Dissociation dynamics of a diatomic molecule in an optical cavity Subhadip Mondal1Derek S. Wang2and Srihari Keshavamurthy1 1Department of Chemistry Indian Institute of Technology Kanpur Uttar Pradesh 208 016 India

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Dissociation dynamics of a diatomic molecule in an optical cavity

Subhadip Mondal,1Derek S. Wang,2and Srihari Keshavamurthy1, ∗

1Department of Chemistry, Indian Institute of Technology, Kanpur, Uttar Pradesh 208 016, India

2Harvard John A. Paulson School of Engineering and Applied Sciences,

Harvard University, Cambridge, MA 02138, USA

(Dated: December 19, 2022)

We study the dissociation dynamics of a diatomic molecule, modeled as a Morse oscillator, cou-

pled to an optical cavity. A marked suppression of the dissociation probability, both classical and

quantum, is observed for cavity frequencies signiﬁcantly below the fundamental transition frequency

of the molecule. We show that the suppression in the probability is due to the nonlinearity of the

dipole function. The eﬀect can be rationalized entirely in terms of the structures in the classical

phase space of the model system.

I. INTRODUCTION

Recent experiments in polariton chemistry [1–6] sug-

gests that the quantum nature of light in the cavity

quantum electrodynamics (cQED) regime [7] may play

a crucial role in controlling chemistry. These experi-

ments show modiﬁed ground-state chemical reactivity of

molecules in cavities in the vibrational strong coupling

(VSC) regime by tuning the mode frequency of an optical

Fabry-Per´ot cavity. An important goal then is to over-

come what is believed to be the bane of mode-speciﬁc

chemistry — intramolecular vibrational energy redistri-

bution (IVR) [8–10]—by bringing the cavity mode fre-

quency into resonance with speciﬁc vibrational modes of

the reactant molecules. Indeed, several recent studies in

the context of VSC have emphasized the role of IVR. We

mention a few examples. Sch¨afer et al. have shown[11]

that the cavity mode can alter the cavity-free IVR path-

ways leading to the inhibition of a reaction. Chen et al.

have argued[12] that exciting polariton modes can lead

to an acceleration of IVR, whereas there is little change

in the dynamics upon exciting the dark modes. In the

collective regime, Wang et al. demonstrate[13] that un-

der suitable conditions there can be enhanced vibrational

energy ﬂow into the cavity mode and thus resulting in

slowing down of unimolecular reactions. The importance

of vibrational anharmonicity has been emphasized[14] by

Hern´andez and Herrera in terms of the formation of vi-

brational polaritons exhibiting a bond strengthening ef-

fect. Although the crucial role of IVR in VSC is being in-

creasingly appreciated, the mechanism by which the cav-

ity modulates the free molecule IVR pathways is not yet

clear. Therefore, given that the cavity mode corresponds

to a harmonic oscillator, one expects that our current

understanding[15–18] of IVR in isolated molecules will

be relevant in the context of polariton chemistry as well.

A ﬁrm theoretical understanding of VSC, particularly

in the experimentally relevant limit of a large number

of molecules in the cavity, is still far from established.

Nevertheless, several theoretical studies have provided

∗srihari@iitk.ac.in

insights into the possible mechanisms by which the reac-

tion rates may get inﬂuenced in the VSC regime [11, 19–

25]. It is now well understood that the transition state

theory (TST) without dynamical corrections is not capa-

ble of explaining the experimental observations [26–29].

However, a comparison of the TS recrossings in terms

of the dynamical correction factor κ(transmission co-

eﬃcient) in the presence (κc) and absence (κ0) of the

cavity indicates κc< κ0. Thus, reaction rates are typ-

ically reduced upon tuning the cavity mode frequency,

while some experiments demonstrate rate acceleration.

Given the form of the Pauli-Fierz Hamiltonian (see be-

low) and that the cavity mode is a simple harmonic oscil-

lator, conventional multidimensional reaction rate theo-

ries can also be brought to bear on the issue [20]. Indeed,

tools and concepts based on gas phase TST to the con-

densed phase rate theories of Grote-Hynes and Kramers

have been invoked [20, 30–32]. Studies utilizing mod-

els based on the quantized Jaynes-Cummings [33] and

Tavis-Cummings model [34–36] that treat molecules as

harmonic oscillators, classical molecular dynamics simu-

lations [37–39] and rigorous ab initio path integral stud-

ies [40] have also been performed to uncover the potential

mechanisms. Nevertheless, despite the large number of

studies, the theoretical results are still inconclusive; we

refer the reader to the recent reviews [6, 41–45] for a sum-

mary of the progress till now. Note that even experimen-

tally there are concerns about the correct interpretation

of the observed eﬀects [46].

A promising approach for further study is quantum dy-

namics simulations of cavity-molecule systems that fully

describe the anharmonic nature of molecular vibrations.

While these single-molecule models do not capture the

complexity of collective coupling in VSC experiments,

similar models have proven invaluable for understanding

molecule-light interactions and can shed light on polari-

ton chemistry. For instance, in 1977, Miller introduced

the Hamiltonian for a single cavity mode interacting with

a diatomic represented by a Morse oscillator to provide

a consistent semiclassical description for absorption, in-

duced emission, and spontaneous emission processes [47].

The diatomic molecule has a single vibrational degree of

freedom and hence issues associated with IVR within the

arXiv:2210.00470v2 [physics.chem-ph] 16 Dec 2022

molecule do not arise, allowing one to focus solely on the

inﬂuence of the molecule-cavity energy ﬂow dynamics on

the reactivity. Along these lines, the Morse oscillator-

cavity model provided valuable classical and semiclas-

sical insights into the excitation and dissociation in di-

atomic systems [48–50], although these studies do not

include the crucial dipole self energy (DSE) term. Re-

cently, Fischer and Saalfrank performed a detailed quan-

tum study of the Morse oscillator-cavity system using

the Pauli-Fierz Hamiltonian with the DSE included [51].

They conclude that, despite the formation of vibrational

polaritons, there is no substantial change in the disso-

ciation energies and bond lengths for coupling strengths

even beyond the VSC threshold. However, dissociation

dynamics of the diatomic molecule was not studied.

Therefore, in the present work we study a simple model

system consisting of a single diatomic molecule, mod-

eled as a Morse oscillator, coupled to a harmonic cav-

ity mode. The aim is to explore whether the disso-

ciation dynamics is inﬂuenced by coupling to the cav-

ity. Both classical and quantum dynamical studies are

done in the VSC regime to assess the relevance of quan-

tum eﬀects. Our results demonstrate signiﬁcant cavity

mode frequency-dependent suppression of the dissocia-

tion probability. Notably, maximal suppression occurs

when the cavity mode frequency is tuned not to the fun-

damental transition frequency of the diatomic molecule,

but rather to far red-shifted frequencies. Interestingly,

both the classical and the quantum dissociation proba-

bilities exhibit this modulation around the same cavity

mode frequency range At these red-shifted frequencies,

certain key nonlinear resonances in the classical phase

space of the molecule disappear. Such resonances, in-

volving the molecular vibration and the cavity mode,

are responsible for energy exchange between the molecule

and the cavity i.e., molecule-cavity IVR. Our analysis re-

veals that the nonlinearity of the molecular dipole func-

tion plays a crucial role. In fact, within the linearized

dipole approximation the dissociation dynamics shows

very little modulation over a wide range of the cavity

frequencies.

The paper is organized as follows. In Sec. II we give

details on the model Hamiltonian along with the rele-

vant parameters utilized in this work. The classical and

quantum dissociation probabilities in the VSC regime are

compared in Sec. III A, illustrating the essential role of

the dipole function. A classical phase space based under-

standing of the results is presented in Sec. III B, followed

by an analysis of the results in Sec. III C and conclusions

and future directions in Sec. IV.

II. MODEL HAMILTONIAN

Our model system corresponds to a diatomic molecule

coupled to a quantized electromagnetic ﬁeld mode of a

Fabry-P´erot cavity. According to Fischer and Saalfrank

[51], the Pauli-Fierz Hamiltonian within the dipole ap-

TABLE I. Parameters for the HF molecule [52]

Symbol Description Value (in au)

αMorse parameter of HF Bond 1.174

DDissociation energy 0.225

qeEquilibrium bond length 1.7329

mReduced mass 1744.59

ADipole moment parameter 0.4541

BDipole moment parameter 0.0064

µ0Dipole moment derivative 0.33

proximation and in the length gauge can be expressed as

H=HM+HC+HMC

HM(q, p) = 1

2mp2+D1−e−α(q−qe)2(1a)

HC(qc, pc) = 1

2p2

c+ω2

cq2

c(1b)

HM C (q, qc) = ωcλcqcµ(q) + 1

2λ2

cµ2(q) (1c)

where Q= (q, qc) and P= (p, pc) are the dynamical

position and conjugate momentum variables. We have

assumed a single cavity mode, described by the har-

monic Hamiltonian HCwith frequency ωc, that is po-

larized along the molecular axis. In addition, we take

an ideal cavity with no loss. The molecular Hamilto-

nian is denoted by HMand the vibration of the diatomic

molecule is modeled by a Morse oscillator with qeand

Ddenoting the equilibrium bond length and the disso-

ciation energy respectively. The ﬁrst term of HMC is

the molecule-cavity coupling (Hint), characterized by a

coupling strength λc≡(0rV)−1/2with 0, r, and V

being the vacuum dielectric constant, dielectric permit-

tivity,and cavity volume respectively. In this work, we set

r= 1. The second term in HM C is the dipole self en-

ergy (DSE). We include the DSE in all our computations,

classical and quantum, since several studies have estab-

lished its importance for a proper analysis of the coupled

cavity-molecule dynamics [51, 53]. Note that the Hamil-

tonian H(Q,P) is a two degrees-of-freedom autonomous

system that conserves the total cavity-molecule energy.

Following the earlier work we also deﬁne the parame-

ters

g=r~ωc

2λc; ΩR=2g√N|dfi|

~(2)

with g(V/m in SI units) being a measure of the cavity-

molecule interaction strength. The Rabi frequency ΩR

for Nmolecules (N= 1 in this work) in the cavity is ex-

pressed in terms of the transition dipole moment dfi be-

tween the initial (i) and ﬁnal (f) vibrational states of the

molecule. The dimensionless parameter η≡ΩR/2ωc≈

µ0(qe)λc(4mω0ωc)−1/2(within the linearized dipole ap-

proximation) determines the speciﬁc coupling regime we

are in; by convention η≥0.1 marks the transition from

(a)

(c)

(b)

0.3

0.4

0.5

ωc , cm-1

2000 3000 4000

QM 2300 cm-1

CM 2300 cm-1

QM 3966 cm-1

CM 3966 cm-1

0.3

0.4

0.5

λc , au

0.1 0.2 0.3 0.4 0.5 0.6

λc , au

0.2

0.3

0.4

0.5

0.6

ωc, cm-1

2000 2500 3000 3500 4000

FIG. 1. (a) Classical and quantum dissociation probabilities PDas a function of the cavity frequency ωcand the molecule-

cavity coupling strength λc. The shaded regions indicate the variation of the dissociation probabilities for λc∈(0.1,0.6) au

incremented in steps of 0.05. (b) The values (ωc, λc) for which maximum suppression in PDoccurs. (c) Classical and quantum

PDversus λcfor two example cavity frequencies ωc= 2300 cm−1and ωc=ω01 ≈3966 cm−1.

the vibrational strong coupling (VSC) to the vibrational

ultrastrong coupling (VUSC) regime. In this work we use

the following functional form [54] of the dipole moment

function

µ(q) = Aqe−Bq4(3)

For future reference we provide the linear approximation

to the dipole function:

µ(q) = µ(qe) + dµ(q)

dq qe

(q−qe) (4)

The parameters for the diatomic molecule, taken from

an earlier work [52] by Brown and Wyatt, are chosen to

represent the hydrogen ﬂuoride (HF) molecule and are

given in Table I.

III. RESULTS AND DISCUSSION

A. Dissociation probability: classical and quantum

In order to study the dissociation dynamics of the di-

atomic molecule, the initial state is chosen to be a po-

lariton wavepacket [51]

Ψ0(q, qc;t= 0) = ψG(q;q0)φ0(qc) (5)

with

ψG(q;q0) = 1

πσ21/4

exp −(q−q0)2

2σ2(6)

being a displaced ground state wavefunction of the har-

monized HF bond of frequency ω0≈4139 cm−1and

width σ≡(~/mω0)1/2. We consider the cavity to have

no photons initially and hence the state φ0(qc) is taken

to be the ground harmonic eigenstate of the cavity mode.

The center of the wavepacket q0is then chosen such that

hHiΨ0≡ hΨ0|H|Ψ0icorresponds to the desired total en-

ergy. Here we ﬁx hHiΨ0= 0.25 au which is above the

dissociation energy of the HF bond. Note that quali-

tatively similar results are obtained for other values of

hHiΨ0as well (see Fig. 2 and Fig. 3 below).

The time evolved quantum state Ψ(q, qc;t) is obtained

by numerically solving (see Appendix) the Schr¨odinger

equation and the quantum dissociation probability is

then calculated as

PQM

D(t) = 1 − hΨ(q, qc, t)|Ψ(q, qc, t)i(7)

The corresponding classical dissociation probabilities

PCM

D(t) are computed (see Appendix) by choosing an en-

semble of initial conditions Ntot sampled from the classi-

cal density ρcl(q, p, qc, pc,0) corresponding to the initial

quantum polaritonic wavepacket. For this study, we take

Ntot = 50000 and time evolve each initial phase space

point by integrating the Hamiltonian’s equations of mo-

tion. A trajectory is considered to be dissociated when

the displacement (q−qe)≥7.5 au. The classical disso-

ciation probability is calculated as

PCM

D(t) = Ndiss(t)

Ntot

(8)

with Ndiss being the number of dissociated trajectories.

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DissociationdynamicsofadiatomicmoleculeinanopticalcavitySubhadipMondal,1DerekS.Wang,2andSrihariKeshavamurthy1,1DepartmentofChemistry,IndianInstituteofTechnology,Kanpur,UttarPradesh208016,India2HarvardJohnA.PaulsonSchoolofEngineeringandAppliedSciences,HarvardUniversity,Cambridge,MA02138,USA(Dated:De...

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Dissociation dynamics of a diatomic molecule in an optical cavity Subhadip Mondal1Derek S. Wang2and Srihari Keshavamurthy1 1Department of Chemistry Indian Institute of Technology Kanpur Uttar Pradesh 208 016 India

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