In the field of
computational chemistry
Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into computer programs, to calculate the structures and properties of m ...
, 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
A potential energy surface (PES) describes the energy of a system, especially a collection of atoms, in terms of certain parameters, normally the positions of the atoms. The surface might define the energy as a function of one or more coordinates; ...
(PES) is a stationary point (described later). The collection of atoms might be a single
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
, an
ion
An ion () is an atom or molecule with a net electrical charge.
The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conven ...
, a
condensed phase
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the sub ...
, a
transition state
In chemistry, the transition state of a chemical reaction is a particular configuration along the reaction coordinate. It is defined as the state corresponding to the highest potential energy along this reaction coordinate. It is often marked wi ...
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
Water () is a polar inorganic compound that is at room temperature a tasteless and odorless liquid, which is nearly colorless apart from an inherent hint of blue. It is by far the most studied chemical compound and is described as the "uni ...
, 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
A chemical structure determination includes a chemist's specifying the molecular geometry and, when feasible and necessary, the electronic structure of the target molecule or other solid. Molecular geometry refers to the spatial arrangement of at ...
,
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
,
chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in wh ...
,
spectroscopy
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
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
In chemistry, the transition state of a chemical reaction is a particular configuration along the reaction coordinate. It is defined as the state corresponding to the highest potential energy along this reaction coordinate. It is often marked wi ...
, 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
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
problem, in which it is desired to find the value of for which is at a
local minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
, 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
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
, which describes the curvature of the PES at , has all positive
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
(is
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite f ...
).
A special case of a geometry optimization is a search for the geometry of a
transition state
In chemistry, the transition state of a chemical reaction is a particular configuration along the reaction coordinate. It is defined as the state corresponding to the highest potential energy along this reaction coordinate. It is often marked wi ...
; this is discussed below.
The computational model that provides an approximate could be based on
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
(using either
density functional theory
Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
or
semi-empirical methods),
force fields
Force field may refer to:
Science
* Force field (chemistry), a set of parameter and equations for use in molecular mechanics simulations
* Force field (physics), a vector field indicating the forces exerted by one object on another
* Force field ( ...
, or a combination of those in case of
QM/MM
The hybrid QM/MM (quantum mechanics/molecular mechanics) approach is a molecular simulation method that combines the strengths of ''ab initio'' QM calculations (accuracy) and MM (speed) approaches, thus allowing for the study of chemical processes ...
. Using this computational model and an initial guess (or
ansatz
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the ...
) 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
Practical aspects of optimization
As described above, some method such as quantum mechanics can be used to calculate the energy, , the gradient of the PES, that is, the derivative of the energy with respect to the position of the atoms, and the second derivative matrix of the system, , also known as the
Hessian matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
, which describes the curvature of the PES at .
An
optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
algorithm can use some or all of , and to try to minimize the forces and this could in theory be any method such as gradient descent, conjugate gradient or Newton's method, but in practice, algorithms which use knowledge of the PES curvature, that is the Hessian matrix, are found to be superior. 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 for performing a successful optimization. Cartesian coordinates, for example, are redundant since a non-linear molecule with atoms has vibrational
degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
whereas the set of Cartesian coordinates has 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. Many modern computational chemistry software packages contain automatic procedures for the automatic generation of reasonable coordinate systems for optimization.
Degree of freedom restriction
Some degrees of freedom can be eliminated from an optimization, 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 optimization
Transition state
In chemistry, the transition state of a chemical reaction is a particular configuration along the reaction coordinate. It is defined as the state corresponding to the highest potential energy along this reaction coordinate. It is often marked wi ...
structures can be determined by searching for
saddle points
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the functi ...
on the PES of the chemical species of interest. A first-order saddle point is a position on the PES corresponding to a minimum in all directions except one; a second-order saddle point is a minimum in all directions except two, and so on. Defined mathematically, an ''n''th order saddle point is characterized by the following: and the Hessian matrix, , has exactly ''n'' negative eigenvalues.
Algorithms to locate transition state geometries fall into two main categories: local methods and semi-global methods. 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 below), fall into both categories.
Local searches
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 process 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.
Dimer 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 “dimer” is formed by two images very close to each other on the PES. 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).
Activation Relaxation Technique (ART)
The Activation Relaxation Technique (ART) is also an open-ended method to find new transition states or to refine known saddle points on the PES. The method follows the direction of lowest negative curvature (computed using the
Lanczos algorithm
The Lanczos algorithm is an iterative method devised by Cornelius Lanczos that is an adaptation of power iteration, power methods to find the m "most useful" (tending towards extreme highest/lowest) eigenvalues and eigenvectors of an n \times n ...
) on the PES to reach the saddle point, relaxing in the perpendicular hyperplane between each "jump" (activation) in this direction.
Chain-of-state methods
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 local search, which was described above.
Chain-of-state methods use a series of vectors, that is points on the PES, connecting the reactant and product of the reaction of interest, and , thus discretizing the reaction pathway. Very commonly, these points are referred to as ''beads ''due to an analogy of a set of beads connected by strings or springs, which connect the reactant and products. The series of beads is often initially created by interpolating between and , for example, for a series of beads, bead might be given by
where . Each of the beads has an energy, , and forces, and these are treated with a constrained optimization process that seeks to get an as accurate as possible representation of the reaction pathway. For this to be achieved, spacing constraints must be applied so that each bead 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 , or alternatively the movement of each bead during optimization, that are tangential to the reaction path. For example, if for convenience, it is defined that , then the energy gradient at each bead minus the component of the energy gradient that is tangential to the reaction pathway is given by
where is the identity matrix and is a unit vector representing the reaction path tangent at . 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.
Synchronous transit
The simplest chain-of-state method is the linear synchronous transit (LST) method. It operates by taking interpolated 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 path, with optimization of the highest energy point orthogonally to the parabola.
Nudged elastic band
In Nudged elastic band (NEB) method, the beads along the reaction pathway have simulated spring forces in addition to the chemical forces, , to cause the optimizer to maintain the spacing constraint. Specifically, the force on each point ''i'' is given by
where
is the spring force parallel to the pathway at each point (''k'' is a spring constant and , as before, is a unit vector representing the reaction path tangent at ).
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 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. There have also been extensions to include
Gaussian process regression
In statistics, originally in geostatistics, kriging or Kriging, also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging giv ...
for reducing the number of evaluations. For systems with non-Euclidean (R^2) geometry, like magnetic systems, the method is modified to the geodesic nudged elastic band approach.
String method
The string method uses splines connecting the points, , to 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, if the equidistance constraint between the points is no-longer satisfied, the points can be redistributed, using the spline representation of the path to 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.
Comparison with other techniques
Geometry optimization is fundamentally different from a
molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the ...
simulation. The latter simulates the motion of molecules with respect to time, subject to temperature, chemical forces, initial velocities,
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
of a solvent, and so on, via the application of
Newton's laws of Motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in moti ...
. This means that the trajectories of the atoms which get computed, have some physical meaning. Geometry optimization, by contrast, does not produce 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.
See also
*
Constraint composite graph The constraint composite graph is a node-weighted undirected graph associated with a given combinatorial optimization problem posed as a weighted constraint satisfaction problem. Developed and introduced by Satish Kumar Thittamaranahalli (T. K. Sati ...
*
Graph cuts in computer vision As applied in the field of computer vision, graph cut optimization can be employed to efficiently solve a wide variety of low-level computer vision problems (''early vision''), such as image smoothing, the stereo correspondence problem, image segm ...
– apparatus for solving
computer vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the hum ...
problems that can be formulated in terms of energy minimization
*
Energy principles in structural mechanics Energy principles in structural mechanics express the relationships between Stress (physics), stresses, Strain (materials science), strains or Deformation (mechanics), deformations, displacement (vector), displacements, material properties, and stru ...
References
{{reflist, 2
External links
Numerical Recipes in Fortran 77
Additional references
* Payne et al., "Iterative minimization techniques for ab initio total-energy calculations: Molecular dynamics and conjugate gradients", ''Reviews of Modern Physics'' 64 (4), pp. 1045–1097. (1992
(abstract)
* Stich et al.,
Conjugate gradient minimization of the energy functional: A new method for electronic structure calculation, ''Physical Review B'' 39 (8), pp. 4997–5004, (1989)
* Chadi,
Energy-minimization approach to the atomic geometry of semiconductor surfaces, ''Physical Review Letters'' 41 (15), pp. 1062–1065 (1978)
Mathematical optimization
Computational chemistry