Project 1:  Connecting microscopic proton transfer mechanisms to proton transport kinetics in hydrogen-bonded networks

Proton transfer (PT),  including proton holes,  and the migration of positive and negative topological charge defects in hydrogen--bonded (HB) networks, is central to a myriad of processes in chemistry, physics, materials science, and biology. Although simulations ^1,2,3 have elucidated many details of  such structural diffusion mechanisms, a theory connecting a particular defect's solvation structure and microscopic PT mechanism to macroscopic charge transport kinetics is lacking. The PI introduced a general statistical-mechanical framework that achieves the aforementioned goal within a chemical master equation (CME) approach⁴. The theory employs the concept of population correlation functions⁵ and associated rate equations⁶ as commonly applied to HB kinetics,  but extends these essential ideas in key ways.

Suppose the HB donors/acceptors (A) are identical, i.e. A—H^……..A.  Addition or removal of a proton creates a  cationic or anionic defect site A*, e.g. H₃O⁺ or OH^-  Define two charge defect population functions:  h(t)=1 if a tagged A atom is A* at time t (h(t)=0 otherwise),  whereas H(t)=1 if A*  retains its identity continuously up to time t.  These functions describe topological defect patterns as opposed to their counterparts for HB kinetics in pure liquids^5,6.  Now introduce an intermittent PT correlation function C_i^AA (t)= <h(0)h(t)>/<h> that yields the probability of finding the same A* at times t=0 and t, irrespective of an identity change in the interim and a continuous PT correlation function C_c^AA (t)=<h(0)H(t)> /<h> that does not allow for identity change of A* in [0,t].

Suppose there are p solvation patterns, each with a coordination number n₁,…,n_p. Then, C_c^AA (t)=<h(0)H(t)> /<h> is particularized by introducing C_c^AA-ni(t) describing the probability of finding the same A* continuously up to time t given that its coordination number is n_i at time t=0.  Lifetimes are obtained from

which is the average time needed for PT from  A* to a first solvation shell member and 1/τ_exch is the average PT rate, with an analogous definition of τ_exch,ni.  `Proton rattling' events, in which the defect returns to its original A* site after two successive PTs, can be either included or excluded. Including allows comparison to femtosecond experiments⁷.  Excluding them provides a clearer picture of the transport process. 

The decay of C_i^AA (t)depends not only on 1/τ_exch but also on the reverse process.   In order to disentangle the coupled forward/backward kinetics⁶, a nearest--neighbor PT correlation function C_nn^AA (t) is introduced, which measures the probability of finding an A atom in the hydration shell of A* at time t given that it was the A* site at t=0; an analogous definition holds for C_nn^AA-ni (t). Including the reverse reaction leads to the rate equation for PT  

with k₁^PT and k_-1^PT being the forward and backward PT rate constants, respectively.  This defines a general formalism, independent of the particular HB network that allows the kinetics of PT and structural diffusion to be investigated theoretically in a quantitative manner. Akin to other CME approaches⁹, the formalism permits one to connect long--time processes to specific microscopic motions. The theory was applied to the well-understood problem of hydronium diffusion in water, yielding time scales in agreement with those of ultrafast spectroscopy studies⁷ and was subsequently used to decide among three proposed mechanisms of hydroxide diffusion in water⁸.

Project 2:  Development of force field parameters for imidazole attached to a polymeric spacer

We have now developed a set of force field parameters for the attachment of imidazole to a polyethyleneoxide spacer, as discussed in the original proposal.  Starting parameters were developed from those of Krishnan and Balasubramanian¹⁰ for polyethyleneoxide and CHARMM22 for imidazole¹¹. The figure and tables below show the atom names and the force field parameters, including charges and intramolecular potential parameters.  All calculations were performed using Gaussian03¹² at the Hartree-Fock level of theory with a 6-31G* basis set.

References:

1. D. Marx, ChemPhysChem 7, 1848 (2006).
2. G. A. Voth, Acc. Chem. Res. 39, 143 (2006).
3. B. Roux, Acc. Chem. Res. 35, 366 (2002).
4. A. Chandra, M. E. Tuckerman and D. Marx, Phys. Rev. Lett. (in press).

5. F. H. Stillinger, Adv. Chem. Phys. 31, 1 (1975).

6. A. Luzar and D. Chandler, Phys. Rev. Lett. 76, 928 (1996); Nature (London) 379, 55 (1996).
7. S. Woutersen and H. J. Bakker, Phys. Rev. Lett. 96, 138305 (2006).
8. M. E. Tuckerman, A. Chandra and D. Marx, Acc. Chem. Res. 39, 151 (2006).
9. J. D. Chodera, et al. J. Chem. Phys. 126, 155101 (2007).
10. M. Krishnan and S. Balasubramanian.  Chem. Phys. Lett. 385, 351 (2004).

11. A. MacKerell, Jr., et al. J. Phys. Chem. B 102, 3586 (1998).

12. M. J. Frisch, et al. Gaussian 03, revision C.02; Gaussian, Inc.: Wallingford, CT, 2004.