Ab Initio Direct (Chemical) Dynamics Nonadiabatic Reactions X Statistical Methods X

2. Ab Initio Direct (Chemical) Dynamics Nonadiabatic Reactions X Statistical Methods X Adiabatic Reactions

3. What do we want to know? Products Rate coefficients, reaction cross sections, initial state dependence, final state distribution, state to state cross sections, Mechanism

4. Remember what? The Potential Energy Surface, of course! A brief reminder of the interpolated PES method

7. More learning as we go As the size of the data set grows, we can use the data itself to do a statistical (Bayesian) analysis of the errors that result from each Taylor series. So we learn how much “trust” we can have in each Taylor series. So we know how to more accurately calculate the interpolation weights.

8. H3O+ R. P. A. Bettens, T. A. Hansen, and M. A. Collins, JCP 111 (1999) 6322

9. M. A. Collins and D. H. Zhang, JCP 111 (1999) 9924.

10. H + H2O

11. Rate coefficientOH + H2a H2O + H M. Yang, D. H. Zhang, M. A. Collins and S.-Y. Lee, JCP 114 (2001) 4759. D. H. Zhang, M. A. Collins and S.-Y. Lee, Science 290 (2000) 961.

12. How many data points does Grow need? (1) You can get an answer with 20-100 mep points (2) You can get the “right” answer with: 4 atoms BeH3, BH3+, CH3+, NH3+, OH3+, NeCOH+,ArCOH+, etc 400-1500 OH3 , OH3- (overkill) 2000 5 atoms FHCOH+ 1500 6 atoms H2OCOH+, CH5 2000-3000

13. Adiabatic Reactions Solve dynamical Equations Born-Oppenheimer Non Born-Oppenheimer (semi-)Classical Quantum Car-Parinello Extended Lagrangian

14. Car-Parinello Extended Lagrangian Benefits: Automated. “Cheap” electronic calculation. Potential Limitations: DFT New calculation at every configuration (use it and lose it).

15. Born-Oppenheimer Methods “Traditional quantum” Uses a preset “grid”, so needs to know the mechanism. Probably needs a PES. Can be “exact”. Diffusion Monte Carlo - bound states only?

16. How many geometries for a “simple” quantum case? Quantum grid 3 atoms ≈ 104 4 atoms ≈ 107 - 108 5 atoms ? (≈ 109 in 7 dimensions) N atoms ≈> 103N-6 x the number of attempts The number is proportional to the relevant volume.

17. Born-Oppenheimer Methods Classical, semiclassical, spawning Can use any ab initio method. Can be direct or use a PES. Do not need to know the mechanism. Dynamical limitations. How many geometries in a classical study? (103 - 105 trajectories) x (103 - 105 - 106 time steps) x (the number of different initial “states”) Is this nearly independent of dimensionality?

18. Direct Methods vs PES (1) Both methods can be automated, but the direct method is simpler. (2) If the dynamics is “complicated” in the sense that many molecular configurations are involved, then a PES may be much less expensive. (3) Point (2) is true a fortiori if high level ab initio methods are necessary. (4) PES method benefits from CNP symmetry.

19. The products and mechanism may be unclear DHO + OCH+ DHO + HOC+ H2O + DOC+ DH2O+ + OC H2O + OCD+

20. How many geometries? Classical, spawning, extended Lagrangian Direct: Number µ Number of trajectories x collision time x number of initial states PES based: Number = Number of data points needed to span the relevant volume

21. How big is the relevant volume? E(R) < Emax Volume µ D3N-6 If there are nS “spectator degrees of freedom” and nL “large amplitude degrees of freedom” with nS + nL = 3N-6, then Volume µ dnS DnL nL may be ≈ 6 + the number of torsions. The identity of the large amplitude degrees of freedom changes from region to region.

22. DHO + OCH+ 6 DHO + HOC+ 6 H2O + DOC+ 5 DH2O+ + OC 6 H2O + OCD+

23. H2O + DOC+

24. Interpolated PES If, on average, just one second-order Taylor expansion is sufficiently accurate over lengths of order d, then the number of data points required for an interpolated PES is Ndata: Ndataµ (D/d)nL (where nL ≈ 6 + the number of torsions)

25. Direct dynamics If a direct method Monte Carlo samples 3N - 6 - 1 degrees of freedom in the initial configurations, then the number of gradient calculations is Ndirect: Ndirectµ Ntrajx the average number of steps per trajectory. Ntraj may thus be virtually independent of nS.

26. However, the more complex the dynamics and the less probable the observables, the larger Ntraj must be. Moreover, it is not clear how the time scale for the collisions scales with the number of atoms. While the direct method can give results for large systems, the question is whether feasible calculations can yield results which are both converged and accurate.

27. Clusters and Fluids Both direct and PES approaches can be applied to clusters and fluids via a many-body expansion. However, a PES for (say) water is transferable to many different problems, while the direct method must be repeated.

28. Conclusions? The direct method has the advantage of simplicity, and may scale well with dimensionality. Storing and re-using the ab initio data in an interpolated PES is more efficient for small systems, and is currently practical for high levels of electronic structure. It is unclear how these two methods will compare For large systems.