This journal is c the Owner Societies 2013 Phys. Chem. Chem. Phys., 2013, 15, 12591--12601 12591 Cite this: Phys. Chem. Chem. Phys., 2013, 15, 12591 Computing Wigner distributions and time correlation functions using the quantum thermal bath method: application to proton transfer spectroscopy Marie Basire, Daniel Borgis* and Rodolphe Vuilleumier* Langevin dynamics coupled to a quantum thermal bath (QTB) allows for the inclusion of vibrational quantum effects in molecular dynamics simulations at virtually no additional computer cost. We investigate here the ability of the QTB method to reproduce the quantum Wigner distribution of a variety of model potentials, designed to assess the performances and limits of the method. We further compute the infrared spectrum of a multidimensional model of proton transfer in the gas phase and in solution, using classical trajectories sampled initially from the Wigner distribution. It is shown that for this type of system involving large anharmonicities and strong nonlinear coupling to the environment, the quantum thermal bath is able to sample the Wigner distribution satisfactorily and to account for both zero point energy and tunneling effects. It leads to quantum time correlation functions having the correct short-time behavior, and the correct associated spectral frequencies, but that are slightly too overdamped. This is attributed to the classical propagation approximation rather than the generation of the quantized initial conditions themselves. 1 Introduction The calculation of quantum time correlation functions (TCFs) for complex molecular systems remains a challenging problem in statistical mechanics. A few approximate methods have been proposed in the literature. One obvious route is to try to relate quantum TCF’s to their classical analogs through various relations in frequency space which have the property, at least, of yielding the exact detailed balance condition. Those rela- tions, which have been used for a long time in spectroscopy, 1 have been reviewed recently for model systems, 2,3 and some extensions have been proposed too. 4 In a more direct quantum simulation perspective, the most successful approaches are the centroid molecular dynamics 5,6 or ring-polymer molecular dynamics 7 and the semi-classical initial value representation (SC-IVR) of Miller and collaborators, 8,9 including its linearized version (LSC-IVR). 10 This latter approach is also termed linearized path integral (LPI) representation of correlation functions. 11–13 This LPI approximation leads to a rather transparent interpretation and has already proved its accuracy for vibrational energy relaxa- tion 14 and static and dynamic quantum effects in liquids. 15–17 In short, it amounts to replacing operators by their Weyl transform and computing the CF using classical trajectories sampled initially from the quantum Wigner distribution rather than the classical Boltzmann distribution. 11,18 For a set of coordinates q ={q i } and for a given temperature b = 1/k B T, the Wigner distribution is related to the quantum density matrix r(q,q 0 )= hq|e bH |q 0 i by Wðp; qÞ¼ 1 ð2phÞ N Q Z du e ipu=h r q u 2 ; q þ u 2 ; (1) where Q is the normalization factor (or partition function). The generation of classical trajectories from the quantum Wigner distribution is also referred to as the classical Wigner model (CWM). 10 The exact calculation of the Wigner distribution of a many- body system still remains a formidable task, and although the theoretical expressions to be evaluated are known, 19,20 no practical calculation has been performed so far. Recently, Poulsen and Rossky have proposed a practical approximation to multi- variate Wigner distributions, based on a Feynman–Kleinert variational principle. 11 The approximation amounts at the end to the convolution of a centroid surface with a variationally optimized local Gaussian distribution. Following the initial ideas of Hellsing et al. 21 and Frantsuzov et al. 22 for quantum sampling in configurational space, and starting from the Bloch Theoretical Physical Chemistry Group, UMR 8640 CNRS-ENS-UPMC, E ´ cole Normale Supe ´rieure, De ´partement de Chimie, 24 rue Lhomond, 75005 Paris, France. E-mail: [email protected], [email protected]Received 2nd February 2013, Accepted 28th May 2013 DOI: 10.1039/c3cp50493j www.rsc.org/pccp PCCP PAPER Published on 29 May 2013. Downloaded by Mississippi State University Libraries on 09/10/2013 18:04:20. View Article Online View Journal | View Issue
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This journal is c the Owner Societies 2013 Phys. Chem. Chem. Phys., 2013, 15, 12591--12601 12591
Cite this: Phys. Chem.Chem.Phys.,2013,15, 12591
Computing Wigner distributions and time correlationfunctions using the quantum thermal bath method:application to proton transfer spectroscopy
Marie Basire, Daniel Borgis* and Rodolphe Vuilleumier*
Langevin dynamics coupled to a quantum thermal bath (QTB) allows for the inclusion of vibrational
quantum effects in molecular dynamics simulations at virtually no additional computer cost. We
investigate here the ability of the QTB method to reproduce the quantum Wigner distribution of a
variety of model potentials, designed to assess the performances and limits of the method. We further
compute the infrared spectrum of a multidimensional model of proton transfer in the gas phase and in
solution, using classical trajectories sampled initially from the Wigner distribution. It is shown that for
this type of system involving large anharmonicities and strong nonlinear coupling to the environment,
the quantum thermal bath is able to sample the Wigner distribution satisfactorily and to account for
both zero point energy and tunneling effects. It leads to quantum time correlation functions having the
correct short-time behavior, and the correct associated spectral frequencies, but that are slightly too
overdamped. This is attributed to the classical propagation approximation rather than the generation of
the quantized initial conditions themselves.
1 Introduction
The calculation of quantum time correlation functions (TCFs)for complex molecular systems remains a challenging problemin statistical mechanics. A few approximate methods have beenproposed in the literature. One obvious route is to try to relatequantum TCF’s to their classical analogs through variousrelations in frequency space which have the property, at least,of yielding the exact detailed balance condition. Those rela-tions, which have been used for a long time in spectroscopy,1
have been reviewed recently for model systems,2,3 and someextensions have been proposed too.4 In a more direct quantumsimulation perspective, the most successful approaches are thecentroid molecular dynamics5,6 or ring-polymer moleculardynamics7 and the semi-classical initial value representation(SC-IVR) of Miller and collaborators,8,9 including its linearizedversion (LSC-IVR).10 This latter approach is also termed linearizedpath integral (LPI) representation of correlation functions.11–13 ThisLPI approximation leads to a rather transparent interpretation andhas already proved its accuracy for vibrational energy relaxa-tion14 and static and dynamic quantum effects in liquids.15–17
In short, it amounts to replacing operators by their Weyltransform and computing the CF using classical trajectoriessampled initially from the quantum Wigner distribution ratherthan the classical Boltzmann distribution.11,18 For a set ofcoordinates q = {qi} and for a given temperature b = 1/kBT,the Wigner distribution is related to the quantum densitymatrix r(q,q0) = hq|e�bH|q0i by
Wðp; qÞ ¼ 1
ð2p�hÞNQ
Zdu eip�u=�hr q� u
2; qþ u
2
� �; (1)
where Q is the normalization factor (or partition function). Thegeneration of classical trajectories from the quantum Wignerdistribution is also referred to as the classical Wigner model(CWM).10
The exact calculation of the Wigner distribution of a many-body system still remains a formidable task, and although thetheoretical expressions to be evaluated are known,19,20 nopractical calculation has been performed so far. Recently, Poulsenand Rossky have proposed a practical approximation to multi-variate Wigner distributions, based on a Feynman–Kleinertvariational principle.11 The approximation amounts at theend to the convolution of a centroid surface with a variationallyoptimized local Gaussian distribution. Following the initialideas of Hellsing et al.21 and Frantsuzov et al.22 for quantumsampling in configurational space, and starting from the Bloch
Theoretical Physical Chemistry Group, UMR 8640 CNRS-ENS-UPMC, Ecole Normale
Superieure, Departement de Chimie, 24 rue Lhomond, 75005 Paris, France.
12592 Phys. Chem. Chem. Phys., 2013, 15, 12591--12601 This journal is c the Owner Societies 2013
equation in Wigner space, Marinica et al. have proposed anothergeneral method to generate the Wigner distribution usingGaussian phase-space packet (GPSP) propagation in imaginarytime, starting from the known classical Maxwell–Boltzmann athigh temperatures.23 This method coupled to the classicalWigner model to compute time correlation functions is closelyrelated to the thermal Gaussian approximation/linearized semi-classical initial value representation (TGA/LSC-IVR) procedureproposed at about the same time by Liu and Miller,24 and laterapplied successfully to the study of several condensed phasesystems.25 Finally, the recent work of Bonella et al.26–28 deservesspecial mention. There the symmetrized time correlation func-tions are expressed in complex time and systematic correctionsbeyond the classical Wigner approximation, which are amenableto practical computation, are proposed.
Recently, a new approach to include quantum effects hasbeen proposed that amounts to coupling the dynamics ofthe system to a heat bath with an appropriate colored noise,29
a so-called quantum thermal bath. This method adds virtuallyno extra computer cost to a molecular dynamics simulation andcan be employed for small to large systems, and even with first-principle molecular dynamics. This quantum thermal bath hasalready been applied to include quantum effects in a variety ofsituations: study of vibrational spectra of polyatomic mole-cules,30 isotope effects in lithium hydride,31 heat transport innanosystems,32,33 methane under shock34 and nuclear quantumeffects on clusters stability.35 Here, we will study the ability ofthe quantum thermal bath not only to reproduce the quantumdistribution of energy in the vibrational modes, but also toreproduce the exact Wigner distribution for a set of test systems.Then we will use the Wigner distribution generated from thequantum thermal bath to sample initial conditions for the calcula-tion of vibrational time-correlation functions for a model of protontransfer in the gas phase and in solution.36–38 This system exhibitsa very anharmonic character and standard quantum corrections onIR intensities are insufficient to account for tunneling and zeropoint energy effects. There too, the accuracy of the procedure willbe checked against exact results.
The paper is organized as follows. In Section 2 we firstbriefly summarize the quantum thermal bath methodology andperform numerical tests for a harmonic oscillator, a Morseoscillator and a particle in a square box. Section 3 then presentsthe application of the quantum thermal bath to the evaluationof the Wigner distribution and time-correlation function of amodel of proton transfer, first in the gas phase, then in solution.Section 4 concludes the paper.
2 Quantum thermal bath method2.1 Methodology
Here, the quantum thermal bath (QTB) method of Dammak et al.29
is briefly summarized. This technique accounts for quantumstatistics by simply coupling the motion of an ensemble ofcoordinates interacting through a given force field to a fictitiousquantum thermal bath. It is in principle valid at any temperatureand for any interatomic potential as well as for ab initio schemes.
It is based on the general linear response theorem that links thegeneralized resistance to the fluctuations of the generalized forcein quantum dissipative systems and its application to Brownianmotion.39 The equation of motion of the ith coordinate qi,associated with the mass mi, is supposed to obey the followingLangevin-like equation:
miqi = Fi({qi}) � mig:qi + Ri, (2)
where Fi represents the force on coordinate i. g is an effective,and at this point arbitrary, frictional coefficient acting on themotion of every coordinate. The specification of QTB resides inthe definition of the associated random force Ri. Followingthe prescription of Callen and Welton,39 its time correlationfunction is taken to fulfill the following properties:
RiðtÞRjðtþ tÞ� �
¼Z þ1�1
IijðoÞe�iotdo2p; (3)
where Iij(o) denotes the power spectral density of the stochasticforce and relates to the frictional coefficient g by the quantummechanical fluctuation–dissipation theorem:40,41
Iij(o,T) = 2migdijy(o,T) (4)
with dij the Kronecker symbol and
yðo;TÞ ¼ 1
2�hoþ �ho
expðb�hoÞ � 1; (5)
b = 1/kBT. Eqn (2) corresponds to a stochastic equation with acolored noise, instead of the white noise, which would prevail fora simple Langevin equation. The white-noise Langevin process isrecovered in the classical limit, b�ho - 0. In that case it is well-known that particles explore the canonical ensemble.
Note that the same fundamental idea, to employ a colorednoise satisfying the quantum fluctuation–dissipation theorem,was also proposed by Ceriotti et al.42–44 These authors moreoverused a non-Markovian generalized Langevin equation in whichthe noise kernel has a retarded component. This allows greaterflexibility on the type of thermostat, for example to optimizethe sampling. The non-Markovian kernel and the noise arise inthat case from the coupled evolution of a set of additionalcoordinates, and the parameters characterizing the dynamicsof these additional degrees of freedom are fitted to reproduce atbest the quantum fluctuation–dissipation theorem. In contrast,the correct energy distribution for a quantum harmonic oscillatoris readily enforced in the QTB method by specifying the power-spectrum of the colored noise.
2.2 Computational details and numerical tests
The way of numerically solving eqn (2) can be summarized asfollows. We rephrase here the original method suggested byDammak et al.,29 in which the random force acting on eachparticle is prepared in advance, rather than the on-the-flymethod of Barrat et al.45 Following their prescriptions, and theequations derived by Maradudin et al.,46 we begin by definingthe simulation time range interval t A [0,tmax], corresponding toa (periodic) grid that spans from 0 to N � 1 with a time step dt.The corresponding frequencies o = l do, with do = 2p/tmax, are
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labelled in the range l A [0,N/2]. For each coordinate i and foreach frequency l do, two independent Gaussian random num-bers r1 and r2 are generated, providing:
Riðl doÞ ¼ffiffiffiffiffiffiffiffitmax
2
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIijðl doÞ
qðr1 þ ir2Þ (6)
¼ R�i ð�l doÞ (7)
A particular realization of the random force Ri(t) is given over alltime ranges by backward fast Fourier transform of Ri(o). This isdone using the efficient FFTW package,47 and this calculation isrepeated for all coordinates. With the knowledge of the set {Ri(t)},the coupled equations of motion are solved using the standardvelocity Verlet algorithm over the range [0,tmax]. Dammak et al. havepointed out that g should be considered as a technical dispersiveparameter of the molecular dynamics,29 which is introduced tofulfill the quantum fluctuation–dissipation theorem.40,41
To better understand the role played by this key parameterand the values to be recommended, we have first considered thesempiternal case of a one-dimensional harmonic oscillator atfrequency o. Its Wigner distribution function is indeed known:
Wðp; qÞ ¼ tanhðf Þp�h
e� tanhðf Þ mo�h q2þ 1
�hmop2
� �(8)
with f = 12b�ho. This function can obviously be separated into
normalized q- and p-contributions, denoted Wq(q) and Wp(p)respectively. The total energy at a given temperature is equalto E = kbT*, where T* = �ho/2kb tanh(f), and in the classicallimit, obviously E = kbT. To test our code and infer the bestchoices for g, we have computed E, Wq(q) and Wp(p) in the QBTframework, for different temperatures and frequencies, andhave analyzed the convergence of the results with respect to theexact theoretical values. For further applications to protontransfer potentials, we have taken a mass of 1 amu, a frequencyo = 3000 cm�1 (0.565 fs�1) and various temperatures. For suchfrequencies, a time step dt = 0.5 fs, reminiscent of the standardintegration step in molecular dynamics codes, and lying at thesuperior limit of the condition odt/2p { 1, is appropriate. InFig. 1, we present an analysis of the convergence of the oscillatorenergy at T = 100 K for various friction parameters g, as obtainedfrom trajectories of total duration 2 � 107 dt. The error bars wereestimated by a block analysis with a time window t going up totmax = 107 dt with a 95% confidence interval. It can be seen thatfor a large friction g = 0.2 fs�1 (g/o = 0.35), the convergence is fastbut the converged value deviates from the exact one. As noted byBarrat and Rodney,45 the exact energy is only recovered in thelimit g/o - 0. For smaller frictions, the asymptotic value doesget closer to the exact result but the convergence is slower, due toa smaller interaction with the bath and a longer equilibrationtime. For g = 2 � 10�5 fs�1, error bars are still of the order of25% after 10 million time steps. We find a good compromise forg = 2 � 10�3 fs�1, i.e. g/o = 3.5 � 10�3, which is compatible withthe value prescribed in ref. 45.
The same conclusion can be reached from the approximateprojected Wigner distribution calculated by direct sampling of thepositions and velocities along the stochastic trajectories; see Fig. 2.
Substantial deviations from the exact curves are observed forg = 0.2 fs�1, whereas they have disappeared at the scale ofthe figure for g = 2 � 10�5 fs�1. The intermediate value g = 2 �10�3 fs�1 thus appears again as an acceptable compromise,with small although sizable differences, and it will be retainedfor the remaining of this work. It can already be learned fromthis paradigmatic example of proton potential that convergenceof the results requires a few million time steps and simulationtimes of the order of a few hundred picoseconds. This isrequired by the fact that the chosen friction should be relativelysmall to generate the correct quantum properties (in the white-noise classical case, the friction can be increased to acceleratethe convergence, since any friction will generate the correctMaxwell–Boltzmann distribution). This gives some clues forfurther application to condensed phase systems.
The influence of anharmonicity is explored next. For this weintroduce a generic ‘‘bound’’ proton potential in the form of a
Fig. 1 Cumulated average of the total energy of a harmonic oscillator as a functionof simulation time. Exact numerical results (plain line) and QTB results (square) witherror bars (triangles up and down) are figured for several friction coefficients g.
Fig. 2 (left) Integrated Wigner function Wq(q) for a harmonic oscillator at T =100 K, computed exactly (plain line) and with QTB for different g values (squares).(right) Same for the integrated Wigner function Wp(p).
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Morse potential plus a confining potential on the dissociatingside in order to prevent dissociation at high temperatures:
V(q) = D � [e�2aq � 2e�aq] + Dz (9)
z denotes a factor being equal to 1 for q r 2.5 Å and toe10�(q�2.5) if q > 2.5 Å. For such potential, anharmonicity canbe measured by the parameter w ¼ �ha=
ffiffiffiffiffiffiffiffiffiffi2mDp
. Taking typicallya = 2.5 �1, we have compared two values of the potential depthparameter D, 80 kcal mol�1 and 20 kcal mol�1, correspondingto wC 0.11, close to the harmonic case, and wC 0.22, to study amore anharmonic case.
In Fig. 3, the total energies obtained by averaging theinstantaneous energy over a 2 ns-long QTB trajectory, Etotal =hV(q)i + hp2/2mi, are compared to exact numerical calculationsfor a range of temperature going from the classical (T = 3000 K) tothe quantum (T = 100 K) regime. Those values were obtained bysolving the Schrodinger one-dimensional differential equationusing Numerov’s method.48 It can be observed that QTB resultsare following closely the exact ones for both cases and over theentire range of temperatures. The error bars were again computedusing a block analysis with a time window of 1 ns. The largestdiscrepancies appear for the D = 20 kcal mol�1 barrier, which isconsistent with the larger anharmonicity and the fact that noisefunction and friction parameter g are strictly defined for a harmonicoscillator. The systematic error does not exceed 8.5%, however,whereas it is 4% in the more harmonic case. Those errors remain inthe range of the estimated error bars at all temperatures. Energiesseem thus to be calculated correctly for moderately anharmonicpotentials (for more pronounced anharmonicities, see below). Suchconclusion were already reached in the literature,41 and even formultidimensional molecular systems.30
In complement to that, let us examine the Wigner distribu-tions. W(p,q) cannot be derived analytically for our modifiedMorse potential but can be computed numerically from theeigenvectors and eigenvalues determined by the Numerovmethod.48 This function is represented in 2D and compared tothe QTB results in Fig. 4 for the quantum case, T = 100 K, and
anharmonic regime, D = 20 kcal mol�1. The 2D-maps do comparewell, with a maximum of the function centered at q C 0 Å, andsimilar expansions in both q and p directions. Even visually,however, differences can be seen, especially in the large q tail.
To be more quantitative a comparison can be done for the qand p integrated Wigner functions. The numerical exact calcu-lations (dashed lines) of Wq(q) are compared to the QTB results(plain lines) in Fig. 5(a), for both the quantum and classicalregimes. As expected, a very good agreement is obtained in theclassical case, as already mentioned in Fig. 3, even in the long-qtail corresponding to the high energy Morse plateau and the zrepulsive part that are explored at 3000 K (see inset of Fig. 5). Inthe quantum regime, corresponding to a much lower tempera-ture, the dissociative part of the Morse potential is not reached.It is seen however that in the QTB calculation the long-distancetail appears slightly above the exact one, which seems to be ageneral trend of QTB: penetration in repulsive potential regionsis overestimated. The maximum of Wq(q) in QTB is also down-shifted by 0.03 Å. As for the integrated p-distribution, Wp(p), it is
Fig. 3 Total energy of a confined Morse oscillator with respect to temperature.Exact numerical results (dashed lines) and QTB results (squares) are figured forboth D = 20 kcal mol�1 (black) and D = 80 kcal mol�1 (red) barrier.
Fig. 4 2D contours of the Wigner distribution W(p,q) at 100 K as functions ofposition q and momentum p. Top: QTB result, bottom: exact calculation.
Fig. 5 (a) Integrated Wigner function Wq(q) for a confined Morse potential withD = 20 kcal mol�1, computed exactly (dashed lines) and from QTB simulations(plain), for T = 100 K (black) and T = 3000 K (red). The inset presents anenlargement of the very low probability region. (b) Same for the integratedWigner function Wp(p).
This journal is c the Owner Societies 2013 Phys. Chem. Chem. Phys., 2013, 15, 12591--12601 12595
seen in Fig. 5(b) that QTB is doing a very good job even in thequantum case.
This statement is lessened, however, when one examines theconditional Wigner distributions Wc(p;q) = W(p,q)/Wq(q) fordifferent values of q; see Fig. 6. In the exact calculation, theprofiles keep a pronounced Gaussian character but their widthvaries with q, whereas they are independent of q for QTB:in that case, the probability W(p,q) is exactly factorized intoWq(q) � Wp(p), which is obviously an approximation and alimitation of the method. The case of a flatter q4 symmetricpotential was already studied in ref. 41 and will be somehowreproduced below by a typical proton transfer potential alonga strong H-bond. We will also face incidentally the case of adouble-well potential.
Here, to push the QTB theory even further into its limits, farfrom any harmonic reference, we have studied the simplisticcase of a quantum particle (a proton) in a one-dimensional boxof size 2 Å. The results are presented in Fig. 7. Of course, in thehigh temperature regime at T = 3000 K, the colored noisesimulation yields the classical limit given by the Maxwell–Boltzmann distribution. It should be noted in passing thateven for this very high temperature, the exact Wq(q) departsfrom the strictly classical behavior very close to the walls, whichyields a sizable difference in the value of the probabilitymeasured inside the box. The situation is very different inthe quantum regime, at low temperature. Wq(q) computed byQTB turns out to be identical to the classical Boltzmanndistribution, with a uniform probability inside the box,whereas the exact function presents a bell shape with a slightpenetration beyond the infinite walls. Regarding Wp(p), theQTB distribution does exhibit the correct Gaussian-likeshape but it is broader than both the Boltzmann and the exactones and thus allows momenta that are too large. This differ-ence is further emphasized by the q-dependent conditionalprobabilities Wc(p;q) presented in Fig. 8. The QTB results areas before independent of q, whereas the exact ones show a largevariation and exhibit oscillations that QTB cannot capture.
It seems then that limits of the QTB method are reached here.Obviously, such a flat potential model is irrelevant to describeatomic motions in molecules. Nevertheless, the failure ofQTB in this case is a useful lesson for further applications tosystems with soft translational modes. We now go back to themain purpose of this paper and turn to a (multidimensional)generic model for proton transfer dynamics in the gas phaseand solution.
3 Application to proton transfer systems3.1 Gas-phase model
For the intramolecular part, we consider the following genericpotential for symmetric/asymmetric A–H–B hydrogen bonds. Itsform is a mixture of the H-bond potential introduced by
Fig. 6 p conditional Wigner distributions for different values of q for a Morsepotential with D = 20 kcal mol�1 at T = 100 K. Exact functions (dashed lines) arecompared to QTB calculations (solid lines).
Fig. 7 Integrated Wigner functions for a particle in a square box as a function ofq (resp. p) for T = 100 K (a) (resp. (c)) and T = 3000 K (b) (resp. (d)). The Boltzmanndistribution (dashed black line) is compared to exact (plain black line) and QTB(red line) calculations.
Fig. 8 p conditional Wigner distributions for different values of q for a 1Dsquare box of size 2 Å at T = 100 K. Exact functions (dashed lines) are comparedto QTB calculations (solid lines).
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Lippicott and Schroeder49,50 and the double-Morse model ofMatsushita and Matsubara51 and is given by:
Vðq;QÞ ¼ D e�2aQ2þq�d
� 2e�a
Q2þq�d
þ 1
� �
þDx2 e�2ax
Q2�q�d
� 2e
�axQ2�q�d
� �
þ Ae�BQ � C
Q6:
(10)
q denotes the distance of the proton from the center of the A–Bbond, whereas Q denotes the A–B distance. We take belowparameters that are typical of O–H–O proton transfer systems.For x = 1, the proton potential is symmetric and the parameterschosen for the Morse potentials (D = 60 kcal mol�1, d = 0.95 Å,a = 2.52 Å�1) yield a barrierless potential for Q o 2.45 Å and aproton frequency of 3750 cm�1 at infinite O–O distances. Theother parameters (A = 2.32 � 105 kcal mol�1, B = 3.15 Å�1, C =2.31 � 104 kcal mol�1 Å6) are adjusted to yield an equilibriumO–O distance of 2.4 Å, an energy of formation of 35 kcal mol�1,and a O–O force constant of 320 kcal mol�1 Å�2, as for a H5O2
+
dimer.52 These parameters are typical of a O–H+–O protontransfer system.36,37 Various cuts of the potential at fixed Qare plotted in Fig. 9 for a Q-value close to equilibrium (Q =2.45 Å), for a shorter and for a longer distance. For x = 0.707 andthe same other parameters, the potential mimics a protontransfer along a weaker asymmetric O–H–N with an equili-brium O–N distance of 2.75 Å and a charge separation processO–H–N - O�–H+–N,38 or an asymmetric intramolecular O–H+–O proton transfer, in which the molecular frame maintains theequilibrium O–O distance beyond its natural, intermolecularvalue. In any case, the above potential reproduces the strongmodulation of the A–H frequency by the A–B distance, anessential ingredient for describing the important band broad-ening of A–H vibrators observed in condensed phases.
We will mainly focus below on the symmetric case, encom-passing the paradigmatic example of the protonated waterdimer H5O2
+, and at the origin of spectacular band broadening,
covering the whole spectrum domain from 1000 to 4000 cm�1,that can be observed for proton transfer systems involvingstrong H-bonds.
Before proceeding to a multidimensional model in solution,we begin by testing the performances of QTB for fixed Qconfigurations, corresponding to the very flat proton potential(Q = 2.45 Å) or the double-well potential (Q = 2.75 Å) of Fig. 9.For those situations, in addition to the classical simulation andexact quantum reference, we can also compare the QTB one-dimensional simulation results to those of the Gaussian phasespace packet (GPSP) propagation method of Marinica et al., alsobased on statistics over phase-space trajectories (although inimaginary time).23 This comparison is illustrated in Fig. 10and 11. For the flat potential, in between the q4-potential andthe confinement in a box, not surprisingly QTB yields a Wq(q)that is too flat and lacks density at the center, whereas Wp(p)appears slightly overestimated. GPSP appears somehow betterfor both functions. For the double-well case, not studied yet inthe literature to our knowledge, we find that QTB does repro-duce quite correctly the penetration across the B10 kcal mol�1
barrier, which is attributed to tunneling effects since, as seen inFig. 11, the classical Boltzmann distribution per se is close tozero in the barrier region. As mentioned earlier for other cases,the quantum penetration tends to be overestimated by QTB.The marginal p-distribution presents, again, the correctGaussian-like shape but appears slightly too wide. Here again,GPSP seems to do, overall, a bit better, with the right shift of thedouble peaks for Wq(q) (but a density that is underestimated atthe barrier), and a better reproduced peak for Wp(p). At thisstage however, no spectacular difference makes it possible todraw a conclusion in favor of one method or the other. GPSPdoes include by essence the variation of the p conditionaldistribution with q that is not accounted for in QTB (see above).It will be interesting in the future to continue such comparisonfor other properties and for multidimensional systems, as wellas in terms of numerical efficiency.
Fig. 9 Potential energy as a function of proton position q for a fixed O–Odistance close to equilibrium (Q = 2.45 Å), and for a shorter and a longer one.
Fig. 10 Integrated Wigner functions as a function of q (a) and p (b) for theproton potential of eqn (10) with Q = 2.45 Å: classical Boltzmann (dashed, black)and exact numerical (plain, black) results are compared to GPSP (green) and QTB(red) approaches.
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3.2 Proton transfer in solution: a 3-dimensional model
3.2.1 Hamiltonian. We now consider the coupling of theproton transfer system described by the gas-phase two-dimensional potential V(q,Q) to a polar solvent. To this end,another important ingredient of the model is the variation ofthe charge distribution when the proton is displaced from Ato B. As in ref. 36 and 38, the charge variation is obtained byassigning a set of coordinate-dependent point charges on thesites a = A, H, B, in the form
ea(q) = (1 � f (q,Q))eRa + f (q,Q)eP
a (11)
where eR,Pa is the value of the charge for the reactant (AH+–B) or
the product species (A–H+B) and f (q,Q) is a switching functiondefined by
f ðq;QÞ � f ðqÞ ¼ 1þ tanhðq� qyðQÞÞ=DqÞ2
: (12)
q†(Q) represents the maximum (for a double well) or minimum(for a single well) of V(q,Q) for a given Q. Typically, Dq = 0.125 Å.The associated dipole is defined by
mðq;QÞ ¼ ðeBðqÞ � eAðqÞÞQ
2þ eHðqÞq: (13)
For a symmetric O–H+–O transfer, one has by symmetry q†(Q) =0 for all Q’s and eP
A = eRB, eP
H = eRH and eP
B = eRA. As in ref. 36, we will
use below three different sets of charges, denoted T1 to T3 anddescribed in Table 1. When going from charges T1 to T3, thestrength of the electrostatic coupling to the solvent is increased.In agreement with previous studies on proton transfer pro-cesses in solution,37,53,54 the solvent is described by a singlevariable E that can be identified to the fluctuating electric fieldexerted by the solvent at the center of the complex in thedirection of the A–H–B dipole. It has been verified many timesby molecular dynamics that such a variable has remarkableGaussian properties.
Thus, including the electrostatic coupling of the solute tothe solvent, the Hamiltonian of the system can be decomposedinto a molecular, vibrational part, and a solvent part:
H ¼ PE2
2mEþ 1
2mEoE
2E2 þHvibðEÞ
HvibðEÞ ¼pq
2
2mqþ PQ
2
2mQþ Vðq;QÞ � mðq;QÞE:
(14)
Here mq = 1 amu, mQ = 16 amu; mE and oE define a mass andfrequency associated with the solvent electric field. Thoseparameters were adjusted as follows. We took oE = 30 cm�1,a typical value for slow orientational modes in water. Wefurther varied the mass mE to get, in either the QTB simulationwith the T1 set of charge or the corresponding exact referencecalculations, a mean-square fluctuation of the electric field
hdE2i12 equal to 5.75 MV m�1, the value estimated recently by
Spezia et al.55 by molecular dynamics at the center of a neonatom in water.
For this generic Hamiltonian describing environment-inducedproton transfer, and with those parameters, we have computedbelow the corresponding infrared spectra using QTB simulations.We compare them again to exact calculations in order to measurethe ability of such a simple method to capture rather strongquantum effects in systems of experimental relevance, wherethe modulation by the environment has a key effect on theobserved spectroscopic properties.
3.2.2 Calculation of IR spectra using QTB simulations. Forthe implementation of the QTB method, only very few changesare necessary with respect to the one-dimensional simulations ofSection 2.2. The following equations of motion are considered:
mq€q ¼ f qðq;Q;EÞ þ Ri �mqg _q
mQ€Q ¼ f Qðq;Q;EÞ þ RQ �mQgQ _Q
mE€E ¼ f Eðq;Q;EÞ þ RE �mEgE _E
8>>><>>>:
(15)
fq,Q,E denotes the force acting on each coordinate, thus thepartial derivative of H with respect to that coordinate. We didnot try to optimize the friction coefficient for each coordinateand took the unique value gq = gQ = gE = 1/500 fs�1, which wasprescribed in Section 2.2 and shown to cover both the classicaland quantum regimes. On the other hand, the electric field E,characterized by a very low frequency, can be considered as avery slow modulation variable and taken in the classical limitwith a strictly delta-correlated white noise RE. For this coordi-nate, the recent implementation of the Verlet equations withfriction from ref. 56 was used. The other parameters chosen forthe colored noise simulations are those presented in Section 3.1.
Fig. 11 Integrated Wigner functions as a function of q (a) and p (b) for theproton potential of eqn (10) with Q = 2.75 Å: classical Boltzmann (dashed, black)and exact numerical (plain, black) results are compared to GPSP (green) and QTB(red) approaches.
Table 1 Sets of charges corresponding to complexes with increasing dipolechange Dm upon proton transfer
12598 Phys. Chem. Chem. Phys., 2013, 15, 12591--12601 This journal is c the Owner Societies 2013
Those simulations make possible the sampling of the Wignerdistribution for the 3 variables, W(pq,q,PQ,Q,PE,E). Since E isclassical, this function can be recast in terms of a Boltzmanndistribution in (PE,E):
Wðpq; q;PQ;Q;PE;EÞ ¼1
Z
ffiffiffiffiffiffiffiffiffiffiffiffiffib
2pmE
se�bP2
E2mEe
�bE2
hE2i
�Wcðpq; q;PQ;Q;EÞ;
(16)
where Wc denotes the conditional Wigner distribution for theinternal coordinates q,Q at fixed solvent configuration E, andZ ¼
RdEZE with
ZE ¼ e� bE2
2hE2iZ
dqdpq dQ dPQ Wcðpq; q;PQ;Q;EÞ
¼ e� bE2
2hE2iXn
e�benðEÞ
(17)
where en(E) are the eigenenergies of the Hamiltonian Hvib(E)in eqn (15). Those, and the associated wave-functions, werecomputed with high numerical precision on a predefined gridof E-values by solving the corresponding 2D Schrodingerequation using a generalized Gauss quadrature method.57 Inorder to test and compare the various procedures, we report inFig. 12 the classical electric field distributions P(E) = ZE/Zobtained, for the three sets of charges T1–T3 of Table 1, eitherfrom the knowledge of the exact Wc(pq,q,PQ,Q;E) distribution ineqn (16) (or eigenvalues in eqn (17)) or sampled directly fromthe QTB simulations. It is seen that those distributions are invery close agreement for the lowest solute–solvent coupling, T1,and remain in fair agreement for the stronger ones, on accountof the sampling uncertainties. The figure illustrates also theinterplay between solvent fluctuations and proton-solvent cou-pling: increasing the coupling makes the solvent more tightlybound and decreases its fluctuations.
We focus from now on the calculation of the infraredspectrum of the system, defined as the Fourier transform ofthe dipole operator time correlation function (TCF), hm(0)m(t)i,where the dipole operator depends on the internal variables(q,Q), as defined in eqn (13). The calculations were performedin the so-called classical Wigner model (CWM) approximationin which classical dynamics trajectories are generated on top ofthe (approximate or exact) quantum Wigner distribution.10,11
Quantum dynamical effects are thus neglected. To do so withinthe QTB formalism, initial configurations, including coordi-nates (q,Q,E) and momenta pq,PQ,PE, were extracted every10 000 steps of 50 independent 1 ns-long stochastic simula-tions, and from each of them short classical dynamics wererun, keeping only the white-noise friction on the classicalvariable E. The dipole correlation function was computed byaveraging over initial conditions and classical trajectories. Foreach initial condition, we have also run classical trajectorieswith the electric field E kept frozen at its initial value. Theobtained dipole TCF’s are absolutely identical, confirming thatE can be considered as a slow modulation variable for q and Q(the so-called slow modulation limit, or inhomogeneous broad-ening limit, in vibrational spectroscopy).58,59 The equivalent‘‘exact’’ TCF’s were then obtained as
mð0ÞmðtÞh i ¼ 1
Z
ZdE e
� bE2
2hE2i mð0ÞmðtÞh iE (18)
where h�iE indicates the exact quantum correlation functionscomputed from the energies and wave-functions resulting fromthe solution of the two-dimensional Schrodinger equation forHvib(E) (here obtained with a Gauss quadrature,57 with 8quadrature points along the Q-direction and 32 points alongthe q-direction).
In Fig. 13, we have plotted the dipole TCF’s obtained withQTB and with the reference quantum calculation describedabove, and this for different degrees of coupling to the solvent(charges T1 to T3). The initial values at t = 0 are seen to beidentical at the scale of the figure, meaning that QTB offers a
Fig. 12 Normalized probability of E for three different couplings to solvent.Distributions in plain lines correspond to the reference quantum formula ZE/Zdefined in eqn (17). Dotted curves result from the sampling of the three-variableQTB trajectories.
Fig. 13 Dipole time correlation functions for a symmetric proton transfer systemand a variety of electrostatic couplings to solvent. Exact correlation functions areplotted as dashed lines, while functions obtained from QTB simulations are inplain lines.
This journal is c the Owner Societies 2013 Phys. Chem. Chem. Phys., 2013, 15, 12591--12601 12599
suitable exploration of configurational space and, overall,includes quantum effects correctly. The initial time decays upto 10 fs are identical too, meaning that initial velocities aregenerated correctly too, on average. The further evolution is thesame for all couplings: the subsequent oscillations appearoverdamped in the QTB-CWM calculations and they areimpaired by a progressive dephasing with respect to the quan-tum results. This shortcoming cannot be attributed to QTB,which seems to generate the Wigner distribution with sufficientaccuracy to yield the correct short time limit of the TCF’s (wherethe CWM approximation is known to be accurate), but is ratherdue to the CWM, classical dynamics approximation itself, thatincorporates the dynamical anharmonic effects only partially.This is therefore the key weak point to be improved in futurework, and indeed the source of several recent efforts.26–28 Wehave also found that, the friction coefficient adopted for theQTB dynamics being small, it has little effect on the correlationfunctions at the picosecond time scale displayed in Fig. 13, aswell as on the associated spectra described below. As a con-sequence, the correlation functions can be computed indiffer-ently on-the-fly, in the course of the colored noise dynamics.This does not correct however the lack of quantum coherencesinherent to the CWM method.
The performance and shortcomings can also be discussed interms of the IR spectra. For the symmetric proton transfer casejust discussed, the IR spectrum is displayed in Fig. 14 for theset of charges T2. In conformity to experimental spectra forsolvated complexes involving a strong symmetrical O–H+–Obond, a continuum appears to cover the range 1000–2000 cm�1.Compared to the exact spectrum computed from the Hamiltonianeigenenergies and eigenfunctions, the QTB/CWM spectrumexhibits obviously the correct width in this region, but it doesnot quite present the correct double-band structure with a finemain peak emerging at B1100 cm�1 and a secondary onearound B1500 cm�1. The low frequency band at B400 cm�1,associated to the Q-motion, is well reproduced, as are also
the small combination bands in the B600–800 cm�1 inter-mediate region.
Fig. 15 illustrates the case of an asymmetric O–H+–O protontransfer system, corresponding to x = 0.707 in the definition ofthe gas-phase potential V(q,Q) of eqn (10), and otherwise thesame charges and solvent parameters. Here the proton signatureappears as for typical weakly H-bonded complexes in solutionas a broad band in the range B2000–3000 cm�1, and the samekind of conclusions can be drawn, not to mention an evenbetter reproduction of the low frequency Q-band: the associatedmotions at C200 cm�1 fall in the classical regime for which theclassical Wigner method becomes exact.
We note finally that for both the symmetric and asymmetriccases of Fig. 14 and 15, the calculation of the IR spectra usinga purely classical white-noise Langevin dynamics gives high-frequency bands that cover identical regions but with a muchtoo low intensity.
4 Concluding remarks
The quantum thermal bath method proposed by Dammaket al.29 is a very appealing numerical procedure, based on aclear (although not easily improvable) quantum linear responseapproximation, for incorporating quantum nuclear effects inmany body molecular dynamics simulation in a straightforwardmanner, and at basically the cost of classical trajectories. It wasalready tested successfully for morse or quartic one-dimensionalpotentials,29,41 as well as for several many-body molecularsystems presenting moderate anharmonicities, such as solids31,45
or non-dissociative polyatomic molecules.30 Applications to lowtemperature clusters are promising too.35 In this paper, we haveextended the tests of QTB to systems involving large anharmonicitiesand a strong, nonlinear coupling to an external bath: this is the casefor proton-transfer complexes in solution that exhibit well-known,although still intriguing, vibrational spectroscopic features, due toenhanced proton mobility. We have also focused on a property thatwas not explored thoroughly in the past: the ability of the method togenerate the correct quantum Wigner distribution in phase space,
Fig. 14 IR spectrum for a symmetric proton transfer complex deduced byFourier transform of dipole correlation functions of Fig. 13. The quantum-mechanical spectrum is plotted in black, whereas the one obtained from QTBsimulations is in red.
Fig. 15 Same as Fig. 14 for an asymmetric proton transfer complex (see text).
12600 Phys. Chem. Chem. Phys., 2013, 15, 12591--12601 This journal is c the Owner Societies 2013
and thus, at least, the correct initial phase-space distribution for thecomputation of correlation functions and associated spectra. Simpletests to Morse-like or quartic-like one-dimensional potential confirmthe overall ability to yield correct averaged quantities, such asenergies, as well as correct marginal (integrated) Wigner distribu-tions. An approximation inherent to QTB is, however, the factoriza-tion of the total Wigner distribution into position and momentumcomponents, so that the p-conditional distribution does not dependon q, as it should. Furthermore the convergence times inherent tothe equilibration of proton motions, limited by the conditiong/o { 1, appear to be rather long, in the nanosecond range, andmay counter-balance, in terms of numerical efficiency, the extremesimplicity of the method. We have also pushed the method tointrinsically non-harmonic potentials: a double-well or an infinitelyrepulsive square well. In the former case, QTB is doing surprisinglywell and does describe properly proton tunneling across the potentialbarrier, although the penetration in the barrier region is slightlyover-estimated. For the square well, the method does reach its limit,with no harmonic reference justifying the approach whatsoever.
We have then studied a three-variable model pertinent toproton-transfer complexes in a fluctuating polar environment. Weconsidered either the weak H-bond case with a strong modulationof the O–H frequency by both the H-donor–acceptor distance andthe solvent, or the strong H-bond case with a delocalized protonand solvent-induced proton transfer. Focusing on the computationof the dipole operator time correlation function, and associated IRspectrum, we have coupled QBT simulations for generating initialphase-space conditions to classical trajectories for the time evolu-tion (QTB-CWM approximation). Compared to exact quantumcalculations, the initial value and the short time behavior of thecorrelation function are reproduced accurately, proving thatthe QTB trajectories provide on average, an accurate sampling ofthe quantum initial conditions. Disagreements appearing at longertimes come obviously from the classical dynamics approximation.The introduction of semi-classical dynamical corrections (like inthe initial value representation10), which will be made consistentwith the generation of the initial Wigner distribution usingQTB trajectories, will thus be our next step of research. Anotherone will be the application of this consistent formalism to thestudy of quantum vibrational effects in water and for protontransfer processes in water.
Acknowledgements
The authors are grateful to H. Dammak and M. Hayoun forsharing their notes and codes concerning the generation ofthe Gaussian colored noise. They also acknowledge financialsupport from the Agence Nationale de la Recherche under grantANR-12-BS08-0010-02.
References
1 J. Borysow, M. Moraldi and L. Frommhold, Mol. Phys., 1985,56, 913.
2 S. A. Egorov, K. F. Everett and J. L. Skinner, J. Phys. Chem. A,1999, 103, 9494.
3 R. Ramirez, T. Lopez-Ciudad, P. Kumar and D. Marx,J. Chem. Phys., 2004, 121, 3973.
4 H. Kim and P. J. Rossky, J. Phys. Chem., 2002, 106, 8240.5 J. Cao and G. Voth, J. Chem. Phys., 1994, 100, 5106.6 S. Jang and G. A. Voth, J. Chem. Phys., 1999, 111, 2357.7 I. R. Craig and D. E. Manolopoulos, J. Chem. Phys., 2004,
125, 3368.8 H. Wang, M. Thoss and W. H. Miller, J. Chem. Phys., 1999,
112, 47.9 M. Thoss, H. Wang and W. H. Miller, J. Chem. Phys., 2001,
114, 9220.10 W. H. Miller, J. Phys. Chem., 2001, 105, 2942.11 J. A. Poulsen, G. Nyman and P. J. Rossky, J. Chem. Phys.,
2003, 119, 12179.12 S. Bonella and D. Coker, J. Chem. Phys., 2005, 122,
194102.13 M. S. Causo, G. Ciccotti, D. Montemayor, S. Bonella and
D. Coker, J. Phys. Chem. B, 2005, 109, 6855.14 Q. Shi and E. Geva, J. Chem. Phys., 2003, 107, 9070.15 J. A. Poulsen, G. Nyman and P. J. Rossky, J. Phys. Chem. A,
2004, 108, 8743.16 J. A. Poulsen, G. Nyman and P. J. Rossky, J. Phys. Chem. B,
2004, 108, 19799.17 J. A. Poulsen, G. Nyman and P. J. Rossky, Proc. Natl. Acad.
Sci. U. S. A., 2005, 102, 6709.18 M. S. Causo, G. Ciccotti, D. Montemayor, S. Bonella and
D. Coker, J. Phys. Chem. B, 2005, 109, 6855.19 V. Filinov, S. Bonella, Y. Lozovik, A. V. Filinov and I. Zacharov,
Classical dynamics in condensed phase simulations, WorldScientific, Singapore, 1998.
20 V. Filinov, P. Thomas, M. Bonitz, V. Fortov, I. Varga andT. Meier, Phys. Status Solidi, 2004, 241, 40.
21 B. Hellsing, S. I. Sawada and H. Metiu, Chem. Phys. Lett.,1985, 122, 303.
22 P. A. Frantsuzov and V. A. Mandelstham, J. Chem. Phys.,2004, 121, 9247.
23 D. C. Marinica, M.-P. Gaigeot and D. Borgis, Chem. Phys.Lett., 2006, 423, 390–394.
24 J. Liu and W. H. Miller, J. Chem. Phys., 2006, 125, 224104.25 J. Liu and W. H. Miller, J. Chem. Phys., 2007, 127, 114506.26 S. Bonella, M. Monteferrante, C. Pierleoni and G. Ciccotti,
J. Chem. Phys., 2010, 133, 164104.27 S. Bonella, M. Monteferrante, C. Pierleoni and G. Ciccotti,
J. Chem. Phys., 2010, 133, 164105.28 M. Monteferrante, S. Bonella and G. Ciccotti, Mol. Phys.,
2011, 109, 3015.29 H. Dammak, Y. Chalopin, M. Laroche, M. Hayoun and
J.-J. Greffet, Phys. Rev. Lett., 2009, 103, 190601.30 F. Calvo, N.-T. Van-Oanh, P. Parneix and C. Falvo, Phys.
Chem. Chem. Phys., 2012, 14, 10503–10506.31 H. Dammak, E. Antoshchenkova, M. Hayoun and
F. Finocchi, J. Phys.: Condens. Matter, 2012, 24, 435402.32 G. Y. Panasyuk, G. A. Levin and K. L. Yerkes, Phys. Rev. E:
Stat., Nonlinear, Soft Matter Phys., 2012, 86, 021116.33 A. V. Savin, Y. A. Kosevich and A. Cantarero, Phys. Rev. B:
This journal is c the Owner Societies 2013 Phys. Chem. Chem. Phys., 2013, 15, 12591--12601 12601
34 T. Qi and E. J. Reed, J. Phys. Chem. A, 2012, 116, 10451–10459.35 F. Calvo, F. Y. Naumkin and D. J. Wales, Chem. Phys. Lett.,
2012, 551, 38–41.36 D. Borgis, G. Tarjus and H. Azzouz, J. Phys. Chem., 1992,
96, 3188.37 D. Borgis, G. Tarjus and H. Azzouz, J. Chem. Phys., 1992,
97, 1390.38 H. Azzouz and D. Borgis, J. Chem. Phys., 1993, 98, 7361.39 H. B. Callen and T. A. Welton, Phys. Rev., 1951, 83, 34.40 A. H. Barrozo and M. de Koning, Comment, Phys. Rev. Lett.,
2011, 107, 198901.41 H. Dammak, Y. Chalopin, M. Laroche, M. Hayoun and
J.-J. Greffet, Reply, Phys. Rev. Lett., 2011, 107, 198902.42 M. Ceriotti, G. Bussi and M. Parrinello, Phys. Rev. Lett., 2009,
102, 020601.43 M. Ceriotti, G. Bussi and M. Parrinello, Phys. Rev. Lett., 2009,
103, 030603.44 M. Ceriotti, G. Bussi and M. Parrinello, J. Chem. Theory
Comput., 2010, 6, 1170–1180.45 J.-L. Barrat and D. Rodney, J. Stat. Phys., 2011, 144, 679–689.46 A. Maradudin, T. Michel, A. McGurn and E. Mendez, Ann.
Phys., 1990, 203, 255–307.
47 M. Frigo and S. G. Johnson, Proc. IEEE, 2005, 93, 216–231.48 E. Hairer, S. P. Nørsett and G. Wanner, Solving ordinary
differential equations I: Nonstiff problems, Springer-Verlag,Berlin, New York, 1993.
49 E. R. Lippincott and R. Shroeder, J. Chem. Phys., 1955,23, 1099.
50 C. Reid, J. Chem. Phys., 1959, 30, 182.51 E. Matsushita and T. Matsubara, Prog. Theor. Phys., 1982,
67, 1.52 R. Vuilleumier and D. Borgis, J. Chem. Phys., 1999,
111, 4251.53 H. Azzouz and D. Borgis, J. Mol. Liq., 1995, 63, 89.54 J. D. Eaves, A. Tokmakoff and P. L. Geissler, J. Phys. Chem. A,
2005, 109, 9424.55 R. Spezia, C. Nicolas, A. Boutin and R. Vuilleumier, Phys.
Rev. Lett., 2003, 91, 208304.56 E. Vanden-Eijnden and G. Ciccotti, Chem. Phys. Lett., 2006,
429, 310–316.57 B. I. Schneider and N. Nygaard, J. Phys. Chem. A, 2002, 106,
10773–10776.58 S. Bratos, J. Chem. Phys., 1975, 63, 3499.59 G. N. Robertson and J. Yarwood, Chem. Phys., 1978, 32, 267.
