In the field of computational chemistry, energy minimization (also called energy optimization, geometry minimization, or geometry optimization) is the process of finding an arrangement in space of a collection of atoms where, according to some computational model of chemical bonding, the net inter-atomic force on each atom is acceptably close to zero and the position on the potential energy surface (PES) is a stationary point (described later). The collection of atoms might be a single molecule, an ion, a condensed phase, a transition state or even a collection of any of these. The computational model of chemical bonding might, for example, be quantum mechanics. As an example, when optimizing the geometry of a water molecule, one aims to obtain the hydrogen-oxygen bond lengths and the hydrogen-oxygen-hydrogen bond angle which minimize the forces that would otherwise be pulling atoms together or pushing them apart. The motivation for performing a geometry optimization is the physical significance of the obtained structure: optimized structures often correspond to a substance as it is found in nature and the geometry of such a structure can be used in a variety of experimental and theoretical investigations in the fields of chemical structure, thermodynamics, chemical kinetics, spectroscopy and others. Typically, but not always, the process seeks to find the geometry of a particular arrangement of the atoms that represents a local or global energy minimum. Instead of searching for global energy minimum, it might be desirable to optimize to a transition state, that is, a saddle point on the potential energy surface. Additionally, certain coordinates (such as a chemical bond length) might be fixed during the optimization. Molecular geometry and mathematical interpretation The geometry of a set of atoms can be described by a vector of the atoms' positions. This could be the set of the Cartesian coordinates of the atoms or, when considering molecules, might be so called internal coordinates formed from a set of bond lengths, bond angles and dihedral angles. Given a set of atoms and a vector, , describing the atoms' positions, one can introduce the concept of the energy as a function of the positions, . Geometry optimization is then a mathematical optimization problem, in which it is desired to find the value of for which is at a local minimum, that is, the derivative of the energy with respect to the position of the atoms, , is the zero vector and the