How Quantum is the Resonance Behavior in Vibrational Polariton Chemistry?

Recent experiments in polariton chemistry have demonstrated that reaction rates can be modified by vibrational strong coupling to an optical cavity mode. Importantly, this modification occurs only when the frequency of the cavity mode is tuned to closely match a molecular vibrational frequency. This sharp resonance behavior has proved to be difficult to capture theoretically. Only recently did Lindoy et al. [Nat. Commun.2023, 14, 273337173299] report the first instance of a sharp resonant effect in the cavity-modified rate simulated in a model system using exact quantum dynamics. We investigate the same model system with a different method, ring-polymer molecular dynamics (RPMD), which captures quantum statistics but treats dynamics classically. We find that RPMD does not reproduce this sharp resonant feature at the well frequency, and we discuss the implications of this finding for future studies of vibrational polariton chemistry.


Computational details 1.RPMD rate theory
The RPMD approximation to the rate constant, k, is defined via the flux-side correlation function: where N is the number of ring-polymer beads, β = 1/k B T is the inverse temperature, β N = β/N , f is the number of physical degrees of freedom, θ(•) is the Heaviside step function, and s(q) defines the dividing surface; in this work, we use a dividing surface based on the centroid of the ring polymer, i q i /N .Note that q(t) in Eq.S1 is obtained by classically evolving q forward in time according to Hamilton's equations of motion generated by the N -bead ring-polymer Hamiltonian, The flux through the dividing surface is given by Using this, we can calculate the rate constant by converging the expression of the rate: In practice, the t → ∞ limit means that one needs to run the simulation for long enough such that the right-hand side of Eq.S4 reaches a plateau, i.e. does not change considerably when further increasing the length of the simulation.Finally, Q r and Q p are the reactant and product partition functions, defined by For the symmetric systems we study in this work, The RPMD rate constant is obtained in the N → ∞ limit, whereas for N = 1, the above equations reduce to classical rate theory.

Explicit and implicit baths
In our system we have two harmonic baths, which are characterized by their spectral densities.Within RPMD, we face a choice as to how to treat the bath degrees of freedom.

Explicit bath
The most straightforward option is to discretize the bath into a finite set of harmonic oscillators.One then converges the rate constant not only with the number of ring-polymer beads, but also with the number of bath modes.This however comes with a number of disadvantages, one being that a large number of modes are needed, which increases the computational cost considerably.Additionally, including more bath modes also implies that the difference in time scale between the slowest and fastest modes grows, which in turn results in the need for shorter time steps (for the high-frequency modes) and longer simulation times (for the low-frequency modes).In this work, we used 500 modes for each bath with the discretization scheme described in Ref. 2. To ensure stable integration of high-frequency modes, we used a Cayley integration algorithm in the normal modes of all harmonic degrees of freedom, as described in Appendix A of Ref. 3. The dividing surface was chosen such that it is perpendicular to the unstable mode at the transition state.
Implicit bath It is possible to analytically integrate out the bath degrees of freedom within the ring-polymer framework.This procedure is outlined in the Appendix of Ref. 4. Static effects of the bath are captured by an effective ring-polymer Hamiltonian in the phase space of the system only, with the bath having the effect of making the ring-polymer springs stiffer.Dynamic friction effects are captured by a Generalized Langevin Equation (GLE).
For an Ohmic bath, we can evaluate the GLE friction kernel analytically.However, as integrating the GLE is still very expensive, we choose to approximate the friction kernel by its Markovian short-and long-time limits.If simulations in both of these limits give identical results, which is usually the case, we can infer that the dynamical friction on the internal modes of the ring polymer does not play an important role, and thereby avoid having to integrate the GLE.
For describing a Debye bath, we can avoid evaluation of the friction kernel altogether by noticing that: 1) by coupling the system to a harmonic oscillator, and this harmonic oscillator in turn to an Ohmic bath, we have effectively coupled the system to a Brownian oscillator; and 2) in the limit of an overdamped oscillator, the Brownian oscillator spectral density reduces to a Debye spectral density. 5Hence, by tuning the frequency of the added harmonic oscillator and the Ohmic friction (which we already know how to treat implicitly), we can mimic the behavior of a specific Debye bath.This approach is illustrated in Figure S1.We found that for describing the Debye bath coupled to the molecular coordinate J R (ω), a harmonic oscillator with Ω = 2000 cm −1 coupled to an Ohmic bath with γ = 20 000 cm −1 is sufficient.For the Debye bath coupled to the cavity mode J L (ω), we need to go to γ = 100 000 cm −1 , turning the harmonic oscillator's frequency up to Ω = 10 000 cm −1 .Due to this large frequency, we would require a much smaller time step, making the simulations more expensive.As the calculations in the low-friction regime are already very demanding, we opted for replacing the Debye bath entirely with its Ohmic equivalent (via Eq. ( 6) of the main text).Results supporting this replacement are presented in Section 2.
For the implicit-bath calculations, we choose the dividing surface to be in the double-well coordinate R only, i.e. s(q) = i R i /N .We evaluated C fs (t) using the Bennett-Chandler approach. 6,7ypes of bath used for the figures in the main text In this study, we used both of the approaches outlined above, as each performs well in a different regime.
For the Kramers curve for system I (Figure 1c of the main text), the implicit-bath approach is more computationally efficient in the low-friction regime.We therefore used this method for all calculations shown in Figure 2.For the higher friction values in Figure 1c, approximating the friction kernel by its two limits gives slightly different results; importantly, when taking the long-time limit of the friction kernel, Eq.S4 does not plateau, so the rate constant becomes ill-defined with this method.Fortunately, in this regime the explicitbath approach becomes cheaper, so we calculated the Kramers curve in Figure 1c using this method.Note that both approaches agree for the lower friction values in this figure, where we deem our approximation to the friction kernel to be valid.
For the Kramers curve for system II (Figure 3b), we found that our approximation to the friction kernel remains valid for all frictions shown, the implicit-and explicit-bath methods give the same curve.For the cavity-modified rates in Figure 3c, we employed the implicitbath approach, as it has one fewer convergence parameters (the number of bath modes).
These details are summarized in Table 1.Additionally, the most important parameters used for converging the transmission coefficient are given in Table 2.As noted in the main text, we replaced the Debye spectral density for the cavity mode used for the HEOM calculations 8 with an Ohmic spectral density, to make the computational costs more tractable.In the following, we justify this by showing that even for the strongest coupling between the cavity mode and its bath, this replacement does not change our results within the error bars.This is shown in Figure S2 for τ c = 100 fs, η s = 0.1ω b and η = 0.00125 for system I (c.f. Figure 2 of the main text).Both classically and within RPMD, the results for a Debye bath agree well with those for an Ohmic bath, although due to the high computational cost, fewer trajectories were run such that the error bars on the RPMD results with the Debye bath remain rather large.

Figure S1 :
Figure S1: Schematic diagram of the relation between a Debye bath, a Brownian bath, and an extra system harmonic oscillator coupled to an Ohmic bath.Here it is illustrated for the cavity mode and its bath, but the same procedure holds for the matter part of the system.For further details, see e.g.Refs. 4 and 5.

Figure S2 :
Figure S2: Comparison of the cavity-induced rate modification for the cavity coupled to an Ohmic bath vs. a Debye bath.