What Microscopic Solvent Dynamics Most Influence Reacting Solute Molecules?

Brief Overview:

My goal with this research program is to investigate the microscopic mechanisms by which the venerable transition-state theory is violated.1 Transition-state theory yields an entirely equilibrium prediction for the rate of barrier crossing (see the picture; right). Of course, the transition-state rate is an overestimate — a reactant that crosses the transition state may fall into the product well, but it might also recross the transition state to become a reactant again. The true rate, k = kkTST, is therefore the transition-state rate modified by the transmission coefficient k, a number between zero and one, which denotes the fraction of successful crossings. A non-unit transmission coefficient occurs because solvent dynamics (Fig. 2) happens on the same time scale as the reactive barrier crossing, thereby coupling the solute and bath dynamics. What solvent dynamics most strongly influence chemical reactions?

First stage: Use Instantaneous Normal Modes2 to compute the generalized Langevin equation friction kernel3 at the barrier to provide an INM Grote-Hynes prediction1 for the transmission coefficient. Can this short-time method really capture barrier dynamics adequately? If so, use the instantaneous normal modes to identify the solvent motions that lead to recrossings. Can we identify the theoretical solvent+solute coordinate4 that minimizes recrossings? In fact, this project is already underway (see Ref. 5).

Second stage: SN2 reaction, Using Steel theory,6 compute averaged reaction dynamics that can be resolved into specific solute-solvent contributions. These averaged mechanistic results erase the correlation between instantaneous solvent environment and reactant motion; is this correlation important?

Note: Both of these projects can be implemented by graduate students or undergrads.

Introduction

My goal with this research program is to investigate the microscopic mechanism by which the venerable transition-state theory is violated. Transition-state theory yields an entirely equilibrium prediction for the rate of barrier crossing:1

,

with the free energy of activation (which contains the equilibrium effect of the solvent on the reaction), T the absolute temperature, and the reactant-well frequency.  Of course, the transition-state rate is an overestimate — a reactant that crosses the transition state may fall into the product well, but it might also recross the transition state to become a reactant again. The true rate is given by1

kexact = kkTST,

where the transmission coefficient k, a number between zero and one, denotes the fraction of successful crossings. A non-unit transmission coefficient occurs because solvent dynamics happens on the same time scale as the reactive barrier crossing, thereby coupling the solute and bath dynamics. This dynamical coupling suggests that the solvent can drive the reactant towards, or away from, the product. This coupling is what I wish to study.

Traditionally, this coupling has been studied mainly, and somewhat indirectly, by computer simulation.1,7,8 A standard molecular dynamics (MD) simulation is impractical for computation of the rate because barrier crossing is such a rare event (the simulation spends essentially all of its time in the reactant well), but the truly interesting reaction dynamics embodied by the transmission coefficient can be probed by starting the reactant at the top of the barrier and evolving the system from there (this method is the so-called reactive flux7 technique for simulating reactions). Clearly one can easily observe recrossings with such a technique (and thus determine k), but the actual process by which the bath induces these recrossings remains obscure due to the enormous number of degrees of freedom present in the system.

Essentially, the problem is that in the computer simulation all degrees of freedom are treated equally, yet from a conceptual standpoint we want to separate the reaction coordinate from the bath degrees of freedom. The goal, then, is to inquire about only those bath motions that actually influence the reaction coordinate. The requisite separation can be made exactly through the use of a Mori-Zwanzig generalized Langevin equation,9 where the solvent affects the reaction coordinate through the time-dependent friction

with the force felt by the reaction coordinate (frozen at the transition state) due to the solvent.10 Grote-Hynes theory1,11 can then be used to express the transmission coefficient in terms of the barrier frequency and a particular Fourier component of the time-dependent friction. Again, though, it is difficult to identify the precise solvent motions that contribute most to the time correlation function in Eq. (1.3), much less determine the motions responsible for a single Fourier frequency.

My program for extending these methods follows.

Explicit Simulation Techniques

The backbone of any condensed-phase theory must be computer simulation, since within the limits of classical mechanics it is here that one obtains exact results for the model employed. Although the simulation methods themselves have not changed much, there are now some quite useful techniques for analyzing their results that would be useful in the study of kinetics. Steel theory,6 which has proven quite effective in studies of solvation dyamics,12 allows one to resolve time correlation functions [such as the one in Eq. (3)] into contributions due to certain classes of solvent motion: solvent translations versus rotations, for instance, or first solvation shell vs. the remaining solvation shells. The technique is straightforward, but so far it has not been applied to kinetic systems.

Steel theory provides averaged mechanistic results, but that averaging necessarily erases the correlation between the instantaneous solvent environment around the solute and the reactant’s motion. Such “inhomogeneous” effects have proven quite important in vibrational relaxation in liquids, and could well be important here.12 Recent simulations of near-critical fluids, which are heterogeneous because they contain domains with gas- or liquid-like structures, have engendered just the methodologies necessary for assessing dynamics associated with a particular solvent environment.13

A simple system such as the trans to gauche isomerization reaction of n-butane3 in a rare-gas liquid would serve as a useful starting point for such techniques. From that simple system it would be natural to revisit more interesting reactions; the well-studied SN2 reaction4

would certainly be a good example, and the presence of hydrogen bonding and long-ranged (Coulombic) forces make it as different as possible from the first system. It would be interesting to see what, if anything, two such disparate systems have in common.

Instantaneous Normal Modes and the Generalized Langevin Equation

My primary focus, however, would be on probing the barrier crossing with the instantaneous normal modes (INMs) of the system.2 Conventional reactive flux trajectories launched from the barrier top often run for only about half a picosecond,8 a time short enough that the INMs should provide a valid description of the dynamics. The essential notion is that in the time for which they are valid, the INMs lead to an exact generalized Langevin equation (GLE).10 Thus, the collective molecular motions that comprise the solvent friction [Eq. (3)] are known explicitly, along with a distinct measure of which motions are most effective at influencing the solute. The net result is that for short (hopefully long enough) times the INMs easily provide considerable mechanistic detail. Indeed, recent results (Fig. 3 and Ref. 5) for an isomerizing diatomic14 in a Lennard-Jones solvent show surprisingly good agreement between INM theory and Molecular Dynamics simulation (except at low-s, where INM friction kernels are required to diverge).

Besides what is becoming a standard use of the INMs, there also exists the intriguing possibility of relating the microscopic solvent environment to the transition-state topology. For some time now GLE-based kinetics has relied on an isomorphism between the GLE and a single degree of freedom coupled to a bath of microscopically-ambiguous harmonic oscillators.1,4,12 Despite the fictitious nature of the oscillators, however, some very real insight has been generated into how the solvent modifies transition-state theory. Two quite useful notions are that in the presence of the solvent the reaction coordinate should be replaced with a collective solute-solvent coordinate,1 and that the solvent rotates the transition state so that it is no longer parallel to the reaction coordinate.4 Since the INMs also map to a bath of harmonic oscillators, only now the oscillators have molecular definitions,10 there appears to be a real hope that the molecular description of the renormalized (collective) reaction coordinate could be found.

References:

  1. It is worth noting that I have restricted the discussion here to classical dynamics. Useful reviews are: P. Hanggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990), R. M. Whitnell and K. R. Wilson, Rev. Comp. Chem. 4, 67 (1993); and B. J. Berne, M. Borkovec, and J. E. Straub, J. Phys. Chem. 92, 3711 (1998).
  2. R. M. Stratt, Acc. Chem. Res. 28, 201 (1995).
  3. G. Goodyear and R. M. Stratt, J. Chem. Phys. 105, 10050 (1996); 107, 3098 (1997).
  4. E. Pollak, J. Chem. Phys. 85, 865 (1986); E. Pollak, in Activated Barrier Crossing, edited by G. R. Fleming and P. Hanggi (World Scientific, Singapore, 1993), A. M. Frishman & E. Pollak, JCP 98, 9532 (1993).4. J. P. Bergsma, B. J. Gertner, K. R. Wilson, and J. T. Hynes, J. Chem. Phys. 86, 1356 (1987).
  5. G. Goodyear, work presented at the 1999 Instantaneous Normal Modes CECAM in Lyon, France.
  6. W. Steele, Mol. Phys. 61, 1031 (1987).
  7. D. Chandler, J. Chem. Phys. 68, 2959 (1978).
  8. J. P. Bergsma, B. J. Gertner, K. R. Wilson, and J. T. Hynes, J. Chem. Phys. 86, 1356 (1987).
  9. B. J. Berne and R. Pecora, Dynamic Light Scattering (Krieger, Malabar, 1990), Chap. 11.
  10. Actually, the force Fx in Eq. (3) ought to be the fluctuating force that shows up in the GLE. In practice, though, it suffices to replace the fluctuating force with the true force on the reaction coordinate. See Ref. 4 and G. Goodyear and R. M. Stratt, J. Chem. Phys. 105, 10050 (1997).
  11. R. F. Grote and J. T. Hynes, J. Chem. Phys. 73, 2715 (1980).
  12. S. C. Tucker, J. Phys. Chem. 97, 1596 (1993).
  13. G. Goodyear and S. C. Tucker, J. Chem. Phys. 110, 3643 (1999).
  14. J. E. Straub, M. Borkovec, and B. J. Berne, J. Chem. Phys. 89, 4833 (1988).