Mode-specific versus local heating effects in infrared-laser-driven reactions

Abstract

Using an infrared laser to control a molecule’s reactivity by targeting specific vibrational modes has long been of interest to chemists. Rapid intramolecular vibrational energy redistribution (IVR) in molecules poses significant challenges to achieving this, as it quickly transfers the pumped energy to other molecular degrees of freedom (${\tau }_{\mathrm{IVR}}=1\mathrm{p}\mathrm{s}$). With recent advances in femtosecond pulsed laser capabilities, however, infrared-laser-driven vibrationally assisted reactivity is worth revisiting. In this work, we theoretically quantify the contributions of both mode-specific assistance and laser-induced heating to reaction rate enhancements of laser-driven molecules. Notably, reactions with lower activation barriers exhibit smaller relative rate enhancements. Furthermore, local heating contributions dominate for low-barrier reactions while the vibrationally-assisted component is more prominent for high-barrier reactions (precise statements of what low- and high-barriers depend on thermal diffusivity of the solvent). We obtain approximate bounds for these rate enhancements. While pulsed laser driving yields rate enhancements several orders of magnitude greater than continuous-wave driving for the same absorbed power, the overall increments remain modest under typical experimental conditions, except for low-frequency modes, where they can be substantial.

I. INTRODUCTION

The past century of synthetic chemistry has contributed a wide-range of chemical transformations that, today, enables the deconstruction of a target molecule into simpler, accessible precursors [1, 2]. However, deleterious side reactions remain possible at every step of the synthetic sequence, leading to material loss that can quickly compound in a multi-step synthesis [3, 4]. The reaction selectivity is often tuned by evaluating different reaction conditions such as choice of solvent, temperature, and/or applicable reagents and catalysts. Introduction of additional reagents and specific parameters can lead to more challenging and expensive experimental protocols [3, 4].

In recent years, light has taken the spotlight as an environmentally friendly activator of chemical reactions [5–7]. It does not introduce additional chemicals, provides more energy directly to the substrate compared to heat, and is often highly selective. Most photochemical transformations are conducted with optical light by exciting the molecular electronic states [5–7]. The ensuing reaction path can in principle be optimised by coherent control of the excitation pulses [8–12]. By contrast, infrared (IR) light can directly populate a specific vibrational mode [13–19]. Here, the allure is the possibility of targeting the reaction coordinate, which is the vibrational path that directly connects the reactant to the product via the transition state. If there exists multiple reaction coordinates, each leading to a distinct product, then, by exciting the desired reaction coordinate (or, more specifically, the vibrational mode that dominates the desired reaction coordinate), one can directly influence the reaction selectivity. This phenomenon, known as mode-selective chemistry [20–22], has been supported by multiple computational studies [23–33], mostly focusing on the microscopic reaction rate of a model potential energy surface (PES) following IR excitation.

Unfortunately, the experimental implementation of IR-laser-driven synthetic protocols pales in comparison to its optical counterpart, most likely because any excitation of the vibrational states will be lost within picoseconds to heat through intramolecular vibrational energy redistribution (IVR), thereby hampering the mode specificity [34–40]. This limitation implies that, for the IR laser to be meaningful, it must accelerate the reaction by an order of the IVR rate (typically ps⁻¹) divided by the laser pulse rate (typically ms⁻¹). As an example, for a thermally activated reaction that completes within an hour, the IR laser must drive the reaction strongly enough that it occurs within microseconds.

Recent advancements in the field of ultrafast IR lasers promise to alleviate this problem, as evident from multiple experimental realisation of femtosecond-IR-initiated reactions [41–43]. This has culminated in a seminal work by Heyne and co-workers, where a bimolecular reaction between an alcohol and an isocyanate (forming a carbamate) was accelerated in the condensed phase by shining ultrafast IR laser pulses on the alcohol O–H stretching vibration [44]. Notably, despite supplementing a comprehensive ab initio study of the reaction free energy landscape, as far as we are aware, no quantitative kinetic model of the IR-induced rate change has been provided. We find this imperative given the aforesaid constraints on the reaction classes and laser designs required for IR-induced mode-selective chemistry to occur.

In this work, we present a theoretical analysis of the potential for femtosecond-IR-driven reactions, focusing on the extent of rate enhancement experienced by a realistic ensemble of reactant molecules in solution. Importantly, we account for the increase in local temperature of the molecule due to the energy dissipated through IVR and its implications on the overall macroscopic reaction rate, which is expected to be the experimental observable. We relate the rate enhancements to experimental parameters, such as the pulse energy E_pulse for a pulsed laser and the power P_cw for a continuous-wave (CW) laser. Motivated by long-standing interest for selective activation of amides for development of novel transformations [45–48], we investigated the lactonization of N-benzyl-2-(hydroxymethyl)-N-methylbenzamide (Figure 1). We anticipated activation of this amide to result in rate acceleration for lactonization to give isobenzofuran-1(3H)-one and release of N-benzyl-N-methyl amine as byproduct. However, based on our theoretical analysis, under a typical pulse repetition period of 1 ms, the femtosecond IR pulse only accelerates the reaction rate for a small fraction (~ 10⁻⁹) of it. Therefore, the overall rate enhancement remains low; for instance, a reaction mixture undergoing a chemical transformation of 14 kcal mol⁻¹ activation barrier must receive at least one vibrational quantum of IR excitation per molecule to evoke a modest ~ 1% overall rate increase.

FIG. 1.

Illustration of N-benzyl-2-(hydroxymethyl)-N-methylbenzamide (1) in toluene solution.

II. RESULTS

We treat an IR-laser driven reaction using a modified one-dimensional transition state theory and obtain bounds on the maximum achievable rate enhancement for a given power absorbed by the molecule for both a pulsed laser and a CW laser. We also account for changes in the local temperature of the molecule T_loc due to energy dissipated via IVR. This energy is then lost to the bulk through vibrational cooling (VC). We decompose the total rate enhancement under IR-laser driving into temperature-induced and vibrationally-assisted components. We note that our model predictions negate the conclusions of Heyne and co-workers [44]. This discrepancy may be due to specific features of their experimental setup not considered by our model, a model designed to be as general and broadly applicable as possible. Further consideration for the explicit presence of multiple vibrational modes that participate beyond IVR, non-adiabatic couplings, and other quantum-mechanical effects may be required for direct analysis of their findings.

A. Rate constant

Reaction rates can typically be expressed as $k=A\mathrm{e}\mathrm{x}\mathrm{p}(-B)$ when $V_B\gg \hslash {\omega }_0$ and $V_B\gg k_{B}T$, where the prefactor A and the exponential term B depend on the barrier height $V_B$, the reactive coordinate real and imaginary frequencies in the potential well ${\omega }_0$ and barrier ${\omega }_B$, respectively, the temperature $T$, and the damping rate in the reactant well $\gamma$ [49]. The rate of a reaction from classical one-dimensional transition state theory (TST) is given by the flux

$$k=A_0{\int }_0^{\mathrm{\infty }}dpP\left(x_B,p\right)\frac{p}{m}=A_0\frac{{\omega }_0}{2\pi }{e}^{-\frac{V_B}{k_{B}T}}$$ (1)

where $x_B$ is the location of the barrier, and $P(x,p)$ is the joint probability density in position $x$ and momentum $p$. Here, $A_0$ includes the entropic contribution to the rate constant. In 1932, Wigner proposed a quantum transition state theory based on heuristic arguments, where the classical probability density $P(x,p)$ is replaced by the Wigner function $W_T(x,p)$ [50]

$$k_0\left(T\right)=A_0{\int }_0^{\mathrm{\infty }}dp{W}_T\left(x_B,p\right)\frac{p}{m}.$$ (2)

Although this expression is known to break down in specific regimes due to its lack of positive definiteness [51], its applicability to parabolic potentials and our primary interest in obtaining estimates allow us to confidently employ the Wigner expression for the rate constant.

Importantly, the Wigner rate expression captures both the low and high temperature limits, $\hslash {\omega }_0\gg k_{B}T$ and $\hslash {\omega }_0\ll k_{B}T$, correctly for a parabolic potential. For instance, the Wigner function evaluated at the barrier for a harmonic potential $V(x)=\frac{1}{2}m{\omega }_0^2{x}^2$ is given by

$$\begin{gathered} W_T(x_B,p)=\frac{1}{\pi \hslash \text{coth}(\hslash {\omega }_0/2k_{B}T)} \\ \text{exp}\left[-\frac{1}{\frac{\hslash {\omega }_0}{2}\text{coth}(\hslash {\omega }_0/2k_{B}T)}\left(\frac{p^2}{2m}+V_B\right)\right] \end{gathered}$$ (3)

where $V_B=\frac{1}{2}m{\omega }_0^2{x}_B^2$. Substituting this into Eq. 2 and integrating over momenta, we find that the resulting expression of the rate captures both the high temperature thermal activation limit $e^{-V_B/k_{B}T}$ and the low temperature quantum tunneling limit $e^{-V_B/\hslash {\omega }_B}$ of the quantum rate if ${\omega }_B~{\omega }_0$ [52, 53].

We analyze two laser driving scenarios: pulsed and continuous-wave (CW), and compute the Wigner functions appropriately. We take the average power absorbed by a molecule $P_{\text{abs}}$ to be the same in both cases for a fair comparison. For a pulsed laser, $P_{\text{abs}}=E_{\text{abs}}/t_{\text{rep}}$ where $E_{\text{abs}}$ is the energy delivered to a molecule by a single pulse and $t_{\text{rep}}$ is the pulse repetition period. We define the average rate constant $k_{\text{avg}}$ of the reaction under laser driving using the time-averaged Wigner function

$$\begin{gathered} k_{\text{avg}}[T]=A_0\underset{0}{\overset{\infty }{\int }}dp\frac{p}{m}\left[\frac{1}{t_{\text{period}}}\underset{0}{\overset{t_{\text{period}}}{\int }}d{t}^{\prime }{W}_T(x_B,p;t^{\prime })\right] \\ =\frac{1}{t_{\text{period}}}\underset{0}{\overset{t_{\text{period}}}{\int }}d{t}^{\prime }k(t^{\prime };T), \end{gathered}$$ (4)

and reordering the integrals allows us to define a time-dependent flux $k(t;T)=A_0{\int }_0^{\mathrm{\infty }}dp\frac{p}{m}{W}_T\left(x_B,p;t\right)$ where we recall that the local temperature $T=T(t)$ is a time-dependent variable itself and the square brackets in $k_{\text{avg}}[T]$ denote that $k_{\text{avg}}$ is a functional of $T(t)$. In the case of a pulsed laser, $t_{\text{period}}$ equals the pulse repetition period $t_{\text{rep}}$ and for driving with a CW laser, $t_{\text{period}}$ equals the period of the driving laser $t_{\omega }=\frac{2\pi }{\omega }$. The Wigner function of a driven, lossy, Brownian oscillator is known [54]. We use this in Eq. 4, to calculate the rate constant (see Supplementary S1). The physics embodied in Eq. 4 is that the IR laser drives the Wigner function away and close to the barrier $x_B$ (Fig. 2). This coherent driving does not even out the flux, the later enjoys substantial enhancement when it is close to $x_B$, and it is this enhancement what we aim to explore in the following sections. Quantum mechanically, this picture translates into the IR laser-driving adding quanta to the reaction coordinate with the concomitant result of either effectively reducing the amount of quanta needed to reach the barrier at $V_B$ or alternatively, enhancing the tunneling probability from reactant to product.

FIG. 2.

(a,d) Sketches showing the trajectory of the phase space probability density when the system is pumped with (a) a pulsed laser and (d) a continuous-wave (CW) laser. The barrier position is indicated by $x_B$. (b,e) The relative change in the rate constant (see Eq. 7) is plotted as a function of (b) the energy absorbed per molecule, $E_{\mathrm{a}\mathrm{b}\mathrm{s}}$, for a pulsed laser, and (e) the power absorbed per molecule, $P_{\mathrm{a}\mathrm{b}\mathrm{s}}$, for a CW laser, when exciting a mode of frequency ${\omega }_0/(2\pi c)=200{\mathrm{c}\mathrm{m}}^{-1}$. For reference, ${10}^{-19}\mathrm{J}\approx 25\hslash {\omega }_0$. The total change is shown in black; the vibrationally-assisted component is shown as a dashed yellow curve, and the temperature-induced contribution as a dashed blue curve. The vertical gray line marks the activation energy $V_B=14\mathrm{k}\mathrm{c}\mathrm{a}\mathrm{l}.{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$. The pulse energy is related to the absorbed energy via $E_{\text{pulse}}=\frac{E_{\text{abs}}A{N}_A}{ϵ\mathrm{l}\mathrm{n}10}\approx E_{\text{abs}}\times 4\times {10}^{15}$, where the laser spot size is $A=\pi r^2=3.14\times {10}^{-8}{\mathrm{m}}^2$ with $r=100\mu \mathrm{m}$, the molar extinction coefficient is $ϵ=200{\mathrm{L}\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}{\mathrm{c}\mathrm{m}}^{-1}$, and $N_A$ is Avogadro’s number. The absorbed power in the CW case is calculated as $P_{\text{abs}}=E_{\text{abs}}/t_{\text{rep}}$, where the pulse repetition period for the pulsed laser is $t_{\text{rep}}=1\mathrm{m}\mathrm{s}$ and the corresponding CW laser power $P_{\mathrm{c}\mathrm{w}}=E_{\text{pulse}}/t_{\text{rep}}$. (c,f) The local temperature change $\mathrm{\Delta }{T}_{\text{loc}}=T_{\text{loc}}-T_{\text{out}}$ of the reacting molecule is shown when it is excited by (c) a pulsed laser and (f) a CW laser, with the energy and power values marked by a star in (b) and (e), respectively. The vibrational cooling time constant is taken to be ${\tau }_{\mathrm{V}\mathrm{C}}=R^2/3\alpha =1.7\mathrm{p}\mathrm{s}$ where $R=7\text{Å}$ is the size of the molecule and $\alpha =0.095{\mathrm{m}\mathrm{m}}^2{\mathrm{s}}^{-1}$.

B. Temperature

We follow an approach similar to [55, 56] to model the temperature change upon laser excitation of the molecule. More sophisticated models for vibrational energy and heat flow in molecules in condensed phase have been discussed elsewhere [57–59]. We divide the system into three parts: (a) the specific vibrational mode that is driven with the laser (red in Fig. 1), (b) the ‘local environment’ that consists of other vibrational modes of molecule (blue in Fig. 1), and (c) molecules further away from the excited molecules (yellow in Fig. 1). The local environment is assumed to be at a uniform temperature $T_{\text{loc}}$. The bulk material is at temperature $T_{\text{out}}$. The average energy of the driven vibrational mode, $⟨E_{\text{mode}}(t)⟩$, varies as

$$\frac{d⟨E_{\text{mode}}(t)⟩}{dt}=-\frac{1}{{\tau }_{\mathrm{I}\mathrm{V}\mathrm{R}}}\frac{⟨p(t){⟩}^2}{m}+F(t)\frac{⟨p(t)⟩}{m}$$ (5)

[60] (see Supplementary S1). Here, the first term is the energy lost from the driven mode to the local environment and the second term the energy supplied by the external force F(t). The local temperature changes according to

$$\frac{d{T}_{\mathrm{l}\mathrm{o}\mathrm{c}}}{dt}=\frac{1}{C_{\mathrm{l}\mathrm{o}\mathrm{c}}}\left[\frac{1}{{\tau }_{\mathrm{I}\mathrm{V}\mathrm{R}}}\frac{⟨p(t){⟩}^2}{m}\right]-\frac{1}{{\tau }_{\mathrm{V}\mathrm{C}}}\left(T_{\mathrm{l}\mathrm{o}\mathrm{c}}-T_{\mathrm{o}\mathrm{u}\mathrm{t}}\right)$$ (6)

where $C_{\text{loc}}$ is the heat capacity of the local environment. The first term on the right-hand side of Eq. 6 quantifies $t$ he energy gained by the local environment from the pumped mode through IVR and the second term is the energy lost by it to the bulk through a process known as vibrational cooling (VC) with the time scale ${\tau }_{\text{VC}}$ [55]. The expression ${\tau }_{\text{VC}}=R^2/3\alpha$ can be obtained from Fourier’s equation assuming a sphere of radius $R$ at temperature $T_{\text{loc}}$ with the temperature at $r\to \mathrm{\infty }$ being $T_{\text{out}}$ where $\alpha$ is the thermal diffusivity of the bulk [56]. For a laser-driven molecule in a dilute solution, ${\tau }_{\text{VC}}=R^2/3\alpha ~1\mathrm{p}\mathrm{s}$ where $R\approx 7\text{Å}$ for our molecule ($\mathbf{1}$, see Fig. 1 for structure) if we take the local environment to be the driven solute molecule and $\alpha =\mathrm{the\; thermal\; diffusivity\; of\; the\; solvent}$ (we use the value for toluene $\alpha =0.095{\mathrm{m}\mathrm{m}}^2{\mathrm{s}}^{-1}$). In Figs. 2c and 2f, we plot the temperature as a function of time when the molecule is driven with a pulsed laser and a CW laser, respectively. In the case of the CW laser, the local temperature change is negligible. However, for a pulsed laser, the local temperature increases transiently over a short timescale of ${\tau }_{\mathrm{V}\mathrm{C}}=1.7\mathrm{p}\mathrm{s}$ before returning to $T_{\text{out}}$. Note that $T_{\text{loc}}$ can only be taken seriously as a local temperature if it is averaged over a thermalization timescale (which we have taken to be $2{\tau }_{\text{IVR}}$; see Supplementary S1); all calculations hereafter take this averaging into account.

Instead of driving vibrational modes of a solute in a dilute solution, if we drive modes in a thin film or the solvent itself, then at high photon fluxes a substantial number of molecules within the laser spot may be excited. In this regime, the assumption that each excited molecule is surrounded by material at temperature $T_{\text{out}}$ extending infinitely in all directions breaks down, and our model of temperature change will not apply. However, in the extreme case where all molecules within the laser spot are excited, we can again apply our model with $R$ equal to the laser spot radius ($r=100\mu \mathrm{m}$). This leads to ${\tau }_{\mathrm{V}\mathrm{C}}~100\mathrm{m}\mathrm{s}$ and can result in a prolonged temperature increase in both the CW and pulsed laser cases.

C. Rate change

The fractional change in the rate constant is

$$\begin{gathered} \frac{\text{Δ}k}{k}=\frac{k_{\text{avg}}[T_{\text{loc}}]-k_0[T_{\text{out}}]}{k_0[T_{\text{out}}]} \\ =\underset{\text{vibrationally–assisted}}{\underbrace{\frac{k_{\text{avg}}[T_{\text{loc}}]-k_0[T_{\text{loc}}]}{k_0[T_{\text{out}}]}}}+\underset{\text{temperature–induced}}{\underbrace{\frac{k_0[T_{\text{loc}}]-k_0[T_{\text{out}}]}{k_0[T_{\text{out}}]}}} \end{gathered}$$ (7)

where the first term is the vibrationally-assisted contribution and the second one the temperature-induced contribution. If the molecule is in a thermal state at temperature $T_{\text{loc}}$, then the vibrationally-assisted contribution will be zero. From Eq. 6, we know that $T_{\text{loc}}(t)$ is a function of time, so while computing the rate constants $k_{\text{avg}}\left(T_{\text{loc}}\right)$ or $k_0\left(T_{\text{loc}}\right)$, as an approximation, we use the instantaneous temperature to calculate the rate and average this over the duration $t_{\text{period}}$. For instance, we use $k\left(t^{\prime };T_{\text{loc}}\left(t^{\prime }\right)\right)$ in Eq. 4 (see Supplementary S1).

In Figs. 2b and 2e, we plot the relative change in the rate constant, $\mathrm{\Delta }k/k$, for a reaction with a low-frequency reaction coordinate ${\omega }_0/(2\pi c)=200{\mathrm{c}\mathrm{m}}^{-1}$. For the same average absorbed power, $P_{\text{abs}}=E_{\text{abs}}/t_{\text{rep}}$, the rate enhancement is several orders of magnitude larger under pulsed laser excitation compared to CW driving. For a short vibrational cooling timescale, ${\tau }_{\mathrm{V}\mathrm{C}}=1.7\mathrm{p}\mathrm{s}$, that is smaller or comparable in magnitude with ${\tau }_{\text{IVR}}$, we find that the vibrationally-assisted mechanism is the dominant contributor to this enhancement, while the temperature-induced contribution is negligible. However, if vibrational cooling is slower, the molecule would remain hot for a longer period, increasing the temperature-induced contribution to the rate enhancement. To facilitate comparison with experimental parameters, we include an additional $x$-axis in these plots indicating the pulse energy of the pulsed laser, $E_{\text{pulse}}$, and the laser power of the CW laser, $P_{\mathrm{c}\mathrm{w}}$. These quantities are related to the energy and power absorbed per molecule through $E_{\text{pulse}}=\frac{E_{\text{abs}}A{N}_A}{ϵ\mathrm{l}\mathrm{n}10}\approx E_{\text{abs}}\times 4\times {10}^{15}$ and $P_{\text{cw}}=\frac{P_{\text{abs}}A{N}_A}{ϵ\mathrm{l}\mathrm{n}10}\approx P_{\text{abs}}\times 4\times {10}^{15}$ where the laser spot size is taken to be $A=\pi r^2=3.14\times {10}^{-8}{\mathrm{m}}^2$ with $r=100\mu \mathrm{m}$, the molar extinction coefficient $ϵ=200{\mathrm{L}\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}{\mathrm{c}\mathrm{m}}^{-1}$, and $N_A$ is Avogadro’s number [61]. Results for a reaction with a high-frequency reaction coordinate, ${\omega }_0/(2\pi c)=1700{\mathrm{c}\mathrm{m}}^{-1}$, showing the corresponding rate enhancement and temperature change, are presented in Supplementary Fig. S1. The rate enhancements are much smaller when such a high frequency mode is driven.

We derive upper bounds for rate enhancement under pulsed laser excitation. If ${\tau }_{\mathrm{I}\mathrm{V}\mathrm{R}}<{\tau }_{\mathrm{V}\mathrm{C}}$, upon excitation with a pulsed laser the molecule initially enters a highly non-thermal state for a short duration ${\tau }_{\text{IVR}}$, before relaxing to a higher-temperature thermal state for a time ${\tau }_{\text{VC}}-{\tau }_{\text{IVR}}$. It then cools to room temperature and remains there for the remainder of the pulse interval, $t_{\text{rep}}-{\tau }_{\text{VC}}$, until the next pulse arrives (see Fig. 2a). On the other hand, when driven with a CW laser, the system is in a non-thermal state for the entire duration, however, this non-thermal state is very close to a thermal state (see Fig. 2d). To obtain a better understanding of the pulsed laser case, we divide the time integral in Eq. 4 into three intervals: IVR period, $\left[0,{\tau }_{\text{IVR}}\right]$, vibrational cooling (VC) period $\left[{\tau }_{\mathrm{I}\mathrm{V}\mathrm{R}},{\tau }_{\mathrm{V}\mathrm{C}}\right]$, and after VC, $\left[{\tau }_{\mathrm{V}\mathrm{C}},t_{\mathrm{r}\mathrm{e}\mathrm{p}}\right]$,

$$\begin{gathered} k_{\text{avg},\text{pulse}}=\frac{1}{t_{\text{rep}}}[\underset{0}{\overset{{\tau }_{\text{IVR}}}{\int }}d{t}^{\prime }k(t^{\prime };T_{\text{loc}})+\underset{{\tau }_{\text{IVR}}}{\overset{{\tau }_{\text{VC}}}{\int }}d{t}^{\prime }k(t^{\prime };T_{\text{loc}}) \\ +\underset{{\tau }_{\text{VC}}}{\overset{t_{\text{rep}}}{\int }}d{t}^{\prime }k(t^{\prime };T_{\text{loc}})] \\ \approx \frac{{\tau }_{\text{IVR}}}{t_{\text{rep}}}{k}_{\text{IVR}}+\frac{({\tau }_{\text{VC}}-{\tau }_{\text{IVR}})}{t_{\text{rep}}}{k}_{\text{VC}} \\ +\frac{(t_{\text{rep}}-{\tau }_{\text{VC}})}{t_{\text{rep}}}{k}_0(T_{\text{out}}). \end{gathered}$$ (8)

Here, the transiently enhanced rate constant before IVR sets in is defined as $k_{\mathrm{I}\mathrm{V}\mathrm{R}}=\frac{1}{{\tau }_{\mathrm{I}\mathrm{V}\mathrm{R}}}{\int }_0^{{\tau }_{\mathrm{I}\mathrm{V}\mathrm{R}}}d{t}^{\prime }k\left(t^{\prime };T_{\text{loc}}\right)$, and the rate constant during the VC period $k_{\mathrm{V}\mathrm{C}}=\frac{1}{{\tau }_{\mathrm{V}\mathrm{C}}-{\tau }_{\mathrm{I}\mathrm{V}\mathrm{R}}}{\int }_{{\tau }_{\mathrm{I}\mathrm{V}\mathrm{R}}}^{{\tau }_{\mathrm{V}\mathrm{C}}}d{t}^{\prime }k\left(t^{\prime };T_{\mathrm{l}\mathrm{o}\mathrm{c}}\right)\approx \frac{1}{{\tau }_{\mathrm{V}\mathrm{C}}-{\tau }_{\mathrm{I}\mathrm{V}\mathrm{R}}}{\int }_{{\tau }_{\mathrm{I}\mathrm{V}\mathrm{R}}}^{{\tau }_{\mathrm{V}\mathrm{C}}}d{t}^{\prime }{k}_0\left(T_{\mathrm{l}\mathrm{o}\mathrm{c}}\right)$.

The rate constant during the IVR period, $k_{\text{IVR}}$, can be substantially larger than $k_0\left(T_{\text{out}}\right)$ because the IR laser induces coherent oscillations of $W_T(x,p,t)$ away from its equilibrium geometry at $x=0$, giving it better access to regions close to the barrier $x_B$ (Fig. 2a,d). However, it is bounded by the barrierless rate $k_{\text{max}}=A_0\frac{{\omega }_0}{2\pi }$. On the other hand, the rate constant during the vibrational cooling period, $k_{\mathrm{V}\mathrm{C}}$, is subject to an even tighter upper bound determined by the maximum local temperature reached, $T_{\text{loc},\mathrm{m}\mathrm{a}\mathrm{x}}=\mathrm{m}\mathrm{a}\mathrm{x}\left[T_{\text{loc}}(t)\right]$,

$$\frac{k_{\mathrm{I}\mathrm{V}\mathrm{R}}}{k_0\left(T_{\mathrm{o}\mathrm{u}\mathrm{t}}\right)}\le e^{\frac{V_B}{k_B{T}_{\mathrm{o}\mathrm{u}\mathrm{t}}}},$$ (9a)

$$\frac{k_{\mathrm{V}\mathrm{C}}}{k_0\left(T_{\mathrm{o}\mathrm{u}\mathrm{t}}\right)}\le e^{\frac{V_B}{k_B{T}_{\mathrm{o}\mathrm{u}\mathrm{t}}}\left(1-\frac{T_{\mathrm{o}\mathrm{u}\mathrm{t}}}{T_{\mathrm{l}\mathrm{o}\mathrm{c},\mathrm{m}\mathrm{a}\mathrm{x}}}\right)}.$$ (9b)

Clearly, larger activation barriers have larger possible relative enhancements.

The transiently enhanced rate constant before IVR, $k_{\text{IVR}}$, is shown in Fig. 3. Taken alone, these results would suggest an appreciable rate enhancement exceeding ${10}^8$ times, a value that scales with the pulse energy. However, we expect this rate acceleration to be short-lived due to the fast IVR and VC timescales. As we shall see, the macroscopic reaction rate $k_{\text{avg,pulse}}$ is typically dominated by the rate following thermalization with the bulk environment $k_0\left(T_{\text{out}}\right)$. In particular, $k_{\text{avg},\text{pulse}}$ is substantially different from $k_0\left(T_{\text{out}}\right)$ only when the non-thermal rate exceeds the thermal rate, weighted by their durations of validity. This can be quantified by obtaining approximate upper bounds on the rate enhancements using Eq. 7, 8, and 9,

$${\left(\frac{\mathrm{\Delta }k}{k}\right)}_{\text{vibrationally–assisted}}\lesssim \frac{{\tau }_{\mathrm{I}\mathrm{V}\mathrm{R}}}{t_{\mathrm{r}\mathrm{e}\mathrm{p}}}\left[e^{\frac{V_B}{k_B{T}_{\mathrm{o}\mathrm{u}\mathrm{t}}}}-1\right]$$ (10a)

$${\left(\frac{\mathrm{\Delta }k}{k}\right)}_{\text{temperature–induced}}\lesssim \frac{{\tau }_{\mathrm{V}\mathrm{C}}}{t_{\mathrm{r}\mathrm{e}\mathrm{p}}}\left[e^{\frac{V_B}{k_B{T}_{\mathrm{o}\mathrm{u}\mathrm{t}}}\left(1-\frac{T_{\mathrm{o}\mathrm{u}\mathrm{t}}}{T_{\mathrm{l}\mathrm{o}\mathrm{c},\text{max}}}\right)}-1\right].$$ (10b)

FIG. 3.

The transiently enhanced reaction rate constant $k_{\text{IVR}}$ relative to the bare rate constant $k_0\left(T_{\text{out}}\right)$ before intramolecular vibrational energy redistribution (IVR), i.e., within ${\tau }_{\mathrm{I}\mathrm{V}\mathrm{R}}=1\mathrm{p}\mathrm{s}$, given it has energy $E_{\mathrm{a}\mathrm{b}\mathrm{s}}$ pumped into it initially. Here, the vertical gray line indicates the activation energy $V_B=14\mathrm{k}\mathrm{c}\mathrm{a}\mathrm{l}{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$. This calculation was performed taking parameters with reaction coordinate ${\omega }_0/(2\pi c)=200{\mathrm{c}\mathrm{m}}^{-1}$ and ${\tau }_{\mathrm{V}\mathrm{C}}=1.7\mathrm{p}\mathrm{s}$. For reference, ${10}^{-19}\mathrm{J}\approx 25\hslash {\omega }_0$. The pulse energy is related to the absorbed energy via $E_{\text{pulse}}=\frac{E_{\mathrm{a}\mathrm{b}\mathrm{s}}A{N}_A}{ϵ\mathrm{l}\mathrm{n}10}\approx E_{\mathrm{a}\mathrm{b}\mathrm{s}}\times 4\times {10}^{15}$, where the laser spot size is $A=\pi r^2=3.14\times {10}^{-8}{\mathrm{m}}^2$ with $r=100\mu \mathrm{m}$, and the molar extinction coefficient is $ϵ=200{\mathrm{L}\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}{\mathrm{c}\mathrm{m}}^{-1}$.

We use the approximately less-than-or-equal-to symbol here because the exponential function $e^{-t/{\tau }_{\text{IVR}}}$ decays only to $1/e=0.37$ of its initial value at time $t={\tau }_{\text{IVR}}$. Therefore, the second and third terms in Eq. 8 will also contribute a little bit to the vibrationally-assisted component. Similar ideas hold for the temperature-induced component. Typically, ${\tau }_{\text{IVR}}=1\mathrm{p}\mathrm{s}$ and $t_{\text{rep}}=1\mathrm{m}\mathrm{s}$, so we need $k_{\text{IVR}}/k_0\left(T_{\text{out}}\right)\gtrsim {10}^9$ to observe substantial enhancements in the reaction rate.

In all calculations in Fig. 2–4, we have used ${\tau }_{\mathrm{V}\mathrm{C}}=1.7$ ps, which is the typical value expected based on the size of molecule $\mathbf{1}$ and the thermal diffusivity of the solvent. Therefore, the molecule is at a higher temperature only for a period of time comparable to the IVR period ${\tau }_{\text{VC}}~{\tau }_{\text{IVR}}$. The temperature-induced contribution becomes large when the maximum temperature achieved, $T_{\text{loc,max}}$, is large. This can be seen from Eq. 9b, as the maximum possible enhancement of $k_{\mathrm{V}\mathrm{C}}$ is smaller than that of $k_{\mathrm{I}\mathrm{V}\mathrm{R}}$ by a factor of $e^{-\frac{V_B}{k_B{T}_{\text{out}}}\frac{T_{\text{out}}}{T_{\text{loc},\text{max}}}}$. This factor tends to 1 in the limit when the maximum achieved temperature is much larger than the barrier height $k_B{T}_{\text{loc,max}}\gg V_B$, i.e., the reaction effectively becomes ‘barrier-less’. This condition is easiest to achieve for reactions with lower activation barriers. Temperature-induced enhancements can also be large when ${\tau }_{\mathrm{V}\mathrm{C}}\gg {\tau }_{\mathrm{I}\mathrm{V}\mathrm{R}}$ as seen from Eq. 10b, because the molecule remains at a higher temperature for a longer period of time. To explore this further, for the calculations in Fig. 5, we use a longer vibrational cooling time of ${\tau }_{\mathrm{V}\mathrm{C}}=1\mathrm{n}\mathrm{s}$. Examples of systems with longer vibrational cooling times include $\mathrm{W}(\mathrm{C}\mathrm{O}{)}_6$ in ${\mathrm{C}\mathrm{C}\mathrm{l}}_4$ with ${\tau }_{\mathrm{V}\mathrm{C}}~700\mathrm{p}\mathrm{s}$ [62], and diatomic and triatomic molecules, such as HCl in ${\mathrm{C}\mathrm{C}\mathrm{l}}_4$ with ${\tau }_{\mathrm{V}\mathrm{C}}~4$ ns [63]. In Fig. 5, we plot $\mathrm{\Delta }k/k$ for different barrier heights $V_B$ when a low-frequency reaction coordinate ${\omega }_0/(2\pi c)=200{\mathrm{c}\mathrm{m}}^{-1}$ is driven with a pulsed laser and we take ${\tau }_{\text{IVR}}=1\mathrm{p}\mathrm{s}$. Here, the energy absorbed per molecule $E_{\text{abs}}=2.4\times {10}^{-20}\mathrm{J}$ is held fixed; this corresponds to a pulse energy $E_{\text{pulse}}=100\mu \mathrm{J}$ when the laser parameters and molar extinction coefficient are taken to be the same as in Fig. 2–4. For larger activation barriers, we find larger relative enhancements, consistent with Eq. 10. Furthermore, in agreement with the bounds in Eq. 10, the temperature-induced component dominates for lower barrier reactions, whereas the vibrationally-assisted part is more significant for higher-barriers cases.

FIG. 4.

Probability that a molecule reacts within ${\tau }_{\text{IVR}}=1\mathrm{p}\mathrm{s}$ given it has energy $E_{\text{abs}}$ pumped into it initially. Here, the vertical gray line indicates the activation energy $V_B=14\mathrm{k}\mathrm{c}\mathrm{a}\mathrm{l}{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$. This calculation was performed taking parameters with reaction coordinate ${\omega }_0/(2\pi c)=200{\mathrm{c}\mathrm{m}}^{-1}$ and ${\tau }_{\mathrm{V}\mathrm{C}}=1.7\mathrm{ps}$. For reference, ${10}^{-19}\mathrm{J}\approx 25\hslash {\omega }_0$. The pulse energy is related to the absorbed energy via $E_{\text{pulse}}=\frac{E_{\mathrm{a}\mathrm{b}\mathrm{s}}A{N}_A}{ϵ\mathrm{l}\mathrm{n}10}\approx E_{\text{abs}}\times 4\times {10}^{15}$, where the laser spot size is $A=\pi r^2=3.14\times {10}^{-8}{\mathrm{m}}^2$ with $r=100\mu \mathrm{m}$, and the molar extinction coefficient is $ϵ=200{\mathrm{L}\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}{\mathrm{c}\mathrm{m}}^{-1}$.

FIG. 5.

The relative change in the rate constant $\mathrm{\Delta }k/k$ (black line) for reactions with different activation barriers $V_B$ when excited with a pulsed laser with $t_{\text{rep}}=1\mathrm{m}\mathrm{s}$ and $E_{\text{pulse}}=100\mu \mathrm{J}$ where the corresponding energy absorbed per molecule $E_{\text{abs}}=2.4\times {10}^{-20}\mathrm{J}$. The reactions for all $V_B$ have the same rate constant $k_0\left(T_{\text{out}}\right)=4.0\times {10}^{-7}{\mathrm{s}}^{-1}$, low-frequency reaction coordinate ${\omega }_0/(2\pi c)=200{\mathrm{c}\mathrm{m}}^{-1}$, and ${\tau }_{\mathrm{I}\mathrm{V}\mathrm{R}}=1\mathrm{p}\mathrm{s}$. The vibrational cooling time was taken to be ${\tau }_{\mathrm{V}\mathrm{C}}=1\mathrm{n}\mathrm{s}$. The vibrationally-assisted (yellow) and temperature-induced (blue) components are indicated with dashed lines.

Eq. 9 and 10 are the main qualitative conclusions of our work. They are constraints that establish that higher laser powers (either through more energy per pulse or higher pulse repetition rates) are required to obtain vibrationally-assisted modifications to chemical reactions that feature low barriers. In the work of Heyne and co-workers [44] exploring the carbamate formation reaction between cyclohexanol and phenylisocyanate in tetrahydrofuran, the activation energy for the reaction was experimentally measured to be $V_B=6.7\pm 0.2\mathrm{k}\mathrm{c}\mathrm{a}\mathrm{l}.{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$, and the pulse repetition period was $t_{\text{rep}}=0.5\mathrm{m}\mathrm{s}$; this limits their enhancement over the IVR period to $k_{\text{IVR}}/k_0\left(T_{\text{out}}\right)\le e^{V_B/k_B{T}_{\text{out}}}\approx 8.2\times {10}^4$ at $T_{\text{out}}=298\mathrm{K}$. Taking ${\tau }_{\text{IVR}}=1\mathrm{p}\mathrm{s}$, we compute $\frac{{\tau }_{\text{IVR}}}{t_{\text{rep}}}\frac{k_{\text{IVR}}}{k_0\left(T_{\text{out}}\right)}=1.6\times {10}^{-4}$ for this reaction, which according to Eq. 10 clearly means there should be no substantial increase in the reaction rate under driving with a pulsed laser $k_{\text{avg,pulse}}\approx k_0\left(T_{\text{out}}\right)$. For future work, it is therefore important to understand the limitations of our model and whether inclusion of more sophisticated versions of chemical dynamics can explain the experimental report.

D. Probability of a molecule reacting

Assuming first-order reaction kinetics, the probability that a molecule reacts within ${\tau }_{\mathrm{IVR}}$ when it is excited with a pulsed laser is

$$P_{\text{react}}=1-e^{-{\tau }_{\mathrm{I}\mathrm{V}\mathrm{R}}{k}_{\mathrm{I}\mathrm{V}\mathrm{R}}}.$$ (11)

In the absence of driving, as the rate constant is $k_0=4\times {10}^{-7}{\mathrm{s}}^{-1}$, the probability a molecule reacts $P_{\text{react}}{\mathrm{s}}^{-1}$ equals $4\times {10}^{-19}$. We plot this probability in Fig. 4.

III. CONCLUSIONS

In summary, in this work we compute the reaction rate enhancement under driving vibrational modes with a laser. We compare the case of pulsed laser excitation with CW driving and demonstrate that pulsed laser excitation provides orders of magnitude more efficient enhancement; however, this relative enhancement is still small for typical experimental parameters. We provide a simple condition for rate enhancement under pulsed excitation, $\frac{t_{\text{rep}}}{{\tau }_{\text{IVR}}}\le \frac{k_{\text{IVR}}}{k_0\left(T_{\text{out}}\right)}\le e^{\frac{V_B}{k_B{T}_{\text{out}}}}$, which is valid when ${\tau }_{\text{VC}}\le {\tau }_{\text{IVR}}$ (in fact, the common situation is ${\tau }_{\text{VC}}~{\tau }_{\text{IVR}}$), and which offers a useful back-of-the-envelope estimate relating the pulse repetition period $t_{\text{rep}}$, IVR timescale ${\tau }_{\text{IVR}}$, and rate constants. These inequalities establish that reactions with low barriers demand higher laser powers to showcase significant rate enhancements. We a lso find the vibrationally-assisted component of the rate enhancement to be the dominant contribution for higher-barrier reactions and the temperature-induced contribution to be prominent for lower-barrier reactions when the vibrational cooling timescale ${\tau }_{\mathrm{V}\mathrm{C}}=1\mathrm{n}\mathrm{s}$ and ${\tau }_{\mathrm{I}\mathrm{V}\mathrm{R}}=1\mathrm{p}\mathrm{s}$. In future studies, we will aim to identify specific reactions where laser-driven rate enhancement is large.

CODE AVAILABILITY

The code accompanying this manuscript is available at https://github.com/SindhanaPS/Vibrationally-assisted_chemistry