Nature acts to form molecular structures that minimize strain. This strain can be expressed in terms of an energy function.
The best arrangement of atoms that minimizes strain is defined in terms of an minimum (extremum) of a energy (cost) function.
For classical solutions, the energy expression contains no explicit separation of electronic and nuclear degrees of freedom.
For quantum mechanical solutions of molecular systems, geometries are typically determined within the framework of the Born-Oppenheimer approximation. [Separating the nuclear and electronic degrees of freedom.] This typically requires an iterative solution to determine the electronic energies, alter the nuclear center geometry, re-compute the electronic energies, etc. That is, the electronic energy is parameterized with respect to the nuclear coordinates and the energy obtained is for a fixed configuration of nuclear coordinates. The process of the geometry optimization is to optimize the total energy with respect to the nuclear coordinates.
There are two primary methods ;
An overview of the iterative improvement scheme is as follows:
There are two different general procedures:
These are optimization methods that do not use derivatives, that is, local descriptions of the changes in geometry. One solves a system of simultaneous equations.
Consider an energy function
Let
Geometric Rearrangement is
where X1 is the new geometric
configuration.
Implementations of function methods.
These methods are useful when derivatives are not available (easily or otherwise). The drawback is these methods can have very slow convergence. Rarely used in quantum mechanical calculations.
(There is a variant of a Fletcher-Powell method sometimes used.)
Consider an energy function
Let
be the initial geometry.
Geometric Rearrangement is
where X1 is the new geometric
configuration.
A Taylor series of the energy function using this rearrangement is;
where
is called the gradient
is called the Hessian
Implementation of Derivative Methods
Gradient Methods
The gradient optimization approximate energy surface at step k
can be expressed in terms of the position vector
,
the computed energy
,
the gradient
and the approximate Hessian
Quasi-Newton methods Estimating the Hessian
Implementation of Derivative Methods
Broyden Family of algorithms;
where
Implementation of Derivative Methods
Note;
Other families of algorithms exist, like the Huang family. Special cases of this are the Conjugate Gradient and Murtagh-Sargent methods.
These rely on statistical mechanics to choose the next configuration. The improvement comes from averaging the results from each iteration.
(The ergodic hypothesis connects the statistical forms of Monte-Carlo to the molecular dynamics.)
Monte Carlo methods
Typically using the Metropolis method. Randomly choose a point in
phase space biased by the Boltzman factor
( ).
This limits the number of configurations to choose from, but these are
the statistically significant ones.
Simulated Annealing - the Metropolis algorithm can get trapped
in a local minima due to the biased selection of new configurations.
A solution is to periodically heat the system ( to move out of
the local minima) and then quench the system back to a lower
temperature. (Just like the annealing process). This will help in
locating a global minima.