Energy minimization

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. 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. 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, r, describing the atoms' positions, one can introduce the concept of the energy as a function of the positions, E(r). Geometry optimization is then a mathematical optimization problem, in which it is desired to find the value of r for which E(r) is at a local minimum, that is, the derivative of the energy with respect to the position of the atoms, ∂E/∂r, is the zero vector and the second derivative matrix of the system, ∂∂E/∂ri∂rj, also known as the Hessian matrix, which describes the curvature of the PES at r, has all positive eigenvalues (is positive definite). A special case of a geometry optimization is a search for the geometry of a transition state; this is discussed below. The computational model that provides an approximate E(r) could be based on quantum mechanics (using either density functional theory or semi-empirical methods), force fields, or a combination of those in case of QM/MM. Using this computational model and an initial guess (or ansatz) of the correct geometry, an iterative optimization procedure is followed, for example: As described above, some method such as quantum mechanics can be used to calculate the energy, E(r) , the gradient of the PES, that is, the derivative of the energy with respect to the position of the atoms, ∂E/∂r and the second derivative matrix of the system, ∂∂E/∂ri∂rj, also known as the Hessian matrix, which describes the curvature of the PES at r.