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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.〔(【引用サイトリンク】title=Input reference of CP2K version trunk, Section GEO_OPT, Keyword TYPE )〕 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 second derivative matrix of the system, , also known as the Hessian matrix, which describes the curvature of the PES at , has all positive Eigen values (is positive definite). A special case of a geometry optimization is a search for the geometry of a transition state, and this is discussed in a dedicated section below. The computational model that provides an approximate 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: # calculate the force on each atom (that is, ) # if the force is less than some threshold, finish # otherwise, move the atoms by some computed step that is predicted to reduce the force # repeat from the start 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Energy minimization」の詳細全文を読む スポンサード リンク
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