Cavity Molecular Dynamics Simulations of Liquid Water Under Vibrational Ultrastrong Coupling

Cavity molecular dynamics simulations of liquid water under vibrational ultrastrong coupling

Tao E. Lia , Joseph E. Subotnika, and Abraham Nitzana,b,1

aDepartment of Chemistry, University of Pennsylvania, Philadelphia, PA 19104; and bSchool of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel

Contributed by Abraham Nitzan, June 11, 2020 (sent for review May 11, 2020; reviewed by Angel Rubio and George C. Schatz)

We simulate vibrational strong coupling (VSC) and vibrational The first step toward proving the above hypothesis is to ascer- ultrastrong coupling (V-USC) for liquid water with classical molec- tain whether or not any dynamical property of molecules is ular dynamics simulations. When the cavity modes are resonantly actually changed for a realistic experiment, a goal that forms coupled to the O−H stretch mode of liquid water, the infrared the central objective of this manuscript. In order to investi- spectrum shows asymmetric Rabi splitting. The lower polari- gate whether such modification occurs, below we will model ton (LP) may be suppressed or enhanced relative to the upper VSC and V-USC using cavity molecular dynamics (MD) sim- polariton (UP) depending on the frequency of the cavity mode. ulation, where the nuclei are evolved under a realistic elec- Moreover, although the static properties and the translational dif- tronic ground-state potential surface. Such an approach is an fusion of water are not changed under VSC or V-USC, we do find extension of the usual simplified 1D models where the mat- the modification of the orientational autocorrelation function of ter side is evolved as two-level systems (24–26) or coupled H2O molecules especially under V-USC, which could play a role in harmonic oscillators (16, 17, 27, 28). Although such simplified ground-state chemistry. models are adequate enough for studying Rabi splitting qualitatively by fitting experimental parameters, these models usually vibrational strong coupling | ultrastrong coupling | molecular dynamics | ignore translation, rotation, and collision, as well as the intri- liquid water cate structure of molecular motion, all of which are crucial for determining the dynamic properties of molecules. There- fore, explicit cavity MD simulations become a more appropriate trong light-matter interactions between a vibrational mode of approach for studying all dynamic properties. Moreover, even Smolecules and a cavity mode have attracted great attention of though one can find a Rabi splitting from 1D models, perform- late (1). The signature of strong interactions is the formation of ing cavity MD simulations is also very helpful for providing lower polariton (LP) and upper polariton (UP), which are mani- more details about the IR spectrum, and this approach can be fested in the Rabi splitting of a vibrational peak in the molecular used to benchmark the validity of 1D models under various infrared (IR) spectrum. According to the normalized ratio (η) conditions. between the Rabi splitting frequency (ΩN ) and the original There have been a few flavors of cavity MD schemes for vibrational frequency (ω0), or η = ΩN /2ω0, one often classifies electronic strong coupling (29–31). For example, Luk et al. 0 < η < 0.1 as vibrational strong coupling (VSC) and η > 0.1 as (30) applied multiscale quantum mechanics/molecular mechan- vibrational ultrastrong coupling (V-USC) (2). The investigation ics simulation for studying the dynamics of electronic polaritons of VSC or V-USC in liquid phase was initially suggested by for Rhodamine molecules. By contrast, MD simulations for Ebbesen and coworkers (3–5), and it was later found experimen- VSC and V-USC, to our best knowledge, have not been exten- tally that VSC or V-USC can modify the ground-state chemical sively studied before. Therefore, below we will first establish a reaction rates of molecules even without external pumping (6). framework for cavity MD simulation including implementation This exotic catalytic effect provides a brand new way to con- trol chemical reactions remotely. As such, there has been a recent push to understand the origins and implications of VSC Significance and V-USC. While the experimental side has focused on the search for Strong coupling between a vibrational molecular peak and large catalytic effects (7–10) as well as understanding polari- a cavity mode is known to lead to interesting spectroscopic ton relaxation dynamics through two-dimensional (2D)-IR spec- features and even the modification of ground-state chem- troscopy (11, 12), on the theoretical side, the nature of VSC and istry from small molecules to enzyme. Here, we develop a V-USC remains obscured. On the one hand, Rabi splitting can simulation tool for modeling such vibrational strong (and be easily modeled by, for example, diagonalizing a model Hamil- ultrastrong) coupling at the interface between optics, chem- tonian in the singly excited manifold (13–15) or solving equations istry, and biology. With liquid water as an example, our of motion classically for a set of one-dimensional (1D) harmonic simulation not only reveals an asymmetric Rabi splitting in the oscillators (16, 17). On the other hand, a robust explanation infrared spectrum, but also captures cavity modification of the of the catalytic effect of VSC or V-USC remains elusive (18– dynamic properties of water. 21). For example, as recently shown by us and others (21–23), the classical potential of mean force along a reaction pathway Author contributions: T.E.L., J.E.S., and A.N. designed research, performed research, is not changed by usual VSC or V-USC setups for standard analyzed data, and wrote the paper.y experiments of interest. Moreover, as demonstrated below, any Reviewers: A.R., Max Planck Institute for the Structure and Dynamics of Matter; and static equilibrium property of a molecule is not changed under G.C.S., Northwestern University. y VSC or V-USC when nuclei and photons are treated classi- The authors declare no competing interest.y cally. Unfortunately, these findings show that one cannot explain Published under the PNAS license.y the observed effect under VSC or V-USC from a static and Data deposition: The code and simulation data are available on GitHub (https://github. classical point of view. From such a conclusion, one possible com/TaoELi/cavity-md-ipi).y hypothesis of the manifestations of VSC or V-USC effect on 1 To whom correspondence may be addressed. Email: [email protected] chemical rates should arise from the modification of nonequi- This article contains supporting information online at https://www.pnas.org/lookup/suppl/ librium, or dynamical, properties of molecules under VSC doi:10.1073/pnas.2009272117/-/DCSupplemental.y or V-USC. First published July 17, 2020.

18324–18331 | PNAS | August 4, 2020 | vol. 117 | no. 31 www.pnas.org/cgi/doi/10.1073/pnas.2009272117 Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 lcrncgon tt,teHmloinEq. 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PHYSICS expensive to evolve exactly. The simplest approximation we can For the case that the frequency of the two photon modes (with make is the classical approximation (i.e., all quantum operators polarization directions perpendicular to the cavity direction) are are mapped to the corresponding classical observables). After both set to be at resonance with the O−H stretch (3550 cm−1), applying the periodic boundary condition for the molecules, the Fig. 1 B–D plots the simulated IR spectrum; the effective cou- −4 −4 −4 equations of motion for the coupled nuclei–photonic system pling strength εe is set as 2 × 10 , 4 × 10 , 6 × 10 , and become (SI Appendix, section 1) 8 × 10−4 a.u. (atomic units), respectively. Clearly, when the cavity modes are coupled to the H2O molecules, the broad O−H 2 Nsub ! ≈ ε ∂d stretch peak is split into a pair of narrower LP and UP peaks. ¨ (0) X q ek,λ X ng,λ Mnj Rnj = Fnj − εk,λ k,λ + 2 dlg,λ This result agrees with the previous theoretical and experimen- e mk,λω ∂Rnj k,λ k,λ l=1 tal work that the inhomogeneous broadening of the vibrational [4a] peak does not lead to the broadening of the polariton peaks (46, 47). More interestingly, our simulation results also suggest Nsub ≈¨ 2 ≈ X that the UP and LP peaks can be largely asymmetric especially m q = −m ω q −ε d . [4b] k,λ k,λ k,λ k,λ k,λ ek,λ ng,λ when εe is large, which agrees with experimental findings at least n=1 qualitatively (10). In Fig. 2A, we plot the Rabi splitting frequency (the differ- (0) (n) P (nl) Here, Fnj = −∂Vg /∂Rnj − l6=n ∂Vinter /∂Rnj denotes ence between the UP and LP frequencies or ω+ − ω−) as a ≈ function of ε. The simulation data (black triangles) can be fit the cavity-free force on each nuclei. We have defined q = e √ k,λ with a linear ansatz (gray line) very well. As mentioned above, qk,λ/ Ncell and the effective coupling strength εk,λ = √ √ e e ε = N ε ∝ N A q 2 because e cell , Fig. 2 demonstrates that the Rabi Ncellmk,λωk,λ/Ω0, where Ncell denotes the number of the splitting is proportional to the square root of the total num- periodic simulation cells for H2O molecules. Nsub denotes ber of molecules, which agrees with theoretical expectation and the number of molecules in a single simulation cell, and the experimental observation (32, 48): total number of molecules is N = NsubNcell. More details on implementation and simulations are explained in Materials and √ Methods and SI Appendix. ω+ − ω− = ΩN ≡ 2g0 N , [7] Results where g0 denotes the coupling constant between a single Asymmetric Rabi Splitting. The signature of VSC is the collec- molecule and the photon mode. tive Rabi splitting in the IR spectrum. In our MD simulations, Of particular interest is the asymmetric nature of the LP and the IR spectrum is calculated by linear response theory. For UP: this asymmetry is manifest in two aspects. As shown in Fig. 2 isotropic liquids, the absorption coefficient α(ω) is expressed as B and C, both the polariton frequencies and the integrated peak the Fourier transform of the autocorrelation function of the total areas of the LP (blue stars) and UP (red circles) show asym- dipole moment µ (42–45): S metric scaling as a function of the normalized Rabi frequency 2 Z +∞ (ΩN /2ω0, where ΩN is taken from Fig. 2A), especially in the V- πβω 1 −iωt n(ω)α(ω) = dt e hµS (0)µS (t)i . [5] USC limit (the red-shadowed region). Note that the standard 30Vc 2π −∞ treatment of collective Rabi splitting does not account for this asymmetry, and the observation of the suppression (or enhance- Here, n(ω) denotes the refractive index, and V denotes the 2 ment) of the LP (or the UP) in ref. 10 was explained by the higher volume of the system (i.e., the simulation cell). The factor ω absorption of water and gold cavity mirrors in the LP region. arises from the energy of the photon that was absorbed by the Some insight into the origin of this asymmetry can be obtained liquid [this expression reflects one of several suggestions that from a simple 1D model where N independent harmonic oscilla- were made for a correction factor that relates the quantum tors interact with a single-photon mode. By taking the self-dipole time-correlation function to its classical counterparts (43)]. SI term into account (to describe V-USC), we obtain (SI Appendix, Appendix, section 1 shows calculation of µS . For VSC and V- section 2) USC experiments, however, because the experimental setups usually detect an IR spectrum by sending light along the cavity direction (which means the k direction of light is along the z axis) 1 q ω2 = ω2 + Ω2 + ω2 ± (ω2 + Ω2 + ω2)2 − 4ω2ω2 [8a] (32), we need to modify the above equation to ± 2 0 N c 0 N c 0 c r 2 +∞ 2 2 πβω 1 Z 2 ΩN 2 ΩN −iωt = ω + ± ΩN ω + (when ωc = ω0), [8b] n(ω)α(ω) = dt e 0 2 0 4 20Vc 2π −∞ * + [6] X × (µS (0) · ei )(µS (t) · ei ) , where ω0 and ωc denote the frequencies of the harmonic oscil- i=x,y lators and the photon mode, respectively. Given ω0 = ωc = −1 3,550 cm and ΩN in Fig. 2A, we have plotted Eq. 8 (the where ei denotes the unit vector along direction i = x, y. Eq. black dashed lines) in Fig. 2B. We see that this analytical result 6 states that the average is performed only along the polar- already shows some asymmetry in the positions of the polari- ization directions of the detecting signal (i.e., the x and y ton peaks when plotted vs. ΩN . While Eq. 8 agrees with our directions here). When the incident light is unpolarized, these simulation data very well in the VSC limit (the green-shadowed two directions are of course equivalent. region), the simulation data seem to be more asymmetric than Fig. 1A plots the simulated IR spectrum of liquid water out- Eq. 8 in the V-USC limit. Such disagreement may arise from −1 side the cavity. The O−H stretch peaks around ∼ 3550 cm , the strong intermolecular interactions between H2O molecules, which is slightly different from experiment (∼ 3400 cm−1). Note which is completely ignored in the simplified 1D model of SI that a more accurate O−H stretch peak can be simulated by per- Appendix. forming path-integral calculations instead of a classical simula- Likewise, the simplified 1D model in SI Appendix also suggests tion (34). that the integrated peak areas of the LP and UP are

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PHYSICS A the relative strength of polaritons by tuning the cavity mode frequency. Furthermore, Fig. 3B, Inset plots the cavity mode frequency (for which the polariton intensities become symmetric) as a function of ΩN /2ω0. Again, we ﬁnd that for large ΩN /2ω0, detecting polaritons with symmetric intensities requires a very large off-resonant cavity mode frequency.

Static Equilibrium Properties of a Single Molecule. Rabi splitting represents the collective optical response of liquid water. As shown above, although MD simulations can obtain the IR spectrum of the polaritons in a straightforward way, one can argue that since most important features of the IR spectrum can be qualitatively described by the 1D harmonic model (SI Appendix, section 2), there is little advantage to perform expensive MD simulations. As has been argued above, the real advantage B of the MD simulations is that one can simultaneously obtain many other physical properties of molecules alongside with the IR spectrum. Below, we will investigate whether any property of individual H2O molecules can be changed under VSC or U-VSC. First, let us consider the static equilibrium properties of H2O molecules. We recently argued that the classical potential of mean force for a single molecule is not changed by the cavity (21) under typical VSC or V-USC setups. In fact, with the same proof procedure, it is easy to show that any static thermodynamic quan- tity of the molecules is not changed by the cavity when nuclei and photons are treated classically. This can be illustrated as follows. Given an observable O = O({Pnj }, {Rnj }), which is a function of the molecules only, the thermodynamic average for this variable

inside the cavity (hOiQED) is calculated by

R −βH G d{R }d{P }d{q }d{p }Oe QED hOi = nj nj ek,λ ek,λ [10a] Fig. 3. (A) Polariton frequency and (B) integrated peak area of polari- QED R −βH G tons as a function of the cavity mode frequency. Note that the energy d{Rnj }d{Pnj }d{eqk,λ}d{pek,λ}e QED splitting between polaritons is minimized when the cavity mode has fre- R −βH G − d{R }d{P }Oe M quency 3,550 cm 1, but the UP and LP become symmetric in intensity at a = nj nj = hOi , [10b] −1 R −βH G M different cavity mode frequency (4,250 cm ). The simulation data for liq- d{Rnj }d{Pnj }e M uid water (scattered points) are compared with the analytical expressions of the simpliﬁed 1D model (black lines) (Eqs. 8 and 9). For simulation param- −4 which is identical to the average outside the cavity (hOiM) after eters, εe= 5 × 10 a.u. and all other parameters are the same as in Fig. 2. −1 G G For parameters of the analytical expressions, we take ω0 = 3,550 cm and the integration over the photon modes, where HQED and HM are −1 ΩN = 937 cm , which corresponds to the resonant Rabi frequency when deﬁned in SI Appendix, section 1. −4 εe= 5 × 10 a.u. (Fig. 2A). The gray solid lines in A represent the uncou- Even though the mathematical proof guarantees that the static pled O−H stretch mode frequency and the cavity mode frequency. B, Inset thermodynamic properties are not changed inside the cavity, it is plots the cavity mode frequency corresponding to the case of symmetric still very helpful to check some static properties by simulation as polaritons (i.e., the crossing point frequency in B) as a function of ΩN/2ω0. it provides a tool for checking the numerical convergence. Fig. 4

between the polaritons is minimal at resonance (∼ 3,550 cm−1), in which the uncoupled O−H stretch mode frequency crosses with the cavity mode frequency (gray solid lines). Such a cross corresponds to the maximally hybridized light-matter state. By contrast, when the cavity mode frequency is larger (smaller) than the molecular frequency, the LP (UP) becomes increas- ingly dominated by the O−H stretch mode (as evident from the uncoupled case for which this mode is represented by the gray horizontal line). Our model assumes that for the uncoupled molecule–cavity case, only the molecular optical transition is coupled to the farfield. This assumption implies that in contrast to the resonance case when the UP peak is larger than the LP peak, when the cavity mode frequency becomes sufficiently large (i.e., the LP is mostly constituted by the matter side), the LP should have a larger peak size than the UP. This finding is confirmed by B B Fig. 3 . More interestingly, Fig. 3 also shows that the sym- Fig. 4. Normalized bond length distribution of O−H in liquid water. The metric peak size of polaritons occurs when the cavity mode −1 result outside the cavity (solid black) is compared with that inside the cavity frequency is ∼ 4,250 cm , which is far beyond the O−H stretch (with effective coupling strength ε = 4 × 10−4 a.u.; cyan dashed). All other −1 e frequency of liquid water (∼ 3,550 cm ). Therefore, in prin- parameters are set the same as Fig. 1. Note that the bond length distribution ciple, from this fact, one would predict that one can engineer is not changed by VSC or V-USC.

18328 | www.pnas.org/cgi/doi/10.1073/pnas.2009272117 Li et al. Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 where autocorrelation orientational the by compute denoted [OACF; must function one theory, n to similar is modification a Such 7B bond). H Fig. O−H individual the for of VACF modes the spec- internal VACF the the of for modification cavity trum small a find do we H V-USC, of motion VACF by center-of-mass the conclusion the although this for that confirmed Note have strengths. coupling we other Again, checking 6B. Fig. in transform shown Fourier is the at looking by corroborated dashed; (cyan cavity the C strength inside coupling that effective and with solid) (black cavity the H of mass n a aclt h ifso constant diffusion the calculate can One the inside strength results outside the coupling with results effective exactly the (with agree properties, cavity black) static (solid two cavity these the oxy- For the between atoms. function distribution gen bond. pair O−H radical the the plots of 5 distribution Fig. length bond normalized the plots that Note 1. V-USC. Fig. or as VSC same by the set are parameters other inside strength All that coupling with effective compared (with is black) cavity (solid the cavity the outside result The water. 5. Fig. ie al. et Li [VACF; function H of sion rotational or translational H the single whether a of in motion interested are H we individual lar, of properties dynamical Molecule. Single a of Properties Dynamical of modification cavity investigation. the further in needs role which a with properties, play static associated may of effects photons treatment quantum and classical course, nuclei a Of within photons. cavity and the nuclei properties inside thermodynamic numeri- changed static not and the are analytical that both suggest Hence, treatments cal changed. this not and strengths, is coupling conclusion other under results the checked have 3 1 vv R ttime at sfrterttoa eair codn olna response linear to according behavior, rotational the for As i.6A Fig. codn olna epneter,tetasainldiffu- translational the theory, response linear to According 0 +∞ (t ) u aia ardsrbto ucin[g(r function distribution pair Radical sntcagdb S rVUC hsfidn a lobe also can finding This V-USC. or VSC by changed not is C eto 3 ). section Appendix, (SI intense less is but n vv (t t 2 2 plots and , a edsrbdb h eoiyautocorrelation velocity the by described be can O ) (t .Teeatareetbtentersl outside result the between agreement exact The O. eoe h he rnia nrilae fmolecule of axes inertial principal three the denotes )dt . P C C l vv C vv eoe h eedeplnma findex of polynomial Legendre the denotes l (t 2 (t (t oeuei hne ne VSC. under changed is molecule O C ) = ) ftecne fms fec molecule: each of mass of center the of )] vv safnto ftm o h etrof center the for time of function a as (t hP = ) l C [ u l hv(t (t n ε e (0) 4–1,wihi endas defined is which (49–51), )] 4 = 2 )v(0)i sntcagdb S or VSC by changed not is O · ε e 2 × u = oeue.I particu- In molecules. O n ]o xgnaosi liquid in atoms oxygen of )] 10 4 (t et e smv othe to move us let Next, ε e D . × 4 = )] −4 10 i from 2 , −4 oeue (e.g., molecules O .. ugssthat suggests a.u.) × g(r .. yndashed). cyan a.u.; 10 C C sntchanged not is ) −4 vv vv (t (ω ...We a.u.). ) which ), by [12] [11] D = l . eut fteisd aiy(ihteefcieculn strength coupling with effective agrees the largely (with as cavity dashed) inside (black the of result results cavity outside time when The 7A, process Fig. ps. relaxation rotation time. initial of function the a in zooms as OACF, first-order the plot of we nent 7A, Fig. In direction. moment dipole 10 which OACF, of order first the only means study will we simplicity, For spectrum te aaeesaesttesm sFg .Nt htteVC snot is VACF the that Note 1. Fig. as V-USC. same or the VSC by the set changed inside are those parameters with compared other strength are coupling The effective solid) transform. (with (black cavity Fourier cavity corresponding the the outside (B) results and results time-domain the 6. Fig. eepaieta pr rmteeadtoa ek,tewidth unchanged. the mostly is peaks, peaks additional bare-molecule these Lastly, the from of future. apart sce- the that in many emphasize studied for we extensively molecules be chemistry should individual ground-state which of the narios, behavior change identifiable. rotational possibly hardly the may is of and peak change bare-molecule The large the by ered for that have Note V-USC. modi- peaks smaller under the small rotation demonstrating additional single-molecule peaks, of UP these fication the 1, as Fig. frequencies in same spectrum IR the water the liquid with Compared the molecule. of bare a from intensities peak Fig. with the emerges in peak small shown additional an clearly a.u.), As environment. single the 7B, a in how rotates describes which molecule direction, dipole-motion the along I B A 1 z o H For (ω −4 4 o large-enough for Inset, × ) ..[ledse-otd) i.7B Fig. dashed-dotted]). [blue a.u. 10 a ergre stesnl-oeueI spectroscopy IR single-molecule the as regarded be can P AFo h etro aso niiulH individual of mass of center the of VACF ε e 1 2 −4 ie,i h S ii) h diinlpa ilb cov- be will peak additional the limit), VSC the in (i.e., I ,the O, [u 1 z n ..[ynsolid], [cyan a.u. (ω (0) hc sdfie as defined is which ), · z u PNAS n xso h rnia xscicdswt the with coincides axes principal the of axis (t )]= I | 1 z (ω uut4 2020 4, August u ε e n = ) i h -S ii or limit V-USC the (in 6 (0) × ω · 10 2 u ε C e n −4 = 1 z (t (ω 4 ). ..[e ahd,and dashed], [red a.u. × | ). lt h corresponding the plots 10 o.117 vol. −4 C .. yndse) All dashed). cyan a.u.; 1 z (t 2 | oeue:(A) molecules: O the ), o 31 no. ε e ≥ 2 z 4 ∼ compo- × | t 8% < 10 18329 Inset [13] 8 0.1 −4 of × ε e

PHYSICS A O−H stretch peak in the IR spectrum where the LP is suppressed and the UP is enhanced. Such asymmetry can be inverted (i.e., the LP is enhanced, and the UP is suppressed) by increasing the cavity mode frequency. Moreover, with a classical treatment of nuclei and photons, while we have found no modification of the static equilibrium properties as well as the translational diffusion of liquid water, we have observed that the OACF of H2O molecules is modified under V-USC. Such observation may perhaps help us understand the catalytic effect of VSC or V-USC. Based on the current framework of cavity MD, future directions should focus on 1) path-integral calculations to study quantum effects in the modification of the molecular dynami- B cal properties and 2) ab initio cavity MD simulations of chemical reactions under VSC or V-USC. This cavity MD framework can also be used to simulate recently reported 2D-IR spectroscopy studies (11, 12) on polariton relaxation dynamics. At the same time, obtaining analytical solutions for the cavity modification of the dynamical properties would also be very helpful. We hope such studies will help solve the mystery of the catalytic effects underlying VSC or V-USC in the near future.

Materials and Methods We calculate several equilibrium and linear response observables of water (0) [Fnj , dng,λ, and ∂dng,λ/∂Rnj in Eq. 4; µS in Eq. 6; bond length, g(r), and VACF in Eq. 11; and OACF in Eq. 12] by a classical force ﬁeld—the q-TIP4P/F water model (34)—which provides the simplest description of both the equilibrium and dynamic properties of liquid water. Coupling to an optical cavity mode is included by modifying an open-source MD package I-PI (52). Fig. 7. The z component of ﬁrst-order OACF of individual H2O molecules. As detailed in SI Appendix, section 1, the cavity is placed along the z A plots the time-domain results [Cz(t)], and B plots the corresponding spec- 1 axis. A pair of thick SiO2 layers is placed between the cavity mirrors so z 2 z trum [I1(ω) = ω C1(ω)]. (A, Inset and B, Inset) A zoomed-in inset is also that the water molecules can move freely only in a small region (but still plotted in each subplot. The results outside the cavity (black dashed) are on the order of micrometers) near the cavity center. Such additional SiO2 compared with those inside the cavity (with effective coupling strength εe layers are used 1) to ensure that the intermolecular interactions between as 4 × 10−4 a.u. [cyan solid], 6 × 10−4 a.u. [red dashed], and 8 × 10−4 a.u. H2O molecules are the same as those in free space and 2) to validate the [blue dashed-dotted]). All other parameters are set the same as Fig. 1. Note long-wave approximation that we have taken from the very beginning. that OACF is changed by V-USC. We consider only two cavity modes polarized along x and y directions, both of which are resonant with the O−H stretch mode. We set the

auxiliary mass for the two photons as mk,λ = 1 a.u. Using periodic bound- Effects of a Multimode Cavity. Note that all of the results pre- ary condition as detailed in SI Appendix, section 1, we simulate 216 H2O sented above consider only a single cavity mode frequency, which molecules in a cubic cell with length 35.233 a.u., so that the water density is valid when the fundamental cavity mode is near resonance is 0.997 g cm−1. At 300 K, we first run the simulation for 150 ps to guar- with the highest molecular vibrational frequency (i.e., the O−H antee thermal equilibrium under a canonical (or NVT) ensemble where a stretch mode ∼ 3,500 cm−1 for liquid water). However, for a cav- Langevin thermostat is added on the momenta of all particles (nuclei + ity with a larger length, the fundamental cavity mode frequency photons). The resulting equilibrium configurations are used as starting can be much smaller than that of the O−H stretch mode. In points for 80 consecutive microcanonical-ensemble (NVE) trajectories of length 20 ps. At the beginning of each trajectory, the velocities are resam- such a case, many cavity modes must be taken into account. In pled by a Maxwell–Boltzman distribution under 300 K. The intermolecular SI Appendix, section 4, we show the results when liquid water Coulombic interactions are calculated by an Ewald summation. The simula- is coupled to a multimode cavity. When different cavity modes tion step is set as 0.5 fs, and we store the snapshots of trajectories every are resonantly coupled to the vibrational modes, we observe a 2 fs. SI Appendix, section 1 has details of the q-TIP4P/F force field and multimode Rabi splitting in the IR spectrum (i.e., several Rabi the implementation details. The code and simulation data are available on splittings are formed for different vibrational modes). At the GitHub (53). same time, however, the above findings regarding the single- molecule properties are not changed when a multimode cavity ACKNOWLEDGMENTS. This material is based upon work supported by US Department of Energy, Office of Science, Office of Basic Energy Sciences is considered. Award DE-SC0019397 (to J.E.S.); US Department of Energy, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Divi- Conclusion sion (to A.N.); and Israel–US Binational Science Foundation Grant 2014113 In conclusion, we have performed classical cavity MD simula- (to A.N.). T.E.L. acknowledges the Vagelos Institute for Energy Science and Technology at the University of Pennsylvania for a graduate fellowship. This tions under VSC or V-USC. With liquid water as an example, research also used resources of the National Energy Research Scientific Com- when the cavity modes are resonantly coupled to the O−H puting Center, a US Department of Energy Office of Science User Facility stretch mode, we have found asymmetric Rabi splitting of the operated under Contract DE-AC02-05CH11231.

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PHYSICS