Finding a minimum of a function is a frequent problem in computational chemistry. One reason for this is that spontaneous processes drive the system toward the 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. At real temperatures, thermal energy and entropy also contribute but many molecular properties can still be rationalized well based on the features of potential energy surfaces. Some examples in which minimization is used are:
The richness of examples above hints that functions that need to be minimized can be quite different. For example, the 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 the known equilibrium distance and the spring force constant. The minimum energy distance between 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 a 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 as Hooke's law, we can easily write down the 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 to find saddle points. For molecules, a first order saddle point corresponds to top of the rotational barrier of an internal rotor or to the transition state of the chemical 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.
Finally, it is important to recognize that several computational methods exist in which 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.