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In the field of computational chemistryenergy minimization also called energy optimizationgeometry minimizationor geometry optimization is the process of energy optimization meaning 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 moleculean iona condensed phasea energy optimization meaning 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 energy optimization meaning, 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: 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. Energy optimization meaning of searching for global energy minimum, it might be desirable to optimize to a transition statethat is, a saddle point on the potential energy surface.
The geometry energy optimization meaning 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 energy optimization meaning lengths, bond angles and dihedral angles.
Given a set of atoms and a vector, rdescribing the atoms' positions, one can introduce the concept of the energy as a function of the positions, E r. A special case of a geometry optimization is a search for the geometry of a transition state ; this is discussed below. Using this energy optimization meaning model and an initial guess or ansatz of the correct geometry, an iterative optimization procedure is followed, for example:. For most systems of practical interest, however, it may be prohibitively expensive to compute the second derivative matrix, and it is estimated from successive values of the gradient, as is typical in a Quasi-Newton optimization.
The choice of the coordinate system can be crucial energy optimization meaning performing a successful optimization. Cartesian coordinates, for example, are redundant since a non-linear molecule with N atoms has 3 N —6 vibrational degrees of freedom whereas the set of Cartesian coordinates has 3 N dimensions. Additionally, Cartesian coordinates are highly correlated, that is, the Hessian matrix has many non-diagonal terms that are not close to zero. This can lead to numerical problems in the optimization, because, for example, it is difficult to obtain a good approximation to the Hessian matrix and calculating it precisely is too computationally expensive.
However, in case that energy is expressed with standard force fields, computationally efficient methods have been developed  able to derive analytically the Hessian matrix in Cartesian coordinates while preserving a computational complexity of the same order to that of gradient computations. Internal coordinates tend to be less correlated but are more difficult to set-up and it can be difficult to describe some systems, such as ones with symmetry or large condensed phases.
Some degrees of freedom can be eliminated from an energy optimization meaning, for example, positions of atoms or bond lengths and angles can be given fixed values. Sometimes these are referred to as being frozen degrees of freedom. Figure 1 depicts a geometry optimization of the atoms in a carbon nanotube in the presence of an external electrostatic field.
In this optimization, the atoms on the left have their positions frozen. Their interaction with the other atoms in the system are still calculated, but alteration the atoms' position during the optimization is prevented. Transition state structures can be determined by searching for saddle points on the PES of the chemical species of interest. Defined mathematically, an n th order saddle point is characterized by the following: Algorithms to locate transition state geometries fall into two main categories: Local methods are suitable when the starting point for the optimization is very close to the true transition state very close will be defined shortly and semi-global methods find application when it is sought to locate the transition state with very little a priori knowledge of its geometry.
Some methods, such as the Dimer method see belowfall into both categories. A so-called local optimization requires an initial guess of the transition state that is very close to the true transition state. Very close typically means that the initial guess must have a corresponding Hessian matrix with one negative eigenvalue, or, the negative eigenvalue corresponding to the reaction coordinate must be greater in magnitude than the other negative eigenvalues.
Further, the eigenvector with the most negative eigenvalue must correspond to the reaction coordinate, that is, it must represent the geometric transformation relating to the energy optimization meaning whose transition state is sought. Given the above pre-requisites, a local optimization algorithm can then move "uphill" along the eigenvector with the most negative eigenvalue and "downhill" along all other degrees of freedom, using something similar to a quasi-Newton method.
The dimer method  can be used to find possible transition states without knowledge of the final structure or to refine a good guess of a transition structure. The method works by moving the dimer uphill from the starting position whilst rotating the dimer to find the direction of lowest curvature ultimately negative.
The method follows the direction of lowest negative curvature computed using the Energy optimization meaning algorithm on the PES to reach the saddle point, relaxing in the perpendicular hyperplane between each "jump" activation in this direction.
Chain-of-state  methods can be used to find the approximate geometry of the transition state based on the geometries of the reactant and product. The generated approximate geometry can then serve as a starting point for refinement via a energy optimization meaning search, which was described above.
Chain-of-state methods use a series of energy optimization meaning, that is points on the Energy optimization meaning, connecting the reactant and product of the reaction of energy optimization meaning, r reactant and r productthus discretizing the reaction pathway. Very commonly, these points are referred to as beads due energy optimization meaning an analogy of a set of beads connected by strings or springs, which connect the reactant and products.
For this to be achieved, spacing constraints must be applied so that each bead r i does not simply get optimized to the reactant and product geometry. Often this constraint is achieved by projecting out components of the force on each bead r ior alternatively the movement of each bead during optimization, that are tangential to the reaction path. By projecting out components of the energy gradient or the optimization step that are parallel to the reaction path, an optimization algorithm significantly reduces the tendency of each of the beads to be optimized directly to a minimum.
The simplest chain-of-state method energy optimization meaning the linear synchronous transit LST method. It operates energy optimization meaning taking energy optimization meaning points between the reactant and product geometries and choosing the one with the highest energy for subsequent refinement via a local search. The quadratic synchronous transit QST method extends LST by allowing a parabolic reaction energy optimization meaning, with optimization of the highest energy point orthogonally to the parabola.
Specifically, the force f i on each point i is given by. In a traditional implementation, the point with the highest energy is used for subsequent refinement in a local search.
There are many variations on the NEB nudged elastic band method,  such including the climbing image NEB, in which the point with the highest energy is pushed upwards during the optimization procedure so as to hopefully give a geometry which is even closer to that of the transition state.
The string method    uses splines connecting the points, r ito measure and enforce distance constraints between the points and to calculate the tangent at each point. In each step of an optimization procedure, the points might be moved according to the force acting on them perpendicular to the path, and then, energy optimization meaning the equidistance constraint between the points is no-longer satisfied, the points can be redistributed, using the spline representation of the path energy optimization meaning generate new vectors with the required spacing.
Variations on the string method include the growing string method,  in which the guess of the pathway is grown in from the end points that is the reactant and products as the optimization progresses. Geometry optimization is fundamentally different from a molecular dynamics energy optimization meaning.
The latter simulates the motion of molecules with respect to time, subject to temperature, chemical forces, initial velocities, Brownian motion of a solvent, and so on, via the application of Newton's laws of Motion.
This means that the trajectories of the atoms which get energy optimization meaning, have some physical meaning. Geometry optimization, by contrast, does not produced a "trajectory" with any physical meaning — it is concerned with minimization of the forces acting on each atom in a collection of atoms, and the pathway via which it achieves this lacks meaning.
Different optimization algorithms could give the same result for the minimum energy structure, but arrive at it via a different pathway. From Wikipedia, the free encyclopedia. For the method in optimal control theory, see Shape optimization. For the general physical principle, see Principle of minimum energy. Retrieved 30 April Journal of Computational Chemistry. Introduction to Computational Chemistry. John Wiley and Sons Ltd. Barkema; Normand Mousseau Introduction to Computational Chemistry; Wiley: Archived from the original on Bell; Arup Chakraborty Comparison to the nudged elastic band and string methods".