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Letter pubs.acs.org/JPCL
Vibrational Quantum Decoherence in Liquid Water Tatsuya Joutsuka,†,¶ Ward H. Thompson,*,‡ and Damien Laage*,† École Normale Supérieure-PSL Research University, Département de Chimie, Sorbonne Universités - UPMC, Univ Paris 06, CNRS UMR 8640 PASTEUR, 24 rue Lhomond, 75005 Paris, France ‡ Department of Chemistry, University of Kansas, Lawrence, Kansas 66045, United States †
S Supporting Information *
ABSTRACT: Traditional descriptions of vibrational energy transfer consider a quantum oscillator interacting with a classical environment. However, a major limitation of this simplified description is the neglect of quantum decoherence induced by the different interactions between two distinct quantum states and their environment, which can strongly affect the predicted energy-transfer rate and vibrational spectra. Here, we use quantum−classical molecular dynamics simulations to determine the vibrational quantum decoherence time for an OH stretch vibration in liquid heavy water. We show that coherence is lost on a sub-100 fs time scale due to the different responses of the first shell neighbors to the ground and excited OH vibrational states. This ultrafast decoherence induces a strong homogeneous contribution to the linear infrared spectrum and suggests that resonant vibrational energy transfer in H2O may be more incoherent than previously thought.
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decoupled from the other modes in the liquid through isotopic dilution of a HOD molecule in D2O. We determine the vibrational decoherence time, identify the molecular origin of this decoherence, and study its impact on the calculated linear infrared spectrum. Our methodology follows the approach pioneered for electronic transitions in refs 13−16. We now describe its key features, keeping the notations of refs 15 and 16. The nonadiabatic transition rate constant between the initial vibrational state 0 and the final vibrational state 1 is determined within perturbation theory from Fermi’s Golden Rule, and the nuclear wave function of the bath is described with frozen Gaussian wavepackets that evolve along classical trajectories without changing their shape.18 This approximation was shown to be acceptable in the short time limit.18 The resulting transition rate constant between states 0 and 1 is
nergy transfers between molecular vibrational modes are ubiquitous in condensed-phase systems.1,2 Their dynamics govern the pathways and kinetics of energy flow at the molecular level involved, for example, in the thermal activation of a reacting solute in a solvent,3,4 in signal propagation within proteins,5 and in the heat protection of a biomolecule through the rapid dissipation of the excess energy in its hydration shell.6 These ultrafast energy transfers can now be probed experimentally through linear and especially nonlinear vibrational spectroscopy techniques (see, e.g., refs 7−10). Theoretical and computational studies of these phenomena usually employ a mixed quantum−classical scheme in which the part of the system containing the oscillator is treated quantum mechanically while the rest is described as a classical bath.1,11 A resulting key limitation lies in the neglect of the bath wave function. In particular, the loss of phase coherence within the quantum system due to interactions with the bath is not properly described. This quantum decoherence process converts the system from an intrinsically quantum superposition of states into a classical distribution of populations.12 We stress that this quantum decoherence is distinct from frequency dephasing, that is, fluctuations of the energy gap, which is already described in the quantum−classical approaches. The importance of decoherence was first demonstrated for electronic nonadiabatic transitions, where its proper inclusion was shown to yield large differences in the energy and electron-transfer rate constant.13−17 However, the impact of quantum decoherence on vibrational transitions and spectroscopy has remained unexplored so far. We present here a study of vibrational quantum decoherence in liquid water. We focus on the OH stretch vibration, © XXXX American Chemical Society
k 01 =
1 ⟨ ℏ2
∞
∫−∞ V01(t )V10(0)J(t ) dt ⟩thermal
(1)
where V01(t) is the vibrational coupling along the classical trajectory propagated on the initial potential surface and ⟨...⟩thermal denotes a thermal average. J(t) is the time-dependent overlap of the two nuclear bath wave functions interacting with the oscillator respectively in the ν = 0 and 1 states. J(t) = ⟨G0(t)|G1(t)⟩, where the nuclear wave functions are approximated by a product of Gaussian wavepackets Received: November 26, 2015 Accepted: January 25, 2016
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DOI: 10.1021/acs.jpclett.5b02637 J. Phys. Chem. Lett. 2016, 7, 616−621
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The Journal of Physical Chemistry Letters |Gν(t )⟩ =
⎡ aj ⎛ aj ⎞3/4 ⎟ exp⎢ − [x − x νj(t )]2 ⎣ 2 π⎠
the classical bath are used to propagate the trajectory. Simulations are performed in the microcanonical ensemble with a 1 fs time step. We verified that the explicit description of the quantum character for the OH vibration does not significantly disturb the local water structure. The average decay of quantum coherence ⟨D(t)⟩thermal is calculated from eq 4 for a series of 960 independent initial configurations selected from a 16 ns equilibrium QCMD trajectory with the OH vibration in the ground state. For each configuration, two trajectories are then propagated for 400 fs from the same initial positions and momenta, respectively in the ground and excited vibrational states. (Simulation details are provided in the SI.) The real and imaginary parts of the quantum coherence decay ⟨D(t)⟩thermal for the OH vibration of HOD in D2O at 300 K are shown in Figure 1. The real part can be approximated
∏ ⎜⎝ j
+
⎡i ⎤ i pνj(t ) ·[x − x νj(t )]⎥ exp⎢ ⎦ ⎣ℏ ℏ
∫0
t
⎤ L ν (τ ) d τ ⎥ ⎦ (2)
a−1/2 j
where Lν is the Lagrangian in the vibrational state ν, is the width of the jth nuclear wavepacket18,19 (see the SI), and xνj(t) and pνj(t) are the position and momentum of the jth nuclear mode at time t, which are propagated classically. Within this framework, the nonadiabatic transition rate constant is15,16 k 01 =
1 ℏ2
⎡
∞
∫−∞ V01(t )V10(0) exp⎢⎣ ℏi ∫0
t
⎤ ΔE01(τ ) dτ ⎥D(t ) dt ⎦
thermal
(3)
The loss of quantum coherence arises both from the fluctuations of the instantaneous energy gap between the two vibrational levels ΔE01(τ) = E0(τ) − E1(τ), leading to the ⎡i t ⎤ dephasing exp⎣ ℏ ∫ (ΔE01(τ ) − ⟨ΔE01⟩) dτ ⎦ , and from the 0 solvent quantum decoherence D(t), defined as15,16 D(t ) = Dphase(t )Dx (t )Dp(t )Dxp(t )
(4)
with ⎡ i Dphase(t ) = exp⎢ − ⎣ ℏ Dx (t ) =
∏ j
Dp(t ) =
∏ j
Dxp(t ) =
∏ j
∫0
t
⎤ ΔK 01(τ ) dτ ⎥ ⎦
⎡ aj ⎤ exp⎢ − [x 0j(t ) − x1j(t )]2 ⎥ ⎣ 4 ⎦
Figure 1. Real (blue) and imaginary (red) parts of the OH stretch vibration coherence decay for HOD in D2O, together with the Gaussian fit of the real part (dashes).
⎡ ⎤ 1 exp⎢ − [p (t ) − p1j(t )]2 ⎥ 2 0j ⎢⎣ 4ajℏ ⎥⎦ ⎡ i ⎤ exp⎢ − [x1j(t ) − x 0j(t )]·[p0j(t ) + p1j(t )]⎥ ⎣ 2ℏ ⎦
with a Gaussian decay exp[−t2/(2τ2)] with τ = 53 fs and yields a decoherence time of τD = ∫ ∞ 0 dt ⟨Re[D(t)]⟩thermal ≃ 68 fs (as shown in the SI, this τD value varies very little with the chosen simulation parameters). This decoherence time is much longer than the 100 fs).31 Population relaxation brings a minor contribution for the OH stretch of HOD in D2O because of its much slower (T1 ≃ 700 fs24) time scale. We note that a prior theoretical study35,36 that employed a numerically approximate solution of a semiclassical Liouville description observed decay of the off-diagonal density matrix element within ∼100 fs. However, while this approach included quantum-state-dependent interactions, it did not include a quantum representation of the bath, which is essential to describe the quantum decoherence due to diverging bath trajectories, which is our focus here. The fast quantum decoherence time scale has critical consequences for the nature of vibrational energy transfer in water because the decoherence time found here is faster than the 200 fs vibrational population transfer time recently calculated when decoherence is neglected.37 This suggests that OH vibrational energy relaxation in liquid water is mostly incoherent and that prior theoretical treatments37−43 where quantum decoherence was ignored may have overestimated the extent of OH vibrational excitation delocalization and the rate of vibrational energy transfer. Our results therefore call for an improved description of linear and nonlinear vibrational spectroscopy experiments and of vibrational energy relaxation in water where the fast vibrational quantum decoherence is explicitly included. Before examining the impact of decoherence on the linear infrared spectrum, we determine which solvent molecules and which motions cause the fast loss of vibrational coherence. Following the approach successfully applied to electronic decoherence,16 at every step of the QCMD trajectories, we transform the Cartesian positions and momenta of all water molecules into single-molecule quasinormal modes,16 that is, 618
DOI: 10.1021/acs.jpclett.5b02637 J. Phys. Chem. Lett. 2016, 7, 616−621
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The Journal of Physical Chemistry Letters
short. As shown in the SI, a modest increase of τD by a few tens of femtoseconds already leads to a significantly improved agreement of the IR line width with experiment. While we found that different choices of QCMD parameters and frozen nuclear Gaussian wavepackets yield very similar τD values (see the SI), a moderate increase in τD may be obtained with, for example, a quantum description of the hydrogen angular motion, which was recently shown44 to oppose the hydrogenbond strengthening induced by the quantum description of the stretch mode and may affect the QCMD in ν = 0 and 1. This will be studied in future work. We finally note that quantum decoherence further induces a small frequency red shift due to the imaginary part of the coherence decay. Interestingly, the line shape obtained from classical molecular dynamics simulations with the popular quadratic frequency map11 is already slightly broader than the experimental spectrum (Figure 4) even with decoherence neglected. Whether this may result from an overestimation of the homogeneous or inhomogeneous contributions to the line width cannot be determined from the linear IR spectra, and future work on two-dimensional infrared spectra will address this point. Note that the present QCMD simulations underestimate the experimental line width when decoherence is neglected and lead to a narrower line width than the frequency map approach (our simple Morse vibrational potential certainly provides a more approximate description than the frequency map, which is calibrated against ab initio calculations11). We finally examine the line shape obtained when quantum decoherence is assumed to be decorrelated from the transition dipole fluctuations
Figure 3. Real (solid lines) and imaginary (dashed lines) parts of the thermally averaged coherence decay for the full decay ⟨D(t)⟩thermal (black) and for its Dx (red), Dp (green), Dxp (blue), and Dphase (violet) contributions defined in eq 5.
semiclassical infrared line shape where bath decoherence is ignored34 can be extended to include this decoherence +∞
I(ω) ∝
∫−∞
dt e i(ω −⟨ω10⟩)t e−t /(2T1) t
× ⟨D(t )μ10 ⃗ (0)·μ10 ⃗ (t )e i ∫0 dτ δω10(τ)⟩
(6)
where ω10(t) is the transition frequency, δω10(t) = ω10(t) − ⟨ω10⟩ is the instantaneous fluctuation with respect to the average frequency, μ⃗ 10(t) is the transition dipole (calculated with the quadratic empirical map of ref 11), and T1 is the vibrational lifetime. The line shape, eq 6, calculated from the 960 nonequilibrium QCMD trajectories is shown in Figure 4
+∞
I(ω) ∝
∫−∞
dt e i(ω −⟨ω10⟩)t e−t /(2T1)⟨D(t )⟩ t
× ⟨μ10 ⃗ (0)·μ10 ⃗ (t )e i ∫0 dτ δω10(τ)⟩
(7)
Figure 4 shows that this result yields a (crude) approximation of the line shape. This suggests that an inexpensive but quite approximate way to include quantum decoherence would be to first compute the coherence decay D(t) from QCMD simulations and then use it as a correction to line shapes calculated from classical molecular dynamics simulations. In summary, our results have shown that vibrational quantum decoherence due to the different responses of the bath to the quantum oscillator ground and excited states is ultrafast in liquid water. This induces a large homogeneous broadening of the linear infrared line shape. This first determination of the decoherence time remains approximate but already shows that quantum decoherence cannot be ignored. Future work will extend this study to the effect of decoherence in nonlinear spectroscopy, including two-dimensional infrared photon echo spectra, and will investigate the role of decoherence in vibrational energy transfer and the recently suggested9 transition between coherent and incoherent transfer regimes when temperature is increased in liquid water.
Figure 4. Infrared line shape for the OH stretch of HOD/D2O, calculated from our QCMD simulations when quantum decoherence is fully included (solid red, eq 6), decorrelated from the transition dipole fluctuations (dashed red, eq 7), and completely neglected (dotted−dashed red, eq 6 with D(t) = 1). The line shape from the quadratic empirical map without decoherence (dashed blue)11 and the experimental measurement (solid black)24 are also shown for comparison.
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and compared with the traditional semiclassical line shape where the solvent coherence loss is ignored, that is, calculated from an equilibrium 16 ns QCMD trajectory propagated in the ground vibrational state and using D(t) = 1 in eq 6. The inclusion of the fast quantum coherence decay is found to lead to a large homogeneous broadening of the line shape, increasing the fwhm from ≃170 to ≃380 cm−1. The resulting line width is overestimated with respect to the experimental value. This suggests that our calculated τD might be slightly too
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b02637. Vibrationally adiabatic QCMD simulation methodology, widths of nuclear wavepackets, and sensitivity of τD and the IR line width to the choice of parameters (PDF) 619
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AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected] (W.H.T.). *E-mail:
[email protected] (D.L.). Present Address ¶
T.J.: Department of Chemistry, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan.
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank Casey Hynes (ENS, Paris and Univ. Colorado, Boulder) and Peter Rossky (Rice Univ.) for stimulating discussions and acknowledge support from the Agence Nationale de la Recherche (Grant No. ANR-2011BS08-010-01) to T.J. and D.L. and from NSF Grant CHE1012661 to W.H.T.
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