# Nonequilibrium molecular dynamics simulations with a backward-forward trajectories sampling for multidimensional infrared spectroscopy of molecular vibrational modes.

**ABSTRACT** A full molecular dynamics (MD) simulation approach to calculate multidimensional third-order infrared (IR) signals of molecular vibrational modes is proposed. Third-order IR spectroscopy involves three-time intervals between three excitation and one probe pulses. The nonequilibrium MD (NEMD) simulation allows us to calculate molecular dipoles from nonequilibrium MD trajectories for different pulse configurations and sequences. While the conventional NEMD approach utilizes MD trajectories started from the initial equilibrium state, our approach does from the intermediate state of the third-order optical process, which leads to the doorway-window decomposition of nonlinear response functions. The decomposition is made before the second pump excitation for a two-dimensional case of IR photon echo measurement, while it is made after the second pump excitation for a three-dimensional case of three-pulse IR photon echo measurement. We show that the three-dimensional IR signals are efficiently calculated by using the MD trajectories backward and forward in time for the doorway and window functions, respectively. We examined the capability of the present approach by evaluating the signals of two- and three-dimensional IR vibrational spectroscopies for liquid hydrogen fluoride. The calculated signals might be explained by anharmonic Brownian model with the linear-linear and square-linear system-bath couplings which was used to discuss the inhomogeneous broadening and dephasing mechanism of vibrational motions. The predicted intermolecular librational spectra clearly reveal the unusually narrow inhomogeneous linewidth due to the one-dimensional character of HF molecule and the strong hydrogen bond network.

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**ABSTRACT:**We present the wavefunction (WF) version of the equation-of-motion phase-matching approach (EOM-PMA) for the calculation of four-wave-mixing (4WM) optical signals. For the material system, we consider a general electronic-vibrational Hamiltonian, comprising the electronic ground state, a manifold of singly-excited electronic states, and a manifold of doubly-excited electronic states. We show that the calculation of the third-order polarization for particular values of the pulse delay times and in a specific phase-matching direction requires 6 independent WF propagations within the rotating wave approximation. For material systems without optical transitions to doubly-excited electronic states, the number of WF propagations is reduced to 5. The WF EOM-PMA automatically accounts for pulse-overlap effects and allows the efficient numerical calculation of 4WM signals for vibronically coupled multimode material systems. The application of the method is illustrated for model systems with strong electron-vibrational and electronic inter-state couplings.Chemical Physics 08/2013; · 1.96 Impact Factor - SourceAvailable from: Shinji Saito
##### Article: Fluctuations and Relaxation Dynamics of Liquid Water Revealed by Linear and Nonlinear Spectroscopy.

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**ABSTRACT:**Many efforts have been devoted to elucidating the intra- and intermolecular dynamics of liquid water because of their important roles in many fields of science and engineering. Nonlinear spectroscopy is a powerful tool to investigate the dynamics. Because nonlinear response functions are described by more than one time variable, it is possible to analyze static and dynamic mode couplings. Here we review the intra- and intermolecular dynamics of liquid water revealed by recent linear and nonlinear spectroscopic experiments and computer simulations. In particular, we discuss the population relaxation, anisotropy decay, and spectral diffusion of the intra- and intermolecular motions of water and their temperature dependence, which play important roles in ultrafast dynamics and relaxations in water. Expected final online publication date for the Annual Review of Physical Chemistry Volume 64 is March 31, 2013. Please see http://www.annualreviews.org/catalog/pubdates.aspx for revised estimates.Annual Review of Physical Chemistry 12/2012; · 13.37 Impact Factor - SourceAvailable from: Xiang Sun[Show abstract] [Hide abstract]

**ABSTRACT:**The workhorse spectroscopy for studying liquid-state solvation dynamics, time-dependent fluorescence, provides a powerful, but strictly limited, perspective on the solvation process. It forces the evolution of the solute-solvent interaction energy to act as a proxy for what may be fairly involved changes in solvent structure. We suggest that an alternative, a recently demonstrated solute-pump∕solvent-probe experiment, can serve as a kind of two-dimensional solvation spectroscopy capable of separating out the structural and energetic aspects of solvation. We begin by showing that one can carry out practical, molecular-level, calculations of these spectra by means of a hybrid theory combining instantaneous-normal-mode ideas with molecular dynamics. Applying the resulting formalism to a model system displaying preferential solvation reveals that the solvent composition changes near the solute do indeed display slow dynamics similar to, but measurably different from, that of the solute-solvent interaction - and that this two-dimensional spectroscopy can effectively single out those local structural changes.The Journal of Chemical Physics 07/2013; 139(4):044506. · 3.12 Impact Factor

Page 1

Nonequilibrium molecular dynamics simulations with a backward-forward

trajectories sampling for multidimensional infrared spectroscopy

of molecular vibrational modes

Taisuke Hasegawaa?and Yoshitaka Tanimura

Department of Chemistry, Graduate School of Science, Kyoto University, Sakyoku,

Kyoto 606-8502, Japan

?Received 29 October 2007; accepted 3 December 2007; published online 14 February 2008?

A full molecular dynamics ?MD? simulation approach to calculate multidimensional third-order

infrared ?IR? signals of molecular vibrational modes is proposed. Third-order IR spectroscopy

involves three-time intervals between three excitation and one probe pulses. The nonequilibrium

MD ?NEMD? simulation allows us to calculate molecular dipoles from nonequilibrium MD

trajectories for different pulse configurations and sequences. While the conventional NEMD

approach utilizes MD trajectories started from the initial equilibrium state, our approach does from

the intermediate state of the third-order optical process, which leads to the doorway-window

decomposition of nonlinear response functions. The decomposition is made before the second pump

excitation for a two-dimensional case of IR photon echo measurement, while it is made after the

second pump excitation for a three-dimensional case of three-pulse IR photon echo measurement.

We show that the three-dimensional IR signals are efficiently calculated by using the MD

trajectories backward and forward in time for the doorway and window functions, respectively. We

examined the capability of the present approach by evaluating the signals of two- and

three-dimensional IR vibrational spectroscopies for liquid hydrogen fluoride. The calculated signals

might be explained by anharmonic Brownian model with the linear-linear and square-linear

system-bath couplings which was used to discuss the inhomogeneous broadening and dephasing

mechanism of vibrational motions. The predicted intermolecular librational spectra clearly reveal

the unusually narrow inhomogeneous linewidth due to the one-dimensional character of HF

molecule and the strong hydrogen bond network. © 2008 American Institute of Physics.

?DOI: 10.1063/1.2828189?

I. INTRODUCTION

Multidimensional

spectra are obtained by recording the signals as a function of

the time durations between the trains of pulses, have the

potential to reveal the detailed static and dynamical features

of molecular interactions in liquids.1,2Techniques such as

two-dimensional Raman3–7and two- and three-dimensional

infrared ?IR? spectroscopes8–13are now being used to inves-

tigate the interactions between the inter- and intramolecular

modes14–22as well as the dephasing mechanism of vibra-

tional motions23–26that relate to the homogeneity versus in-

homogeneity of liquids. The homogeneous broadening or vi-

brational dephasing of signals arise from the solvent motion

occurring on time scales comparable to the time scales of

vibration, while the inhomogeneous broadening is arising

from the liquid features that change so slowly and look es-

sentially static.27,28The anharmonic coupling between the

inter- and intramolecular modes as well as dipole and in-

duced dipole interactions also play an important role for en-

ergy and coherent relaxation processes in liquids.29–32The

contributions to the signals from them can be distinguished

as a difference of profiles or different spectral peaks in mul-

vibrationalspectroscopies,whose

tidimensional spectroscopy, which is similar to the spin echo

measurement for dephasing and the two-dimensional nuclear

magnetic resonance measurement for the well-known cou-

pling effects between spins. The advanced experiments such

as multidimensional spectroscopies need to be guided by the-

oretical calculations and simulations, since a relationship be-

tween the molecular motion and signals is not simple for

such complex systems as molecular liquids, and other non-

linear processes may overlap with the signal of interest due

to the complexity of the experimental setups.

For analyzing the detail of experimental results, molecu-

lar dynamics ?MD? simulations are helpful means since they

can predict not only the line shape and intensities of the

signals but also the role of molecular interactions through the

capability of changing the interaction potentials. Previous

simulations for two-dimensional ?2D? IR spectroscopy have

used the basic strategy to ascertain the importance of the

structure change or vibrational dephasing by partitioning the

role of the environment, as taken from the MD simulation in

the form of the fluctuations of the vibrational energy states,

then calculated the signals by employing the analytical ex-

pression of the third-order response function of infrared echo

measurements.33–41This procedure works fairly well for

cases where the modes to be observed and the solvent or

environment modes are well separated.21This procedure

a?Electronic mail: hasegawa@kuchem.kyoto-u.ac.jp.

THE JOURNAL OF CHEMICAL PHYSICS 128, 064511 ?2008?

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Page 2

does not work, however, if the target modes and the environ-

ment are strongly coupled or the target modes are not well

defined due to the structure or conformal changes of mol-

ecules that are influenced by the bath. In addition, even in the

well separated case, this procedure does not provide a legiti-

mate description of bath effects for a long time period since

the bath modes cause not only fluctuations in the energy

levels but also energy dissipation which leads the system to

thermal equilibrium as described through fluctuation-

dissipation theorem.21,26The most rigorous way to calculate

nonlinear signals of complex system is to perform full MD

calculations to simulate response functions. From the full

MD approach, the absolute magnitude of the response func-

tion in relation to other background signal may be deduced

from this procedure, which is extremely useful to compare

the predicted signals with the experimental results. Note that,

although exact calculations must be based on quantum mo-

lecular dynamics simulation, the quantum effects usually

play a minor role for vibrational spectroscopy in condensed

phases other than the zero point oscillation of high-frequency

intramolecular modes. Therefore, classical molecular dynam-

ics simulations and, if necessary, with some quantum correc-

tions may be accurate enough to simulate most of experi-

mental data of the 2D IR spectroscopies.

While the molecular dynamics simulation techniques

themselves are well established, the calculation of the non-

linear response functions from full MD simulations is not

well explored. This is because nonlinear vibrational spectros-

copy became experimentally possible recently and the calcu-

lation of nonlinear response functions is extremely difficult

due to the high sensitivity of the stability matrix elements

involved in the nonlinear response functions.42–46As was

pointed out, while the conventional one-dimensional Raman

and linear IR spectroscopies analyze the harmonicity of mo-

lecular motions, the multidimensional Raman and IR spec-

troscopies can detect nonharmonic motions of molecules by

eliminating the harmonic contribution to the signals.26This

feature of multidimensional spectroscopies also makes the

simulation difficult since we cannot apply the normal mode

analysis which assumes harmonic motions for inter- and in-

tramolecular modes.

Three MD methods have been developed for 2D Raman

spectroscopy. The first one is the equilibrium approach that

computes, exactly or approximately, a nonlinear optical re-

sponse function expressed in the multiple Poisson brackets

of the equilibrium molecular trajectories.42,44,47–51The sec-

ond method is the nonequilibrium MD ?NEMD? approach

that performs the 2D spectroscopy experiment on the

computer.52–55In the NEMD approach, the Raman polariz-

ability is directly calculated from NEMD trajectories under a

pair of external laser pulses with different time sequences.

The third method hybridizes the first and second methods to

avoid the time-consuming calculations of the stability matri-

ces which are inherent in the equilibrium method with using

nonequilibrium trajectories for a single laser excitation.56,57

In this paper, we use the NEMD approach to calculate

the signals of 2D and three-dimensional ?3D? IR spec-

troscopies. To reduce the number of necessary NEND trajec-

tories, we decompose the third-order response function into

doorway and window parts.58,59While the conventional

NEMD simulation utilizes the trajectories from the initial

equilibrium state to the final state, our approach goes from

the decomposed state to the final state ?forward in time? for

the window part and the decomposed state to the initial state

?backward in time? for the doorway part. It is shown that the

NEMD approach with the forward-backward trajectories

sampling method enhances the efficiency of the numerical

simulation for such multidimensional IR spectroscopy as

three-pulse IR photon echo measurement. To demonstrate the

accuracy and efficiency of the present approach, we have

evaluated the signals of 2D and 3D IR vibrational spec-

troscopies for liquid hydrogen fluoride. The liquid HF forms

the hydrogen bond network in a zigzag chain structure due to

the strong hydrogen bond interactions and quadrupole mo-

ment of HF molecules.60–65Thus, the liquid HF exhibits un-

usually narrow inhomogeneous linewidth in contrast to liq-

uid water64–79

and is an interesting target even for

demonstration. The one-dimensional character of the hydro-

gen bond network also makes the conceptualization of the

spatial correlations simple.

This paper is organized as follows. In Sec. II, we explain

our simulation method for the third-order IR response func-

tion. In Sec. III, we show the results of the simulations for

liquid hydrogen fluoride. Section IV is devoted to concluding

remarks.

II. SIMULATION METHODS

A. Doorway and window functions

Since the extension to N particle systems in three-

dimensional space is straightforward, we start from a one-

dimensional particle system described by the Hamiltonian

H?p,q?, where p and q are the momentum and coordinate of

the particle, respectively. In the third-order IR measurement,

the system interacts with the three IR pump pulses HI

=−??q??k=a,b,cEk?t?, where ??q? is the molecular dipole and

Ek?t? is the time envelopment of the kth pulse, and then the

excited molecular dipole is detected by the probe pulse. If we

consider the impulsive pump pulses, Ea?t?=Ea??t?, Eb?t?

=Eb??t−t1?, and Ec?t?=Ec??t−t1−t2?, and probe the dipole at

t=t1+t2+t3, the third-order IR response is expressed in terms

of the molecular dipole moments as66

R?t3,t2,t1? =?i?3

?3????? ˆ?t1+ t2+ t3?,? ˆ?t1+ t2??,? ˆ?t1??,? ˆ?0???,

?2.1?

where ? ˆ?t? is the Heisenberg operator of the dipole moment

??q ˆ?. The molecular states in the response functions are ex-

pressed in the Liouville space brackets ?q,q†????q??q†? and

??q,q†?? ˆ??q?? ˆ?q†? for any density operator ? ˆ. The initial

equilibrium distribution and the trace operation are then ex-

pressed respectivelyas

???dqdq†??q−q†???q,q†?, where ?=1/kBT and Z is the par-

tition function. We denote the commutation operator of the

Hamiltonian ?the quantum Liouvillian? and the dipole by

e−iLˆtAˆ?e−iHˆt/?AˆeiHˆt/?with iLˆAˆ?i?Hˆ,Aˆ?/? and ? ˆ?Aˆ?? ˆAˆ

e−?Hˆ/Z=??eq??

and

?d?tR????

064511-2 T. Hasegawa and Y. TanimuraJ. Chem. Phys. 128, 064511 ?2008?

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Page 3

−Aˆ? ˆ for a right-hand side operator Aˆ. The response functions

can be rewritten as

?3?d?tR????? ˆe−iLˆt3? ˆ?e−iLˆt2? ˆ?e−iLˆt1? ˆ???eq??.

R?t3,t2,t1? =?i?3

?2.2?

By

=??dq?dq?†?q?,q?†????q?,q?†???d????????????, we can de-

compose the response functions into the doorway and win-

dow functions as58,59

R?t3,t2,t1? =?d??Wcd?t2+ t3;???Dab???;t1?.

insertingthe completenessrelation1

?2.3?

The window function is defined by

Wcd?t2+ t3;??? ?i

??d?tR????? ˆe−iLˆt3? ˆ?e−iLˆt2?????,

?2.4?

and is interpreted as the expectation value of ? ˆ with the

densityoperator

? ˆc?t2+t3?=exp?−iLˆt3?? ˆ?exp?−iLˆt2??????

that was initially in the uniform ?flat? distribution ????? then

interacted with the third-pump pulse ? ˆ?between the time

evolutions exp?−iLˆt2? and exp?−iLˆt3?. The doorway function

is defined by

Dab???;t1? ??i?2

?2?????? ˆ?e−iLˆt1? ˆ???eq??.

?2.5?

This is the probability distribution of the uniform distribution

?????

for the densityoperator

? ˆ???eq??. This operator corresponds to the expectation value

of ? ˆ?for density operator at time t1started from the initial

state with the first pump excitation ? ˆ???eq??.

In the Wigner representation, the distribution function is

expressed as67,68

2???

−?

? ˆab?t1?=? ˆ?exp?−iLˆt1?

P?p,q? =

1

?

dxeip·x/???q −x

2,q +x

2????.

?2.6?

The

=??dp?dq??p?,q?????p?,q??. Then, the doorway and window

functions are, respectively, rewritten as

Wcd?t2+ t3;p?,q?? ???dpdq??q?exp?− LW?p?,q??t3?

completenessrelation isnowgivenby1

? ?W

???p − p????q − q??

??q??exp?− LW?p?,q??t2?

?2.7?

and

Dab?p?,q?;t1? ???dp0dq0??p0− p????q0− q??

? ? ˆW

??q0?exp?− LW?p0,q0?t1?

?? ˆW

??q0?Peq?p0,q0?,

?2.8?

where −LW?p,q? and ? ˆW

the dipole operator i? ˆ?/? in the Wigner representation,26

respectively, and Peq?p0,q0?=??p0,q0??eq??=e−?H?p0,q0?/Z.

??q? are the quantum Liouvillian and

To obtain the above expression, we inserted the com-

pleteness relation right after the second pump interaction. We

can break the theoretical description at arbitrary point during

the course of the time evolution by assuming the laser fields

do not overlap at the assigned point, and rigorously recast the

response function in terms of the two decomposed

functions.59For the 2D IR case, it is convenient to decom-

pose the response function into the doorway and window

functions before the second pump excitation as

Wbcd?t2+ t3;p?,q?? ???dpdq??q?exp?− LW?p?,q??t3?

? ?W

??q??exp?− LW?p?,q??t2?

? ? ˆW

??q????p − p????q − q???2.9?

and

Da?p?,q?;t1? ???dp0dq0??p0− p????q0− q??

?exp?− LW?p?,q??t1?? ˆW

??q0?Peq?p0,q0?.

?2.10?

In both cases, the response function is calculated by integrat-

ing over p? and q? as

R?t3,t2,t1? =?dp?dq?Wcd?t2+ t3;p?,q??Dab?p?,q?;t1?

?2.11?

or

R?t3,t2,t1? =?dp?dq?Wbcd?t2+ t3;p?,q??Da?p?,q?;t1?.

?2.12?

Either in the quantum or classical case, the above ex-

pression indicates that the response function can be obtained

from the separately calculated doorway and window func-

tions for any p? and q?. Expressions for N particles system

canbeobtainedby

??p1

−p????q−q??→??p−p???q−q?? ? ?j=1

??qj

etc. The quantum and classical Liouvillians are given by

a=x,y,z?pj

?qj

−1

− Ujk?qj

2i

?pj

replacing

?z??,

?p,q?→?p,q?

??q?→? ˆW

N?a=x,y,z??pj

N?a=x,y,z??dpj

?x?,p1

?y?,...,pN

?z?,q1

?x?,q1

?y?,...,qN

? ˆW

??q???p

?a?−pj??a??

?a?dqj

?a?−qj??a??, ??dpdq→??dpdq??j=1

?a?,

− LW?p,q? = −?

j=1

N

?

?a??

?a?

i??

k?j?Ujk?qj

?x?−?

?a?,qk???

2i

?

?pj

?a?,qk?

?a?+?

?

?2.13?

and

064511-3Simulation of multidimensional IR spectra J. Chem. Phys. 128, 064511 ?2008?

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Page 4

− Lcl?p,q? = −?

j=1

N

?

a=x,y,z?pj

?a??

?qj

?a?

+?

k?j

?Ujk?qj

?a?,qk?

?qj

?a?

?

?pj

?a??,

?2.14?

respectively, where Ujk?qj,qk? is the interaction potential be-

tween the jth and kth particles. In this paper, we employ the

classical Liouvillian to calculate the response function by

means of MD simulations.

B. Nonequilibrium molecular dynamics simulation

We use the nonequilibrium ?finite field? simulation ap-

proach to evaluate the doorway and window functions. The

nonequilibrium approach allows us to calculate the physical

observable from nonequilibrium simulations for different

pulse configurations and sequences.52,53

In order to evaluate Wcd?t;p?,q??, conventional finite

field approaches utilize the unperturbed and perturbed

Hamiltonians, H and Hc???=H−Ec??q????−t1−t2?, respec-

tively. Instead, we have employed the inverted force

method,54which considers Hc??? and the perturbed Hamil-

tonian with the opposite sign Hc ¯???=H+Ec??q????−t1−t2?.

Using Hc ¯??? instead of H, we can efficiently eliminate the

higher order contribution of the perturbation from the expec-

tation values. We denote the Liouvillian for Hc??? and Hc ¯???

by Lcl

bution functionsareexpressed

=exp?−?t1

t?=exp?−?t1

By treating the pump excitation Ecperturbatively, we

can express the window function as

2Ec?t??dpdq??q??Pc?p,q;p?,q?;t?

− Pc ¯?p,q;p?,q?;t??

c?p?,q?;?? and Lcl

c ¯?p?,q?;??. The corresponding distri-

as

c?p?,q?????p−p????q−q?? and Pc ¯?p,q;p?,q?;

td?Lcl

Pc?p,q;p?,q?;t?

td?Lcl

c ¯?p?,q?;?????p−p????q−q??, respectively.

Wcd?t;p?,q?? ?

1

?

1

2Ec?t??c?p?,q?;t? − ?c ¯?p?,q?;t??,

?2.15?

where t=t2+t3and ?c?p?,q?;t? and ?c ¯?p?,q?;t? mean the

expectation values of the dipole for the perturbation

?Ec??q????−t1−t2?. Since the even order contribution of

the perturbation will be cancel out by adding ?c?p?,q?;t?

and ?c ¯?p?,q?;t?, the inverted method is quite efficient at

calculating the third-order response functions.

C. Forward and backward propagations

Equations ?2.7?–?2.12? indicate that the response func-

tion can be obtained from the separately calculated doorway

and window functions for any p? and q?. First, we consider

Wcd?t2+t3;p?,q?? and Dab?p?,q?;t1?, which are both in the

second order of the dipole interactions. As explained above,

Wcd?t2+t3;p?,q?? is evaluated by the NEMD simulation

from two nonequilibrium trajectories in phase space that start

from initial values p? and q? in a uniform distribution and

evolve forward in time for the perturbed Hamiltonians Hc???

and Hc ¯???. Accordingly, the doorway function Dab?p?,q?;t1?

is calculated from the nonequilibrium trajectories that start

with initial values p0and q0in the equilibrium distribution

and propagate forward in time for the Hamiltonians Hb???

=H−Eb??q????−t1? and Hb¯???=H+Eb??q????−t1? after op-

erating the first pump excitation denoted by i? ˆW

natively, we can evaluate the doorway function using the

trajectories corresponding to the backward propagation in

time. This can be done without considering the phase-space

compression factor since we do not have any dissipation pro-

cesses and the system dynamics is time reversible.59,69We

rewrite Eq. ?2.8? as

??q?/?. Alter-

Dab?p?,q?;t? ???dp0dq0Peq?p0,q0?

?? ˆW

??q??exp?−?

t1

0

d?LW?p?,q?;???

?? ˆW

??q????p0− p????q0− q??.

?2.16?

The initial state of the backward trajectories p? and q? are in

the uniform distribution. After operating the second pulse

i? ˆW

and nonequilibrium trajectories from time t1to 0 for the

Hamiltonians

Ha ¯???=H+Ea??q?????

−Ea??q????? backward in time. The final state, which is the

initial equilibrium state in the forward case, is defined by

integration over p0 and q0 with the weight function

Peq?p0,q0?.

As Eq. ?2.11? indicates, we need to generate trajectories

from a sampling point in the uniform distribution of ?p?,q??.

In practice, however, we do not have to generate points of

?p?,q?? that are far from the equilibrium distribution

Peq?p?,q??. This is because the trajectories from those points

give very small contributions to Dab?p?,q?;t1? due to the

factor Peq?p??−t1?,q??−t1??, where ?p??0?,q??0????p?,q??

and ?p??−t1?,q??−t1?? are the backward trajectories from t

=0 to t=−t1for the perturbed Hamiltonians Ha??? and Ha ¯???

with the second pump excitation. Thus, instead of using uni-

form distribution for ?p?,q??, we can generate the trajecto-

ries from the sampling points in the equilibrium distribution

Peq?p?,q?? and calibrate the weight of probability by multi-

plying exp??E?p?,q???, where E?p?,q?? is the total energy

of molecules for the configuration ?p?,q??. In this scheme,

Eq. ?2.11? is rewritten as

??q?/? to the initial state, we calculate the equilibrium

and

Ha???=H

R?t3,t2,t1? =?dp?dq?Wbcd?t2+ t3;p?,q??Peq?p?,q??

?Da??p?,q?;t1?,

?2.17?

where

Da??p?,q?;t1? = e?E?p?,q??Da?p?,q?;t1?.

?2.18?

The classical doorway function for the canonical distribution

is expressed as

064511-4 T. Hasegawa and Y. TanimuraJ. Chem. Phys. 128, 064511 ?2008?

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Page 5

Da??p?,q?;t1? ?

1

2Ea?t?

−exp?− ??Ha ¯?p?,q?;− t1??

Za ¯

exp?− ??Ha?p?,q?;− t1??

Za

?,

?2.19?

where ?Ha=Ha?p,q;−t1?−H0?p,q;0? is the work done to

the molecular system by the first laser interaction, and Zaand

Za ¯are the partition functions for Ha?p,q;−t1? and Ha ¯?p,q;

−t1?, respectively. Since we have the relation

pi?− t1?

m

?Ha?p?,q?;− t1? ??

i

???− t1?

?qi

????− t1?Ea?t

= ? ˙?− t1?Ea?t,

?2.20?

by expanding Eq. ?2.19? in terms of laser interactions and by

collecting the terms in the order of Ea, we can obtain the

simple expression

2?? ˙?p?,q?;− t1? −1

?exp?− ?H0?p,q;0??? ˙?p,q;− t1??,

Da??p?,q?;t1? ? −?

Z?dpdq

?2.21?

which enables us to calculate the doorway function from

equilibrium trajectories. This expression is convenient to cal-

culate the two-dimensional third-order response function.

The window function in Eq. ?2.17? is expressed as

Wbcd?t2+ t3;p?,q?? ?

1

4EbEc??t?2??bc?p?,q?;t2+ t3?

+ ?b¯c ¯?p?,q?;t2+ t3? − ?b¯c?p?,q?;t2

+ t3? − ?bc ¯?p?,q?;t2+ t3??,

?2.22?

and the response function is evaluated as

R?t3,t2,t1? =

− ?

4EbEc??t?2???bc?t2+ t3? + ?b¯c ¯?t2+ t3?

− ?b¯c?t2+ t3? − ?bc ¯?t2+ t3??

??? ˙?− t1??NVT,adiabatic,

?2.23?

where ?A?t??NVT,adiabaticdenotes the average value of the func-

tion A?t? in initially the canonical and then the adiabatically

evolving ensemble and ?? ˙?−t1?=? ˙?−t1?−?? ˙?−t1??NVT,adiabatic.

This expression is equivalent to the equilibrium and nonequi-

librium hybrid MD method developed to simulate fifth-order

two-dimensional Raman spectroscopy.56

Alternatively, we can rewrite Eq. ?2.11? as

R?t3,t2,t1? =?dp?dq?Wcd?t2+ t3;p?,q??

?Peq?p?,q??Dab? ?p?,q?;t1?,

?2.24?

where Dab? ?p?,q?;t1?=e?E?p?,q??Dab?p?,q?;t1?. The classical

doorway function in Eq. ?2.24? is written as

4EaEb??t?2?

+exp?− ??Ha ¯b¯?p?,q?,− t1??

Za ¯b¯

Dab? ?p?,q?;t1? ?

1

exp?− ??Hab?p?,q?,− t1??

Zab

−exp?− ??Hab¯?p?,q?,− t1??

Zab¯

−exp?− ??Ha ¯b?p?,q?,− t1??

Za ¯b

?,

?2.25?

where ?Hab=Hab?p,q;−t1?−H0?p,q;0? is the work done to

the molecular system by the first and second laser interac-

tions and Zabis the partition function for Hab?p?,q?,−t1?.

Using the relation

pi?0?

m

?qi

?Hab?− t1? ??

i

???0?

????0?Eb?t

+?

i

pi,b?− t1?

m

??b?− t1?

?qi,b

????− t1?Ea?t

= ? ˙?0?Eb?t + ? ˙b?− t1?Ea?t,

?2.26?

and expanding Eq. ?2.25? to collect the terms in the order of

EaEb, we obtain

Dab? ?p?,q?;t1? ?1

2Eb?t?? ˙b?p?,q?;− t1? − ? ˙b¯?p?,q?;

− t1?? + ?2? ˙?p?,q?;− t1?? ˙?p?,q?;0??

1

Z?

− ?

−

Z2?dpdq?

− ? ˙b¯?p,q;− t1?? + ?2? ˙?p,q;− t1?

?? ˙?p,q;0??exp?− ?H0?p,q;0??.

− ?

2Eb?t?? ˙b?p,q;− t1?

?2.27?

The corresponding classical window function is given by Eq.

?2.15?. Then, the response function is expressed as

R?t3,t2,t1? =?

1

2Ec?t??c?t2+ t3? − ?c ¯?t2+ t3???

−?

− ?

2Eb?t?? ˙b?− t1? − ? ˙b¯?− t1?? + ?2? ˙?− t1?? ˙?0???

− ?

NVT,adiabatic

1

2Ec?t??c?t2+ t3? − ?c ¯?t2+ t3???

NVT,adiabatic?

2Eb?t?? ˙b?− t1? − ? ˙b¯?− t1?? + ?2? ˙?− t1?? ˙?0??

NVT,adiabatic

.

?2.28?

064511-5Simulation of multidimensional IR spectra J. Chem. Phys. 128, 064511 ?2008?

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Page 6

This expression is convenient to calculate the 3D third-order

response function. The number of necessary nonequilibrium

simulations for the third-order response function depends on

the calculation schemes. The simulation processes for the

conventional NEMD method and the developed two schemes

are schematically illustrated in Fig. 1. Notice that the second

term in Eq. ?2.28? will vanish if we sample the whole phase

points because this term arises from the difference between

the partition functions involved in Eq. ?2.28?. To obtain ac-

curate signals from the finite number of the phase points, we

need to include this term.

In the next section, we examine the accuracy and effi-

ciency of our approach by calculating the 2D and 3D signals

for liquid hydrogen fluoride.

III. TWO-AND THREE-DIMENSIONAL IR SIGNALS

OF HYDROGEN FLUORIDE LIQUID

A. Computational details

To simulate 2D and 3D IR signals for liquid hydrogen

fluoride, we used the FF?64,0.78? intermolecular potential

expressed in the Chebyshev expansion form.64The Ewald

method was used for the long range Coulomb interactions.

The simulation was carried out with 64 HF molecules. The

total dipole moment was calculated by the first-order dipole-

induced-dipole approximation and the dipole moment and

the molecular polarizability were taken from experimental

values.70–73The simulation box was a 1.29 nm3and the short

range forces were cut off at the half length of the simulation

box. The equations of motion were solved by the velocity-

Verlet integrator of rigid bodies74,75with the time step of

0.1 fs for canonical ?NVT? calculations and 2 fs for micro-

canonical ?NVE? calculations. The Nose-Hoover chain with

20 thermostats was used to generate the NVT ensemble. The

NVT simulations were performed at 273 K. We set E?t

=1.0?10−5V m−1s in the NEMD calculations. The laser

fields are applied during very short time step of ?t=1.0

?10−40fs to obtain the impulsive responses. We first made

3?104temporary configurations from a NVT trajectory at

4 ps intervals in order to prepare the NVT ensemble. We then

took fragments of the 4 ps NVE trajectory from the tempo-

rary configurations after 4 ps equilibrations. The initial con-

figuration of the NEMD calculations was sampled at 4 fs

intervals from each fragment of the trajectories. Then, the

signals were calculated by averaging over the 3?107initial

configurations.

B. The accuracy and efficiency of the simulation

First, we compared the accuracy and efficiency of third-

order 2D IR signals for fixed t2=0 calculated from the two

FIG. 1. The simulation processes are illustrated. The thick lines correspond

to adiabatic NEMD trajectories. The dotted lines correspond to equilibrium

NVE ?constant energy? trajectories. White circles are the initial configura-

tions of the adiabatic NEMD calculations. The dotted lines or equilibrium

trajectories should be generated from canonical ensembles. The exponential

terms can be calculated by the time derivative of the dipole moment ?Eqs.

?2.22? and ?2.27??. ?a? Conventional NEMD method. Eight NEMD trajecto-

ries are used to extract the third-order response from the total response. ?b?

The simulation process using Eq. ?2.22?. Four NEMD trajectories and one

equilibrium trajectory are employed. ?c? The simulation process using Eq.

?2.27?. Two NEMD trajectories, two backward NEMD trajectories, and one

equilibrium trajectory are used. The trajectory in the t2period corresponds to

the equilibrium one.

FIG. 2. ?Color? The time domain third-order response

function Rzzzz?t1,0,t3? for liquid HF at t2=0 fs. Panels

?a? and ?b? are calculated from Eqs. ?2.22? and ?2.27?,

respectively. The signal strengths are normalized by the

peak strength of the signal in panel ?a?.

064511-6 T. Hasegawa and Y. TanimuraJ. Chem. Phys. 128, 064511 ?2008?

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Page 7

different schemes of the doorway and window decomposi-

tions. The first scheme is expressed as Eq. ?2.23? and is

based on the decomposition before the second pump excita-

tion, while the second scheme is expressed as Eq. ?2.28? and

is decomposed after the second pump excitation. We calcu-

lated the third-order response functions for all parallel polar-

ization Rzzzz?t1,0,t3? based on the two schemes. The results

of the first and second schemes are given in Figs. 2?a? and

2?b?, respectively. The signal profiles and strength are almost

the same. However, there are some disagreements in the re-

gion t3?150 fs, which is depicted in Fig. 3 as the difference

between the two profiles. This is because the signal from the

first scheme converges slower than that from the second

scheme in the region of t3?50 fs. In the first scheme, the

window function is defined by the second-order difference of

the dipole moment, so that the true signal is very weak com-

pared to the thermal fluctuation noise in the region of t3

?150 fs and the signal converges slower in the first scheme

than the second scheme in this region.

The second scheme allows us to calculate the 3D signals

for different t2much faster than the fast scheme. This is

because the second scheme uses equilibrium trajectories in

the t2period and we can calculate the third order responses

for different t2by choosing pairs of initial configurations on

one NVE fragment for the doorway and window functions.

In the first scheme, however, we need to repeat the NEMD

calculation for each t2to have 3D signals as a function of t1,

t2, and t3.

C. The time-domain and frequency-domain signals

Figures 4?a?–4?f? illustrate the time-domain 2D IR sig-

nals for liquids HF with all parallel polarizations. The oscil-

lations of the period of 60 fs are induced by the intermolecu-

lar vibrational motions of the hydrogen bonds. The signal

profile in Fig. 4?a? resembles the 2D IR signal calculated

from Gaussian-Markovian Fokker-Planck equation with a

linear-linear ?LL? and square-nonlinear ?SL? system-bath

couplings.26,28,76,77The conventional Brownian oscillator

model considers the LL interaction which describes the

damping and thermal activation processes of the Langevin

dynamics. The SL interaction changes the curvature of the

system potential,78which in turn causes the vibrational

dephasing process. Analogous to the case of electronically

resonant spectroscopy, LL interactions induce the longitudi-

nal ?T1? and transverse ?T2? relaxations, whereas SL interac-

tions inducesimilareffects

broadening.26Figure 4?a? exhibits similar profiles, as shown

in Fig. 15?v? of Ref. 26, which corresponds to a slow modu-

lation ?inhomogeneous? case of an anharmonic Brownian os-

cillator system. The phase difference of the signals in Fig.

4?a? and Fig. 15?v? in Ref. 26 may be attributed to the non-

linear coordinate dependence of the dipole moment of liquid

hydrogen fluoride. The echo like profile along t1=t3vanishes

around t2=200 fs, which is the same order as the noise cor-

relation time estimated from the results of the LL+SL model

?Fig. 15?v? in Ref. 26? as 1/?=1/?0.1?0??500 fs.

asthe inhomogeneous

FIG. 3. ?Color? The difference in the third-order response functions calcu-

lated from Eqs. ?2.22? and ?2.27?.

FIG. 4. ?Color? The time domain

third-orderresponse

Rzzzz?t1,t2,t3? at ?a? t2=0 fs, ?b? t2

=20 fs, ?c? t2=100 fs, ?d? t2=200 fs,

?e? t2=300 fs, and ?f? t2=400 fs calcu-

lated from Eq. ?2.27?. The signal

strengths are normalized by peak

strength of the original signal shown

in ?a?.

functions

064511-7Simulation of multidimensional IR spectra J. Chem. Phys. 128, 064511 ?2008?

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Page 8

We next present the frequency domain correlation spec-

tra for different t2to discuss the role of dephasing and an-

harmonicity on the vibrational modes.10,21We calculated the

correlation spectra in the following procedure. First we have

performed a 3D Fourier transformation of the signal with all

parallel polarization in t1, t2, and t3. We then took the signal

only in the range of ?2?200 cm−1and reconverted it into a

time domain in t2. The correlation spectra was obtained by

adding the imaginary part of ??1,?3?=?+,+? quadrant and

??1,?3?=?−,+? quadrant. Figures 5?a?–5?f? show the corre-

lation spectra with the waiting time t2=0, 20, 100, 200, 300,

and 400 fs, respectively. In each figure, we see a positive

peak around ??1,?3?=?550,550? cm−1corresponding to the

vibrational coherence from the fundamental oscillation, and

a negative peak around ??1,?3?=?550,400? cm−1corre-

sponding to the vibrational coherence from an anharmonic

oscillation. These peaks are induced by the vibrational mo-

tion of the hydrogen bonds network in the liquid HF. The

width of the peak at ??1,?3?=?550,550? is about 50 cm−1

and is much narrower than the inhomogeneous linewidth ob-

served in liquid water.79This is due to the formation of the

strong hydrogen bonds and one-dimensional character of HF

molecule.60–65This narrow band feature is also supported by

the recent neutron scattering experiment.80

As expected for intermolecular vibrational motions, a

large anharmonicity ??150 cm−1? is observed. Since we ne-

glected the intramolecular vibrational motion of HF, no en-

ergy transfer take place between the intermolecular and in-

tramolecular motions. To have effects from intramolecular

motions, one has to employ a multibody intramolecular po-

tential instead of a simple harmonic potential.65In addition,

since quantum effects may play some roles in the high fre-

quency intramolecular vibrational mode, one has to involve a

quantum correction in the simulations. These two challeng-

ing tasks are left for further studies.

IV. CONCLUSIONS

The present paper describes a methodology to efficiently

calculate the signals of 2D and 3D third-order IR spec-

troscopies by means of the NEMD simulation. The key fea-

tures of this approach are the doorway-window decomposi-

tion of the response function and the backward-forward

sampling of the NEMD trajectories. The efficiency and ac-

curacy of the simulation were tested to calculate 2D and 3D

IR signals of liquid HF using two different schemes of the

doorway-window decompositions. The scheme represented

by Eq. ?2.28? makes the calculation of 3D IR signals more

efficient than the conventional NEMD method and the

scheme represented by Eq. ?2.23?. Using the scheme repre-

sented by Eq. ?2.28?, we calculated the 3D signals for differ-

ent t2. The calculated signal exhibits vibrational dephasing

within a correlation time of ??200 fs, which may be de-

scribed by an anharmonic Brownian oscillator model with

linear-linear and square-liner system-bath couplings. The an-

harmonicity of intermolecular vibrational modes was also

confirmed by the frequency domain correlation spectra. We

presented the efficient algorithm to calculate the 3D IR sig-

nals for the intermolecular vibrational motions, without in-

cluding the intramolecular contribution. The developments

of the multibody intramolecular potential and the calculation

strategy of the quantum corrections are necessary for further

studies.

ACKNOWLEDGMENTS

The authors thank A. Ishizaki, S. Saito, T. Yagasaki, and

R. J. D. Miller for their helpful discussions. Y.T. is thankful

for the financial support from a Grant-in-Aid for Scientific

Research B19350011 from the Japan Society for the Promo-

tion of Science. T.H. is supported by the research fellowship

of Global COE program, International Center for Integrated

Research and Advanced Education in Material Science,

Kyoto University, Japan. A part of the computations was per-

formed at Research Center for Computational Science, Oka-

zaki, Japan.

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FIG. 5. ?Color? The correlation spec-

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=20 fs, ?c? t2=100 fs, ?d? t2=200 fs,

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064511-8T. Hasegawa and Y. Tanimura J. Chem. Phys. 128, 064511 ?2008?

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