Geometry Optimization

Pertemuan VI

Geometry Optimization

Backgrounds Real molecules vibrate thermally about their

equilibrium structures. Finding minimum energy structures is key to

describing equilibrium constants, comparing to experiment, etc.

Before GO After GO (PM3-Steepest Descent)

C-C Bond 1.34 Å 1.32197 Å

C-H Bond 1.08 Å 1.08604 Å

C-C-H Angle 120° 123.034°

Geometry Optimization

In its essence, geometry optimization is a problem in applied mathematics.

How does one find a minimum in an arbitrary function of many variables?

Example of paths taken when an angle changes in a geometry optimization.(a) Path taken by an optimization using a Z-matrix or redundant internalcoordinates. (b) Path taken by an optimization using Cartesian coordinates.

Optimization Algorithms Non Derivative methods

Simplex Method The Sequential Univariate Method

Derivative Methods First order derivative

Steepes Descent Conjugate gradient (The Fletcher-Reeves Algorithm) Line Search in One Dimension Arbitrary Step Approach

Second Order derivative Newton Raphson Quasy Newton

Simplex Method

The Sequential Univariate Method

Steepest Descent

Conjugate Gradient

Line Search in one Dimension

Newton-Raphson

Which minimization should I use?

The choice of minimisation algorithm should consider: Storage and computational requirements The relative speed The availability of analytical derivatives and the

robustness of the method

Convergence Criteria In contrast to the simple analytical functions thet we

have used to illustrate the operation of the various minimisation methods, in real molecular modelling applications it is rarely possible to identify the exact location of minima.

We can only ever hope to find an approximation to the true minima.

Instruction to stop the minimisation step = convergence criteria Energy gradient Coordinate gradient Root Mean Square gradient

Application of Minimization

Normal Mode Analysis The Study of Intermolecular Processes Determination of Transitions Structure and

Reaction Pathways