Quantum Dynamics of Vibrational Polariton Chemistry Lachlan P. Lindoy1Arkajit Mandal1and David R. Reichman1 1Department of Chemistry Columbia University

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Quantum Dynamics of Vibrational Polariton Chemistry

Lachlan P. Lindoy,1Arkajit Mandal,1and David R. Reichman1, ∗

1Department of Chemistry, Columbia University,

3000 Broadway, New York, New York, 10027, U.S.A

We employ an exact quantum mechanical simulation technique to investigate a model of cavity-

modiﬁed chemical reactions in the condensed phase. The model contains the coupling of the reaction

coordinate to a generic solvent, cavity coupling to either the reaction coordinate or a non-reactive

mode, and the coupling of the cavity to lossy modes. Thus, many of the most important features

needed for realistic modeling of the cavity modiﬁcation of chemical reactions are included. We ﬁnd

that when a molecule is coupled to an optical cavity it is essential to treat the problem quantum

mechanically in order to obtain a quantitative account of alterations to reactivity. We ﬁnd sizable

and sharp changes in the rate constant that are associated with quantum mechanical state splittings

and resonances. The features that emerge from our simulations are closer to those observed in

experiments than are previous calculations, even for realistically small values of coupling and cavity

loss. This work highlights the importance of a fully quantum treatment of vibrational polariton

chemistry.

I. INTRODUCTION

A series of recent experiments [1–8] have suggested

that when molecular vibrations are coupled to the radi-

ation modes inside an optical cavity, ground state chem-

ical kinetics can be both enhanced [1, 2, 8] or sup-

pressed [3, 6, 8]. Such eﬀects are purported to operate in

the absence of external optical pumping [9] and have been

hypothesized to arise from the hybridization of molecular

vibrational states and the photon (Fock) states of a cavity

radiation mode [1, 2]. The interpretation of these exper-

iments is still a matter of debate, and thus the viability

of marked alterations in chemical reactivity remains an

open topic. Indeed, while the spectroscopic ﬁngerprints

of light-matter hybridization such as the Rabi-splitting

observed in the IR spectra are manifest, the signiﬁcance

of the coupling to radiation modes for markedly changing

chemical reactivity is unclear.

Theoretical studies that describe both the radiation

modes as well as the molecular vibrations using classi-

cal mechanics have had limited success in describing cur-

rently available experiments [10–18]. Speciﬁcally, previ-

ous work [10–12] using the Grote-Hynes (GH) rate the-

ory [19–21], applicable in the limit of strong molecule-

bath interactions, do show cavity frequency-dependent

chemical kinetics modiﬁcation. However these stud-

ies [10–12] predict that the chemical reaction rate is sup-

pressed most strongly when the cavity frequency is near

to the barrier frequency as opposed to molecular vibra-

tional frequencies as seen in experiments, and that the

rate proﬁle is only weakly modiﬁed in an extremely broad

manner with respect to the cavity frequency ωc, span-

ning thousands of wavenumbers (∼5000 cm−1). This

is in stark contrast to experimental observations, where

the width of the changes in the rate proﬁle are on the

order of ∼100 cm−1[1–6]. On the other hand, a recent

∗drr2103@columbia.edu

study [13] using the Pollak-Grabert-H¨anggi (PGH) the-

ory [22], as well as direct trajectory-based computational

work [16, 23], have predicted enhancement of chemical

rates when the molecule-bath coupling is relatively weak.

Interestingly, these studies have predicted a signiﬁcantly

sharper rate proﬁle than that which emerges at strong

coupling, have demonstrated that the eﬀect is more siz-

able, and have revealed that the chemical rate is most

strongly enhanced when the cavity frequency is close to

the reactant vibrational frequency [13, 16, 23]. While

these studies move theory closer to laboratory observa-

tions, there is still a substantial gulf between experiments

and our theoretical understanding.

A major missing component in the theoretical work

discussed above is the quantum nature of problem. Thus,

a direct account of even the formation of Rabi-split po-

laritonic states is omitted. Simple quantum corrections

to the GH theory, such as found using quantum transition

state theory [24] or zero-point energy corrections to the

energy barrier [25] have been carried out, but these ap-

proximate calculations diverge from experimental expec-

tations even more than their fully classical counterparts,

showing, for example, an even broader range of alteration

of the rate proﬁle than that seen in classical calculations.

Recent fully quantum dynamical studies which ignore the

explicit interactions of the molecule with the solvent de-

grees of freedom also do not ﬁnd a resonant structure in

the cavity frequency dependence of chemical rate [25, 26].

Taken as a whole, these studies point to the clear pressing

need to perform exact quantum calculations on models

that include the relevant molecular, solvent, and cavity

degrees of freedom.

In this work, we use a customized version of the hi-

erarchical equations of motion (HEOM) approach [27]

(see Methods for further details) to exactly simulate cav-

ity modiﬁed chemical kinetics of a single molecule cou-

pled to a radiation mode as well as dissipative molec-

ular and solvent modes. The need to modify how the

HEOM calculations are carried out is crucial for obtain-

ing converged exact quantum dynamical rates. As in

arXiv:2210.05550v1 [quant-ph] 11 Oct 2022

some recent studies [12, 23], we also include the cou-

pling of the cavity mode to a bath which mimics cavity

loss [28]. We show that coupling molecular vibrations

to a cavity radiation mode can both enhance or suppress

chemical reactivity, with the largest eﬀect occurring when

the cavity mode is near resonant with speciﬁc molecu-

lar vibrational modes. Crucially, we ﬁnd that the cav-

ity frequency-dependent rate shows a much sharper pro-

ﬁle (∼100 cm−1) compared to what is predicted from

classical rate theories [10, 13] or with quantum correc-

tions [17, 25], even when compared to recent theoretical

results in the classical weak-coupling regime [16, 23]. Our

results also demonstrate that the details of the solvent-

molecule interactions are extremely important, and that

even realistically small rates of cavity loss can play a

crucial role in enabling cavity modiﬁcation of chemical

reactivity.

Speciﬁcally, we ﬁnd that the extent and the nature (en-

hancement or suppression) of the cavity modiﬁcation to

the reaction rate depends sensitively on (a) the details of

the molecular system, such as the potential energy sur-

face and the vibrational eigenspectrum, (b) the details of

the solvent, as encoded in the spectral density and solvent

friction and (c) the details of the cavity (e.g. the light-

matter coupling strength and the rate of cavity loss) and

how it couples to matter. Our results reveal that the cav-

ity modiﬁcation of chemical rates can largely be rational-

ized by considering how molecular vibrational states are

altered by hybridization with cavity photon states (form-

ing so-called vibrational polaritons) to eﬀectively increase

or decrease the interaction of the molecule with its envi-

ronment. The resonant structure in the cavity-modiﬁed

reaction rate naturally arises from the hybridization of

light and matter, which occurs most strongly when the

cavity and matter states are in resonance. These eﬀects

emerge from fundamental quantum light-matter interac-

tions and cannot be fully captured with simple classical

or semiclassical descriptions of light and matter.

In essentially all previous experimental work [1–8] a

large ensemble of molecules (∼1010 molecule per cav-

ity mode [29, 30]) are collectively coupled to the cavity

radiation modes. In contrast, most theoretical work, in-

cluding the calculations we present here, operate in the

single molecule limit and do not address collective ef-

fects in a direct manner. Despite some studies address-

ing collective polaritonic behavior [12, 31, 32], a detailed

theoretical explanation for such collective eﬀects remains

elusive. We would like to point out, however, that sin-

gle molecule studies are possible in principle. In this

regard, as will be discussed below, it is crucial to note

that our quantum calculations suggest that alterations

to reaction rates may be observed with cavity coupling

strengths that are orders of magnitude smaller than sug-

gested in other recent single molecule studies. This fact

highlights the potential feasibility of experimental obser-

vation of modiﬁcation of rates even in the single molecule

limit.

II. RESULTS

Theoretical Model. The model quantum electrody-

namics (QED) Hamiltonian used in this work is based

on the Pauli-Fierz (PF) light-matter Hamiltonian in the

dipole gauge in the single mode and long-wavelength lim-

its, and is written as [33–35]

H=ˆ

Hmol +ˆ

Hsolv +ˆ

Hcav +ˆ

Hloss,(1)

where ˆ

Hmol is the molecular Hamiltonian, ˆ

Hsolv describes

solvent as well as molecule-solvent interactions, ˆ

Hcav is

the cavity Hamiltonian describing a radiation mode and

its interaction to matter in the dipole gauge, and ˆ

Hloss

describes the cavity loss term.

In this work we consider a molecular Hamiltonian

Hmol =ˆ

TR+V(ˆ

R) that contains a one-dimensional re-

action coordinate R. The ground state potential en-

ergy surface along this reaction coordinate, V(ˆ

R) =

ω4

16Eb·ˆ

R4−1

2ω2

b·ˆ

R2−c·R3, takes the form of a double

well potential. In the main text we consider a barrier fre-

quency ωb= 1000 cm−1, barrier Eb= 2250 cm−1, and a

symmetric double potential with c= 0 (see Supplemen-

tary Note 2 for results with c6= 0), as shown in Fig. 1a

(black solid line). The molecular Hamiltonian ˆ

Hmol can

be equivalently represented using the vibrational states,

Hmol =X

Ei|viihvi|

≡¯

E0|vRihvR|+|vLihvL|+X

i≥2

Ei|viihvi|

+ ∆|vRihvL|+|vLihvR|,(2)

where {|vii} are the vibrational eigenstates of the molec-

ular Hamiltonian ( ˆ

Hmol|vii=Ei|vii). In the second line

we have introduced localized states |vLi=1

√2(|v0i+|v1i)

and |vRi=1

√2(|v0i−|v1i), with an energy ¯

E0=1

2(E0+

E1) and a coupling ∆ = 1

2(E1−E0). These states are

the localized ground states of the left and the right wells

(blue and red wavefunctions in Fig.1a), respectively. We

deﬁne the well frequency ω0=E2−¯

E0≈1140 cm−1.

Hsolv is taken as

Hsolv =P2

2+1

2ω2

QQ+CQR

ω2

Q2

2+1

2Ω2

jˆ

Xj+Cjˆ

Ω2

j2

ˆp2

2+1

2ω2

jˆxj+cjˆ

ω2

j2

.(3)

The ﬁrst line describes a spectator mode with coordinate

Q(or equivalently a collective solvent mode [12, 36, 37])

coupled to the reaction coordinate. The second line de-

scribes a set of dissipative solvent modes described by

(a)

𝐽!(𝜔)

𝐽"(𝜔)

(b)

Tunneling

Dominated

Regime

Energy-Diffusion

Limited Regime

Spatial-Diffusion

Limited Regime

Kramers

Turnover

(c)

Unstructured solvent

Structured solvent

(d)

Unstructured solvent

Structured solvent

𝐽"(𝜔)

𝐽!(𝜔)

(e)

(f)

𝐽 𝜔 (×10-7 a.u.)

𝐶!(×10-5 a.u.)

𝜂"/𝜔#(a.u.)

𝜅(x10-6 fs-1)

FIG. 1. Eﬀect of molecule-solvent coupling on chemical kinetics. (a) Potential energy surface and vibrational eigenstates of

model molecular system. (b) Eﬀective spectral density for unstructured (JU(ω), black dashed line) and structured (JS(ω), red solid line)

solvent environment as schematically depicted in (e) and (f), respectively. The peak at 1200 cm−1in JS(ω) arises due strong coupling

(CQ) between the molecule and a spectator mode Q. (c) Chemical rate constant κas a function of solvent friction ηswhen the molecule

is embedded in an unstructured environment, as illustrated in (e). Chemical rate constant κas a function of molecule-spectator mode

coupling CQwhen the molecule is embedded in an structured environment as illustrated in (f)

a broad spectral density JU(Ω) = π

2Pj

Ωjδ(Ω −Ωj) =

2ΛsΩΓ/(Ω2+ Γ2) = ηsΩΓ2/(Ω2+ Γ2) (black dashed line

in Fig. 1a). The third line describes a set of secondary

solvent modes ˆ

Xjthat couple to the spectator mode co-

ordinate Qand are also described with a broad spectral

density Ju(ω) = π

2Pj

ωjδ(ω−ωj)=2λsωγ/(ω2+γ2).

The cavity Hamiltonian ˆ

Hcav describing a single cavity

mode and its coupling to matter is given by

Hcav =ˆp2

2+1

2ω2

cˆqc+r2

ωc

ηc·ˆµ2

.(4)

Here, ωcis the cavity photon frequency and ηc=

ωcq~ωc

20Vˆeis the light-matter coupling strength with vac-

uum permittivity 0, quantization volume Vand the di-

rection of polarization ˆe. Further, ˆpc=iq~ωc

2(ˆa†−ˆa)

and ˆqc=q~

2ωc(ˆa†+ ˆa), where ˆa†and ˆaare the photon

creation and annihilation operators,and ˆµis the matter

dipole operator.

Finally ˆ

Hloss describes the bath that is coupled to the

cavity mode which enables cavity loss,

Hloss =X

Π2

2+1

2˜ω2

kˆ

Qk+Ckˆqc

˜ω2

k2

,(5)

where ˜ωkand Ckwhich control cavity leakage are de-

scribed via the spectral density JL(ω) = π

2PkC2

˜ωkδ(ω−

˜ωk) = 2λLωγL/(ω2+γ2

L). With this spectral den-

sity, the cavity loss rate is deﬁned as Γc= 1/τc=

2J(ωc)/(1 −e−βωc) where τcis the cavity lifetime and

β= 1/kBTwith the Boltzmann constant kBand tem-

perature T. Physically, our model Hamiltonian ﬂexibly

contains nearly all essential ingredients assumed to in-

ﬂuence chemical reactions in a cavity. Further details of

the model parameters are provided in the Supplementary

Note 1.

Chemical Kinetics Outside a Cavity. The chem-

ical kinetics of a molecular system in the absence of the

cavity (setting ηc= 0) embedded in an unstructured sol-

vent (setting CQ= 0) depends on the molecule-solvent

Cavity

Loss

Cavity

Radiation

Molecule

Solvent

(a)

!!"#$ %&

"#$ '&

photon

matter

no coupling

(d)

Quantum

Classical

(c)(b)

()*+ (Arb. Units.)

1.8

1.6

1.4

1.2

1.0

𝜅(𝜔,)/𝜅-

(e)

."/ .#

Absorption

Rate

Constant

FIG. 2. Resonant cavity modiﬁcation of ground state chemical kinetics. (a) Schematic illustration of a molecule coupled to a

lossy cavity radiation mode as well as to other solvent molecules. (b) Potential energy surfaces for a molecular ground adiabatic state with

0 photons |G, 0i(black solid line) and with 1 photon (blue solid line) |G, 1i, as well as the corresponding vibrational eigenstates of |G, 0i

and |G, 1i(horizontal sold lines). (c) Vibrational polariton eigenspectrum as a function of cavity photon frequency ωc. (d) Absorption

spectrum (violet shaded area) showing a polaritonic Rabi-splitting of ∼30 cm−1and the cavity modiﬁed rate constant κ(normalized with

the rate constant κ0outside of the cavity) with a cavity lifetime τc= 100 fs (blue solid line). The linewidth of the absorption spectrum

and chemical rate constant are similar in magnitude. (e) Comparing the cavity modiﬁed chemical rate constant computed using exact

quantum (blue solid line) and classical (green solid line) dynamical simulations, showing the failure of classical description in quantitatively

capturing the eﬀects of quantum light-matter interactions. Here, the light-matter coupling is ηc= 0.00125 and solvent friction ηs= 0.1ωb.

interaction strength. Such a setup is schematically il-

lustrated in Fig. 1e and the corresponding bath spectral

density JU(ω) is presented in Fig. 1b (black dashed line).

The chemical rate constant κ(see Methods), obtained

from exact quantum dynamics simulation using a spe-

cialized HEOM approach (see Methods) as a function

of ηsis presented in Fig. 1b, and shows three distinct

regimes. For very low ηsthe chemical kinetics is domi-

nated by direct nuclear tunneling, that is the transition

|vLi → |vRivia the tunnel-coupling ∆ in Eq. 2. We

refer to this regime as the tunneling-dominated regime.

In this regime, an increase in the molecule-solvent inter-

action (by increasing ηs) leads to a sharp decline in the

reaction rate [38] as the bath (solvent) degrees of freedom

eﬀectively renormalize and lower ∆.

While the increase in ηsreduces the direct nuclear tun-

neling, an alternate reaction pathway involving thermal

excitations, that is |vLi → {|vii} → |vRi, starts to play

an increasingly important role. For ηs>0.1ωb, the later

pathway becomes the dominant one. For 1.8ωb> ηs>

0.1ωb, the overall reaction rate is limited by the equilibra-

tion rate of the vibrational states in the left well. In this

regime, akin to the energy diﬀusion-limited regime in the

Kramers turnover problem [20] [39], the overall reaction

rate increases with increasing ηsas can be seen in Fig. 1c.

Finally, further increases in ηsdrives the system into the

spatial diﬀusion-limited regime where the reaction rate

decreases with increasing solvent friction ηs. In classical

rate theory this transition from energy diﬀusion-limited

regime to the spatial diﬀusion-limited regime is referred

to as the Kramers turnover [20].

For a molecular system embedded in a structured sol-

vent outside of the cavity (ηc= 0 in ˆ

H), the chemical rate

as a function of the molecule-spectator mode coupling

CQis presented in Fig. 1d. The corresponding molec-

ular system is also schematically illustrated in Fig. 1f.

The eﬀective bath spectral density [36, 37] JS(ω) =

2Pj

˜ωjδ(ω−˜ωj), for the eﬀective (and equivalent) sol-

vent Hamiltonian ˆ

Hsolv ≡Pj

˜p2

2+1

2˜ω2

j(˜xj+˜

Cj˜

Q/˜ω2

j)2,

is presented in Fig. 1b (red solid line). In comparison to

the unstructured environment, the spectral density JS(ω)

for the structured environment shows a sharp spike at

1200 cm−1, which originates from the spectator mode Q

with frequency ωQ= 1200 cm−1, and the width of this

peak originates from the secondary solvent spectral den-

sity Ju(ω). This is characteristic of complex molecular

environment [40–42] where the spectral density contains

numerous spikes. It is worth noting that the reaction

coordinate is strongly coupled to other orthogonal vi-

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QuantumDynamicsofVibrationalPolaritonChemistryLachlanP.Lindoy,1ArkajitMandal,1andDavidR.Reichman1,1DepartmentofChemistry,ColumbiaUniversity,3000Broadway,NewYork,NewYork,10027,U.S.AWeemployanexactquantummechanicalsimulationtechniquetoinvestigateamodelofcavity-modiedchemicalreactionsinthecondensedph...

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Quantum Dynamics of Vibrational Polariton Chemistry Lachlan P. Lindoy1Arkajit Mandal1and David R. Reichman1 1Department of Chemistry Columbia University

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