Applicability of the CaldeiraâLeggett Model to ... - ACS Publications

Letter pubs.acs.org/JPCL

Applicability of the Caldeira−Leggett Model to Vibrational Spectroscopy in Solution Fabian Gottwald, Sergei D. Ivanov,* and Oliver Kühn Institute of Physics, University of Rostock, Universitätsplatz 3, 18055 Rostock, Germany S Supporting Information *

ABSTRACT: Formulating a rigorous system−bath partitioning approach remains an open issue. In this context, the famous Caldeira−Leggett model that enables quantum and classical treatment of Brownian motion on equal footing has enjoyed popularity. Although this model is by any means a useful theoretical tool, its ability to describe anharmonic dynamics of real systems is often taken for granted. In this Letter, we show that the mapping between a molecular system under study and the model cannot be established in a self-consistent way, unless the system part of the potential is taken effectively harmonic. Mathematically, this implies that the mapping is not invertible. This “invertibility problem” is not dependent on the peculiarities of particular molecular systems and is rooted in the anharmonicity of the system part of the Caldeira−Leggett model potential.

M

The advantage of this model is the possibility to derive from it a simple reduced equation of motion termed generalized Langevin equation (GLE) that reads

odel systems play an important role for our understanding of complex many-body dynamics. Reducing the description to few parameters can not only ease the interpretation, but enable the identification of key properties.1 In condensed phase dynamics the spin-boson2 and Caldeira− Leggett (CL)3,4 models have been the conceptual backbone of countless studies.5 The latter has been extended to describe linear and nonlinear spectroscopy within the second-order cumulant approximation, termed multimode Brownian oscillator model in this context,6,7 and has become a popular tool for assigning spectroscopic signals in the last two decades. The CL model comprises an arbitrary system (coordinates x, potential VS(x)), which is bilinearly coupled to a bath of harmonic oscillators (coordinates Qi, potential VB({Qi})) via a system−bath coupling potential VS−B(x,{Qi}).3 Later, Caldeira and Leggett extended the model to an arbitrary function of system coordinates in the coupling and motivated the linearity of the coupling on the bath side.4 Here, we limit ourselves to the bilinear version for the reasons that will become apparent later. Restricting ourselves to a one-dimensional case yields5 VB({Q i}) + VS − B(x , {Q i}) ≡

∑ i

1 2 2 ωi Q i − 2

p ̇ (t ) = −

∫0

t

dτ ξ(t − τ )p(τ ) + R(t )

(2)

where the bath is reduced to non-Markovian dissipation and fluctuations represented by the memory kernel ξ(t) and the stochastic force R(t), respectively. In the CL model, the former is connected via a cosine Fourier transform to the spectral density J(ω) =

1 m

∑ i

gi2 ωi2

δ(ω − ωi)

(3)

and the latter is a Gaussian process with zero-mean; m is the system mass. Importantly, the derivation can be performed both in classical9,10 and quantum11,12 domains. For the present purpose, we would limit ourselves to the classical GLE, whose derivation from the CL model requires no further approximations. In this case, the fluctuating force is connected to the memory kernel by means of the so-called fluctuation− dissipation theorem (FDT)

∑ giQ ix i

(1)

⟨R(0)R(t )⟩ = mkTξ(t )

with the bath frequencies ωi, unit bath masses, and the coupling strengths gi. Often the square on the right-hand side of eq 1 is completed causing thereby a harmonic correction to the system potential VS(x): Ṽ S(x) ≡ VS(x) − ∑i g2i /(2ω2i )x2. The CL model imposes no restrictions on the form of the system potential, apart from the presence of the aforementioned trivial correction. Note that this model also excludes pure dephasinglike line broadening, which is due to system−bath coupling, nonlinear in the system part (see, e.g., refs 1 and 8). © 2015 American Chemical Society

∂VS̃ (x) − ∂x

(4)

which establishes the canonical ensemble with the temperature T. We note that according to Ford et al.:11 “... if there is a universal description, then it must be of the form we have obtained [that is, eq 2]”. This generalized equation has been Received: April 7, 2015 Accepted: June 23, 2015 Published: June 23, 2015 2722

DOI: 10.1021/acs.jpclett.5b00718 J. Phys. Chem. Lett. 2015, 6, 2722−2727

Letter

The Journal of Physical Chemistry Letters

GLE form.10,39 The GLEs obtained this way describe a projection-operator modified dynamics where noise and dissipation are formally obtained as projected quantities. The second mapping is based on the linear projection, which yields the GLE (eq 2) and the FDT (eq 4) without any further approximations. The system potential becomes effectively harmonic, that is, Ṽ S(x) = 1/2 mω̃ 2x2 at the price of projecting all the system anharmonicity into the bath. The effective harmonic frequency follows as mω̃ 2 = kT/⟨x2⟩ with ⟨...⟩ denoting canonical averaging. The third route employs a nonlinear projection. Here, the system potential Ṽ S(x) is given by the potential of the mean force felt by the system at coordinate x, averaged over all bath coordinates Qi.10,40,41 In this formalism, the memory kernel carries a functional dependence on the system’s positions and momenta and the fluctuating force has more complicated statistical properties than given by the FDT in eq 4.40 This functional dependence is hardly tractable and an apparent simplification is to linearize the kernel with respect to positions and momenta. In this case the contribution proportional to x vanishes and the resulting integral coincides with that in eq 2.40 These three routes provide us with the three versions of the GLE that are parametrized from explicit MD simulations upon inverting the equation

used, for instance, in the theory of vibrational relaxation for estimating characteristic relaxation times,13−16 reaction rates,17,18 and for thermostatting purposes;19−21 see, for example, refs 22−24 for reviews. The tremendous simplicity of the CL model is, thus, that it reduces the system−bath interaction, VS−B(x,{Qi}) to a single spectral density. However, a real system Hamiltonian might very well differ from the CL one. In practical applications, therefore, one tries to find a mapping between the two in order to make full use of the model’s simplicity. Importantly, if such a mapping can be established, then the full quantum-mechanical treatment of the bath can be performed analytically via the Feynman−Vernon influence functional25,26 without any further approximations. Moreover, numerically exact hierarchy type equation of motion approaches are essentially based on the CL model.27,28 Further, it is ideally suited for numerical methods that solve the Schrödinger equation in many dimensions.29 The CL model can also be taken as a starting point for quantumclassical hybrid simulations, that is by treating only the usually low-dimensional system part quantum-mechanically and the bath (semi)classically.30 Finally, the machinery for a purely classical treatment by means of a GLE is provided by the method of colored noise thermostats.20,21 It is thus the CL model that establishes a unified framework for a quantumclassical comparison of dynamical properties in condensed phase systems provided that the real system can be mapped onto this model. We note in passing that various models based on a coupling that is nonlinear in system coordinates exist, such as, for example, the so-called square-linear CL model.31,32 They, however, correspond to different GLE types, which do not lead to any known mapping (see Supporting Information) and, thus, are not considered here. This puts forward the central question of this Letter, that is, whether such a mapping onto the bilinear CL model exists for an arbitrary system. Surprisingly, the amount of information concerning the applicability of this tremendously popular model is scarce. Makri argued for the applicability if the system−bath coupling can be treated in the linear response regime.33 Goodyear and Stratt derived the model in the short time limit introducing so-called instantaneous normal modes.34 This approach proved valuable for studying, for example, solvation dynamics,35 vibrational relaxation,36 and line-broadening mechanisms37 of solute− solvent systems. However, due to its short-time nature the instantaneous normal mode approximation is particularly suited to describe localized high-frequency vibrations only. In addition, the bath statistics often deviates from the predictions of the CL model15,34 as it was particularly observed for aqueous systems.38 In order to approach the central question, we discuss three possible ways for constructing a mapping onto the CL model. First, one might assume that the real bath can be either exactly or approximately described by the CL Hamiltonian in a certain coordinate system. This case corresponds to a direct representation of the bath in a harmonic form, eq 1. The system potential VS(x) would be then the actual system potential as given, for instance, by an explicit force field; note the presence of the aforementioned harmonic correction in Ṽ S(x). Another possibility is to rigorously derive the GLE from first principles for an arbitrary open system. If the resulting equation coincided with that derived from the CL model, it would then serve as an indirect proof of the model itself. Here, Mori’s projection operator techniques are commonly involved to recast the natural system dynamics into a linear or nonlinear

̇ (t ) = CpF ̃ (t ) − Cpp S

∫0

t

dτ ξ(t − τ )Cpp(τ )

(5)

to obtain the memory kernel ξ(t), see ref 42 and Supporting Information. Here, Cpp(t) is the autocorrelation function of the system’s momentum p and CpF̃S(t) denotes the correlation function of the system’s momentum and the system force F̃S(x) = −∂xṼ S(x) . Thus, the mapping between the explicit system under study and the CL model can be formulated as the mapping ( : {Cpp(t); CpF̃S(t)} → ξ(t). The particular form of the system force F̃S(x) specifies the GLE employed and thus the mapping ( , referred to as ( D, ( LP and ( NLP for the direct mapping, mapping based on linear and on nonlinear projection technique, respectively. We note again that the latter mapping requires a linearization of the memory kernel. It is also worth mentioning that the GLEs stemming from ( D and ( LP coincide up to a constant frequency shift if the system potential is harmonic. If the bath is harmonic enough, for example, when the temperature is sufficiently low, then the coupling on the bath side is linear and the linearized ( NLP GLE coincides with the other two, as the mean force corrections vanish. In order to verify a mapping ( , and thus, to deduce the applicability of the CL model, we compute the linear vibrational spectra from the GLEs and compare them against their explicit MD counterparts. The match between the spectra, thus, describes the success of the system−bath model; see Figure S5 in the Supporting Information. Models and Methods. We have employed three hydrogenbonded systems: HOD in H2O, OH in H2O, and the ionic liquid [C2mim][NTf2] at room temperature. These examples have been chosen because H-bonding is known to come along with a considerable anharmonicity in the potential energy surface (PES). In particular, HOD in H2O or D2O is among the most studied hydrogen-bonded systems24,29,43−45 and the comparison with the OH radical is used to study the effect of completely decoupling the OH stretch from OD stretching and HOD bending motions. Note that, although usually the OD stretch in HOD is investigated in many studies of HOD in 2723


Letter


depicted in the upper panels of Figure 1 with the error bars given in red. It becomes obvious that excellent fit results can be

H2O, we consider the vibrational spectrum of the OH stretch. The reason is that this OH stretch is resonantly coupled to the same vibrations in the surrounding H2O molecules leading to nontrivial spectra thereby providing a nice testing opportunity for the CL model. The ionic liquid represents a system where, in addition to moderate H-bonding, there exists a strong Coulomb interaction between the ion pairs.46 The HOD/OH in H2O systems are comprised of 466 molecules in a periodic box with the length of 2.4 nm interacting according to the force field taken from ref 47. In the latter case, the HOD molecule is substituted by the diatomic OH fragment. The ionic liquid simulations were carried out in a periodic box with the length of 4.5 nm containing 216 ion pairs and the force field described in ref 48. All explicit molecular dynamics (MD) simulations were performed with the GROMACS program package (Version 4.6.5).49 The spectra were computed for the OH stretch in the HOD/OH molecules (i.e., excluding the surrounding water molecules) and for the C(2)−H stretch of the imidazolium ring48 in the ionic liquid, see Supporting Information for details. In the ionic liquid, the potential for the C−H stretching motion was reparametrized to a Morse potential using DFT-B3LYP calculations.50 Note that in the spirit of the system−bath treatment, the O−H/C−H bond lengths were taken as the respective system coordinates, x, whereas all other degrees of freedom constituted the bath; note that the ro-vibrational coupling was neglected.51 We employed the “standard protocol” for calculating IR spectra, that is, a set of NVE trajectories, each 6 ps long (time step 0.1 fs), was started from uncorrelated initial conditions sampled from an NVT ensemble. In order not to violate the detailed balance condition, the Cpp(t) functions for the system coordinate were Fourier-transformed to yield the vibrational spectra;52,53 note that these spectra are proportional to the absorption spectra in the one-dimensional case. Further, all spectra were smoothed by multiplying Cpp(t) with a Gaussian window function before carrying out the Fourier transform. The window width was set to the correlation time in the system. In order to achieve convergence, 1000 trajectories for the considered stretching motion were employed, see Supporting Information for further details. As a numerical solver for the GLE, we adopted the method of colored noise thermostats.20,21 The simulation setup for GLE simulations was the same as that for the explicit ones. Results. Following the three routes described above, we start with checking whether the direct mapping exists, which implies that the actual Hamiltonian of the system studied has the form of eq 1. The linearity of the coupling on the system side can be verified via explicit MD simulations in a straightforward manner. Here, we sampled 5000 uncorrelated configurations and varied the system coordinate, x, within the range accessible due to its thermal fluctuations keeping all bath coordinates fixed, see Supporting Information. The system−bath coupling VS−B(x) (note that the parametric dependence on the bath coordinates will be skipped) probed this way was least-squares fitted to linear functions f(x) = ax + b. In order to quantify the fit error with respect to a reasonable scale, we considered the relative deviation r (x ) =

f (x) − VS − B(x) |VS − B(xi) − VS − B(xf )|

Figure 1. Panels (a)−(c): Averaged r(x) (see eq 6). Panels (d)−(f): Distribution functions of the forces corresponding to the noise term R(t). Left panels correspond to HDO in bulk water; middle, to OH in bulk water; right, to the ionic liquid [C2mim][NTf2]. Note different scales on the y axes.

obtained for the two aqueous systems (panels a and b) where the relative deviations are below 2%. In contrast, the averaged fit errors for the ionic liquid (panel c) are in the range of 50% on the scale of the overall change of the potential on the accessible interval. Hence, we can conclude that for aqueous systems the coupling VS−B(x) is linearly dependent on the bond length, x, as it is assumed in the CL model, whereas it is by no means true for the ionic liquid considered. Given the fact that the CL model is strictly equivalent to the GLE, the assumptions concerning the nonlinearity of the coupling on the bath side and the harmonicity of the bath, being hardly separable, together imply Gaussian statistics of the noise. The Gaussian statistics can be easily probed via explicit MD simulations by calculating the distribution of the environmental forces exerted on the system coordinate, which is kept fixed to remove the dissipative contributions, see Supporting Information. The resulting distributions shown in the lower panels of Figure 1 are clearly asymmetric and centered at positive values for the two aqueous systems (panels d and e). The distribution for the ionic liquid (panel f) is centered almost at zero and is only slightly skewed, thus looking much more similar to a Gaussian. To summarize, checking the linearity on the system side suggests that the model should perform well for aqueous systems and badly for the ionic liquid, whereas the corresponding results for the linearity on the bath side imply the opposite. Another test for the bath’s linearity is provided by probing the temperature dependence of the spectral density, which should be absent if the two are strictly fulfilled. The results lead to similar conclusions; see Figure S7 in the Supporting Information. In order to elucidate the impact of these nonlinearities onto vibrational spectra, we compare the GLE spectra with the ones obtained from explicit MD simulations in Figure 2. The GLE spectra (dotted green curves therein) for aqueous systems (panels a,b) are significantly different from the reference ones (black curves therein) both in shape and position. For the ionic liquid (panel c), these differences are not so pronounced, though they are still noticeable. This comparison supports the conjecture that the Gaussian statistics of the bath is more important for spectra, although it cannot be considered as its proof in general. Still, in all cases considered, the match of

(6)

with xi/xf being the initial/final values of the probed range, respectively. The results averaged over the chosen data set are 2724


Letter


that was used to establish mapping ( . Hence, mathematically, we require ) to be the inverse mapping of ( . Unfortunately, mapping ( is not invertible, since it constitutes a mapping of two functions onto one, unless the two functions are mutually dependent. Thus, ) cannot be the inverse mapping of ( in the general case, and the self-consistency requirement of the formalism is generally corrupted. This problem is termed the “invertibility problem”. To illustrate this point on a particular example, the procedure described above led to two strikingly dif ferent pairs of {Cpp(t); CpF̃S(t)} for mapping ( D, as is shown in Figure 3 for the case of

Figure 2. Explicit MD and GLE spectra for (a) HDO in bulk water, (b) OH in bulk water, and (c) the ionic liquid (black, explicit MD result; red stars, ( LP; dotted green, ( D; dashed orange, linearized ( NLP. See text for definitions).

spectra is not achieved and, thus, the direct mapping ( D fails to exist for the systems studied. We have to stress, that assessing the key assumptions of the CL model directly, can only allow one to draw conclusions about ( D. In contrast, these assumptions might be of no relevance for any mapping other than direct. The second route is based on the mapping ( LP, which is exact and therefore the corresponding spectra (red stars in Figure 2) perfectly match the explicit MD ones (black lines) for all systems considered. Importantly, this mapping is established at the price of putting all the system anharmonicity into the bath, thus giving an effectively harmonic CL model with a frequency of ω̃ (for an application to water; see, e.g., ref 54). As a word of caution, we note that this model is tailored to yield linear response correctly, whereas other observables are not expected to be reproduced. The third route is based on the mapping ( NLP, which employs the nonlinear projection technique. Note that although the derivation is exact, the memory kernel has been linearized to restore the desired equation structure. The resulting spectra, dashed orange curves in Figure 2, improve slightly with respect to that of the direct mapping (dotted green lines) in terms of peak positions but still have a wrong shape compared to the reference spectra (black lines). Hence, the mapping ( NLP is not sufficient to completely describe the spectra for the systems studied, and the only successful approach is the effectively harmonic mapping ( LP. As a sideresult we note that intramolecular forces appear to be not very important, as it follows from the comparison of panels (a) and (b) in Figure 2. To summarize, we have considered three ways to construct a mapping and have shown that only ( LP, based on an effectively harmonic potential, gives correct spectra. However, such an effective harmonic system potential would not be suitable for describing nonlinear infrared spectroscopy since nonlinear response functions will vanish in this case. In principle, there could be a yet unknown mapping yielding anharmonic system potentials. To this end, we would like to consider the problem from a more general perspective without being bound to a particular way of constructing the mapping. We note that so far our assessment was based on vibrational spectra, that is on Cpp(t). In fact, performing the dynamics according to eq 2 yields two presumably independent correlation functions, and thus defines a mapping ) : ξ(t) →{Cpp ′ (t); CpF ′ ̃S(t)}. In order to have a self-consistent formalism, the pair of correlation functions {C′pp(t); C′pF̃S(t)} should coincide with the pair {Cpp(t); CpF̃S(t)} obtained from the explicit MD simulations

Figure 3. Cpp(t) in panel (a) and CpF̃S(t) in panel (b) are shown for the HOD in water case. Black stands for explicit results, green for ( D according to eq 2.

HOD in water. Importantly, in the case of the mapping ( LP, this problem does not surface because for a harmonic force CpF̃S(t) and Cpp(t) are mutually dependent as CpF̃S(t) = −ω̃ 2∫ t0 Cpp(τ)dτ. Thus, the mapping ( LP can be formulated with Cpp(t) alone and the invertibility problem does not occur. This suggests that it is the anharmonicity of the system potential, Ṽ S(x), in the GLE that is at the root of all deficiencies of the applicability of the CL model to real anharmonic molecular systems. Since this consideration is independent on the particular form of the mapping ( , the invertibility problem would surface in general, that is, for an arbitrary anharmonic system. Conclusions and Outlook. In this Letter, the applicability of the CL model to the dynamics of realistic anharmonic systems has been investigated. Three possible mappings between a real system and the model, that is, the direct mapping and the mappings based on (non)linear projection techniques, have been considered. Investigating three representative systems, we obtained numerical evidence that only the mapping based on the linear projection technique (( LP) reproduces the linear absorption spectrum. Importantly, this comes at the price of projecting the system’s anharmonicity into the bath, resulting in an effectively harmonic potential that is not suitable for describing nonlinear spectroscopy. The general argument against a mapping that leads to an anharmonic system potential and gives a reasonable description have been provided and illustrated on the particular example of the direct mapping. This argument, termed invertibility problem, is not bound to the particular way of constructing the mapping and thus can be considered as a strong indication that such a mapping does not exist within the (bilinear) CL model. One might ask whether extending the model by considering a nonlinear coupling on the system side would yield a solution. Unfortunately, the structure of the GLE derived from such an extended model would differ from that obtained from the 2725


Letter


(8) Levine, A. M.; Shapiro, M.; Pollak, E. Hamiltonian Theory for Vibrational Dephasing Rates of Small Molecules in Liquids. J. Chem. Phys. 1988, 88, 1959. (9) Zwanzig, R. Nonlinear Generalized Langevin Equations. J. Stat. Phys. 1973, 9, 215−220. (10) Zwanzig, R. Nonequilibrium Statistical Mechanics; Oxford University Press: New York, 2001. (11) Ford, G. W.; Kac, M. On the Quantum Langevin Equation. J. Stat. Phys. 1987, 46, 803−810. (12) Ford, G. W.; Lewis, J. T.; Oconnell, R. F. Quantum Langevin Equation. Phys. Rev. A 1988, 37, 4419−4428. (13) Whitnell, R. M.; Wilson, K. R.; Hynes, J. T. Fast Vibrational Relaxation for a Dipolar Molecule in a Polar Solvent. J. Phys. Chem. 1990, 94, 8625−8628. (14) Benjamin, I.; Whitnell, R. M. Vibrational Relaxation of I2 in Water and Ethanol: Molecular Dynamics Simulation. Chem. Phys. Lett. 1993, 204, 45−52. (15) Tuckerman, M.; Berne, B. J. Vibrational Relaxation in Simple Fluids: Comparison of Theory and Simulation. J. Chem. Phys. 1993, 98, 7301−7318. (16) Gnanakaran, S.; Hochstrasser, R. M. Vibrational Relaxation of HgI in Ethanol: Equilibrium Molecular Dynamics Simulations. J. Chem. Phys. 1996, 105, 3486. (17) Grote, R. F.; Hynes, J. T. The Stable States Picture of Chemical Reactions. II. Rate Constants for Condensed and Gas Phase Reaction Models. J. Chem. Phys. 1980, 73, 2715. (18) Gertner, B. J.; Wilson, K. R.; Hynes, J. T. Nonequilibrium Solvation Effects on Reaction Rates for Model SN2 Reactions in Water. J. Chem. Phys. 1989, 90, 3537−3558. (19) Ceriotti, M.; Bussi, G.; Parrinello, M. Nuclear Quantum Effects in Solids Using a Colored-Noise Thermostat. Phys. Rev. Lett. 2009, 103, 030603. (20) Ceriotti, M.; Bussi, G.; Parrinello, M. Langevin Equation with Colored Noise for Constant-Temperature Molecular Dynamics Simulations. Phys. Rev. Lett. 2009, 102, 020601. (21) Ceriotti, M.; Bussi, G.; Parrinello, M. Colored-Noise Thermostats à La Carte. J. Chem. Theory Comput. 2010, 6, 1170−1180. (22) Abe, Y.; Ayik, S.; Reinhard, P.-G.; Suraud, E. On Stochastic Approaches of Nuclear Dynamics. Phys. Rep. 1996, 275, 49−196. (23) Tanimura, Y. Stochastic Liouville, Langevin, Fokker-Planck, and Master Equation Approaches to Quantum Dissipative Systems. J. Phys. Soc. Jpn. 2006, 75, 1−39. (24) Hynes, J. T. Molecules in Motion: Chemical Reaction and Allied Dynamics in Solution and Elsewhere. Annu. Rev. Phys. Chem. 2015, 66, 1−20. (25) Feynman, R.; Vernon, F. The Theory of a General Quantum System Interacting with a Linear Dissipative System. Ann. Phys. (N. Y). 1963, 24, 118−173. (26) Caldeira, A.; Leggett, A. Path Integral Approach to Quantum Brownian Motion. Physica A 1983, 121, 587−616. (27) Dijkstra, A. G.; Tanimura, Y. Non-Markovian Entanglement Dynamics in the Presence of System-Bath Coherence. Phys. Rev. Lett. 2010, 104, 250401. (28) Suess, D.; Eisfeld, A.; Strunz, W. T. Hierarchy of Stochastic Pure States for Open Quantum System Dynamics. Phys. Rev. Lett. 2014, 113, 150403. (29) Giese, K.; Petković, M.; Naundorf, H.; Kühn, O. Multidimensional Quantum Dynamics and Infrared Spectroscopy of Hydrogen Bonds. Phys. Rep. 2006, 430, 211−276. (30) Egorov, S. A.; Rabani, E.; Berne, B. J. Nonradiative Relaxation Processes in Condensed Phases: Quantum Versus Classical Baths. J. Chem. Phys. 1999, 110, 5238−5248. (31) Kühn, O.; Tanimura, Y. Two-Dimensional Vibrational Spectroscopy of a Double Minimum System in a Dissipative Environment. J. Chem. Phys. 2003, 119, 2155−2164. (32) Tanimura, Y.; Ishizaki, A. Modeling, Calculating, and Analyzing Multidimensional Vibrational Spectroscopies. Acc. Chem. Res. 2009, 42, 1270−1279.

rigorous projection techniques, see Supporting Information for a particular example of square-linear model. This makes establishing the mapping for an extended model difficult if at all possible. Finally, we would like to emphasize that notwithstanding these technical difficulties, such an extension would still fall short in describing the real dynamics in many solute−solvent systems. In fact, there is ample evidence that, as a consequence of anharmonicity, high-frequency solute modes usually relax via overtone or combination tone transitions, possibly including low-frequency phonon modes;55 for a study of the Fermi resonance in the present ionic liquid example, see ref 46. This has been nicely illustrated using the instantaneous normal mode approach in refs 34 and 56. In principle, it could be possible to solve the aforementioned problems by extending the relevant system and including the most important nonlinearly coupled bath modes, for example, in case of a Fermi resonance.

■

ASSOCIATED CONTENT

S Supporting Information *

Contains additional information on: the multimode Brownian oscillator model, constructing the mappings onto the CL model, calculating classical vibrational spectra, spectral density extraction, calculating the mean field potential for the ( NLP approach, the ionic liquid setup, the logic of the applicability check, checking the direct mapping ( D, and an extension of the CL model to nonlinear system−bath coupling. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b00718.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS

■

REFERENCES

The authors gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (Sfb 652 (O.K.) and IV 171/2-1 (S.D.I.)). We thank Dr. Sergey Bokarev for fruitful discussions.

(1) May, V., Kühn, O. Charge and Energy Transfer Dynamics in Molecular Systems, 3rd Revised and Enlarged ed.; Wiley-VCH: Weinheim, 2011. (2) Leggett, A. J.; Chakravarty, S.; Dorsey, A. T.; Fisher, M. P. A.; Garg, A.; Zwerger, M. Dynamics of the Dissipative Two-State System. Rev. Mod. Phys. 1987, 59, 1−85. (3) Caldeira, A.; Leggett, A. Influence of Dissipation on Quantum Coherence. Phys. Rev. Lett. 1981, 46, 211−214. (4) Caldeira, A. O.; Leggett, A. J. Quantum Tunnelling in a Dissipative System. Ann. Phys. (N. Y). 1983, 149, 374−456. (5) Grabert, H.; Schramm, P.; Ingold, G.-L. Quantum Brownian Motion. The Functional Integral Approach. Phys. Rep 1988, 168, 115− 207. (6) Mukamel, S. Principles of Nonlinear Optical Spectroscopy; Oxford University Press: New York, 1995. (7) Tanimura, Y.; Mukamel, S. Real-Time Path-Integral Approach to Quantum Coherence and Dephasing in Nonadiabatic Transitions and Nonlinear Optical Response. Phys. Rev. E 1993, 47, 118−136. 2726


Letter


(54) Fried, L. E.; Bernstein, N.; Mukamel, S. Simulation of the Femtosecond Optical Response of a Solute in Water. Phys. Rev. Lett. 1992, 68, 1842−1845. (55) Heyne, K.; Nibbering, E. T. J.; Elsaesser, T.; Petković, M.; Kühn, O. Cascaded Energy Redistribution upon O−H Stretching Excitation in an Intramolecular Hydrogen Bond. J. Phys. Chem. A 2004, 108, 6083−6086. (56) Larsen, R. E.; Stratt, R. M. Instantaneous Pair Theory for HighFrequency Vibrational Energy Relaxation in Fluids. J. Chem. Phys. 1999, 110, 1036−1052.

(33) Makri, N. The Linear Response Approximation and Its Lowest Order Corrections: An Influence Functional Approach. J. Phys. Chem. B 1999, 103, 2823−2829. (34) Goodyear, G.; Stratt, R. M. The Short-Time Intramolecular Dynamics of Solutes in Liquids. I. An Instantaneous-Normal-Mode Theory for Friction. J. Chem. Phys. 1996, 105, 10050−10071. (35) Ladanyi, B. M.; Stratt, R. M. Short-time Dynamics of Solvation: Relationship Between Polar and Nonpolar Solvation. J. Phys. Chem. 1996, 100, 1266−1282. (36) Larsen, R. E.; David, E. F.; Goodyear, G.; Stratt, R. M. Instantaneous Perspectives on Solute Relaxation in Fluids: The Common Origins of Nonpolar Solvation Dynamics and Vibrational Population Relaxation. J. Chem. Phys. 1997, 107, 524−543. (37) Ryu, S.; Stratt, R. M. A Case Study in the Molecular Interpretation of Optical Kerr Effect Spectra: Instantaneous-NormalMode Analysis of the OKE Spectrum of Liquid Benzene. J. Phys. Chem. B 2004, 108, 6782−6795. (38) Møller, K. B.; Rey, R.; Hynes, J. T. Hydrogen Bond Dynamics in Water and Ultrafast Infrared Spectroscopy: A Theoretical Study. J. Phys. Chem. A 2004, 108, 1275−1289. (39) Mori, H. Transport, Collective Motion, and Brownian Motion. Prog. Theor. Phys. 1965, 33, 423−455. (40) Kawai, S.; Komatsuzaki, T. Derivation of the Generalized Langevin Equation in Nonstationary Environments. J. Chem. Phys. 2011, 134, 114523. (41) Lange, O. F.; Grubmüller, H. Collective Langevin Dynamics of Conformational Motions in Proteins. J. Chem. Phys. 2006, 124, 214903. (42) Gottwald, F.; Karsten, S.; Ivanov, S. D.; Kühn, O. Parametrizing Linear Generalized Langevin Dynamics from Explicit Molecular Dynamics Simulations. J. Chem. Phys. 2015, DOI: 10.1063/1.4922941. (43) Cho, M. Coherent Two-Dimensional Optical Spectroscopy. Chem. Rev. 2008, 108, 1331−418. (44) Stenger, J.; Madsen, D.; Hamm, P.; Nibbering, E. T. J.; Elsaesser, T. Ultrafast Vibrational Dephasing of Liquid Water. Phys. Rev. Lett. 2001, 87, 027401. (45) Ivanov, S. D.; Witt, A.; Shiga, M.; Marx, D. Communications: On Artificial Frequency Shifts in Infrared Spectra Obtained from Centroid Molecular Dynamics: Quantum Liquid Water. J. Chem. Phys. 2010, 132, 031101. (46) Roth, C.; Chatzipapadopoulos, S.; Kerlé, D.; Friedriszik, F.; Lütgens, M.; Lochbrunner, S.; Kühn, O.; Ludwig, R. Hydrogen Bonding in Ionic Liquids Probed by Linear and Nonlinear Vibrational Spectroscopy. New J. Phys. 2012, 14, 105026. (47) Paesani, F.; Voth, G. A. A Quantitative Assessment of the Accuracy of Centroid Molecular Dynamics for the Calculation of the Infrared Spectrum of Liquid Water. J. Chem. Phys. 2010, 132, 014105. (48) Köddermann, T.; Paschek, D.; Ludwig, R. Molecular Dynamic Simulations of Ionic Liquids: A Reliable Description of Structure, Thermodynamics and Dynamics. ChemPhysChem 2007, 8, 2464− 2470. (49) Hess, B.; Kutzner, C.; van der Spoel, D.; Lindahl, E. GROMACS 4: Algorithms for Highly Efficient, Load-Balanced, and Scalable Molecular Simulation. J. Chem. Theory Comput. 2008, 4, 435−447. (50) Chatzipapadopoulos, S.; Zentel, T.; Ludwig, R.; Lütgens, M.; Lochbrunner, S.; Kühn, O. ChemPhysChem 2015, DOI: 10.1002/ cphc.201500433. (51) Berne, B. J.; Tuckerman, M. E.; Straub, J. E.; Bug, A. L. R. Dynamic Friction on Rigid and Flexible Bonds. J. Chem. Phys. 1990, 93, 5084. (52) Ramírez, R.; López-Ciudad, T.; Kumar P, P.; Marx, D. Quantum Corrections to Classical Time-Correlation Functions: Hydrogen Bonding and Anharmonic Floppy Modes. J. Chem. Phys. 2004, 121, 3973−3983. (53) Ivanov, S. D.; Witt, A.; Marx, D. Theoretical Spectroscopy Using Molecular Dynamics: Theory and Application to CH+5 and Its Isotopologues. Phys. Chem. Chem. Phys. 2013, 15, 10270−10299. 2727


Applicability of the CaldeiraâLeggett Model to ... - ACS Publications

Recommend Documents

Applicability of the CaldeiraâLeggett Model to ... - ACS Publications