describing the bond between the two atoms as a spring that follows Hooke's law with known equilibrium distance and spring force constant. The minimum energy distance between the neon and argon atom, however, requires different potential: one that arises from the London dispersion attraction between instantaneous dipoles on two atoms and is balance by the exchange repulsion between electron clouds at very short distance. Often the functions have enormous number of independent variables and parameters (enzyme with surrounding water) and it is common to find that dependent and independent variables are related by complicated nonlinear relationships (hydrogen atom wave function). For many functions, such has Hooke's law we can easily write down first and second derivatives while for others the task is arduous at best.
Minimization is related to the process of optimization. When we say that we minimize or optimize a molecule, we typically mean that we are seeking a geometry that corresponds to a local energy minimum. For molecules, such geometries correspond to stable structures or metastable reaction intermediates. In optimization, we may also seek for saddle points. For molecules, a first order saddle point corresponds to the transition state of the reaction. Minima and saddle points are known as the stationary points because at these points the net forces with respect to small structural changes are zero. This mathematical property allows us to know if the optimization has found a stationary point. To tell if a stationary point is the minimum or maximum, the signs of second derivatives should be investigated. For minima, all second derivatives are positive while in the saddle point one and only one second derivative component is negative.
Several methods exist for finding a minimum of the function. One way to classify minimization a method is based on what kind of derivatives are used to guide the minimization.
Methods that use only the first derivative are commonly used to locate the minimum energy geometries and optimize wave functions. The first derivative tells us the downhill direction and also suggests how large steps we should take when stepping down the hill (large steps on a steep slope, small steps on flatter areas that are hopefully near the minimum). Methods such as deepest descent and variety of conjugate gradient minimization algorithms belong to this group. If the derivative of a function can be calculated analytically, the time spent at each step is not much higher than the time needed for evaluation of the energy at this step. However, if the derivative must be calculated numerically, we need to carry out additional energy calculations near each point to obtain the derivative.
Methods that use both the fist derivative and the second derivative can reach the minimum in the least number of steps because the curvature information allows estimation of were the minimum is. The simplest method in this category is the Newton-Rhapson minimization. The Newton-Rhapson method used the gradient (F) and the Hessian (H) at the current point R the new point R_newthat is closer to the true minimum. In case of one-dimensional line search, the new position is calculated as R_new = R - F/H. In case of several variables, the coordinates of the new point are calculated from analogous vector-matrix expression R_new = R - H-1*F
Finally, it is important to recognize that several computational methods exist where minimization is not of central importance. For example, if our interest lies mainly in the dynamics of molecules at usual temperatures one can apply methods of molecular dynamics that follow molecular motions away from minimum energy configurations. Somewhat surprisingly, one can use Monte Carlo or molecular dynamics simulation techniques to obtain free energies for equilibrium processes without minimization. Also, calculations of many observable molecular properties, such as infrared spectra, can be carried out without further minimization once the minimum energy geometry and corresponding optimal wave function have been found.