Molecular Modeling: Reaction Rates

1. Molecular Modeling:Reaction Rates C372 Introduction to Cheminformatics II Kelsey Forsythe

2. What’s in a rate? • Chemical Rate Law • Rate depends on: • Anything which changes motion of system • Pressure, temperature • Number of elements (atoms, molecules etc.) • Rate a f(P,T)*g(N)

3. Chemical Intuition • Elementary Reaction Steps? • Deconstructing reaction in terms of simple one or two component reactions

4. Rate Law

5. Rate Law • General

6. Rate Law • Typically measure rate as function of temperature at constant pressure • Note: ‘a’ can have ANY value

7. Rate Laws • Zero Order • First Order in A • Second Order in A

8. Integrating Rate Law • Oft used approximations: • Steady state approximation • Pseudo first order reaction • Identifying slow/rate-determining step • Rapid equilibration step(s) • Equal concentrations of reactants

9. Connections to Thermodynamics • Develop a microscopic picture of how a reaction proceeds (i.e. some wall/barrier must be surmounted)

10. Arrhenius Rate Theory • Based on empirical results • Van’t Hoff plots • Postulated following formulas

11. Transition State Theory • A+B  AB‡ P • AB‡ is intermediate or transition state complex • AB‡ P fast relative to A+BAB‡ • ALL AB‡ reactive

12. Transition State Theory • A+B  AB‡ P • Use MM, Semi-Empirical or Ab Initio to calculate frequencies and estimate thermodynamic values • kTST(T)>kexact(recrossing effects; MD corrections) • kclassical<kquantal (tunneling corrections; QTST, Centroid TST)

13. Transition State Theory • Ex. Michelis-Menton method for enzymatic reactions • E+S  ESP • Assume rate increases linearly w/ E-concentration • Assume S>>E

14. Michelis-Menton method for enzymatic reactions • S approaches infinity • S approaches S<<1

15. Michelis-Menton method Theory vs. Experiment • From R. Lumry, E. L. Smith and R. R. Glantz, 1951, J. Am. Chem. Soc. 73, 4330. • The hydrolysis of carbobenzoxyglycyl-L-tryptophan using pancreatic carboxypeptidase catalyst

16. Michelis-Menton method Theory vs. Experiment • The hydrolysis of carbobenzoxyglycyl-L-tryptophan using pancreatic carboxypeptidase catalyst Least Squared analysis displayed agreement with experimental results

17. Incorporating Dynamics (Recrossings etc.) • Dividing surface • Reaction Coordinate? • Decomposing full N-D space into a single reaction coordinate or minimum energy path through the Born-Oppenheimer surface

18. Reaction Coordinate?

19. Reaction Coordinate • Minimum Energy Path on Born-Oppenheimer surface • Steepest Descent path • Passes through saddle point/transition state

20. Reaction CoordinateSN2 Exchange

21. Rate Simulations • Require knowledge of molecular dynamics • Position of atoms/molecules • Distribution/partition function of species • Environment (Temperature etc.) • Phase (liquid, solid, gas)

22. Molecular Dynamics • Solve Newton’s Equations • Mathematically, if know initial values of forces, momenta and coordinates: • Taylor series expansion

23. Molecular Dynamics • Taylor series expansions • Similar equations for the velocity and acceleration

24. Molecular Dynamics • Various numerical approximations • Predictor-Corrector • Gear • Verlet • Leap Frog Method • Runge-Kutta • Optimal Integrator: • Maximize time step • Minimize strorage/time • Conserve energy

25. Molecular DynamicsPredictor-Corrector • Truncate Taylor Expansions • Predict new values for r,v and a • Calculate “correct” acceleration using equation of motion

26. Molecular DynamicsPredictor-Corrector • Correct predicted values • Modify c’s such that error O((dt)L+1) (Lth order method)

27. Molecular DynamicsVerlet • Solve Newton’s Equation • Velocities eliminated • Simpletic (preserves underlying physics) • Error a (dt)4 (vs. (dt)3 for predictor-corrector at same order) • Larger steps possible • Less storage/time required

28. MD Method ComparisonS. K. Gray, D. W. Noid and B. G. Sumpter, J. Chem. Phys. 101(5) 4062(1994)MD ODE=pc-method Si2 = Position-Verlet Si4 ~ RKNystrom 1000 CH2 tmax=10ps

29. MD Method ComparisonS. K. Gray, D. W. Noid and B. G. Sumpter, J. Chem. Phys. 101(5) 4062(1994)MD ODE=pc-method Si2 = Position-Verlet Si4 ~ RKNystrom 1000 CH2 tmax=10ps

30. Addendum • PCModel (Serena Software) • Utilizies a modified version of Verlet called the Beeman algorithm • Gilbert: Often for large molecular systems when one can separate time scales the larger motions can be sampled less often than the faster time scale (bond vibrations, fs) motions thus making such calculations more computationally feasible

31. Molecular DynamicsQuantum Corrections • ZPE • Isotope effects • Tunneling

32. Collections of Particles • Brownian motion • Non-linear behavior • Characterize • Mean free path • Avearage # collisons • Flux

33. Collections of Particles • Brownian motion • Condensed phase systems • <r2> a Diffusion constant a friction/viscocity MD!!!

34. Applicable to gas phase reactions (di/tri atomics) Solve time-dependent schrodinger equation Determine scattering matrix Determine scattering cross section Calculate rate constant Use k(T) to get thermodynamic quantities Quantum Scattering Theory

35. Quantum Rate Theory • Rate a Flux through hypersurface

36. Other Methods • PST (Phase Space Theory) • RRK/RRKM theory • TST for unimolecular reactions (e.g. no intrinsic barrier) • VTST (Variational Transition State Theory) • Finds (n-1) surface which minimizes the rate • Marcus Theory • Applicable to electron transfer • Oxidation-reduction • Photosynthesis • Centroid Theory • Based on Feynman path integrals (quantum particle = centroid of collection of classical particles)

37. Advanced Simulation Methods • Monte Carlo • Applicable to macro-systems • QM/MD • Use QM for Force evaluation • Use classical MD to propogate atoms/molecules