Lecture: ab initio Methods Optimization Methods for Molecular Geometries

Overview

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.

Computational Methods

There are two primary methods ;

  1. Iterative Improvement schemes.
  2. Divide and Conquer

Iterative Improvement schemes.

An overview of the iterative improvement scheme is as follows:

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There are two different general procedures:

Function methods

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 tex2html_wrap_inline250

Let tex2html_wrap_inline252

Geometric Rearrangement is

tex2html_wrap_inline254 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.)

Derivative Methods

Consider an energy function tex2html_wrap_inline250

Let tex2html_wrap_inline252 be the initial geometry.

Geometric Rearrangement is

tex2html_wrap_inline254 where X1 is the new geometric configuration.

A Taylor series of the energy function using this rearrangement is;

tex2html_wrap_inline262
where

tex2html_wrap_inline264

tex2html_wrap_inline266 is called the gradient

tex2html_wrap_inline268 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 tex2html_wrap_inline254, the computed energy tex2html_wrap_inline272, the gradient tex2html_wrap_inline274 and the approximate Hessian tex2html_wrap_inline276

tex2html_wrap_inline278

tex2html_wrap_inline280

These methods will be discussed below.

Quasi-Newton methods Estimating the Hessian

  1. Unit matrix - (steepest descent) Important structural information is not used or available. Stiff modes and flexible modes are not distinguished.
  2. Empirical guess
  3. Semi-empirical calculation
  4. Numerical calculation
  5. Analytic calculation
  6. Full Hessian computation with updates.

Implementation of Derivative Methods

Broyden Family of algorithms;

tex2html_wrap_inline282

tex2html_wrap_inline284

tex2html_wrap_inline286

where

tex2html_wrap_inline288

tex2html_wrap_inline290

Implementation of Derivative Methods

Note;

  1. tex2html_wrap_inline292 is the DFP method
  2. tex2html_wrap_inline294 is the BFGS method
  3. Optimally conditioned method chooses tex2html_wrap_inline296 to minimize the condition number (ratio of smallest to largest eigenvalue) for the Hessian.

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 (tex2html_wrap_inline298 ). This limits the number of configurations to choose from, but these are the statistically significant ones.

  1. Choose a new configuration
  2. Compute the energy
  3. If the energy is lower than the current value, accept the configuration.
  4. If the energy is higher, randomly accept the new configuration to an average of about 50%
  5. select a new configuration.

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.

Divide and Conquer



Author: Ken Flurchick