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Rates of Molecular Vibrational Energy Transfer in Organic Solutions Stéphanie Essafi, and Jeremy N. Harvey J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b12563 • Publication Date (Web): 16 Mar 2018 Downloaded from http://pubs.acs.org on March 16, 2018
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The Journal of Physical Chemistry
Rates of Molecular Vibrational Energy Transfer in Organic Solutions Stephanie Essafi,a,† Jeremy N. Harveya,b,* a. School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS, UK b. Department of Chemistry, Celestijnenlaan 200F, B-3001 Leuven, Belgium.
[email protected] † Current address: École Normale Supérieure, PSL Research University, UPMC Univ Paris 06, CNRS, Département de Chimie, PASTEUR, 24 rue Lhomond, 75005 Paris, France.
Abstract For condensed-phase reactions, commonly used kinetic models assume that energy exchange from and to solvent molecules is much faster than any reactive steps. However, it is becoming increasingly evident that this does not always hold true. In this work, we use molecular dynamics simulations to explore the timescale for energy transfer between solvent and solute in some typical organic solvents. As a reference, energy transfer between solvent molecules is also considered. The timescale is found to depend most strongly on the identity of the solvent. Energy transfer occurs fastest, with a timescale of roughly ten picoseconds, for ethanol, DMSO or THF, while it is noticeably slower in dichloromethane and especially supercritical argon, where a timescale well in excess of a hundred picoseconds is found. This suggests that the experimental search for non-thermal effects on selectivity and reactivity in organic chemistry should pay special attention to the choice of solvent, as the effects may occur more frequently in some solvents than in others. Introduction The distribution of internal energies in a sample of molecules plays a key role in the theory of rates of chemical reactions. 1,2,3 The section of the Boltzmann distribution corresponding to energies larger than the activation barrier is responsible for observed reactivity. For reactions in the liquid phase, it is 1 often assumed that one is in Kramers’ intermediate friction regime, with energy transfer from and to solvent molecules occurring faster than any reactive steps, such that reactants, intermediates and products are assumed to be present in Boltzmann distributions at all times. Hence the precise rate at which molecules exchange energy with one another is not the main focus of attention. This is different for gas-phase reactions, where there has been a much greater appreciation of the importance of 4 energy transfer dating back to the early work by Lindemann and Hinshelwood. Attention to this aspect naturally occurred earlier than for condensed-phase reactions, because many gas-phase reactions under fairly ordinary conditions display behavior that deviates from the one that would be 5 expected for a fully Boltzmann-distributed system. Nevertheless, energy transfer can play a role in solutions also, and this has been considered in a number of experimental and computational approaches for some time already. Pump-probe laser experiments and related approaches allow molecules to be prepared in high-energy states, and for the decay of the resulting vibrational energy by transfer to solvent to be probed. 6 The rates of vibrational energy decay vary considerably. In some cases, such transfer can occur very slowly, with one notorious case being that of collisionally-induced loss of energy from the v = 1 state of nitrogen in liquid nitrogen, which is slower than radiative emission i.e. is associated with a lifetime great than a minute.7 This is however an extreme case, with experiment suggesting a timescale of the order of picoseconds for many cases.6 An extensive theoretical framework has been assembled for predicting 8 the rate of vibrational energy transfer in solutions, based e.g. on Fermi’s golden rule. Extensive
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computational work has also been carried out to probe the timescales for energy transfer and the 9,10 detailed atomistic pathways that influence transfer. In some recent work, we and others have become interested in the role of energy transfer in solution for defining reactivity, selectivity and post-transition state behaviour.11,12,13,14,15,16,17,18 Commonly used kinetic models such as simple versions of transition-state theory are based on the statistical approximation, i.e. the assumption that energy exchange within the solute or between the solute and surrounding solvent molecules is always much faster than bond-breaking or bond-forming events. 19 However, it is becoming increasingly evident that this assumption should be regarded with caution. For example, reactions that involve a very-short lived, “reactive”, intermediate in the vicinity of a branching point on the potential energy surface (PES) are known to be improperly described by statistical models. In this case, statistical models fail because the intermediate reacts before it has had time to transfer all its excess energy to the solvent. In one interesting example, selectivity of hydroboration of terminal alkenes has been shown to involve dependence on the rate of vibrational cooling of a key intermediate due to energy transfer both intramolecularly from the core reacting region to the rest of the molecule, and intermolecularly, to solvent.18 As mentioned above, there has already been extensive experimental and theoretical work concerning vibrational energy transfer in liquids, both for pure liquids and for solute-solvent systems.8 In order to obtain the insight needed to gauge when vibrational energy transfer effects upon reactivity or selectivity may occur, it is helpful to know how rapid energy transfer is for some typical organic molecules and solvents. This is the aim of the present work. The approach used is computational, based on classical molecular dynamics simulations together with the fluctuation-dissipation theorem to assess rates. The target molecule is dissolved in a periodic box of solvent molecules, and subjected to a long molecular dynamics simulation, during which the internal energy of the molecule is monitored. Calculation of the autocorrelation function of the internal energy provides a timescale for energy transfer. We have considered both transfer between solvent molecules, and transfer between a solute and solvent. Concerning solvents, we have explored non-polar, polar aprotic and polar protic solvents. For the solute, we have considered both protic and aprotic solutes. For example, one system studied, glucose in tetrahydrofuran (THF) solution, is shown in Fig. 1.
Figure 1. Schematic representation of the glucose in THF system. Computational Methods MD simulations were performed using the TINKER 6.0 package 20 along with the MM3 force field unless stated otherwise.21,22,23 Each system consisted of a cubic box of 20 Å per side (Vbox = 8 nm3) with periodic boundary conditions applied in the three directions to simulate an infinite system. A smooth van der Waals cut-off threshold was set at 7 Å. Long-range electrostatic interactions were treated with the Ewald summation method using the default cut-off value (9 Å). The number of molecules per box was determined according to the density of each solvent and are reported in Table
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S1 (in the SI). Initial randomly constructed configurations were energy minimized according to the −1 −1 standard procedure to a 0.1 kcal mol Å rms gradient. This step was followed by a 10 ps NVT equilibration run (1 fs time step, T = 298 K), in order to yield a reasonable randomized starting point for the energy transfer simulation. The main calculation was then a long NVE simulation (between 1 and 20 ns) with a short time-step of 0.2 fs, chosen so as to yield accurate conservation of the total energy, typically to within 0.05 kcal mol−1 for the whole system. By using the NVE ensemble, the simulation does not correspond to a well-defined temperature, but the box size used is large enough that large fluctuations in apparent T are not observed. Following the simulation, the internal energy of each molecule was computed every 20 fs from the positions and velocities recorded during the simulation: the overall kinetic energy of the atoms in each molecule was calculated after subtracting the centre of mass velocity and the internal potential energy was obtained by re-calculating the energy of the isolated molecule at its instantaneous configuration and subtracting the potential energy at the overall minimum. Finally, correlation functions were computed from these data and statistical analyses performed to ensure sustainable convergence. Results The aim of the present study is to determine the characteristic rates at which molecules exchange energy with the environment in typical organic solvents. We refer to this as “vibrational” energy transfer, given that we probe the intramolecular energy of the solute molecules, which is largely vibrational energy. Coupling to solvent may however transfer energy to translational motion of the solvent molecules. Our study is based on classical simulations, in line with extensive previous work 24,25 11,18 both for gas-phase and condensed phase examples. A number of model systems have been chosen, of which the simplest involve exchange of energy for a pure liquid. We considered the cases of ethanol, dimethyl sulfoxide (CH3)2SO (DMSO), tetrahydrofuran (THF), pentane and dichloromethane. As mentioned above, we picked this set of solvents based on their common use in organic synthesis, and based on an attempt to cover a range of types of solvent, including non-polar, polar aprotic and protic cases. We also considered four solute-solvent systems, namely glucose in THF, paniculatine in THF, THF in DCM, and THF in supercritical argon (Scheme 1). Glucose is a typical polar organic molecule, while paniculatine is mostly non-polar, and THF as solute provides a useful point of comparison to the studies of pure solvents. Although supercritical argon is not very commonly used in organic synthesis, it was added to the list for comparison purposes. The density of supercritical argon was arbitrarily set to one typical −3 26 value 750 kg m , as also used in other recent computational work.
Scheme 1. Organic molecules considered in this work. We begin by assessing our methodology. Some aspects of the method have simply been assumed to be approximately valid, as testing is difficult. First of all, we use a classical simulation method, which is in principle a severe approximation given the quantization of vibrational energy, though comparison of quantum mechanical and classical methods suggests that at least for lower-frequency modes, classical approaches are reasonable.27,28,29 While in principle, quantum mechanical corrections could be used, given the general reasonable success obtained with classical methods for the study of vibrational energy transfer, we consider that this the classical approach should yield reasonable results and especially trends therein. A second point that is problematic is the approach we have used
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to compute energies for individual molecules within the solutions. Kinetic energies are trivially separable, so they can be calculated for individual solutes at each point in the simulation. For this reason, in one previous preliminary study of energy transfer,11c only the molecular kinetic energies were used as a proxy for the overall internal energy, relying on overall correlation between kinetic and potential energy in the spirit of the virial theorem. However, the virial theorem is not strictly applicable to anharmonic systems, and it is preferable to have a model of potential energies also. Here, intramolecular potential energies have been computed by comparing the molecular mechanics energy of each molecule taken in isolation at the structure adopted in the simulation to that at the minimum of the potential energy surface for that molecule. This approach thereby neglects all intermolecular potential energy terms. These could in principle vary in such a way that the correlation times computed here are incorrect. However, accurate ways of splitting the intermolecular energy between individual molecules are not easy to define, and we preferred to use the present approach. Another assumption is that the rate of loss of excess internal energy to solvent – the quantity of interest for understanding the reactivity issues that we are interested in19 – can be done by monitoring 11d the rate of decay of random fluctuations in the internal energy. In previous work, we have indeed found that this is a reasonable approximation, and have therefore used it again here. Other aspects are more easy to check. The energy transfer rates discussed below obviously depend on the parameters used in the classical simulations. In order to check the convergence of our results with respect to important parameters, calculations for the pure DMSO solvent box were repeated with respectively a larger box (30 Å per side instead of 20 Å), a longer timestep (1 fs instead of 0.2 fs) and a larger van der Waals cut-off value (9 and 10 Å, depending on the size of the box, instead of 7 Å). The quantities we are interested in, in particular the correlation time, appear to be virtually unaffected by the changes (see Table S2 in the SI), which suggests that the results are indeed converged with respect to these three parameters. Another key parameter is the interaction potential used in the MD simulations. For most of the work, we used the computationally inexpensive MM3 force field. To evaluate the influence of the force field on the energy transfer rates, calculations for the pure DMSO solvent box were also repeated using the polarizable AMOEBA force field as implemented and 30 developed in Tinker. The main difference between the MM3 and AMOEBA force fields concerns the electrostatic interactions, which include a treatment of polarizability in the case of AMOEBA, whereas fixed charges are used in MM3. The two force fields also differ by the detail of bond stretching, bending and torsion terms, as well as non-bonded van der Waals terms. Despite the large difference between the two approaches, the internal energy decorrelation times obtained are reasonably close to one another, differing by less than a factor of 2. This provides one measure of the accuracy of the simulation protocol used. We now turn to describing the main results. For each system of interest, we constructed and equilibrated a box as described in the Computational Methods section. We then monitored the evolution of the internal energy of each solvent or solute molecule as a function of time during long NVE simulations. This yielded time series such as the one presented in Figure 2 (black curve). The particular example shown is the ‘glucose in THF’ system, and corresponds to the internal energy of the glucose molecule alone. As can be seen, the internal energy of the solute varies very rapidly, with strong fluctuations even over short times. The two plots of the internal energy using moving averages show also that the system demonstrates some persistence of fluctuations away from the long-term average even at longer time, e.g. on the 200 ps timescale (red curve). The yellow curve indicates that at longer time (1 ns timescale), only small energy variations persist, though it is clear that they have not yet completely vanished.
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Figure 2. Internal energy of glucose as a function of time for the ‘glucose in THF’ case (black curve). The red and yellow curves correspond to the moving average for periods of respectively 200 ps and 1 ns. Energies extracted from a single 20 ns NVE simulation. The time-series were analysed in terms of the autocorrelation function of the internal energy 〈 0 〉 where is the internal energy of a given solvent molecule at time , is the mean internal energy over all time steps, and 〈… 〉 refers to the average over all molecules and all time origins. The results for the pure liquids are reported in Figure 3 and Table 1. As expected from previous work, the autocorrelation function of the internal energy decays roughly exponentially with (Figure 3). The internal energy decorrelation time is then given by the time required for to reach 1⁄ (Table 1). Care is required in order to reach a well-converged estimate of this decorrelation time. This was done by using the ‘blocking’ method to estimate the variance of the mean internal energy, , and checking that the whole simulation was long enough to ensure that convergence had been reached 31 (see SI for details). In all cases but DCM, a 1 to 2 ns production run was sufficient to reach convergence. In the case of DCM, a much longer run was required (20 ns).
Figure 3. Autocorrelation function of the internal energy for pure liquids. The trend in relaxation time is as follows: ethanol ~ DMSO
