# Can Dark States Explain Vibropolaritonic Chemistry?

Matthew Du, Joel Yuen-Zhou<EMAIL_ADDRESS>Department of Chemistry and
Biochemistry, University of California San Diego, La Jolla, California 92093,
United States

###### Abstract

Collective strong coupling between a disordered ensemble of $N$ localized
molecular vibrations and a resonant optical cavity mode gives rise to 2
polariton and $N-1\gg 2$ dark modes. Thus, experimental changes in thermally-
activated reaction kinetics due to polariton formation appear entropically
unlikely and remain a puzzle. Here we show that the overlooked dark modes,
while parked at the same energy as bare molecular vibrations, are robustly
delocalized across $\sim$2-3 molecules, yielding enhanced channels of
vibrational cooling, concomitantly catalyzing or suppressing a chemical
reaction. As an illustration, we theoretically show a 55% increase in an
electron transfer rate due to enhanced product stabilization.

The past decade has seen much interest in the control of chemical phenomena
via the strong coupling of matter to confined electromagnetic modes [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. An exciting prospect in
this direction is vibropolaritonic chemistry, that is, the use of collective
vibrational strong coupling (VSC) [18, 19, 20] to modify thermally-activated
chemical reactivity without external pumping (e.g., laser excitation) [21].
While collective VSC involves a large number of molecules ($N\approx
10^{6}-10^{12}$) per photon mode, it has been observed to substantially alter
the kinetics of organic substitution [22, 23], cycloaddition [24], hydrolysis
[25], enzyme catalysis [26, 27], and crystallization [28], among other
electronic ground-state chemical processes.

However, such modified reactivity under VSC is still not well understood.
Studies [29, 30, 31, 32] show that the observed kinetics cannot be explained
with transition state theory (TST) [33], the most commonly used framework to
predict and interpret reaction rates. Breakdowns of TST, including recrossing
the activation barrier [34], deviation from thermal equilibrium [31, 35], and
quantum/nonadiabatic phenomena [36, 37, 31, 35, 38], have also been
considered. The aforementioned works regard all $N$ vibrations coupled to the
cavity mode to be identical. Under this assumption, VSC forms two polaritons
and $N-1$ optically dark vibrational modes, where the latter remain unchanged
from the cavity-free system. It follows that VSC-induced changes to thermally-
activated reactivity must arise from the polaritons. In fact, one study from
our group highlighted the molecular parameter space where polaritons dominate
the kinetics [36] with respect to dark modes; however, this hypothesis has
been questioned as entropically unlikely [39].

Disorder, despite its ubiquity in molecular systems, has often been ignored
when modeling molecules under strong light-matter coupling. Only recently has
it been shown that the strong coupling of disordered chromophores to an
optical cavity mode can produce dark states which are delocalized on multiple
molecules [40, 41] (hereafter, referred as semilocalized). This
semilocalization is predicted to improve or even enable coherent energy
transport [40, 42]. Other findings hint at adding sample impurities to help
strong coupling modify local molecular properties [43].

In this Letter, we demonstrate that the VSC of a disordered molecular ensemble
(Fig. 1) can significantly modify the kinetics of a thermally-activated
chemical reaction. The altered reactivity is attributed to the semilocalized
dark modes. The semilocalization affects the reaction rate by changing the
efficiency with which a reactive mode dissipates energy.

Figure 1:  Schematic of the model setup. Inside an optical cavity, a cavity
mode collectively interacts with an energetically disordered ensemble of
molecular vibrations, each belonging to a separate molecule. When a molecule
reacts, its cavity-coupled vibrational mode concomitantly experiences a
displacement in equilibrium geometry. Depicted here are molecules that undergo
intramolecular electron transfer, the reaction studied in this work.

Consider a disordered ensemble of $N$ molecular vibrations, respectively
corresponding to $N$ independent molecules, inside an optical cavity (Fig. 1).
The system is described by the Hamiltonian
$H=\hbar\omega_{c}a_{0}^{\dagger}a_{0}+\hbar\sum_{i=1}^{N}(\overline{\omega}_{v}+\delta\omega_{i})a_{i}^{\dagger}a_{i}+\hbar\sum_{i=1}^{N}g_{i}(a_{i}^{\dagger}a_{0}+\text{h.c.})$.
Vibrational mode $i$ is represented by annihilation operator $a_{i}$ and has
frequency $\overline{\omega}_{v}+\delta\omega_{i}$, where
$\overline{\omega}_{v}$ is the mean vibrational frequency and
$\delta\omega_{i}$ is the frequency offset of mode $i$. Reflecting
inhomogeneous broadening (static diagonal disorder), $\delta\omega_{i}$ is a
normally distributed random variable with mean zero and standard deviation
$\sigma_{v}$. The cavity mode is represented by annihilation operator $a_{0}$,
has frequency $\omega_{c}$, and couples to vibration $i$ with strength
$g_{i}$. For simplicity, we hereafter take $g_{i}=g$ for all $i$.

Using $H$, we investigate the physicochemical properties of a disordered
molecular system under VSC. Unless otherwise noted, calculations assume that
the cavity is resonant with the average vibration,
$\omega_{c}=\overline{\omega}_{v}$, and couples to the vibrations with
collective strength $g\sqrt{N}=8\sigma_{v}$. Numerical values reported below
are obtained by averaging over 5000 disorder realizations, _i.e._ , sets
$\\{\omega_{i}\\}_{i=1}^{N}$. In plots versus the $H$ eigenfrequencies, each
data point is an average over the $H$ eigenmodes—from all disorder
realizations—whose frequency lies in the bin
$\left(\Delta\omega(l-\frac{1}{2}),\Delta\omega(l+\frac{1}{2})\right]$ for
$\Delta\omega=10^{-1}\sigma_{v}$ and $l\in\mathbb{Z}$.

We first study the eigenmodes of $H$. Formally, mode $q=1,\dots,N+1$ is
represented by operator $\alpha_{q}=\sum_{i=0}^{N}c_{qi}a_{i}$ and has
frequency $\omega_{q}$. Figs. 2(a) and 2(b) show the probability distribution
and photon fraction ($|c_{q0}|^{2}$), respectively, of the eigenmodes with
respect to eigenfrequency. The majority of modes form a broad distribution in
frequency around $\overline{\omega}_{v}$ and are optically dark. A minority of
modes are polaritons, which have frequency $\overline{\omega}_{v}\pm
8\sigma_{v}$ and photon fraction $\approx 0.5$. As $N$ rises, the eigenmodes
become increasingly composed of dark modes, whose probability distribution
approaches that of the bare vibrational modes [Fig. 2(a), gray dashed line].

Figure 2: (a) Probability distribution and (b) photon fraction of the
eigenmodes of $H$. Values are plotted versus eigenfrequency $\omega_{q}$ for
various $N$. In (a), the gray dashed line is the probability density function
[$P(\omega_{q})=(\sigma_{v}\sqrt{2\pi})^{-1}\exp\left(-(\omega_{q}-\overline{\omega}_{v})^{2}/(2\sigma_{v}^{2})\right)$,
displayed in units of $1/\Delta\omega$] of the bare vibrational modes.

Next, we examine the delocalization of the dark modes. For the purpose of
studying chemical reactions, it is useful to compute the molecular
participation ratio (PR) [44, 41]. This measure, defined as

$\text{molecular
PR}=1/\sum_{i=1}^{N}\left|\frac{c_{qi}}{\sqrt{\sum_{i=1}^{N}|c_{qi}|^{2}}}\right|^{4}$
(1)

and analogous to the usual PR [45], estimates the number of molecules over
which eigenmode $q$ is delocalized. According to Fig. 3(a), the average dark
mode has molecular PR $\sim$2-3, and this semilocalization persists as $N$
increases. These phenomena were first noted independently by Scholes [41] and
Schachenmayer and co-workers [40]. For additional insight, we plot the squared
overlap [$|c_{qi}|^{2}$, Fig. 3(b)] and frequency difference [$|\omega_{qi}|$,
where $\omega_{qi}=\omega_{q}-\omega_{i}$; Fig. 3(c)] between each dark mode
and each bare vibrational mode. The typical dark mode has sizable overlap with
the bare modes that are nearest to it in frequency. The frequency difference
between the dark mode and any of its major constituents is $O(\sigma_{v}/N)$,
_i.e._ , negligible for large $N$.

Figure 3: (a) Average molecular PR of the dark modes as a function of $N$. (b)
Squared overlap $|c_{qi}|^{2}$ (log10 scale) and (c) frequency difference
$|\omega_{qi}|$ between each eigenmode $q$ and each bare vibrational mode $i$.
Each group of modes is ordered from low to high frequency, i.e., dark
(polariton) modes have index $q=2,\dots,N$ ($q=1,N+1$). The quantities in
(b)-(c) are plotted for various $N$.

We now explore how VSC influences the kinetics of a thermally-activated
chemical reaction. Consider a reactive molecule under collective VSC (Fig. 1).
The molecule undergoes nonadiabatic intramolecular electron transfer, and a
cavity mode interacts collectively with a reactive vibrational mode and $N-1$
nonreactive vibrational modes. The model here considered is general enough
that it should also be applicable to the case of “solvent-assisted VSC” [46,
47, 48, 49].

To model the reaction of one molecule in the ensemble, we employ the
Hamiltonian $H_{\text{rxn}}=H+\sum_{X=R,P}|X\rangle\langle
X|\\{E_{X}+\hbar\omega_{r}[\lambda_{X}(a_{r}+a_{r}^{\dagger})+\lambda_{X}^{2}]\\}+V+H_{s}^{(l)}$.
Note that vibrational and cavity modes are still described by $H$. In writing
$H_{\text{rxn}}$, we have changed the numerical index of the bare vibrational
mode ($i=1$) involved in the reaction to the letter $r$ (i.e.,
$\omega_{1}\rightarrow\omega_{r}$, $a_{1}\rightarrow a_{r}$); hereafter, we
refer to this mode as $v_{r}$. The electronic subspace consists of reactant
$|R\rangle$ and product $|P\rangle$ states. Electronic state $|X\rangle$ has
energy $E_{X}$ and couples to $v_{r}$ with dimensionless strength
$\lambda_{X}$. As a result of the vibronic coupling, $v_{r}$ experiences a
displacement in its equilibrium position upon electron transfer. The
interaction between $|R\rangle$ and $|P\rangle$ is represented by
$V=J_{RP}(|P\rangle\langle R|+\text{h.c.})$, where $J_{RP}$ is the interaction
strength. Through $H_{s}^{(l)}$ [50], Hamiltonian $H_{\text{rxn}}$ also
accounts for low-frequency vibrational modes of the solvent that help mediate
electron transfer.

Since we are considering a nonadiabatic reaction, we treat $V$ perturbatively
and calculate rates of reactive transitions between the zeroth-order
electronic-vibrational-cavity eigenstates of $H_{\text{rxn}}$. These states
take the form $|X,\chi\rangle=|X\rangle\otimes|\tilde{\chi}_{(X)}\rangle$.
Belonging to the subspace of vibrational and cavity modes,
$|\tilde{\chi}_{(X)}\rangle=\left(\prod_{q=1}^{N+1}D_{q}^{\dagger}(\lambda_{Xq})\right)|\chi\rangle$
is a displaced Fock state with $m_{q}^{(\chi)}$ excitations in $H$ eigenmode
$q$. The undisplaced Fock state $|\chi\rangle$ is an eigenstate of $H$, and
$D_{q}(\lambda)=\exp(\lambda\alpha_{q}^{\dagger}-\lambda^{*}\alpha_{q})$ is a
displacement operator. Mode $q$ has equilibrium (dimensionless) position
$\lambda_{Xq}=\lambda_{X}c_{qr}(\omega_{r}/\omega_{q})$ when the system is in
electronic state $|X\rangle$. Returning to the electronic-vibrational-cavity
state $|X,\chi\rangle$, we can write its energy as
$E_{(X,\chi)}=E_{X}+\sum_{q=1}^{N+1}m_{q}^{(\chi)}\hbar\omega_{q}+\Delta_{X}$,
where
$\Delta_{X}=\lambda_{X}^{2}\hbar\omega_{r}-\hbar\sum_{q=1}^{N+1}|\lambda_{Xq}|^{2}\omega_{q}$
is the difference in reorganization energy—namely, that due to vibronic
coupling between $v_{r}$ and $|X\rangle$—with and without VSC.

Following extensions [36, 37, 35] of Marcus-Levich-Jortner theory [51, 52, 53]
to electron transfer under VSC, the rate of the reactive transition from
$|R,\chi\rangle$ to $|P,\chi^{\prime}\rangle$ can be expressed as

$\displaystyle k_{(R,\chi)\rightarrow(P,\chi^{\prime})}$
$\displaystyle=\mathcal{A}F_{\chi,\chi^{\prime}}\exp\left(-\beta
E_{a}^{\chi,\chi^{\prime}}\right),$ (2)

where $\mathcal{A}=\sqrt{\pi\beta/\lambda_{s}}|J_{RP}|^{2}/\hbar$,
$\lambda_{s}$ is the reorganization energy associated with low-frequency
solvent modes [50], and $\beta$ is the inverse temperature For this
transition, the activation energy is
$E_{a}^{\chi,\chi^{\prime}}=(E_{(P,\chi^{\prime})}-E_{(R,\chi)}+\lambda_{s})^{2}/(4\lambda_{s})$.
Through the Franck-Condon (FC) factor
$F_{\chi,\chi^{\prime}}=|\langle\tilde{\chi}_{(R)}|\tilde{\chi}^{\prime}_{(P)}\rangle|^{2}$,
the transition rate depends on the overlap between initial and final
vibrational-cavity states $|\tilde{\chi}_{(R)}\rangle$ and
$|\tilde{\chi}^{\prime}_{(P)}\rangle$, respectively. It can be shown that the
rate $k_{(P,\chi^{\prime})\rightarrow(R,\chi)}$ corresponding to the backward
transition, $|P,\chi^{\prime}\rangle\rightarrow|R,\chi\rangle$, is related to
Eq. (2) by detailed balance [54].

We specifically study a reaction where, in the absence of light-matter
coupling, reactive transitions occur on the same timescale as internal
thermalization (i.e., thermalization of states having the same electronic
component). In such cases, internal thermal equilibrium is not maintained
throughout the reaction, and the reaction rate (_i.e._ , net rate of reactant
depletion) may not be approximated by a thermal average of reactant-to-product
transition rates. Instead, the reaction rate can also depend on, _e.g._ ,
backward reactive transitions (from product to reactant) or vibrational
relaxation.

With this in mind, we numerically simulate the bare ($g=0$) and VSC reactions
using a kinetic model [50, 35], which includes forward and backward reactive
transitions [Eq. (2)], vibrational and cavity decay, and energy exchange among
dark and polariton states [55, 56, 57]. The reaction parameters [50], which
include $\overline{\omega}_{v}=2000$ cm-1 and $\sigma_{v}=10$ cm-1, are such
that the population dynamics proceeds almost completely through states
$|X,\chi\rangle$ with zero or one excitation in the vibrational-cavity modes.
To reduce computational cost, the kinetic model includes only these states,
which are denoted by $\chi=0$ and $\chi=1_{q}$, respectively, where $q$ is an
eigenmode of the vibrational-cavity subspace. The kinetic master equation [50]
is numerically solved with the initial population being a thermal distribution
of reactant states ($|R,\chi\rangle$). Then, the _apparent_ reaction rate is
obtained by fitting the reactant population as a function of time [50].

Even though we run the full numerical simulations as explained above, we now
introduce approximate models that shed conceptual intuition on the calculated
kinetics. First, the bare reaction can be essentially captured by Fig. 4(a),
described as follows. Starting from its vibrational ground state, the reactant
converts to product mainly by a $0\rightarrow 1$ vibronic transition, which
excites the reactive mode and has rate $k_{f}\equiv
k_{(R,0)\rightarrow(P,1_{r})}$, where

$\displaystyle k_{f}$ $\displaystyle=\mathcal{A}F_{0,1_{r}}\exp\left(-\beta
E_{a}\right),$ (3)

$E_{a}\equiv E_{a}^{0,1_{r}}$. The vibrationally hot product either reverts to
the reactant at rate $k_{b}\equiv k_{(P,1_{r})\rightarrow(R,0)}\gg k_{f}$,
where

$\displaystyle k_{b}$
$\displaystyle=k_{f}\exp\left[\beta(E_{P}+\hbar\omega_{r}-E_{R})\right],$ (4)

or decays to its vibrational ground state at rate $\gamma\approx k_{b}$. Once
the product reaches its vibrational ground state, it effectively stops
reacting due to the high reverse activation energy. This kinetic scheme leads
to a bare reaction rate [50]

$k_{\text{bare}}^{(\text{analytical})}=k_{f}\left(\frac{\gamma}{\gamma+k_{b}}\right).$
(5)

Second, under VSC, the primary reaction pathway of the bare case is split into
multiple pathways, each involving the (de)excitation of a dark or polariton
eigenmode $q$. For the VSC reaction channels, the forward and backward rates
take the form

$\displaystyle k_{f}^{(q)}$
$\displaystyle=\mathcal{A}F_{0,1_{q}}\exp\left(-\beta E_{a}^{(q)}\right),$ (6)
$\displaystyle k_{b}^{(q)}$
$\displaystyle=k_{f}^{(q)}\exp\left\\{\beta\left[E_{P}+\hbar\omega_{q}+\Delta_{P}-(E_{R}+\Delta_{R})\right]\right\\},$
(7)

respectively, where $k_{f}^{(q)}\equiv k_{(R,0)\rightarrow(P,1_{q})}$,
$k_{r}^{(q)}\equiv k_{(P,1_{q})\rightarrow(R,0)}$, and $E_{a}^{(q)}\equiv
E_{a}^{0,1_{q}}$. Now, consider the following argument, which holds strictly
for large $N$. As $N$ increases, the average bare mode becomes localized on
dark modes that have essentially the same frequency as it [Figs. 3(b)-3(c)],
and its overlap with the polariton modes vanishes [_i.e._ ,
$c_{qr}\propto\frac{1}{\sqrt{N}}\to 0$ for $q=1,N+1$; see Fig. 3(b)]. These
observations suggest that $c_{qr}\not\approx 0$ only for modes $q$ that are
dark and have frequency $\omega_{q}\approx\omega_{r}$. It is then
straightforward to show that $\left.E_{a}^{(q)}\right|_{c_{qr}\not\approx
0}\approx E_{a}$, $F_{0,1_{q}}\approx|c_{qr}|^{2}F_{0,1_{r}}$, and

$\displaystyle k_{f/b}^{(q)}$ $\displaystyle\approx|c_{qr}|^{2}k_{f/b}.$ (8)

Thus, VSC leads to reaction channels that have lower rates of reactive
transitions, due to changes not in activation energies but in FC factors,
which are smaller as a result of the semilocalization of dark modes. From Eq.
(8), it is evident that the total forward rate is approximately that of the
bare reaction ($\sum_{q=1}^{N+1}k_{f}^{(q)}\approx k_{f}$, since
$\sum_{q=1}^{N+1}|c_{qr}|^{2}=1$). However, once a forward reactive transition
happens—and a dark mode is excited—the product either returns to the reactant
at a reduced rate ($k_{b}^{(q)}<k_{b}$) or, due to the almost fully
vibrational nature of the dark modes, vibrationally decays to its stable form
($|P,0\rangle$) at essentially the same bare rate ($\gamma$). In other words,
VSC suppresses reverse reactive transitions by promoting the cooling of the
reactive mode upon product formation. In analogy to Eq. (5), we determine an
effective rate for the VSC reaction [50]:

$\displaystyle k_{\text{VSC}}^{(\text{analytical})}$
$\displaystyle=k_{f}\left\langle\frac{\gamma}{\gamma+|c_{qr}|^{2}k_{b}}\right\rangle_{\text{dark
modes }q},$ (9)

where $\langle\cdot\rangle_{\text{dark modes }q}$ is a weighted average over
all dark modes $q$, each with weight $|c_{qr}|^{2}$. Since $|c_{qr}|^{2}<1$
for all $q$, then
$k_{\text{VSC}}^{(\text{analytical)}}>k_{\text{bare}}^{(\text{analytical})}$.
We emphasize that the major contributions to the average in Eq. (9) come from
dark modes which are closest in frequency to the bare reactive mode (see
above). Further enhancement of the VSC reaction, beyond that given by
$k_{\text{VSC}}^{(\text{analytical})}$, occurs via dissipative scattering from
these dark modes to those with $c_{qr}\approx 0$. Said differently, the
product is protected from reversion to reactant when dark modes with
relatively more reactive character lose their energy to those with relatively
less. The VSC reaction kinetics, as described above, is summarized in Fig.
4(b).

Fig. 4(e) shows the ratio of VSC reaction rate to bare reaction rate, as
determined from numerical kinetic simulations. As we have shown analytically,
VSC significantly accelerates the reaction compared to the bare case. As $N$
increases, the rate enhancement [Fig. 4(e), purple filled circles] converges
to approximately 55%; hence, despite our computational simulations capping at
$N=2^{8}$, we expect this behavior and the subsequent analysis to hold for the
experimentally relevant $N=10^{6}-10^{12}$. Interestingly, the rate
enhancement follows the same $N$-dependence as the reactive-mode
delocalization, which is shown in Fig. 4(d) and defined as
$1/\sum_{q=1}^{N+1}|c_{qr}|^{4}$ (the PR of $v_{r}$ when the mode is expressed
in the eigenbasis of $H$). This correlation supports that, for many-molecule
VSC, the reaction proceeds through the dark modes and is enhanced by their
semilocalization. The present scenario is quite generic and contrasts with our
previous model where extreme geometric parameters are needed for polaritons to
dominate the VSC kinetics [36]. Indeed, the reaction here is unaffected by
cavity leakage when $N$ is sufficiently large [50].

Figure 4: (a) Schematic of the bare reaction kinetics. Parabolas represent
potential energy surfaces with respect to the effective low-frequency
coordinate $x_{s}$. After a reactive transition
($|R,0\rangle\rightarrow|P,1_{r}\rangle$) with rate $k_{f}$, the product
either reverts to the reactant ($|P,1_{r}\rangle\rightarrow|R,0\rangle$) at
rate $k_{b}$ or vibrationally decays to its stable form
($|P,1_{r}\rangle\rightarrow|P,0\rangle$) at rate $\gamma\approx k_{b}$. (b)
Schematic of the reaction kinetics under VSC. The half-yellow half-gray
parabolas qualitatively represent potential energy surfaces of product states
with one excitation in a dark mode ($|P,1_{q}\rangle$). The reaction proceeds
via multiple reaction channels, each involving a reactive transition
($|R,0\rangle\rightarrow|P,1_{q}\rangle$) with rate $k_{f}^{(q)}<k_{f}$ to the
product with one excitation in a dark mode. The total forward rate is
approximately the bare rate $k_{f}$. In contrast, the vibrationally hot
product formed from each reaction channel either returns to the reactant
($|P,1_{q}\rangle\rightarrow|R,0\rangle$) at rate $k_{b}^{(q)}<k_{b}$ or cools
($|P,1_{q}\rangle\rightarrow|P,0\rangle$) at the _bare_ molecular rate
$\gamma$. There is also scattering (at effective rate $\zeta$) from dark modes
with $c_{qr}\not\approx 0$ to those with $c_{qr}\approx 0$. Overall, VSC
accelerates product thermalization, suppressing backward reactive transitions,
and thus enhancing the net reaction rate. (c) $v_{r}$ delocalization and (d)
$k_{\text{VSC}}/k_{\text{bare}}$ as a function of $N$. (e) Activation enthalpy
$\Delta H^{\ddagger}$ versus activation entropy $\Delta S^{\ddagger}$ for
reactions with $N=2^{8}$: bare (gray circle), VSC (purple circle), and bare
with vibrational decay rate $\gamma$ made 100 times faster (purple diamond).
The black dashed line is a fit to the points shown. In (d) and (e), the
individual rates ($k_{\text{VSC}}$, $k_{\text{bare}}$) and thermodynamic
parameters ($\Delta H^{\ddagger}$, $\Delta S^{\ddagger}$) are averages over
5000 disorder realizations.

For additional mechanistic insight into VSC catalysis and following the
procedures in [22, 46, 24], we plot in Fig. 4(f) the activation enthalpy
($\Delta H^{\ddagger}$) versus activation entropy ($\Delta S^{\ddagger}$) for
multiple cases of VSC and bare reactions. The thermodynamic parameters of
activation are computed by calculating the apparent reaction rate for
additional temperatures and fitting the obtained values to the Eyring-Polanyi
equation [50]. This fit indicates that changes in effective parameters $\Delta
H^{\ddagger}$ and $\Delta S^{\ddagger}$ can result from dynamical effects such
as accelerated vibrational decay, rather than from potential energy changes.

The deduced connection between reactivity under VSC and reactive-mode
delocalization is corroborated by their dependence on cavity detuning
$\delta=\omega_{c}-\overline{\omega}_{v}$ (Fig. 5). We find, for various
light-matter coupling strengths, that the reactive-mode delocalization is
maximum close to resonance and eventually decreases with detuning. The rate
enhancement due to VSC mostly follows the same trend. Any major deviation from
this trend is attributed to the polaritons and expected to vanish if $N$ is
sufficiently increased [50], since extreme molecular geometries are not
considered [36].

Figure 5: (a) $k_{\text{VSC}}/k_{\text{bare}}$ and (b) $v_{r}$ delocalization,
as a function of cavity detuning $\delta$, for various collective light-matter
coupling strengths $g\sqrt{N}$ and fixed $N=2^{8}$. In (a), $k_{\text{VSC}}$
is an average over 5000 disorder realizations, and $k_{\text{bare}}$ is a
value used to obtain Fig. 4(d).

In conclusion, we show that, by forming semilocalized dark modes, the VSC of a
disordered molecular ensemble can modify the kinetics of a thermally-activated
chemical reaction. For a reactive molecule under collective VSC, we find that
the electron transfer rate is significantly increased. The spreading of
reactive character across dark modes, as well as the dissipative scattering
among these modes, allows the reactive mode to thermalize more efficiently
once the product is formed, suppressing dynamical effects, such as reversion
to reactant. Finally, while we study a nonadiabatic reaction, our main
conclusions are general and should be relevant for the understanding of
adiabatic reactions under VSC (work in progress). More broadly, our work
highlights that the previously overlooked dark states are the entropically
likely channels through which collective light-matter interaction can control
chemistry.

###### Acknowledgements.

We are grateful to Jorge Campos-Gonzalez-Angulo, Arghadip Koner, Luis
Martínez-Martínez, Kai Schwennicke, Stephan van den Wildenberg, and Garret
Wiesehan for useful discussions. This work employed computational resources of
the Extreme Science and Engineering Discovery Environment (XSEDE), which is
supported by National Science Foundation Grant No. ACI-1548562, under
allocation No. TG-ASC150024. We also thank Marty Kandes and Nicole Wolter for
assistance in using these resources. Acknowledgment is made to the donors of
The American Chemical Society Petroleum Research Fund for partial support of
this research through the ACS PRF 60968-ND6 award. M.D. is also supported by a
UCSD Roger Tsien Fellowship.

## References

  * Agranovich  _et al._ [2011] V. M. Agranovich, Y. N. Gartstein, and M. Litinskaya, Chem. Rev. 111, 5179 (2011).
  * Coles  _et al._ [2014] D. M. Coles, N. Somaschi, P. Michetti, C. Clark, P. G. Lagoudakis, P. G. Savvidis, and D. G. Lidzey, Nat. Mater. 13, 712 (2014).
  * Ebbesen [2016] T. W. Ebbesen, Acc. Chem. Res. 49, 2403 (2016).
  * Kowalewski  _et al._ [2016] M. Kowalewski, K. Bennett, and S. Mukamel, J. Phys. Chem. Lett. 7, 2050 (2016).
  * Chikkaraddy  _et al._ [2016] R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, Nature 535, 127 (2016).
  * Feist  _et al._ [2017] J. Feist, J. Galego, and F. J. Garcia-Vidal, ACS Photonics 5, 205 (2017).
  * Barnes  _et al._ [2018] B. Barnes, F. García Vidal, and J. Aizpurua, ACS Photonics 5, 1 (2018).
  * Herrera and Spano [2018] F. Herrera and F. C. Spano, ACS Photonics 5, 65 (2018).
  * Baranov  _et al._ [2018] D. G. Baranov, M. Wersäll, J. Cuadra, T. J. Antosiewicz, and T. Shegai, ACS Photonics 5, 24 (2018).
  * Ribeiro  _et al._ [2018] R. F. Ribeiro, L. A. Martínez-Martínez, M. Du, J. Campos-Gonzalez-Angulo, and J. Yuen-Zhou, Chem. Sci. 9, 6325 (2018).
  * Flick  _et al._ [2018] J. Flick, N. Rivera, and P. Narang, Nanophotonics 7, 1479 (2018).
  * Hertzog  _et al._ [2019] M. Hertzog, M. Wang, J. Mony, and K. Börjesson, Chem. Soc. Rev. 48, 937 (2019).
  * Herrera and Owrutsky [2020] F. Herrera and J. Owrutsky, J. Chem. Phys. 152, 100902 (2020).
  * Coccia  _et al._ [2020] E. Coccia, J. Fregoni, C. A. Guido, M. Marsili, S. Pipolo, and S. Corni, J. Chem. Phys. 153, 200901 (2020).
  * Hoffmann  _et al._ [2020] N. M. Hoffmann, L. Lacombe, A. Rubio, and N. T. Maitra, J. Chem. Phys. 153, 104103 (2020).
  * Antoniou  _et al._ [2020] P. Antoniou, F. Suchanek, J. F. Varner, and J. J. Foley, J. Phys. Chem. Lett. 11, 9063 (2020).
  * Haugland  _et al._ [2020] T. S. Haugland, E. Ronca, E. F. Kjønstad, A. Rubio, and H. Koch, Phys. Rev. X 10, 041043 (2020).
  * Shalabney  _et al._ [2015] A. Shalabney, J. George, J. Hutchison, G. Pupillo, C. Genet, and T. W. Ebbesen, Nat. Commun. 6, 6 (2015).
  * Long and Simpkins [2015] J. P. Long and B. S. Simpkins, ACS Photonics 2, 130 (2015).
  * Casey and Sparks [2016] S. R. Casey and J. R. Sparks, J. Phys. Chem. C 120, 28138 (2016).
  * Hirai  _et al._ [2020a] K. Hirai, J. A. Hutchison, and H. Uji-i, ChemPlusChem 85, 1981 (2020a).
  * Thomas  _et al._ [2016] A. Thomas, J. George, A. Shalabney, M. Dryzhakov, S. J. Varma, J. Moran, T. Chervy, X. Zhong, E. Devaux, C. Genet, J. A. Hutchison, and T. W. Ebbesen, Angew. Chem., Int. Ed. 128, 11634 (2016).
  * Thomas  _et al._ [2019] A. Thomas, L. Lethuillier-Karl, K. Nagarajan, R. M. A. Vergauwe, J. George, T. Chervy, A. Shalabney, E. Devaux, C. Genet, J. Moran, and T. W. Ebbesen, Science 363, 615 (2019).
  * Hirai  _et al._ [2020b] K. Hirai, R. Takeda, J. A. Hutchison, and H. Uji-i, Angew. Chem., Int. Ed. 132, 5370 (2020b).
  * Hiura  _et al._ [2019] H. Hiura, A. Shalabney, and J. George, ChemRxiv (2019).
  * Vergauwe  _et al._ [2019] R. M. A. Vergauwe, A. Thomas, K. Nagarajan, A. Shalabney, J. George, T. Chervy, M. Seidel, E. Devaux, V. Torbeev, and T. W. Ebbesen, Angew. Chem., Int. Ed. 58, 15324 (2019).
  * Lather and George [2021] J. Lather and J. George, J. Phys. Chem. Lett. 12, 379 (2021).
  * Hirai  _et al._ [2020c] K. Hirai, H. Ishikawa, J. Hutchison, and H. Uji-i, ChemRxiv (2020c).
  * Galego  _et al._ [2019] J. Galego, C. Climent, F. J. Garcia-Vidal, and J. Feist, Phys. Rev. X 9, 021057 (2019).
  * Campos-Gonzalez-Angulo and Yuen-Zhou [2020] J. A. Campos-Gonzalez-Angulo and J. Yuen-Zhou, J. Chem. Phys. 152, 161101 (2020).
  * Li  _et al._ [2020] T. E. Li, A. Nitzan, and J. E. Subotnik, J. Chem. Phys. 152, 234107 (2020).
  * Zhdanov [2020] V. P. Zhdanov, Chem. Phys. 535, 110767 (2020).
  * Hänggi  _et al._ [1990] P. Hänggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990).
  * Li  _et al._ [2021] X. Li, A. Mandal, and P. Huo, Nat. Commun. 12, 1315 (2021).
  * Du  _et al._ [2021] M. Du, J. A. Campos-Gonzalez-Angulo, and J. Yuen-Zhou, J. Chem. Phys. 154, 084108 (2021).
  * Campos-Gonzalez-Angulo  _et al._ [2019] J. A. Campos-Gonzalez-Angulo, R. F. Ribeiro, and J. Yuen-Zhou, Nat. Commun. 10, 4685 (2019).
  * Phuc  _et al._ [2020] N. T. Phuc, P. Q. Trung, and A. Ishizaki, Sci. Rep. 10, 7318 (2020).
  * Fischer and Saalfrank [2021] E. W. Fischer and P. Saalfrank, J. Chem. Phys. 154, 104311 (2021).
  * Vurgaftman  _et al._ [2020] I. Vurgaftman, B. S. Simpkins, A. D. Dunkelberger, and J. C. Owrutsky, J. Phys. Chem. Lett. 11, 3557 (2020).
  * Botzung  _et al._ [2020] T. Botzung, D. Hagenmüller, S. Schütz, J. Dubail, G. Pupillo, and J. Schachenmayer, Phys. Rev. B 102, 144202 (2020).
  * Scholes [2020] G. D. Scholes, Proc. R. Soc. A 476, 20200278 (2020).
  * Chávez  _et al._ [2021] N. C. Chávez, F. Mattiotti, J. A. Méndez-Bermúdez, F. Borgonovi, and G. L. Celardo, Phys. Rev. Lett. 126, 153201 (2021).
  * Sidler  _et al._ [2021] D. Sidler, C. Schäfer, M. Ruggenthaler, and A. Rubio, J. Phys. Chem. Lett. 12, 508 (2021).
  * Galego  _et al._ [2017] J. Galego, F. J. Garcia-Vidal, and J. Feist, Phys. Rev. Lett. 119, 136001 (2017).
  * Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Rep. Prog. Phys. 56, 1469 (1993).
  * Lather  _et al._ [2019] J. Lather, P. Bhatt, A. Thomas, T. W. Ebbesen, and J. George, Angew. Chem., Int. Ed. 58, 10635 (2019).
  * Schütz  _et al._ [2020] S. Schütz, J. Schachenmayer, D. Hagenmüller, G. K. Brennen, T. Volz, V. Sandoghdar, T. W. Ebbesen, C. Genes, and G. Pupillo, Phys. Rev. Lett. 124, 113602 (2020).
  * Szidarovszky  _et al._ [2020] T. Szidarovszky, G. J. Halász, and A. Vibók, New J. Phys. 22, 053001 (2020).
  * Davidsson and Kowalewski [2020] E. Davidsson and M. Kowalewski, J. Phys. Chem. A 124, 4672 (2020).
  * [50] See Supplemental Material at [URL will be inserted by publisher] for Hamiltonian $H_{s}^{(l)}$, kinetic model, parameters and numerical methods for calculating reaction rates, procedure to determine thermodynamic parameters of activation, derivations of analytical reaction rates, and supplemental figures.
  * Marcus [1964] R. A. Marcus, Annu. Rev. Phys. Chem. 15, 155 (1964).
  * Levich [1966] V. G. Levich, Adv. Electrochem. Electrochem. Eng 4, 249 (1966).
  * Jortner [1976] J. Jortner, J. Chem. Phys. 64, 4860 (1976).
  * Kühn and Naundorf [2003] O. Kühn and H. Naundorf, Phys. Chem. Chem. Phys. 5, 79 (2003).
  * del Pino  _et al._ [2015] J. del Pino, J. Feist, and F. J. Garcia-Vidal, New J. Phys. 17, 053040 (2015).
  * Xiang  _et al._ [2018] B. Xiang, R. F. Ribeiro, A. D. Dunkelberger, J. Wang, Y. Li, B. S. Simpkins, J. C. Owrutsky, J. Yuen-Zhou, and W. Xiong, Proc. Natl. Acad. Sci. U. S. A. 115, 4845 (2018).
  * Xiang  _et al._ [2019] B. Xiang, R. F. Ribeiro, L. Chen, J. Wang, M. Du, J. Yuen-Zhou, and W. Xiong, J. Phys. Chem. A 123, 5918 (2019).