1. Ab initio path integrals and applications of AIMD to problems of aqueous ion solvation and transport Mark E. Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Sciences New York University, 100 Washington Sq. East New York, NY 10003

2. Illustration of hydride transfer in dihydrofolate reductase From Agarwal, Billeter, Hammes-Schiffer, J. Phys. Chem. B (2002). Nuclear quantum effects critical for describing this reaction!

3. Time-dependent quantum mechanics Real-time quantum propagator:Expectation value:

4. Electron source x 1 2 Heuristic Derivation of the Path Integral

5. Electron source x Heuristic Derivation of the Path Integral

6. Electron source x Heuristic Derivation of the Path Integral x0x0

7. Derivation of the path integral Hamiltonian: Trotter Theorem:

8. Derivation of the path integral (cont’d) Coordinate-space completeness relation:

9. Derivation of the path integral (cont’d) Matrix elements of Ω

10. Derivation of the path integral (cont’d) Reassemble: Discrete path integral for the canonical partition function:

11. .............. Classical Isomorphism 1 2 3 P P-1 Chandler and Wolyner, J. Chem. Phys. 74, 4078 (1981) Interaction between two cyclic polymer chains “Classical” cyclic polymer chain in an potential V(x)

12. Ab initio path integrals Partition function for N particles on ground-state surface (Path-integral BO approximation):............. 1 2 3 P-1 P MET, et al. JCP 99, 2796 (1993); Marx and Parrinello, JCP 104, 4077 (1996); MET, et al. JCP 104, 5579 (1996) Trace condition:

13. MET and D. Marx, Phys. Rev. Lett. 86, 4946 (2001)

14. Proton transfer in malonaldehyde MET and D. Marx, Phys. Rev. Lett. 86, 4946 (2001) Classical

15. Proton transfer in malonaldehyde MET and D. Marx, Phys. Rev. Lett. 86, 4946 (2001) Quantum

20. Reaction coordinate:

21. Path integral molecular dynamics Path integrals can be evaluated via Monte Carlo or molecular dynamics. Molecular dynamics offers certain advantages in terms of parallelization since in each step, the entire system is moved. Ab initio path integrals (path integrals with potential energies and forces derived “on the fly” from electronic structure calculations are considerably more efficient with molecular dynamics. Naïve molecular dynamics suffers from severe sampling problems, so how to we create a molecular dynamics approach that is as efficient as Monte Carlo?

22. Time scales in path integral molecular dynamics Write the partition function as follows: Naïve choice of Hamiltonian for molecular dynamics

23. Problems with naïve approach As the system becomes more quantum, P → ∞ and ω P → ∞. However, potential is attenuated by a factor of 1/P, and harmonic term dominates. System will stay close to closed orbits and not sample configuration space. Harmonic term has a spectrum of frequencies. Highest frequency determines the time step, which means slow, large-scale chain motions and breathing modes will not be sampled efficiently. Need to sample the canonical distribution, so, at the very least, the system needs to be coupled to a thermostat.

24. Path integral molecular dynamics Martyna, Tuckerman, Berne JCP 99, 2796 (1993) The path integral is just a bunch of integrals, so we can just change variables. Introduce a linear transformation: whose effect is to diagonalize the harmonic nearest-neighbor coupling: Ensure all modes move on same time scale by choosing: Couple each degree of freedom to a heat bath (Langevin, Nosé-Hoover chains,….)

25. Transformations Staging: Normal Modes:

26. Results for harmonic oscillator No transformations PIMD (staging) PIMC (staging)

27. 1806:

28. PEM vs. AAEM fuel cells (AAEM=Alkali-anion exchange membrane) From Varcoe and Slade, Fuel Cells 5, 198 (2005)

29. Structures of the excess proton in water

30. H9O4+H9O4+ H5O2+H5O2+ H3O+H3O+

31. +++ Grotthuss Mechanism (1806) Vehicle Mechanism

32. DFT (BLYP) proton diffusion constants D(H3O+) = 7.2 x 10-9 m2/s complete DVR basis [Berkelbach, Lee, Tuckerman (in preparation)] D(H3O+) = 6.7 x 10-9 m2/s (Exp: Halle and Karlström, JCSFT II 70, 1031 (1983)) Complete DVR basis set: System specifics: 31 H 2 O + 1 H 3 O+ in a 10 Å periodic box 60 ps simulation DVR grid size = 75 3 Troullier-Martins pseudopotentials

34. The Grotthuss mechanism in water MET, et al,JPC, 99, 5749 (1995); JCP 103, 150 (1995) D. Marx, MET, J. Hutter, M. Parrinello, Nature 397, 601 (1999). N. Agmon, Chem. Phys. Lett. 244, 456 (1995) T. J. F. Day, et al. J. Am. Chem. Soc. 122, 12027 (2000) Solvent coordinate view: P. M. Kiefer, J. T. Hynes J. Phys. Chem. A 108, 11793 (2004)

35. The Grotthuss mechanism in water Second solvation shell H-bond breaking followed by formation of intermediate Zundel complex: P Presolvation Concept: Proton-receiving species must be “pre-solvated” like the species into which it will be transformed in the proton-transfer reaction. MET, et al,Nature 417, 925 (2002)

36. The Grotthuss mechanism in water Computed transfer time τ = 1.5 ps NMR: 1.3 ps Transfer of proton resulting in “diffusion’’ of solvation structure: A. Chandra, MET, D. Marx Phys. Rev. Lett. 99, 145901 (2007)

37. Probability distribution functions Quantum Classical (P=8 Trotter points)

39. Quantum delocalization of structural defect D. Marx, MET, J. Hutter and M. Parrinello Nature 397, 601 (1999)

40. “Proton hole” mechanism of hydroxide mobility N. Agmon, Chem. Phys. Lett. 319, 247 (2000) OH - H+

41. Spectra of 14 M KOH IR Raman Librovich and Maiorov, Russian J. Phys. Chem. 56, 624 (1982)

42. Identified in neutron scattering of concentrated NaOH and KOH solutions: A. K. Soper and coworkers, JCP 120, 10154 (2004); JCP122, 194509 (2005). Also in other CPMD studies: B. Chen, et al. JPCB 106, 8009 (2002); JACS 124, 8534 (2002). And in X-ray absorption spectroscopy: C. D. Cappa, et al. J. Phys. Chem. A 111, 4776 (2007) System specifics: 31 H 2 O + 1 OH - in 10 Å periodic box Plane-wave basis, 70 Ry cutoff Simulation time: 60 ps BLYP functional, Troullier-Martins PPs

43. Weak H-bond donated by hydroxide also identified in neutron scattering of concentrated NaOH and KOH solutions: A. K. Soper and coworkers, JCP 120, 10154 (2004); JCP122, 194509 (2005). M. Smiechowski and J. Stangret, JPCA 111, 2889 (2007). T. Megyes, et al. JCP 128, 044501 (2008). B. Winter, et al. Nature (2008)

44. Hydronium :

45. Water:

46. Hydroxide:

47. d 1 d 2  = d 1 - d 2  > 0.5 Å  < 0.1 Å O* H’ O*H/O*O H’O MET, D. Marx, M. Parrinello Nature 417, 925 (2002) (P=8 Trotter points)

49. 90 o 105 o Geometry of relevant solvation complexes H 9 O 5 - H7O4-H7O4-

50. θ MET, et al. Science 275, 817 (1997) Classical Quantum

52. Selected References 1.M. E. Tuckerman, et al. J. Chem. Phys. 103, 150 (1995); J. Phys. Chem. 99, 5749 (1995) 2.N. Agmon, Chem. Phys. Lett. 244, 456 (1995). 3.M. E. Tuckerman, et al. J. Chem. Phys. 99, 2796 (1993) 4.D. Marx and M. Parrinello, Z. Phys. B 95, 143 (1994) 5.D. Marx and M. Parrinello, J. Chem. Phys. 104, 4077 (1996) 6.M. E. Tuckerman, et al. J. Chem. Phys. 104, 5579 (1996) 7.D. Marx, et al. Nature 367, 601 (1999) 8.N. Agmon, Chem. Phys. Lett. 319, 247 (2000). 9.M. E. Tuckerman, et al. Nature 417, 925 (2002) 10.M. E. Tuckerman, et al. Acc. Chem. Res. 39, 151 (2006) 11.A. Chandra, et al. Phys. Rev. Lett. 99, 145901 (2007) 12.A. K. Soper, et al., J. Chem. Phys. 120, 10154 (2004); J. Chem. Phys. 122, 194509 (2005) 13.M. Smiechowski and J. Stangret, J. Phys. Chem. A 111, 2889 (2007). 14.C. D. Cappa, et al. J. Phys. Chem. A 111, 4776 (2007) 15.T. Megyes, et al. J. Chem. Phys. 128, 044501 (2008). 16.E. F. Aziz, et al. Nature, 455 89 (2008).