2. Geometry Optimisation Modelling

3. OH + C 2 H 4 *CH 2 -CH 2 -OH CH 3 -CH 2 -O* 3D PES

4. What computational chemistry can do for you: - structural properties (bond lengths, bond angles and dihedral) -energetic properties (which isomer is more stable, how fast a reaction should go: reactant and TS energies - chemical reactivity (from electron distribution nucleophilic and electrophilic sites) - spectral properties (IR, UV and NMR spectra) - interaction properties (molecular fitting) Lewars

5. Introduction carbon skeleton functional group the ultimate goal: interconversion of one structure to another one architecture of the molecule stereochemistry

6. property=f(structure) Physical properties Chemical reactivities Biological activities molecular structure activity=f(structure) reactivity=f(structure) property=f(structure) Optimization Geometry

7. Stable structure Multiple stable structures the energy differences (  E) is a measure of relative stability. Stable structures and transition states Stable structures and transition states TS

8. Typical reaction mech. VARIABLES: 3 translational coordinates and 3 rotational coordinates of a general n-atomic molecule leave (3n – 6) internal coordinates. Potential Energy Surface (PES) representation of chemical reaction

9. Nomenclature PES equivalent to Born-Oppenheimer surface Point on surface corresponds to position of nuclei Minimum and Maximum Local Global Saddle point (min and max)

10. Terminology

11. Geometry Optimization Basic Scheme Find first derivative (gradient) of potential energy Set equal to zero Find value of coordinate(s) which satisfy equation

12. Modeling Potential energy (1-d)

13. Modeling Potential energy (>1-d) Hessian

14. Find Equilibrium Geometry for the Morse Oscillator

16. Bottlenecks No Functional Form More than one variable Coupling between variables

17. Geometry Optimization (No Functional Form) Bracketing (w/parabolic fitting) Find energy (E 1 ) for given value of coordinate x i Change coordinate (x i+1 =x i -  x) to give E 2 Change coordinate (x i+2 =x i +  x) to give E 3 If (E 2 >E 1 and E 3 >E 1 ) then x i+1 > x min >x i+2 Fit to parabola and find parabolic minimum Use value of coordinate at minimum as starting point for next iteration Repeat to satisfaction (Minimum Energy error tolerance)

18. Terminology

19. Potential Energy Surface (PES) A force field defines for each molecule a unique PES. Each point on the PES represents a molecular conformation characterized by its structure and energy. Energy is a function of the coordinates. (Next) Coordinates are function of the energy. energy coordinates

20. Goal of Energy Minimization A system of N atoms is defined by 3N Cartesian coordinates or 3N- 6 internal coordinates. These define a multi-dimensional potential energy surface (PES). A PES is characterized by stationary points: Minima (stable conformations) Maxima Saddle points (transition states) Goal of Energy Minimization Finding the stable conformations energy coordinates

21. Classification of Stationary Points 0.0 4.0 8.0 12.0 16.0 20.0 090180270360 transition state local minimum global minimum energy coordinate Type Minimum Maximum Saddle point 1 st Derivative 0 2 nd Derivative* positive negative *Refers to the eigenvalues of the second derivatives (Hessian) matrix

22. Minimization Definitions Given a function: Find values for the variables for which f is a minimum: Functions Quantum mechanics energy Molecular mechanics energy Variables Cartesian (molecular mechanics) Internal (quantum mechanics) Minimization algorithms Derivatives-based Non derivatives-based

23. A Schematic Representation Starting geometry  Easy to implement; useful for well defined structures  Depends strongly on starting geometry

24. Population of Minima Most minimization method can only go downhill and so locate the closest (downhill sense) minimum. No minimization method can guarantee the location of the global energy minimum. No method has proven the best for all problems. Global minimum Most populated minimum Active Structure

25. A General Minimization Scheme Starting point x 0 Minimum? Calculate x k+1 = f(x k ) Stop yes No

26. Two Questions f(x,y) Where to go (direction)? How far to go (magnitude)? This is where we want to go

27. How Far To Go? Until the Minimum Real function Cycle 1: 1, 2, 3 Cycle 2: 1, 2, 4 Line search in one dimension Find 3 points that bracket the minimum (e.g., by moving along the lines and recording function values). Fit a quadratic function to the points. Find the function’s minimum through differentiation. Improved iteratively. Arbitrary Step x k+1 = x k + k s k, k = step size. Increase  as long as energy reduces. Decrease when energy increases. 4 3 2 1 5

28. Where to go? Parallel to the force (straight downhill): S k = -g k How far to go? Line search Arbitrary Step Steepest Descent

29. Steepest Descent: Example -15-10-5051015 -15 -10 -5 0 5 10 15 441 361 289 169 225 121 81 49 25 9 1 Starting point: (9, 9) Cycle 1: Step direction: (-18, -36) Line search equation: Minimum: (4, -1) Cycle 2: Step direction: (-8, 4) Line search equation: Minimum: (2/3, 2/3)

30. Steepest Descent:Overshooting SD is forced to make 90º turns between subsequent steps (the scalar product between the (-18,-36) and the (-8,4) vector is 0 indicating orthogonality) and so is slow to converge.

31. Ligand geometry scoring: -11.2 kcal/molscoring: -5.7 kcal/mol

32. Orientation - interactions scoring: -11.2 kcal/mol scoring: -5.7 kcal/mol

33. KONFORMÁCIÓS TÉR. Protein folding és konformációs tér SZERKEZETI ÉS Polimer molekulák szerkezetei és reakciói Kis molekulák és reakcióik Configuration and conformational space C2H4O2C2H4O2

34. Energy landscape

35. R RC T TC TS 3.ábra ● OH ++ H 2 O C 6 H 13 N 2 O 3

36. Summery I.

37. Summery II.