Reimagining the geometry of transition states (in PRL!)

I'm excited to report that my former graduate student, Dr. Galen Craven, and I just published an article in Physical Review Letters (PRL). The critical question in determining the rate between chemical reactants and products relies on knowing when exactly the reactants become products. This is like asking yourself when did you get sick? You might remember when you were healthy and you know when you definitely have a cold, but do you know when you transitioned from being healthy to sick? Presumably, if you could know when this transition happens, then you would know when to take medicine or when not to. For example, if you start feeling a little off but you haven't hit the transition to being sick, you might still not get sick at all and so there would be no need to take a pre-emptive medication. In the same way, chemists need to know when molecular reactions really take place and when exactly they did so. Transition state theory then provides a way to use that transition to obtain the rate of a reaction. And that's also useful because then we know if it will take place in the same time scale as other events such as being fast enough to finish while you are on a quick break or so slow that it won't happen before the universe has ended.

Specifically, we discovered a new way for obtaining the structure of the transition state between reactants and products when the reaction is in a complex solvent. All of the previous methods had obtained this surface by optimization (using variational transition state theory) or through successive approximations (using perturbation theory).The key is a mathematical tool, called the Lagrangian descriptor, that had been developed earlier by Wiggins and his colleagues in the area of fluid mechanics.We were able to use the Lagrangian descriptor to obtain the transition state geometry directly without either optimizing the rate or employing perturbation theory. And this means that we now have a new tool for obtaining reaction rates in nonequilibrium systems.

As with most articles in PRL, it was a tortuous path through the reviewing process. We were pleased that nearly all of the reviewers (and we had 6 in the end!) saw the work as novel and potentially game-changing. The full reference of the article is: G. T. Craven and R. Hernandez, "Lagrangian descriptors of thermalized transition states on time-varying energy surfaces," Phys. Rev. Lett. 115, 148301 (2015). (doi:10.1103/PhysRevLett.115.148301) I'm happy to acknowledge the support from the Air Force Office of Scientific Research (AFOSR).

Controlling chemical reactions by kicking their environs

Chemists dream of controlling molecular reactions with ever finer precision. At the shortest chemical length scales, the stumbling block is that atoms don’t follow directions. Instead, we “control” chemical reactions by way of putting the right molecules together in a sequence of steps that ultimately produce the desired product. As this ultimate time scale must be sufficiently short that we will live to see the product, the rate of a chemical reaction is also important. If the atoms don’t quite move in the right directions quickly enough, then we are tempted to direct them in the right way through some external force, such as from a laser or electric field. But even this extra control might not be enough to overcome the fact that the atoms forget the external control because they are distracted by the many molecules around them.

In order to control chemical reactions at the atom scale, we are therefore working to determine the extent to which chemical reaction rates can be affected by driven periodic force. Building on our recent work using non-recrossing dividing surfaces within transition state theory, we succeeded in obtaining the rates of an albeit relative simple model reaction driven by a force that is periodic (but not single frequency!) in the presence of thermal noise. It is critical that the external force is changing the entire environment of the molecule through a classical (long-wavelength) mode and not a specific vibration of the molecule through a quantum mechanical interaction. The latter had earlier been seen to provide only subtle effects at best, but the former can be enough to dramatically affect the rate and pathway of the reaction as we saw in our recent work. Thus while our most recent work is limited by the simplicity of the chosen model, it holds promise for determining the degree of control of the rates in more complicated chemical reactions.

This work was performed by my recently graduated student, Dr. Galen Craven, in collaboration with Thomas Bartsch from Loughborough University. The tile of the article is "Chemical reactions induced by oscillating external fields in weak thermal environments."The work was funded by the NSF, and the international partnership (Trans-MI) was funded by the EU People Programme (Marie Curie Actions). It was just released at J. Chem. Phys. 142, 074108 (2015).

LiCN taking a dip in an Ar bath

We all know that it’s easier to move through air than water. Changing the environment to molasses means that you’ll move even slower. Thus it’s natural to think that the thicker (denser) the solvent (bath), the slower a particle will swim through it. More precisely, what matters is not the density but the degree to which the moving particle interacts with the solvent, and this can be described through the friction between the particle and the fluid. Chemical reactions have long known to be increasingly slower with increasing friction. The problem with this seemingly simple concept is that Kramers showed long ago that there exists a regime (when the surrounding fluid is very weakly interacting with the particle) in which the reactions actually speed up with increasing friction. This crazy regime arises because reactants need energy to surmount the barriers leading to products, and they are unable to get this energy from the solvent if their interaction is very weak. A small increase of this weak interaction facilitates the energy transfer, and voila the reaction rate increases. What Kramers didn’t find is a chemical reaction which actually exhibits this behavior, and the hunt for such a reaction has long been on…

A few years ago, my collaborators in Madrid and I found a reaction that seems to exhibit a rise and fall in chemical rates with increasing friction. (I wrote about one of my visits to my collaborators in Madrid in a previous post.) It involves the isomerization reaction from LiCN to CNLi where the lithium is initially bonded to the carbon, crosses a barrier and finally bonds to the nitrogen on the other side. We placed it inside an argon bath and used molecular dynamics to observe the rate. Our initial work fixed the CN bond length because that made the simulation much faster and we figured that the CN vibrational motion wouldn’t matter much. But the nagging concern that the CN motion might affect the results remained. So we went ahead and redid the calculations releasing the constraint on the CN motion. I’m happy to report that the rise and fall persisted. As such the LiCN isomerization reaction rate is fastest when the density of the Argon bath is neither too small nor too large, but rather when it is just right.

The article with my collaborators, Pablo Garcia Muller, Rosa Benito and Florentino Borondo was just published in the Journal of Chemical Physics 141, 074312 (2014), and may be found at this doi hyperlink. This work was funded by the NSF on the American side of the collaboration, by Ministry of Economy and Competiveness-Spain and ICMAT Severo Ochoa on the Spanish side, and by the EU’s Seventh Framework People Exchange programme.

Getting to the shore when riding a wave from reactant to product

Suppose that you were trying to get across a floor that goes side to side like a wave goes up and down. If you started on the start line (call it the reactants), how long would it take you to cross to the finish line (call it the products) on the other side? In earlier work, we found “fixed” structures that could somehow tell you exactly when you were on the reactant and product sides even while the barrier was waving side to side. Just like you and I would avoid getting seasick while riding such a wave, the “fixed” structure has to move, but just not as much as the wave. So the structure of our dividing surface is “fixed” in the sense that our planet is always on the same orbit flying around the sun. This latter analogy can’t be taken too far because we know that our planet is thankfully stable. In the molecular case, the orbit is not stable. We just discovered that the rates at which molecular reactants move away from the dividing surface can be related to the reaction rate between reactant and products in a chemical reaction. (Note that the former rates are called Floquet exponents.) This is a particularly cool advance because we are now able to relate the properties of the moving dividing surface directly to chemical reactions, at least for this one simplified class of reactions.

The work involved a collaboration with Galen Craven from my research group and Thomas Bartsch from Loughborough University. The title of the article is "Communication: Transition state trajectory stability determines barrier crossing rates in chemical reactions induced by time-dependent oscillating fields.” The work was funded by the NSF, and the international partnership (Trans-MI) was funded by the EU People Programme (Marie Curie Actions). It was just released as a Communication at J. Chem. Phys. 141, 041106 (2014). Click on the JCP link to access the article.

When the buffalo roam, do they go over the pass or across the plain?

In one of our papers from last year (recapped in an earlier post), we found that in at least one chemical reaction (the ketene isomerization), the reaction could involve rather distinct pathways. On the one hand, the system could go across the break between the two energetic mountains separating the reactant and product. On the other hand, it could find a flat plain in which it could meander slowly across. The first of these two cases involves a narrow pass that is difficult for it to get through. The other is a wide plain but it costs a lot of energy to get up to it. Chemical reaction rate theory is built on the notion that the reaction always goes across the narrow passage as long as it’s the easiest one to get over. However, in the last decade there has been a lot of work observing that roaming over the flat plain has its privileges.

My postdoctoral student, Inga Ulusoy, and I wondered whether the ketene reaction gave rise to both possible classes of paths. It did! We also wondered the degree to which each path affected the rate in which the molecule reacted. We found earlier that the traditional pathway (over the break between the barriers) was the most important one in a model of the reaction with only two degrees of freedom. This led us and others to question whether our result was an accident of the simplicity of our model. In our recent paper, we extended the model to a larger number of degrees of freedom. Interestingly, the main result was the same. Namely, the reacting partners still have the possibility of roaming, but the direct path over the break between the barriers is still the most important one.

The article, "Revisiting roaming trajectories in ketene isomerization at higher dimensionality,” was recently published at Theoretical Chemistry Accounts 133, 1528 (2014). (doi:10.1007/s00214-014-1528-z) The work was supported by the AFOSR. Equally, importantly, it was a real treat to include our work in an issue published in honor of Greg Ezra's 60th birthday. I have followed his work since I was a graduate student, and have learned much from it. While science is immutable, it’s the fact that people are involved in the discovery that makes it humane. And for this reason, it’s particularly fun to be able to contribute to issues that honor the people involved in advancing science.

Stability within field induced barrier crossing (#APSphysics #PRE #justpublished)

Suppose that a 5' foot wall stood between you and your destination. In order to determine if and when you got to the other side, all you'd have to do is stand at the top of the wall and check when you got there. (Presumably falling down to the other side from the top would be a lot easier than getting to the top.) If, instead, there was a large mob of people trying to get across the wall, we'd have to keep track of all of them, but again only as to when each got to the top of the wall. This kind of calculation is called transition state theory when the people are molecules and the wall is the energetic barrier to reaction. The key concept is that the structure—that is, geometry—of the barrier determines the rate, and this geometry doesn't move.

If the wall were to suddenly start to slide towards and away from where you were first standing, then it might not be so easy to stay on top of it as you tried to cross over. Certainly, an observer couldn't just keep their eyes fixed to a point between the ends of the room because the wall would be in any one spot only for a moment. So is there still a way to follow when the reactants have gotten over the wall—that is, that they are reactants—in the crazy case when the barrier is being driven back-and-forth by some outside force? My student Galen Craven, our collaborator Thomas Bartsch (from Loughborough University), and I found that there is indeed such a way. The key is that you now have to follow an oscillating point at the same frequency as the barrier but not quite that of the top of the barrier. In effect, if the particle manages to cross this oscillating point, even if it hasn't quite crossed over the barrier, you can safely say that it is now a product. There is one crazy path, though, for which the particle follows this point and never leaves it. In this case, it would be like Harry Potter at King's Cross station never choosing to live or die. That's the stable path that we found in the case of field induced barrier crossing.

The title of the article is "Persistence of transition state structure in chemical reactions driven by fields oscillating in time." The work was funded by the NSF, and the international partnership (Trans-MI) was funded by the EU People Programme (Marie Curie Actions). It was released recently as a Rapid Communication at Phys. Rev. E. 89, 04801(R) (2014). Click on the PRE Link to access the article.

Counting to 41,044,208,702,632,496,804 and beyond...

Counting is a rather simple thing. You increment a given number by one just like in the journey of one thousand miles you take a step and repeat. Trouble is knowing when to stop. Ultra runners might not stop until they get to 100 miles (about 180,00 steps), but few will be able to tell you exactly how many steps they took. It's just too annoying to keep the count up and enjoy the view. Regardless, this gives us a good estimate of how high a person might be able to count in a day (if they count really fast). To get to the number in the title, it would take such a person over 600 billion years...

Now consider how long it would take to count the number of possible distinct non-recrossing paths following the edges of a square between the opposing corners along a diagonal. That's two, and it takes you less than a second. Suppose that you make it a two by two square. The number of possible distinct non-recrossing paths now comes up to 12. You might then ask about a 3 by 3 square or larger. Or maybe not as that might seem a little too mathematical. Dr. Minato and his colleagues followed this train of thought and made a YouTube video to illustrate how quickly the count grows. It's in Japanese with English subtitles and already has well over a million page views!

While this is a very fundamental question, it's useful to recognize that knowing how to count paths (particularly using Minato's clever algorithms) on an arbitrary network has lots of cool applications. Among these could be the determination of the sum of chemical pathways between reactants and products. That's the problem that I'm interested in, but it's a little harder because each path has a different weight (or cost.) The cost isn't necessarily the same for each of Minato's subsets and consequently it isn't trivial to reuse his existing algorithms. But here lies a challenge to a possible advance in the field of chemical physics.

Check out "The Art of 10^64 -Understanding Vastness-Time with class! Let's count!" on YouTube.

Advancing Science in Tandem with Colleagues in Japan… (#ACS #PRE #justpublished)

Sometimes features in my projects appear to come in waves. Intellectual discourse with distant groups appears to be a running theme at the moment. As with our recent work on roaming reactions, my recent article in Physical Review E involves a bit of back and forth with colleagues around the world. This time, it's my friends Kawai and Komatsuzaki at Hokaido University.

Since the start of my independent research career, I have been working on developing a series of models to describe the motion of particles in solvents that change with time. It's like trying to describe how a returner will run during a football kick-off without fully specifying the details of where all the blockers are and will be. The truth is that the blockers will move according to how the kick returner moves. This coordinated response between the chosen system—the returner or a molecule—and it's environment is not so easy to describe, and has been the object of much of the NSF-funded work by our group. We have managed to develop several models using stochastic differential equations that allow us to include such coupled interactions to varying degrees. While we were able to describe the response of the blockers to the returner in ever more complicated ways, for the most part they never seemed willing to talk to each other. A few years ago, Kawaii and Komatsuzaki found a formal way to extend the environment—that is, the blockers—so that they are able to affect each other as they respond to the system motion. In order to do this, though, you have to make some strong assumptions about the environment that are hard to satisfy for typical systems. In our latest paper, Alex Popov and I show that the our formalism is able to capture some of the generality of their model while still accommodating more general particles and environments.

The discussion between our two groups is a bit technical, but the back-and-forth is helping us all advance our understanding of the theory as well as enable its applications. The discourse is also not restricted to paper (in ink or bits) as our groups are now meeting regularly at workshops (such as in Telluride), and at our respective institutions (such as in my upcoming visit to Hokkaido university.)  Again, it's the opportunity for open discourse that makes it fun to keep advancing science!

The title of the article is "The T-iGLE can capture the nonequilibrium dynamics of two dissipated coupled oscillators," and the work was funded by the NSF. It was released recently at Phys. Rev. E, 88, 032145 (2013). Click on http://dx.doi.rog/:10.1103/PhysRevE.88.032145 to access the article.

Roaming pathways and rates: A case study on ketene(#ACS #JPC-A #justpublished)

It's no surprise that chemists care about chemical reactions. We want to know both how the reactions take place—mechanism or pathway—and how much time it takes to happen—rates. One of the most successful, though approximate, theories has been transition state theory (TST), in part, because it provides an answer to both questions. You simply need to find the saddle (or col) on the potential energy landscape between reactants and products. That bottleneck, which can be described with varying levels of fanciness, gives you a sense of how the atoms in the reacting system have to distort so as to proceed to products. The energy of the bottleneck can be used in a well-known formula to obtain the rate. Recently, however, Joel Bowman and others have discovered the possibility that the reactants could avoid the bottleneck entirely. These roaming trajectories thus pose a challenge to TST, and have generated a lot of well-deserved buzz.

In those cases when roaming trajectories wander so far away from the transition state (bottleneck) that new product channels (such as radical molecules) become accessible, there is no doubt that everything goes topsy turvy. However, we were curious as to whether roaming trajectories would turn TST upside down even when such channels are not available. In recent work, we studied the ketene isomerization reaction—that is when it interconverts from one form to another—and found that it gave rise to roaming trajectories (such as the one pictured here.) Unfortunately, TST remains reasonable for this system as long as one is careful to generalize the dividing surface associated with the bottleneck so as to appropriately include roaming trajectories. So perhaps all remains good with TST after all?!

The title of the article is "Effects of Roaming Trajectories on the Transition State Theory Rates of a Reduced-Dimensional Model of Ketene Isomerization" and the work was funded by the AFOSR. It was released recently at J. Phys. Chem. A, ASAP (2013). (doi:10.1021/jp402322h) 

Click on http://dx.doi.org/10.1021/jp402322h to access the article.

Doing chemistry on a gondola!? (Parting thoughts on #TellurideSciencestyle)

Practically every TSRC Workshop lecture starts with some quip about how the landscapes of the terrain inspires us to think about our science. In my case, the landscapes are reminiscent of the potential energy surface that governs a chemical reaction. Molecules, like people, look for the shortest path traversing the lowest point on the ridge between the reactant valley and the product valley. Sometimes, the path isn't traversable because there is still snow along it or some other obstacle blocks the way. Or maybe there's something along the path that can accelerate the journey (such as the gondola giving you a lift and sidestepping 1000 feet or so of the vertical rise). Both of these events also occur in chemistry (or biochemistry). You can have steric clashes with other parts of the reacting molecule or the solvent which slows the reaction down, and you can have catalysts (or enzymes) or solvent fluctuations that speed it up. 

More specifically, one of the current debates in chemical reaction rate theory concerns whether transition state theory is sufficiently accurate (or at least, correctable) to make it useful for quantitative description. This debate maps directly to the question of how to compute the rate of crossing from one  valley to another across a torturous mountain range. The mountaineers will likely spend a lot of time discussing whether there is a single col (and which one) affords the best passage. Similarly, chemists debate whether there is a single "transition state" which describes the reaction. Some, including me, argue that accessibility to the transitions state (or col) and the ease to move through it is equally important. That is, it's not just a single point which determines the passage, but also the shape of the surface around it. The debate doesn't stop here because we can start to kibitz about whether or not there exists a second (or more) transition state that is important to describe the passage between the valleys (of the reactants and products). Or perhaps there exist a gondola which takes you over them even faster? While this latter event doesn't actually occur in the chemical context unless you add additional (shuttling) molecules to the system, the analogies between Telluride's landscapes and the energy landscapes over which chemistry takes place are clearly richer than they may appear at first sight. Thus, while this is my last post on Telluride until my next trip there, it will most certainly continue to inspire me.