feynman diagram description of 2d-raman-thz spectroscopy ... · an intuitive feynman diagram...

J. Chem. Phys. 150, 044202 (2019); https://doi.org/10.1063/1.5079497 150, 044202

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Feynman diagram description of 2D-Raman-THz spectroscopy applied to water

Cite as: J. Chem. Phys. 150, 044202 (2019); https://doi.org/10.1063/1.5079497Submitted: 31 October 2018 . Accepted: 18 December 2018 . Published Online: 29 January 2019

David Sidler, and Peter Hamm

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Feynman diagram description of 2D-Raman-THzspectroscopy applied to water

Cite as: J. Chem. Phys. 150, 044202 (2019); doi: 10.1063/1.5079497Submitted: 31 October 2018 • Accepted: 18 December 2018 •Published Online: 29 January 2019

David Sidler and Peter Hamm

AFFILIATIONSDepartment of Chemistry, University of Zurich, Zurich, Switzerland

ABSTRACT2D-Raman-THz spectroscopy of liquid water, which has been presented recently [J. Savolainen et al., Proc. Natl. Acad. Sci. U. S.A. 110, 20402 (2013)], directly probes the intermolecular degrees of freedom of the hydrogen-bond network. However, being arelatively new technique, its information content is not fully explored to date. While the spectroscopic signal can be simulatedbased on molecular dynamics simulation in connection with a water force field, it is difficult to relate spectroscopic signaturesto the underlying microscopic features of the force field. Here, a completely different approach is taken that starts from an assimple as possible model, i.e., a single vibrational mode with electrical and mechanical anharmonicity augmented with homo-geneous and inhomogeneous broadening. An intuitive Feynman diagram picture is developed for all possible pulse sequencesof hybrid 2D-Raman-THz spectroscopy. It is shown that the model can explain the experimental data essentially quantitativelywith a very small set of parameters, and it is tentatively concluded that the experimental signal originates from the hydrogen-bond stretching vibration around 170 cm−1. Furthermore, the echo observed in the experimental data can be quantified byfitting the model. A dominant fraction of its linewidth is attributed to quasi-inhomogeneous broadening in the slow-modulationlimit with a correlation time of 370 fs, reflecting the lifetime of the hydrogen-bond networks giving rise to the absorptionband.

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I. INTRODUCTION

THz spectroscopy of liquids probes their intermolecularmodes directly. Applied to liquid water, it reveals three distinctbands, which are assigned to hydrogen bond bending modes(50 cm−1), hydrogen bond vibrations (170 cm−1), and librations,which are hindered rotations of water molecules (600 cm−1).1These spectra can give us a picture of the motions in water;however, vibrational coherences are very short lived in waterdue to the fast, chaotic dynamics of the hydrogen bondingnetwork, causing very broad and blurred bands. This limits theamount of information that can effectively be extracted fromthese spectra.

Extending the spectroscopy into two dimensions canthin out the information and thus increase the amountof accessible information and can furthermore disentanglehomogeneous from inhomogeneous broadening.2,3 There hasbeen significant effort to extend the very concept of 2D

spectroscopy into the THz regime, either in the form of 2DRaman spectroscopy,4–13 2D-THz spectroscopy,14–19 or hybrid2D-Raman-THz methods.20–23 Due to technical limitations,the latter is as of now the only 2D spectroscopy in the THzrange that has been successfully applied to water and aqueoussalt solutions.22–24 Of particular interest in these experimentsis the observation of a very short-lived echo, whose decay timehas been related to the relevant time scales in water. However,a detailed understanding of the 2D-Raman-THz response isstill lacking.

Unlike conventional 2D IR spectroscopy,25 2D-Ramanspectroscopy and 2D-Raman-THz spectroscopy are describedby second-order perturbation theory, despite the fact that itis a 3rd-order response with regard to the electrical fields.This is since the Raman process is electronically non-resonantand thus instantaneous, and it is assumed that the two fieldinteractions giving rise to the Raman process occur at thesame time. The three instead of four interactions of the

J. Chem. Phys. 150, 044202 (2019); doi: 10.1063/1.5079497 150, 044202-1

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system (including the emission process) bring about that onehas to induce a two-quantum transition, which would be for-bidden in the harmonic approximation. This forbidden transi-tion is a bottleneck of the signal, which is why its very cause(i.e., electrical or mechanical anharmonicity) determines theshape of the signal. Thus, the two-quantum transition must bean integral part of any model used to describe 2D-Raman-THzspectroscopy.

Time dependent perturbation theory is the starting pointto calculate the 2D-Raman-THz response. The response func-tions for the three different time-orderings are26

RRTT = −Tr{µ̂(t1 + t2)[µ̂(t1), [α̂(0), ρeq]]},RTRT = −Tr{µ̂(t1 + t2)[α̂(t1), [µ̂(0), ρeq]]},RTTR = −Tr{α̂(t1 + t2)[µ̂(t1), [µ̂(0), ρeq]]},

(1)

where µ̂ is the dipole operator, α̂ is the polarizability opera-tor, t1 is the time between the first and the second laser pulseinteracting with the sample, and t2 is the time from the secondlaser pulse to the detection process. We have concentratedon the Raman-THz-THz (RTT) and the THz-Raman-THz (TRT)pulse sequences,22–24,27–30 where the detection step measuresan emitted THz field, while Blake and co-workers looked at theTHz-THz-Raman (TTR) pulse sequence with a Raman processfor detection.20,21

The theory of 2D-Raman-THz spectroscopy is similar to2D-Raman spectroscopy,2,3,31–38 as well as hybrid IR-Ramantechniques.26,39–42 In addition, a fair share of theory hasbeen published tailored specifically for 2D-Raman-THz spec-troscopy.27–29,43–48 Since typical THz experiments work ina frequency range equivalent to kBT, the response can bederived in the classical limit from molecular dynamics (MD)simulation.38 This approach appears to be the method ofchoice for complicated systems like water since basically alleffects, apart from possible quantum effects,24 are capturedimplicitly by a MD force field, including anharmonicities, modecoupling, chemical exchange, and orientational averaging. Ithas been shown that the MD approach reveals responses,which strongly depend on the force field used, especially onthe description of polarizability, albeit in a rather nonintu-itive and indirect way.29,44 These MD simulations are largely“black-box,” and it is difficult to disentangle the contributionsto the 2D-Raman-THz response and to relate spectroscopicsignatures to the underlying microscopic features of a waterforce field.

In order to learn more about the microscopic mecha-nism giving rise to the 2D-Raman-THz response, we takea completely different approach here and start from an assimple as possible model, i.e., a single vibrational mode aug-mented with homogeneous and inhomogeneous broadening.To that end, we follow the conceptual framework introducedin Ref. 36 for the description of 2D Raman spectroscopy,starting from a quantum-mechanical harmonic oscillator inan eigenstate representation and adding electrical and/ormechanical anharmonicity. The response functions [Eq. (1)]can then be depicted in a very intuitive way in terms of

Feynman diagrams, from which one can directly read off thepeak position in a 2D spectrum. We will see that the model canexplain the experimental data essentially quantitatively with asmall set of parameters.

II. MODEL SYSTEMA. Zero-order description: Harmonic oscillator

Due to its simplicity and the fact that it is often a verygood approximation for molecular vibrations, the harmonicoscillator is an obvious starting point to set the stage,

Ĥ(0) =~ω

2

(p̂2 + q̂2

). (2)

Analytical solutions of energy levels and eigenfunctions exist.From the thermal population of eigenstates, the equilibriumdensity matrix ρeq in Eq. (1) can readily be constructed. Fur-

thermore, the dimensionless position operator q̂ =√

mω~ x̂ can

be expressed in a harmonic oscillator eigenstate basis |ϕi〉,

(qH)ij ≡〈ϕi |q̂ |ϕj

〉= 1√

2

[√jδi+1,j +

√iδi−1,j

](3)

(where the subscript in qH stands for “harmonic”), which canbe seen from the common creation (b̂) and annihilation (b̂†)operator formalism with q̂ = 1/

√2(b̂†+b̂).49 With that, the dipole

and polarizability operators can be expanded in qH,

α ∝ qH + . . . ,µ ∝ qH + . . . .

(4)

The proportionality factors are irrelevant for the purpose ofthis paper, as they give rise to an overall intensity of the 2D-Raman-THz signal, which however is typically not determinedexperimentally in an absolute sense. The δi+1,j and δi−1,j termsin qH couple states i and i ± 1, which leads to the well-knowni → i ± 1 selection rules of the harmonic oscillator. This levelof description is often sufficient to describe linear (1D) THz orRaman spectroscopy.

The i → i ± 1 selection rules cause coherence pathways,in which the density matrix is alternating between populationand coherence states. As one starts from a thermal populationstate ρeq, an even number of interactions would be neededto return to a population state after the emission of a sig-nal. Therefore, due to the odd number of interactions in 2D-Raman-THz spectroscopy, the harmonic oscillator togetherwith Eq. (4) would predict a vanishing signal. The appearanceof a signal requires zero- or two-quantum transitions, whichcan be accomplished by breaking the symmetry of the system,either by considering the nonlinearity of µ and α (electricalanharmonicity) or by perturbing the potential of the oscilla-tor (mechanical anharmonicity). Softening the selection rulesresults in a bottleneck for the response; hence, the signalshape is very sensitive to the specific nature of the anhar-monicity, which thus is a crucial aspect of the modeling of2D-Raman-THz spectra.

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https://scitation.org/journal/jcp


B. Electrical anharmonicityElectrical anharmonity is introduced by considering

higher order terms in equation Eq. (4),36

α ∝ qH + σαqH2 + . . . ,µ ∝ qH + σµqH2 + . . . ,

(5)

with dimensionless smallness parameters |σµ | � 1 and/or |σα |� 1. In a harmonic eigenstate basis, the quadratic term is

(qH2)ij = 12[(2i + 1)δi,j +

√(i + 1)(i + 2)δi,j−2 +

√( j + 1)( j + 2)δi,j+2

], (6)

which again can be seen from the creation/annihilationoperator formalism.49 In analogy to an electrical quadrupoletransition, the δi ,j-term allows for zero-quantum transitionsi → i, and the δi ,j±2-terms allow for two-quantum transitionsi→ i ± 2.

C. Mechanical anharmonicityMechanical anharmonicity (which has not been con-

sidered in Ref. 36) breaks the symmetry by modifying thepotential energy function. We consider a cubic anharmonicoscillator by adding σM~ωq3 to the harmonic Hamiltonian

Ĥ = Ĥ(0) + σM~ωq3, (7)

with dimensionless smallness parameter |σM| � 1. The eigen-states of the anharmonic oscillator |ϕ(anh)i 〉 can be calculatedperturbatively.50 Although the position operator q̂ per se isnot affected by this addition, its matrix representation inan anharmonic eigenstate basis is. It can be shown (seeAppendix A) that this matrix representation takes the form〈ϕ(anh)i |q̂ |ϕ

(anh)j 〉 = (qH)ij + σM(qM)ij, with

(qM)ij =12

[−3(2i+1)δi,j+

√(i + 1)(i + 2)δi,j−2+

√( j + 1)( j + 2)δi,j+2

], (8)

where the subscript “M” stands for “mechanic anharmonicity.”qM has the same structure as qH2 [Eq. (6)] with the δi ,j-termcausing zero-quantum transitions and the δi ,j±2-terms caus-ing two-quantum transitions; however, the prefactors deter-mining the transition probabilities and the signs of the peaksare different.

For a system with mechanical and electrical anharmonic-ity at the same time, the operators are

α = qH + σMqM + σα(qH + σMqM)2

≈ qH + σMqM + σαqH2,

µ = qH + σMqM + σµ (qH + σMqM)2

≈ qH + σMqM + σµqH2,

(9)

where we neglected in the second step the terms that arehigher than first order in any of the smallness parame-ters σµ , σα , and σM. For the same reason, we also neglectcoherence pathways that contain more than one forbiddentransition.

D. Feynman diagramsBased on Eq. (9), the response functions of Eq. (1) can be

separated into

RRTT ∝ σαRαRTT + σµRµRTT + σMR

MRTT,

RTRT ∝ σαRαTRT + σµRµTRT + σMR

MTRT,

RTTR ∝ σαRαTTR + σµRµTTR + σMR

MTTR,

(10)

with

RαRTT = − Tr{qH(t1 + t2)

[qH(t1),

[qH2(0), ρeq

] ] },

RαTRT = − Tr{qH(t1 + t2)

[qH2(t1),

[qH(0), ρeq

] ] },

RαTTR = − Tr{qH2(t1 + t2)

[qH(t1),

[qH(0), ρeq

] ] },

RµRTT = − Tr{qH(t1 + t2)

[qH2(t1),

[qH(0), ρeq

] ] }

− Tr{qH2(t1 + t2)

[qH(t1),

[qH(0), ρeq

] ] },

RµTRT = − Tr{qH(t1 + t2)

[qH(t1),

[qH2(0), ρeq

] ] }

− Tr{qH2(t1 + t2)

[qH(t1),

[qH(0), ρeq

] ] },

RµTTR = − Tr{qH(t1 + t2)

[qH(t1),

[qH2(0), ρeq

] ] }

− Tr{qH(t1 + t2)

[qH2(t1),

[qH(0), ρeq

] ] },

RMRTT = − Tr{qM(t1 + t2)

[qH(t1),

[qH(0), ρeq

] ] }

− Tr{qH(t1 + t2)

[qM(t1),

[qH(0), ρeq

] ] }

− Tr{qH(t1 + t2)

[qH(t1),

[qM(0), ρeq

] ] },

= RMTRT = RMTTR,

(11)

which allows us to discuss the contributions from electri-cal (σα , σµ ) and mechanical anharmonicity (σM) separately.To this end, we expand the nested commutators in Eqs. (11)and represent all terms in the form of Feynman diagrams(Figs. 1–3). The corresponding 2D peaks are also summa-rized in Table I together with their intensities, which orig-inate from products of the prefactors of the correspondingδ-terms in Eqs. (3), (6), and (8) and their sums when con-tributions overlap in frequency space. The frequency posi-tions of the in total four peaks, (−ω, ω), (ω, ω), (ω, 0), and(ω, 2ω), are the same as in Ref. 36, but their intensities dif-fer since the selection rules of the three pulse sequencesof 2D-Raman-THz spectroscopy differ from those in 2D-Raman spectroscopy. The various spectra are discussed in thefollowing.

In RαRTT, RαTRT, and R

αTTR (Fig. 1), the nonlinearity of α allows

the Raman interaction to induce a zero- or a two-quantumtransition via the contribution of qH2. Starting with RαRTT, fourFeynman diagrams can be constructed (Fig. 1, top left). In thiscase, the first Raman interaction induces a forbidden transi-tion, i.e., a zero-quantum or a two-quantum transition. Thepathways contributing to peak (e) start with a zero-quantumtransition from either the left or the right, which however leadto two equivalent contributions of opposite sign that perfectlycancel out. Peak (f), on the other hand, starts with a two-quantum transition from the left. From there, the system goesinto a |1〉〈0| coherence by the second interaction from the left

J. Chem. Phys. 150, 044202 (2019); doi: 10.1063/1.5079497 150, 044202-3




FIG. 1. Feynman diagrams and 2D frequency-domain spectra of RαRTT (left), RαTRT (middle), and R

αTTR (right), showing the real part. The various 2D peaks are labeled with

letters, and the corresponding Feynman diagrams are shown above the spectra with THz interactions depicted in red arrows and the Raman interaction depicted as bluedouble-arrows. The weighting factor of each pathway is denoted above the Feynman diagram. The response functions were exponentially damped according to Eq. (14) witha time-constant long enough (i.e., T1 = 12ω−1) so that all peaks are resolved. Peak (c∗) is the conjugate complex of peak (c).

or into |2〉〈1| coherence via an interaction from the right. Fora harmonic potential, the energy levels are equally spaced andboth coherences oscillate with the same frequency. If we fur-thermore assume that the 0-1 and the 1-2 dephasing times arethe same [see Eq. (14)],36 both contributions again cancel, andwe have RαRTT = 0 altogether.

By contrast, two peaks appear in the TRT pulse sequence.For peak (a), the first interaction brings the system into a |0〉〈1|coherence, and the second Raman interaction does a two-quantum transition to bring the system into the |2〉〈1| coher-ence; hence, the peak appears at (−ω, ω). Peak (b), by con-trast, is diagonal at (ω, ω) since the second interaction does azero-quantum transition, which does not affect the state. Thezero-quantum transition can either act from the left or theright side, giving rise to two contributions with an oppositesign. However, these contributions do not fully cancel since

the interaction from the left is weighted by the (qH2)1,1 = −3/2element and the interaction from the right is weighted by(qH2)0,0 = +1/2. Finally, the TTR sequence has its forbiddentransition in the emission step with zero-quantum coherencesfor peak (c) and two-quantum coherences for peak (d). Peak(c), with ω2 = 0, appears together with its conjugate com-plex (c∗). The conjugate complex of all other peaks shows upat negative frequencies ω2, but the negative ω2-half-space isnot shown in Fig. 1 for clarity.

There are more peaks in RµRTT, RµTRT, and R

µTTR (Fig. 2) since

there are two interactions with µ, and the forbidden transitioncan occur with either one of it. However, Eq. (11) shows thateach term of Rµ can be written as a sum of two terms that wealready discussed in the context of Rα , e.g., RµRTT = R

αTRT +R

αTTR.

The corresponding 2D spectra in Fig. 2 therefore are overlaysof two spectra, each from Fig. 1.

FIG. 2. Feynman diagrams and 2D frequency-domain spectra of RµRTT (left), RµTRT (middle), and R

µTTR (right); for a description, see the caption of Fig. 1.

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FIG. 3. Feynman diagrams and 2D frequency-domain spectrum of RMRTT= RMTRT = R

MTTR; for a description, see the caption of Fig. 1.

Finally, the responses RMRTT, RMTRT, and R

MTTR originating

from mechanical anharmonicity are shown in Fig. 3. The termσMqM contributes to µ and α equally, and the forbidden tran-sition can occur at any of the three interactions. Since µ = αfor this case, the order of interactions does not matter, andRMRTT = R

MTRT = R

MTTR. The corresponding 2D spectrum is in

essence an overlay of all three spectra shown in Fig. 1, exceptfor the fact that the features that include zero-quantum tran-sitions (b) and (c) change signs and amplitudes due to thedifferent diagonal element of qM. Furthermore, peak (f) is nownonzero due to the anharmonic shift of the energy levels.

As in Ref. 36, we considered so far only Feynman dia-grams starting from the |0〉〈0| element of the density matrix inFigs. 1–3, implicitly assuming a temperature of T = 0 K. It can

TABLE I. Intensities of all non-zero peaks appearing in Eq. (11).

Peak

(a) (b) (c) (d)

Response (−ω, ω) (ω, ω) (ω, 0) (ω, 2ω)

RαRTT 0 0 0 0RαTRT 1 −1 0 0RαTTR 0 0 1 −1RµRTT 1 −1 1 −1RµTRT 0 0 1 −1RµTTR 1 −1 0 0RMRTT 1 3 −3 −1RMTRT 1 3 −3 −1RMTTR 1 3 −3 −1

however be shown that the overall response function does notdepend on the starting level, despite the fact that more Feyn-man diagrams come into play and hence is in fact temperatureindependent (see Appendix B for details). For that, we howeverneed to assume that the line shape functions depend on leveldifferences only [see Eq. (14)] and that the effect of mechan-ical anharmonicity is on the transition probabilities only (viaqM), while we keep the energy spectrum equidistant. Figure 3shows that this is a good approximation. That is, the anhar-monic shift ω21 − ω10 is small, and for any reasonable, nottoo narrow spectral linewidth, both contributions giving riseto peak (f) largely cancel each other. In comparison, the effectof mechanical anharmonicity on the transition probabilities,giving rise to peaks (a)–(e), is much larger. Within that approx-imation, quantum and classical response functions are in factidentical, and simple analytic expressions can be derived forthe three response functions by collecting all terms (withoutthat approximation, an explicit sum over Boltzmann-weightedinitial states would be needed that converges extremely slowlyfor ~ω � kBT),

RRTT(t1, t2) ∝ Θ(t1, t2)((σµ + σM) cos(ωt1 −ωt2)Γ−ω,ω (t1, t2) + (−σµ + 3σM) cos(ωt1 +ωt2)Γω,ω (t1, t2)

+ (σµ − 3σM) cos(ωt1)Γω,0(t1, t2) + (−σµ − σM) cos(ωt1 + 2ωt2)Γω,2ω (t1, t2)),

RTRT(t1, t2) ∝ Θ(t1, t2)((σα + σM) cos(ωt1 −ωt2)Γ−ω,ω (t1, t2) + (−σα + 3σM) cos(ωt1 +ωt2)Γω,ω (t1, t2)

+ (σµ − 3σM) cos(ωt1)Γω,0(t1, t2) + (−σµ − σM) cos(ωt1 + 2ωt2)Γω,2ω (t1, t2)),

RTTR(t1, t2) ∝ Θ(t1, t2)((σµ + σM) cos(ωt1 −ωt2)Γ−ω,ω (t1, t2) + (−σµ + 3σM) cos(ωt1 +ωt2)Γω,ω (t1, t2)

+ (σα − 3σM) cos(ωt1)Γω,0(t1, t2) + (−σα − σM) cos(ωt1 + 2ωt2)Γω,2ω (t1, t2)),

(12)

where Θ(t1, t2) is the Heaviside function ensuring that theresponse vanishes for t1 < 0 or t2 < 0. In second order per-turbation theory, the complex conjugate of a term can be

generated by two permutations in the commutator. Therefore,the two complex conjugate components have the same signand reveal cos-functions upon addition.

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We augment these response functions with dephasingterms Γ(t1, t2) in essence along the lines of Ref. 32, butreplacing the part originating from vibrational T1 relaxationwith Eq. (36), rather than Eq. (37), from Ref. 36. That is, eachresponse function in Eq. (12) contains one term each for peaksat (−ω, ω), (ω, ω), (ω, 0), and (ω, 2ω), respectively, which aremultiplied with the following damping functions:

Γ−ω,ω (t1, t2) = e− t1+t22T1 e−2g(t1)−2g(t2)+g(t1+t2),

Γω,ω (t1, t2) = e− t1+t22T1 e−g(t1+t2),

Γω,0(t1, t2) = e− t1+2t22T1 e−g(t1),

Γω,2ω (t1, t2) = e− t1+2t22T1 e+g(t1)−2g(t2)−2g(t1+t2).

(13)

As discussed in Ref. 36, the damping terms caused by vibra-tional relaxation depend on quantum numbers according to

T(n,n)1 = T1,

T(n,m)2 =2T1|n −m | .

(14)

The line shape function g(t) is

g(t) = ∆ω2τ2c(e−t/τc + t/τc − 1

), (15)

whose origin is Gaussian fluctuations of the transition fre-quency ω(t) with standard deviation ∆ω and correlation timeτc,

〈δω(t)δω(0)〉 = ∆ω2e−t/τc , (16)

where δω(t) = ω(t) − 〈ω(t)〉. In the limes ∆ωτc � 1, Eq. (15)causes pure dephasing with T∗2 = (∆ω

2τc)−1 and in the limes∆ωτc � 1, inhomogeneous dephasing with a Gaussian lineshape with width ∆ω.

E. Inhomogeneous broadening and echoesIn any of the responses, peak (a) with frequencies of oppo-

site signs (−ω,ω) is a rephasing pathway,30 as a coherence that

FIG. 4. Time-domain signals RRTT, RTRT, and RTTR for an inhomogeneously broad-ened system, using the parameters from Table II for the hydrogen-bond stretchingvibration, but setting τc = 100 ps and T1 = 1 ps. The effects of nonlinear polariz-ability are shown in the top row and those of nonlinear dipole moment are shownin the bottom row.

dephased during the first time period might rephase duringthe second, provided the band is inhomogeneously broadenedand maintains some amount of memory on the oscillation fre-quency. Rephasing requires an “inversion of coherence,” andthe only possibility to achieve that is the coherence path-way that starts with a |0〉〈1| coherence which is then broughtinto a |2〉〈1| coherence by the second pulse (see Figs. 1–3).In a time-domain representation, rephasing will generate anecho along t1 = t2 and in a frequency-domain representation,tilted 2D line shapes. Peak (b), by contrast, is a non-rephasingpathway.

To explore the appearance of echoes (Fig. 4), we chooseline shape parameters for the hydrogen-bond stretchingvibration as listed in Table II, except for unrealistically longvalues of τc = 100 ps and T1 = 1 ps. Echoes along t1 = t2 areseen for RαTRT, R

µ

RTT, and Rµ

TTR, i.e., coherence pathways withtheir two-quantum transition for the second pulse to inducethe required two-quantum transition. Experimentally, one candistinguish between RRTT, RTRT, and RTTR by the choice of apulse sequence; hence when observing an echo, one gets ahandle on determining the major source of anharmonicity. Thevertical features in RαTTR, R

µ

RTT, and Rµ

TRT originate from peak(c), which does not experience any inhomogeneous dephas-ing in the t2 direction since its frequency in ω2 is alwayszero.

III. COMPARISON TO WATER EXPERIMENTSA. Instrument response function

We now explore to what extent the model derived herecan explain the experimental data of liquid water from Ref. 22.We start with noting that this experiment measured the RRTTand RTRT responses by interchanging the timing between theRaman and THz-pump pulses, but not the RTTR response,which would require a different detection scheme.20,21 Withthe particular arrangement of the delay lines in the experi-ment of Ref. 22, we measured

R(t1, t2) = RRTT(t1, t2) + RTRT(−t1, t2 − t1), (17)

where time t1 is from the Raman pump pulse to the THz pumppulse and time t2 is from the THz pump pulse to detection[which is why t2 − t1 appears as an argument of RTRT where theTHz pump comes before the Raman pump, while t2 is the timefrom the second Raman interaction to detection in the defini-tion of RTRT in Eq. (12)]. Furthermore, the experimental signalis obtained from the response functions by a convolution withthe instrument response function (IRF),

S(t1, t2) = I(t1, t2) ~ R(t1, t2), (18)

which is calculated from the THz field ETHz and the envelopeof the Raman pulse IRaman,

I(t1, t2) ∝ddt2

ETHz(t2)IRaman(t2 + t1). (19)

Equation (19) is an idealized expression to illustrate the basicidea; in the real experiment, the IRF contains in additiona transfer function describing how the emitted THz field

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reshapes as it propagates from the sample to the detectioncrystal, as well as a correction for a non-perfect Gouy phase(see Refs. 22 and 51 for details).

It is illustrative to look at the IRF in the frequency domain(Fig. 5), as it shows which one of the peaks in the 2D responses(Figs. 1–3) the experiment is sensitive to. The IRF spectrum forthe RTT pulse sequence [Fig. 5(a)] has two nodal lines, one atω2 = 0, which is caused by the derivative d/dt2 in Eq. (19), andone at ω1 = ω2, which caused by the fact that the THz pumppulse does not have any DC component [i.e., ∫ ETHz(t)dt = 0].Peaks (b) and (c) lie on these nodal lines, as they would con-tain zero-quantum transitions for either the THz pump pulseor the THz emission process. We will nevertheless see thesepeaks to a certain extent since the damping in water is veryfast; hence, they have a significant spectral width that extendsinto regions where the IRF is non-zero. Furthermore, a smallerbandwidth is observed in the rephasing quadrant (ω1 < 0,ω2 >0) than in the non-rephasing quadrant (ω1, ω2 > 0), reflectingthe fact that the peak at ω2 = −ω1 includes a two-quantumtransition which requires a higher bandwidth of the THzpulse.

For the RTRT response, the IRF has to be transformed toaccount for the time shift applied in Eq. (17) [Fig. 5(b)]. Theeffect of a time transform t1 = −t′1 and t2 = t

′2 − t

′1 on the fre-

quency domain can be derived by inspecting the effect on asingle basis function of the Fourier transformation

exp(i[ω1t1 +ω2t2]) = exp(i

[ω1(−t′1) +ω2(t

′2 − t

′1)

] )= exp

(i

[ω′1t

′1 +ω

′2t′2

] ), (20)

FIG. 5. The absolute value of the IRF in the frequency domain for (a) the RTTpulse sequence and (b) the TRT pulse sequence, which have been derived fromEq. (19) with the experimentally determined pulses and transfer function.22

with ω′1 = −ω1 − ω2 and ω′2 = ω2. The transformed instrument

response has nodal lines at ω1 = 0 and ω2 = 0. Hence, in thiscase, the diagonal peak (b) lies in a region where the IRF is largesince the envelope of the Raman pulse has a zero-frequencycomponent.

B. 1D spectra and 2D responseFigure 6 shows the 1D Raman spectrum of water in the rel-

evant frequency range, which contains two spectroscopic fea-tures: the hydrogen-bond bending vibration around 50 cm−1

and the hydrogen-bond stretch vibration around 170 cm−1 (thelibrational mode around 600 cm−1 is not considered here sinceit is completely outside our experimental observation win-dow). Also shown in Fig. 6 are fits to these two bands, assuminga linear response function

RR(ω′) = =∫ ∞

0sin(ωt)e−

t2T1 e−g(t)e−iω

′tdt. (21)

The parameters are listed in Table II and will be justified laterbased on the fit of the echo observed in the 2D response.

Starting with the hydrogen-bond stretching vibration,Fig. 7 (left column) shows the corresponding 2D results inthe time domain, i.e., the convolution of Eq. (17) with the IRF.The line shape parameters of Table II were assumed, exceptfor τc = 100 ps and T1 = 1 ps, which in a first step was cho-sen unrealistically slow in order to more clearly demonstratethe resulting echoes (i.e., the same parameters as in Fig. 4).Each one of the three types of anharmonicities gives rise toa different 2D response. The echoes shown in Fig. 4 sur-vive the convolution and are marked by arrows in Fig. 7. Inthe case of a nonlinear dipole moment, the rephasing peakis in RµRTT and the echo is visible in the (upper right) RTTquadrant along t1 = t2. On the other hand, for Sα with thenonlinearity in the polarizability, the rephasing peak is inthe RαTRT response and the echo appears in the upper leftquadrant along the t2 = −2t1 due to the time transforma-tion in Eq. (17). Finally, SM is a mix of both (albeit with dif-ferent signs) since mechanical anharmonicity allows for allcoherence pathways at the same time. Figure 7 (right col-umn) shows the same, but now for τc = 370 fs and T1 = 250 fs

FIG. 6. The experimental anisotropic Raman spectrum of water (blue) togetherwith two line shape functions used to fit the two hydrogen-bond modes with theparameters listed in Table II. The experimental data have been taken from Ref. 52.

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FIG. 7. Time domain signals of the responses Sµ (top), Sα (middle), and SM

(bottom) after convolution of Eq. (17) with the IRF Eqs. (18) and (19). The rightcolumn uses the parameters as in Table II, and the left column sets τc = 100 psand T1 = 1 ps to more clearly demonstrate the echoes. Discernible echoes aremarked by arrows.

(which will be justified later based on a fit to the experimentaldata). The echoes are now masked by the instrument responsefunction.

Modeling the hydrogen-bond bending vibration around50 cm−1 on the same footing reveals similar results; see Fig. 8.The echoes are clearer in this case, even when using realisticparameters for τc and T1 (Fig. 8, right column). In addition, thesignal intensity is significantly larger (Fig. 7 has been upscaledby a factor of 2), reflecting the limited bandwidth of the THz

FIG. 8. Time domain signals of the responses Sµ (top), Sα (middle), and SM

(bottom) after convolution of Eq. (17) with the IRF Eqs. (18) and (19). The rightcolumn uses the parameters as in Table II, and the left column sets τc = 100 psand T1 = 1 ps to more clearly demonstrate the echoes. Discernible echoes aremarked by arrows.

pulses in the experiment, which peak at 50 cm−1 but onlypartially cover the 170 cm−1 band.22

C. Fit of the water responseThe free parameters of the model are σµ , σα , σM, T1, τc,

ω, and ∆ω. In an iterative process, we varied the dephasingparameter T1, τc to reproduce the 2D-response and ω and ∆ωto reproduce the position and width of the hydrogen-bondbending vibration in the 1D spectrum (Fig. 6). In addition, small

FIG. 9. (a) Experimental water responseat room temperature and the signalcalculated from the anharmonic oscil-lator model form, (b) the hydrogen-bond stretching vibrations, and (c) thehydrogen-bond bending vibrations withthe parameters listed in Table II. Dis-cernible echoes are marked by arrows.The experimental data have been takenfrom Ref. 22.

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TABLE II. Parameters obtained from the fit of the hydrogen-bond bending vibrationand hydrogen-bond stretching vibration.

Stretching Bending

ω (cm−1) 165a 45a

∆ω (cm−1) 75 40τc (fs) 370 300b

T1 (fs) 250 300bσµ/σM 1.4 0.96σα/σM −0.1 0.01

aAs a result of Eq. (21), the peak position of the band does not coincideexactly with ω.bIt has not been possible to minimize the RMSD for the hydrogen-bondbending vibration since for any set of parameters, the RMSD is higher thanthat of the instrument response function per se. The optimization algo-rithm therefore converged to very short times τc and T1, in which case, themolecular response becomes δ-shaped. In analogy to the hydrogen-bondbending vibration, we fixed τc = T1 = 300 fs and then optimized all otherparameters.

variations of the delay-time zeros and the correction forthe Gouy phase were allowed, which are not very accuratelydefined in the experiment.22 Once these nonlinear parame-ters are fixed, the 2D responses of Figs. 7 and 8 (right column)can be considered a basis in which the experimental responseis expanded in order to minimize the root-mean-square devi-ation (RMSD) between experimental and fitted spectrum. Fig-ure 9 shows that this procedure results in remarkably goodagreement with the experimental data, despite the simplic-ity of the model and the small number of parameters. Table IIsummarizes the resulting parameters for the two modes.

The RMSD of the fit for the hydrogen-bond stretchingvibration is smaller by a significant factor of 0.7 as com-pared to that of the hydrogen-bond bending vibration. It hasin fact not been possible to find a minimum in the RMSDfor the hydrogen-bond bending vibration (see the footnotein Table II), and in addition, the shift in t2 required to obtainthe best fit for hydrogen-bond bend vibration is larger (i.e.,∆t2 = 170 fs) than what we would think is its experimentaluncertainty (the corresponding value for the hydrogen-bondstretching vibration is ∆t2 = 35 fs). We therefore suggest thatthe observed experimental signal originates predominantlyfrom the hydrogen-bond stretching vibration, despite the factits response is reduced to a certain extent as it is at the veryedge of the experimentally accessible frequency window. Wehave not considered scenarios in which both bands contributein parallel or, even more so, couple with each other becausethe number of fitting parameters would be too large and thefitting problem would be under-determined.

Figure 10 plots the RMSD of the fit for the hydrogen-bond stretching vibration as a function of τc and T1, revealingthat both parameters are not strongly correlated. As thesetwo parameters determine the relative contribution of inho-mogeneous vs homogeneous dephasing, we conclude that thesignatures of the echo are still present in the 2D response. Inthat regard, it is fortunate that the rephasing peak (a) in RµRTT(Fig. 2, bottom left) and RMRTT (Fig. 3, bottom) has the same

FIG. 10. RMSD between the experimental and simulated spectrum as a functionof τc and T1, considering the hydrogen-bond stretching vibration. The parametersfor ω and ∆ω were kept fixed to the values reported in Table II since they aredetermined mostly by the 1D spectrum (Fig. 6), which in turn changes only verylittle when varying τc and T1. The linear parameters σµ , σα , σM, on the otherhand, were optimized for each (τc , T1)-point in this plot.

sign, while all other peaks have the opposite sign (and thecontribution of RµRTT is negligible; see Table II). Together withthe corresponding weights of the two contributions (σµ/σM= 1.4), the rephasing peak will actually dominate in the over-all response. That is illustrated in Fig. 11, which shows theresponse in the frequency domain and without the convo-lution with the instrument response function for the sameparameters as in Fig. 9(b); RRTT is dominated by one rephasingpeak that is strongly elongated along the diagonal due to the

FIG. 11. The same model as in Fig. 9(b), but before convolution with the instru-ment response function and in the frequency domain, plotting RRTT (top) and RTRT(bottom) separately.

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inhomogeneous broadening. As a result, an echo is discerniblein the time-domain data of Fig. 9(b), despite the fact that it isessentially completely masked in the individual contributionsof Fig. 7 (right column).

During the fit, the absolute intensity of the experimentalsignal was not considered. Likewise, the proportionality con-stants in Eq. (4), dµ/dq and dα/dq, are not known. Therefore,the three parameters σµ , σα , and σM are only known moduloan overall scaling factor, and only their relative contributionsto the signal (e.g., σµ/σM and σα/σM) can be extracted. Thesign of the overall signal can however be determined. The signof the overall signal is given by the signs of σM times that ofdα/dq [since dµ/dq enters quadratically in Eq. (10)]. To repro-duce the experimentally observed sign, we either have σMnegative and dα/dq positive (which is what we assumed here),or vice versa.

IV. DISCUSSION AND CONCLUSIONWe have shown that the experimental 2D-Raman-THz

response of liquid water can be explained almost quantita-tively by describing the hydrogen bond bending modes withthe help of a very simple anharmonic oscillator model. Themodel contains only six independent parameters, i.e., σµ/σMand σα/σM as well as ω0, ∆ω, τc, and T1. Despite the factthat the instrument response function masks the informa-tion content of the molecular 2D response to a significantextent, it is still sufficient to determine these parameters withconfidence (Fig. 10), if we assume that the response origi-nates from predominantly one of the two bands in the obser-vation window of the experiment. We tentatively concludethat this dominating band is the hydrogen-bond stretch-ing vibration at 170 cm−1, but we cannot exclude additionalcontributions from the hydrogen-bond bending vibrationaround 50 cm−1, or from couplings between both bands.The librational mode around 600 cm−1, on the other hand,is completely outside our experimental observation window(Fig. 5).

The physical interpretation of σµ , σα , and σM is notstraight forward since the low frequency modes of water arecollective in nature and presumably delocalized to a certainextent. It is nonetheless worth noting that the biggest contri-bution to the signal originates from mechanical anharmonicityσM and the dipole nonlinearity σµ , while the contributionfrom the nonlinearity in the polarizability is negligible withσα = 0.1σM (Table II). This observation is in stark contrastto results from recent MD work, which revealed an echo forthe TRT pulse sequence.28,44,45 That echo originated from thehindered rotation band around 600 cm−1, which we howeverdo not observe in the experiment due to the limited band-width. A large nonlinearity in the polarizability for that modecan be understood from the fact that a strict ∆J = 2 selectionrule would apply for a Raman interaction, together with a ∆J = 1selection rule for a THz interaction, in the limiting case of afree rather than hindered rotor.30 This explanation obviouslydoes not apply for the hydrogen-bond bending and stretchingvibrations.

Regarding the line shape function, on the other hand, theparameters for τc, ∆ω, and T1 quantify the degree of inhomo-geneous broadening of the hydrogen-bond-bending vibration(Table II). That is, with ∆ωτc ≈ 5, the line shape function is inthe “slow-modulation” or “quasi-inhomogeneous” limit.25 Atthe same time, vibrational relaxation with T1 = 250 fs con-tributes only 20 cm−1 to the total linewidth. The correlationtime τc = 370 fs, in turn, is a measure of the lifetime of thehydrogen-bond networks giving rise to the hydrogen-bondstretching vibration. The typical lifetime of a single hydrogenbond is 1 ps;53–56 hence, we conclude that those modes aredelocalized over ≈3 hydrogen bonds.

The quality of the fit of Fig. 9 is much better thanthat of much more sophisticated calculations based on awater force field in connection with MD simulations.29 Theproblem with the MD approach arises from the fact that awater force field needs to describe the thermodynamics anddynamics of water reasonably well in the first place, whilethe anharmonicities, which are the bottleneck of the 2D-Raman-THz signal, result from these constraints only in anindirect way. Anharmonicities are typically not fitted explic-itly, except if very specific effects, such as nuclear quan-tum effects,57 are to be described. Furthermore, in particu-lar, when polarizability is included in a water force field, thenumber of parameters is typically very large, and the prob-lem is underdetermined. Electrical anharmonicity has a lot todo with the redistribution of charges during the motion ofwater molecules, and being able to quantify, it might revealguidelines to design better water models.58 To that end, itwould be important to infer the anharmonicity parametersσµ ,σα , and σM from a MD simulation of a realistic water forcefield.

In conclusion, we think that 2D-Raman-THz spec-troscopy has the highest information content as to date withregard to the intermolecular degrees of freedom water, butlearning how to extract that information from the experimen-tal response is challenging. The present work constitutes asignificant step in that direction.

ACKNOWLEDGMENTSWe thank Yoshitaka Tanimura for very valuable discus-

sions. This work has been supported by the Swiss NationalScience Foundation (SNF) through the NCCR MUST.

APPENDIX A: POSITION OPERATORIN AN ANHARMONIC EIGENSTATE BASIS

In the cubic anharmonic oscillator model, the harmonicoscillator Hamiltonian Ĥ0 is perturbed by σM~ωq,3

Ĥ = Ĥ0 + σM~ωq̂3, (A1)

where q̂ =√mω/~x̂ is the unitless position operator and σM is

the size of the perturbation. To first order in σM, the eigen-functions of Ĥ are expressed as a linear combination of the

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harmonic eigenfunctions |ϕn〉,50

|ϕ(anh)n 〉 = |ϕn〉 + σM(an |ϕn+1〉 + bn |ϕn−1〉 + cn |ϕn+3〉 + dn |ϕn−3〉

),

(A2)

with

an = −3(n + 1

2

)3/2,

bn = 3(n

2

)3/2,

cn = −13

[(n + 3)(n + 2)(n + 1)

8

] 1/2,

dn =13

[n(n − 1)(n − 2)

8

] 1/2.

(A3)

Mechanical anharmonicity does not affect the position oper-ator q̂ per se, but rather changes its matrix representationdue to a change of the basis functions { |ϕi〉} → { |ϕ

(anh)i 〉}.

We will call q̂ in a harmonic eigenfunction basis qH and qAif it is expressed in anharmonic eigenfunctions. Expressing(qA)i,j ≡ 〈ϕ

(anh)i |q̂ |ϕ

(anh)j 〉 in terms of (qH)k,l ≡ 〈ϕk |q̂ |ϕl〉, using

Eq. (A2), we get a large collection of terms

(qA)i,j = (qH)i,j + σM(aj(qH)i,j+1 + ai(qH)i+1,j + bj(qH)i,j−1 + bi(qH)i−1,j

+ cj(qH)i,j+3 + ci(qH)i+3,j + dj(qH)i,j−3 + di(qH)i−3,j), (A4)

where terms higher than first order in σM have been dis-carded. The matrix form of the position operator in harmonicoscillator eigenbasis is [see Eq. (3)]

qH =1√

2

*..............,

0√

1 0 0√

1 0√

2 0

0√

2 0√

3

0 0√

3 0. . .

. . .. . .

+//////////////-

. (A5)

A matrix with (qH)i,j+1 is the matrix qH shifted up by one row,

1√

2

*..........,

√1 0

√2

0√

2 0. . .

0 0√

3. . .. . .

+//////////-

. (A6)

Hence, the elements√jδi+1,j are transformed to

√i + 1δi,j

terms in that shifted matrix and√iδi−1,j of qH to

√iδi−2,j.

Using this procedure, the shifted matrices in Eq. (A4)become

(qH)i,j+1 = 1√2

(√i + 1δi,j +

√iδi−2,j

),

(qH)i+1,j = 1√2

(√j + 1δi,j +

√jδi,j−2

),

(qH)i,j−1 = 1√2

(√iδi,j +

√i + 1δi,j−2

),

(qH)i−1,j = 1√2

(√jδi,j +

√j + 1δi−2,j

),

(qH)i,j+3 = 1√2

(√i + 1δi−2,j +

√iδi−4,j

),

(qH)i+3,j = 1√2

(√j + 1δi,j−2 +

√jδi,j−4

),

(qH)i,j−3 = 1√2

(√iδi,j−2 +

√i + 1δi,j−4

),

(qH)i−3,j = 1√2

(√jδi−2,j +

√j + 1δi−4,j

),

(A7)

which include zero- (δi ,j), two- (δi−2,j and δi ,j−2), and four-quantum (δi−4,j and δi ,j−4) contributions. Collecting all terms,we obtain

1√2

(aj√i + 1 + ai

√j + 1 + bj

√i + bi

√j)δi,j = − 32

(2i + 1

)δi,j,

1√2

(aj√i + bi

√j + 1 + cj

√i + 1 + di

√j)δi−2,j =

12

√i√i − 1δi−2,j,

1√2

(ai

√j + bj

√i + 1 + ci

√j + 1 + dj

√i)δi,j−2 =

12

√j√j − 1δi,j−2,

1√2

(cj√i + di

√j + 1

)δi−4,j = 0.

(A8)

We see that the δi−4,j elements cancel (likewise for δi ,j−4), andonly zero- and two-quantum transitions remain in qA, in addi-tion to the one quantum transitions from qH. Inserting Eq. (A9)into Eq. (A4), we get

qA = qH + σMqM, (A9)

where the second term describes the correction to the posi-tion matrix due to anharmonicity with the final expression ofqM given in Eq. (8).

APPENDIX B: TEMPERATURE INDEPENDENCEOF THE RESPONSE FUNCTION

Temperature enters via the thermal population of theinitial density matrix ρeq. In IR spectroscopy, one usuallyassumes that only the ground state is initially populated due to~ω � kBT, but that assumption no longer holds in THz spec-troscopy, and the contributions from thermally populated ini-tial states have to be considered. Figure 12 exemplifies thatby showing all Feynman diagrams and their intensity con-tributions to the four non-zero peaks in RµRTT. It can beseen that the overall intensities do not depend on the initialstate and hence also not on temperature. This result is uni-versal and applies in the same way for the other responsefunctions. However, a prerequisite for this to work is theassumption that the line shape functions depend on energylevel differences only [Eq. (14)] and that the energy spac-ing between states is equidistant, i.e., that of a harmonicoscillator, and that the effect of mechanical anharmonicity

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FIG. 12. Collection of all Feynman diagrams responsible for the four peaks (ω,ω), (ω, 0), (−ω,ω), and (ω, 2ω) present in RµRTT. The pathways are ordered by their initialstate, and the intensity of each pathway is denoted below the corresponding Feynman diagrams.

only enters via the softened selection rules that allow forzero- and two-quantum transitions. Figure 3 demonstratesthat this is a good approximation (see discussion in the maintext).

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