1. Geometry Optimization Pertemuan VI

3. 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.

4. Before GOAfter GO (PM3-Steepest Descent) C-C Bond1.34 Å1.32197 Å C-H Bond1.08 Å1.08604 Å C-C-H Angle120°123.034°

5. 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?

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

8. 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

9. Simplex Method

11. The Sequential Univariate Method

12. Steepest Descent

14. Conjugate Gradient

15. Line Search in one Dimension

17. Newton-Raphson

18. 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

19. 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

20. Application of Minimization Normal Mode Analysis The Study of Intermolecular Processes Determination of Transitions Structure and Reaction Pathways