Are water simulation models consistent with steady-state and ultrafast vibrational spectroscopy experiments?

Abstract

Vibrational spectroscopy can provide important information about structure and dynamics in liquids. In the case of liquid water, this is particularly true for isotopically dilute HOD/D2O and HOD/H2O systems. Infrared and Raman line shapes for these systems were measured some time ago. Very recently, ultrafast three-pulse vibrational echo experiments have been performed on these systems, which provide new, exciting, and important dynamical benchmarks for liquid water. There has been tremendous theoretical effort expended on the development of classical simulation models for liquid water. These models have been parameterized from experimental structural and thermodynamic measurements. The goal of this paper is to determine if representative simulation models are consistent with steady-state, and especially with these new ultrafast, experiments. Such a comparison provides information about the accuracy of the dynamics of these simulation models. We perform this comparison using theoretical methods developed in previous papers, and calculate the experimental observables directly, without making the Condon and cumulant approximations, and taking into account molecular rotation, vibrational relaxation, and finite excitation pulses. On the whole, the simulation models do remarkably well; perhaps the best overall agreement with experiment comes from the SPC/E model.

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Are water simulation models consistent with steady-state and
ultrafast vibrational spectroscopy experiments?
J.R. Schmidta, S.T. Robertsb, J.J. Loparob, A. Tokmakoffb, M.D. Fayerc, J.L. Skinnera,*
aTheoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, WI 53706, United States
bDepartment of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139, United States
cDepartment of Chemistry, Stanford University, Stanford, CA 94305, United States
Received 1 February 2007; accepted 18 June 2007
Available online 6 July 2007
Abstract
Vibrational spectroscopy can provide important information about structure and dynamics in liquids. In the case of liquid water, this
is particularly true for isotopically dilute HOD/D2O and HOD/H2O systems. Infrared and Raman line shapes for these systems were
measured some time ago. Very recently, ultrafast three-pulse vibrational echo experiments have been performed on these systems, which
provide new, exciting, and important dynamical benchmarks for liquid water. There has been tremendous theoretical effort expended on
the development of classical simulation models for liquid water. These models have been parameterized from experimental structural and
thermodynamic measurements. The goal of this paper is to determine if representative simulation models are consistent with steady-state,
and especially with these new ultrafast, experiments. Such a comparison provides information about the accuracy of the dynamics of
these simulation models. We perform this comparison using theoretical methods developed in previous papers, and calculate the exper-
imental observables directly, without making the Condon and cumulant approximations, and taking into account molecular rotation,
vibrational relaxation, and finite excitation pulses. On the whole, the simulation models do remarkably well; perhaps the best overall
agreement with experiment comes from the SPC/E model.
? 2007 Elsevier B.V. All rights reserved.
Keywords: Water; Vibrational spectroscopy; Dynamics; Simulation models
1. Introduction
The properties of water are of profound importance in
many branches of science, from geology to biology to
oceanography [1]. Water is particularly ubiquitous in
chemistry, where it serves as a common solvent for many
reactions. The polar nature of water indicates that it
strongly solvates many types of charged or polar reactants,
products, or intermediates, thus allowing chemists to carry
out reactions that would not be possible in the gas phase,
or in a non-polar solvent. Consider, for example, the pro-
totypical nucleophilic substitution reaction of organic
chemistry. In the non-concerted form of this reaction,
involving a charged carbo-cation intermediate, a highly
polar solvent such as water can speed the reaction by many
orders of magnitude relative to non-polar solvents, by sol-
vating and stabilizing the cationic intermediate [2]. Water
also plays an essential role as solvent in almost all biolog-
ically relevant reactions and processes [3].
Given the vital role of water as an experimental solvent,
it is not surprising that water plays in equally important
role in molecular dynamics (MD) simulation. This is evi-
denced by the vast numbers of models that have been cre-
ated for use in simulations of neat liquid water or for
aqueous solution [4,5]. Most of the existing water models
have been empirically parametrized to reproduce a subset
of the structural and thermodynamic data available, such
as radial distribution functions, heat of vaporization, freez-
ing/boiling point, or dielectric constant. Agreement with
experimental quantities not used to determine the model
parameters may or may not be quantitative. More recent
0301-0104/$ - see front matter ? 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.chemphys.2007.06.043
*Corresponding author.
E-mail address: skinner@chem.wisc.edu (J.L. Skinner).
www.elsevier.com/locate/chemphys
Available online at www.sciencedirect.com
Chemical Physics 341 (2007) 143–157

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models have been developed from high-level experimental
studies on small water clusters [6,7], or completely from
ab initio calculations [8,9]. Almost none of the existing
water models have been parameterized to reproduce the
dynamical properties of liquid water. These properties
include the rate of hydrogen bond breaking/formation, dif-
fusion constant, rotational correlation times, and spectral
densities of low-frequency intermolecular modes (such as
librations and hydrogen bond vibrations). An accurate
description of these properties would seem to be essential
in order to make quantitative predictions for processes
where the dynamics of the solvent plays an important role.
Much of our microscopic information about liquid
water comes from neutron scattering [10,11] and X-ray dif-
fraction [12,13], X-ray absorption and X-ray Raman scat-
tering [14,15], dielectric relaxation [16], and nuclear
magnetic resonance (NMR) [17,18]. Vibrational spectro-
scopy provides a complementary technique to access struc-
tural and dynamical information about bulk liquid water.
This technique can be particularly illuminating when the
liquid has been isotopically labeled so as to eliminate diffi-
culties due to intermolecular vibrational coupling. For
example, in the case of the OD stretch of dilute HOD in
H2O, vibrational observables can be interpreted in terms
of simple OD stretch local modes. In this case the OD
stretch frequency can be related to the solvation environ-
ment of the local mode. Since the solvation environment
is continually changing in time due to equilibrium molecu-
lar dynamics, in principle vibrational spectroscopy can pro-
vide information about both the structure and dynamics in
the liquid. In the case of water, however, the large breadth
of the absorption spectrum dictates that the line shape is
mostly inhomogeneously broadened, and hence contains
little dynamical information.
Many theoretical papers have been devoted to calcula-
tions of the vibrational absorption spectrum of water (for
recent examples see Refs. [19–23]). Of particular relevance
herein, we note that as early as 1991 Hermansson et al. cal-
culated a static approximation to the absorption spectrum
by performing ab initio calculations of the OH stretch fun-
damental frequencies and transition dipoles for HOD/D2O
clusters, in the field of the point charges of the surrounding
water molecules [24]. The clusters were taken from snap-
shots of a Monte Carlo simulation, and the spectrum was
approximated as the distribution of frequencies weighted
by the square of the transition dipoles. More recently we
have developed a similar theoretical formalism to describe
both the infrared and Raman spectra for both the HOD/
D2O and HOD/H2O model systems, which includes fluctu-
ating transition frequencies and dipoles, and motional nar-
rowing [25,26]. Briefly, we correlate the ab initio calculated
transition frequencies and dipoles for a number of water
clusters to the projection of the electric field on the OH
(OD) bond obtained from the empirical point charges
located on the (solvent) water molecules. This allows us
to approximate rapidly the transition frequencies and
dipoles during a subsequent MD simulation. Application
of this methodology to the SPC/FQ model [27] demon-
strated good agreement with both experimental IR and iso-
tropic Raman spectra over a wide range of temperatures
[26]. This technique has also been applied to solutes in
aqueous solution [28–32].
While linear (steady-state) spectroscopic measurements
of IR and Raman spectra yield little dynamical informa-
tion, recent time-resolved, ultrafast experiments have
begun to appear on the HOD/D2O and HOD/H2O systems
[33–50]. These experiments are capable of accessing the
dynamical information that is hidden (in line shapes) by
inhomogeneous broadening. The latest and most powerful
experiments involve the three-pulse vibrational echo tech-
nique. For example, Fecko et al. carried out three-pulse
echo peak shift experiments on the HOD/D2O system,
which yield high resolution dynamical information on time
scales ranging from well under 100 fs to over 1 ps [41,42].
Asbury et al. measured two-dimensional infrared (2DIR)
spectra of the HOD/H2O system, which the authors ana-
lyzed in terms of a ‘‘dynamical line width’’ to quantify
the dynamics of the liquid [43–45]. Subsequently, Loparo
et al. measured 2DIR spectra of the HOD/D2O system
[46,47,50], and analyzed these results in terms of various
other metrics to discern the underlying dynamics [48].
These ultrafast experiments provide new dynamical
information about water, and so it seems important to test
the predictions of current simulation models against these
experiments. This is in fact the goal of this paper. For such
a comparison we have found that for water it is essential to
calculate experimental observables directly [51], rather
than, for example, comparing experimental and theoretical
results for an experimentally derived quantity such as the
frequency time-correlation function. Thus for such a com-
parison one needs to calculate non-linear response func-
tions [52], which (in the absence of making the cumulant
approximation) are functions of three independent time
variables. In these calculations it is also important to
account for non-Condon (the transition dipole depends
on the molecular environment) effects [51]. Finally, so as
to allow for a quantitative comparison between theory
and experiment, one needs to convolute over the experi-
mental pulse profiles, and include vibrational relaxation
phenomenologically. Others have made the connection
between simulation models and these ultrafast observables
[49,50,53–57], but not at this level of detail.
Of the vast numbers of existing water models, we choose
tofocusonasmallsubsetthatareamongthemostcommonly
utilized and computationally tractable: the SPC/E, TIP4P,
SPC/FQ, TIP4P/FQ, TIP5P/E, and Dang–Chang models.
An overview of the selected water models is given below.
1.1. SPC/E
The SPC/E [58] model is a rigid, non-polarizable water
model that is extremely popular for MD simulation due to
its simplicity. The model consists of three interaction sites,
located on the oxygen and hydrogen atomic centers. The
144
J.R. Schmidt et al. / Chemical Physics 341 (2007) 143–157

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bond length and angles differ from the experimental gas-
phase geometry of water; the O–H bond lengths are set to
1 A˚and the bond angle is set to a tetrahedral angle of
109.47?. This reflects the fact that the average O–H bond
lengthens, and the bond angle expands, upon moving from
the gas phase to solution. The SPC/E model represents a
small reparameterization of the simple point charge (SPC)
model [59], which was performed in order to account for
the contribution of the self-energy of effective polarization
to the calculated heat of vaporization. Both the original
SPC and the modified SPC/E potential were parameterized
to reproduce the experimental room-temperature density
and heat of vaporization of liquid water, but the SPC/E
model ends up producing reasonable radial distribution
functions and an acceptable diffusion constant as well.
1.2. TIP4P
The TIP4P [60] model is a four-site, rigid, non-polariz-
able model that is also extremely popular. The fourth,
off-atom, interaction site is located a small distance from
the oxygen atom, along the H–O–H bisector. The inclusion
of the additional interaction site improved the agreement
with the experimentally measured second peak of the
O–O radial distribution function. The geometry of the
TIP4P model corresponds to the experimentally measured
geometry of a gas-phase water monomer; that is, O–H
bond lengths of 0.9572 A˚and a bond angle of 104.5?.
The resulting parameterization does an excellent job of
reproducing the experimental room-temperature density
of water, as well as the energy and enthalpy of vaporiza-
tion. The model also does a reasonable job reproducing
theexperimentalO–O radial
although the peak positions are somewhat shifted to
slightly smaller distances compared with experimental mea-
surements [10,11]. In terms of computational efficiency, the
model fares quite well. Although an additional interaction
site is introduced, the number of required distance calcula-
tions per water–water interaction increases only from 9 (in
the case of a three-site model) to 10; thus, the additional
computational burden is not great.
distributionfunction,
1.3. TIP5P/E
The TIP5P/E [61] model is a reparameterization of the
TIP5P model [62], which was very slightly modified to
account for the use of the Ewald summation for long-range
electrostatics in the MD simulation; the original TIP5P
model was instead parameterized for use with long-range
cutoffs. Both the TIP5P and TIP5P/E models are rigid,
non-polarizable, five-site models, with two off-atom interac-
tion sites located at the oxygen ‘lone-pair’ tetrahedral posi-
tions. The nuclear positions are in the TIP4P geometry.
TheTIP5P/Emodelwasparameterizedtoreproducedirectly
the experimental O–O radial distribution function. In addi-
tion, the model was parameterized to reproduce the density
andenergyofwater overawiderange oftemperatures,from
?40to100 ?C;additionally,themodelreproducestheexper-
imentally measured density maximum around 4 ?C.
1.4. SPC/FQ
The SPC/FQ [27] model is a rigid, polarizable variant of
the popular SPC/E model. Unlike the original SPC/E
model, the atomic charges on each water molecule can fluc-
tuate in response to its changing electrostatic environment.
In particular, the ‘electro-negativity equalization’ principle
is used to calculate the value of each of the atomic charges
at each time step. Such a polarizable model could be
expected to have significant advantages over similar, non-
polarizable models. Whereas non-polarizable models must
account for an effective averaged polarization, which hin-
ders their transferability, polarizable models could have
the ability to reproduce data over a wide variety of thermo-
dynamic state points. The SPC/FQ model was parameter-
ized to reproduce directly the experimental gas-phase
dipole moment, as well as radial distribution functions,
and the energy and pressure of liquid water. The model
also does a reasonable job of reproducing other quantities
such as the NMR and Debye relaxation times. One partic-
ular shortcoming of the SPC/FQ model is its inability to
describe out-of-plane polarization, due to the fact that all
of the SPC/FQ charges lie in the plane of the molecule.
In contrast, the experimentally measured polarizability of
water is almost spherically symmetric [63].
1.5. TIP4P/FQ
The TIP4P/FQ [27] model is a rigid, polarizable variant
of the popular TIP4P model. It is the fluctuating charge
analogue of the TIP4P model, in the same way that SPC/
FQ is the fluctuating charge analogue of the SPC/E model.
The TIP4P/FQ model was parameterized to reproduce
directly the experimental gas-phase dipole moment, as well
as radial distribution functions and the energy and pressure
of liquid water. The model also does a excellent job of
reproducing other quantities such as the NMR and Debye
relaxation times and dielectric constant. The diffusion con-
stant predicted by the model is slightly too low, but never-
theless an improvement over the original TIP4P model,
which yielded a diffusion constant that was too high.
1.6. Dang–Chang
The Dang–Chang model [64] is also a rigid, polarizable,
four-site model, similar to the TIP4P model. Like TIP4P,
the model is based on the experimental gas-phase water
geometry and contains an off-atom interaction site located
on the H–O–H bisector. Unlike TIP4P, however, the model
includes a point polarizability on the fourth site, which
results in a potential that is no longer pairwise additive.
The model was extensively parameterized to reproduce
not only liquid-state properties, including density, enthalpy
of vaporization, liquid-state dipole, diffusion constant, and
J.R. Schmidt et al. / Chemical Physics 341 (2007) 143–157
145

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radial distribution functions, but also energetic and geo-
metric information from small water clusters. Thus, the
model can be expected to show reasonable properties over
a wide range of densities and temperatures spanning from
the gas phase to the dense liquid [64,65].
In this paper we calculate both the linear (infrared
absorption and isotropic Raman scattering line shapes),
and non-linear ultrafast (three-pulse vibrational echo)
spectroscopic observables for each of these water models.
We fully account for non-Condon effects and relevant
experimental parameters, so as to allow for a quantitative
comparison between theory and experiment. Cognizant
that errors in the structure (rather than dynamics) of the
models will affect the calculated experimental observables,
we also employ a normalization technique that we believe
allows us to focus on dynamical differences between the
models, while minimizing differences that originate from
small differences in the predicted water structure. Based
on these results, we reach general conclusions about which
of the models seem to reproduce best the experimentally
measured water dynamics.
2. Calculation of line shapes and echoes
For both the HOD/H2O and HOD/D2O systems, we
will consider infinitely dilute HOD, with a single relevant
OH or OD vibrational mode. This local mode has high fre-
quency compared to kT, and so we will treat it quantum
mechanically. All other degrees of freedom will be treated
classically. This (anharmonic) local mode has vibrational
eigenstates labeled 0,1,2... For a given configuration of
the classical degrees of freedom, the local mode has a fun-
damental transition frequency denoted by x10, and a tran-
sition dipole moment ~ l10. If the electric field of all incident
pulses of radiation is polarized in the^ ? direction, and detec-
tion is in the ^ ? direction as well, then the only relevant
quantity is the projection of the transition dipole along
the field direction: l10?~ l10?^ ?. Variations in l10 from
changes in the classical degrees of freedom can come from
changes in the magnitude of ~ l10(non-Condon effects), and
from changes in the angle that~ l10makes with ^ ? (rotations).
2.1. Infrared and isotropic Raman line shapes
Within the mixed quantum-classical approximation
described above the infrared absorption line shape for the
OH (OD) stretch is given by [25,26,52]
Z1
IðxÞ ?
?1
dte?ixt
l10ð0Þl10ðtÞexp i
Zt
0
x10ðtÞ
????
e?jtj=2T1:
ð1Þ
The brackets denote a classical equilibrium statistical
mechanical average, and T1is the lifetime of the first excited
vibrational state: T1= 700 (1450) fs for the OH (OD)
stretching mode of HOD in D2O (H2O) [42,43]. The line
shape is related to the absorption cross section r(x) by [66]
rðxÞ ? xIðxÞ:
Similarly, the isotropic Raman line shape is given by
[26,66]
Z1
ð2Þ
IðxÞ ?
?1
dte?ixt
a10ð0Þa10ðtÞexp i
Zt
0
x10ðtÞ
????
e?jtj=2T1;
ð3Þ
where a10is the matrix element of the isotropic polarizabil-
ity operator. The line shape is related to the scattering
intensity S(x) by [66]
SðxÞ ? ðxE? xÞ4IðxÞ;
where xEis the frequency of the exciting light.
ð4Þ
2.2. Three-pulse vibrational echoes
Within the same notation as above, three-pulse echo
observables can be written in terms of 3rd-order non-linear
response functions [51,52], given by
R1ðt3;t2;t1Þ ¼
R2ðt3;t2;t1Þ ¼ R1ðt3;t2;t1Þ;
R3ðt3;t2;t1Þ ¼ ? l10ð0Þl10ðt1Þl21ðt1þ t2Þl21ðt1þ t2þ t3Þexp i
?
R5ðt3;t2;t1Þ ¼ R4ðt3;t2;t1Þ;
R6ðt3;t2;t1Þ ¼ ? l10ð0Þl10ðt1Þl21ðt1þ t2Þl21ðt1þ t2þ t3Þexp ?i
?
R8ðt3;t2;t1Þ ¼ ? l10ð0Þl21ðt1Þl10ðt1þ t2Þl21ðt1þ t2þ t3Þexp ?i
l10ð0Þl10ðt1Þl10ðt1þ t2Þl10ðt1þ t2þ t3Þexp i
Zt1
0
dsx10ðsÞ
??
exp ?i
Zt1þt2þt3
t1þt2
dsx10ðsÞ
????
;
ð5aÞ
ð5bÞ
ð5cÞ
Zt1
Zt1
0
dsx10ðsÞ
??
?
exp ?i
Zt1þt2þt3
Zt1þt2þt3
t1þt2
dsx21ðsÞ
?
?
?
?
?
l10ð0Þl10ðt1Þl10ðt1þ t2Þl10ðt1þ t2þ t3Þexp ?i
?
?
;
R4ðt3;t2;t1Þ ¼
0
dsx10ðsÞ
?
exp ?i
t1þt2
dsx10ðsÞ
;
ð5dÞ
ð5eÞ
ð5fÞ
Zt1
0
dsx10ðsÞ
??
exp ?i
?
?
Zt1þt2þt3
Zt1þt2
Zt1þt2þt3
t1þt2
dsx21ðsÞ
??
?
?
l10ð0Þl21ðt1Þl21ðt1þ t2Þl10ðt1þ t2þ t3Þexp ?i
?
?
?
?
;
R7ðt3;t2;t1Þ ¼
Zt1þt2þt3
Zt1þt2
0
dsx10ðsÞ
?
exp ?i
?
t1
dsx21ðsÞ
?
;
?
ð5gÞ
0
dsx10ðsÞ
?
exp ?i
t1
dsx21ðsÞ
;
ð5hÞ
146
J.R. Schmidt et al. / Chemical Physics 341 (2007) 143–157

Page 5

where l21 is the projection of the 2-1 transition dipole
along the electric field unit vector, and x21is the fluctuating
2-1 transition frequency.
Vibrational relaxation is accounted for phenomenologi-
cally by multiplying these non-linear response functions by
appropriate factors. Using a harmonic scaling relation to
estimate the (unknown) lifetime of the second excited state
[67], the appropriate multiplicative factors are given in
Table 1 [42,68]. Note that since the lifetime of the second
excited state is relevant only during the coherence period
t3, which is very short, it is not essential to have an accurate
value for this lifetime.
The three-pulse echo is generated by exposing the sam-
ple to a sequence of three resonant pulses with wavevectors
~k1,~k2, and~k3 and frequencies x1, x2, and x3. The time
delay between the first two pulses is s, between the second
two pulses is denoted as T, and the time after the third
pulse is given by t. The resulting signal is observed in the
~ks¼ ?~k1þ~k2þ~k3phase-matching direction.
The integrated echo signal, I(s,T), is calculated by inte-
grating the square of the electric field in the~ksdirection,
Eð3Þð~ks;t;s;TÞ, over t:
Iðs;TÞ ?
0
Z1
where the polarization Pð3Þð~ks;t;s;TÞ is given by [52]
i
? h
j¼1
?Rjðt3;t2;t1ÞE3ðt?t3ÞE2ðtþT ?t3?t2Þ
?E?
?exp iðx3þx2?x1Þt3þiðx2?x1Þt2?ix1t1
Z1
dtjEð3Þð~ks;t;s;TÞj2
?
0
dtjPð3Þð~ks;t;s;TÞj2;
ð6Þ
Pð3Þð~ks;t;s;TÞ¼
? ?3X
3
Z1
0
dt3
Z1
0
dt2
Z1
0
dt1
1ðtþT þs?t3?t2?t1Þ
½?;
ð7Þ
where Eiis the experimental pulse profile for the ith pulse.
The above expression for the polarization assumes that
the pulses are well separated in time, and that all delay
times are positive; thus it is valid only for s, T, and t larger
than the pulse durations. Similar expressions, involving
contributions from the remaining non-linear response func-
tions, can be written down for situations where either of
these assumption is violated [68]. The ‘‘echo peak shift’’,
s*(T), is defined to be the value of s that maximizes the inte-
grated echo signal for a given value of waiting time, T.
The vibrational echo signal can also by measured via
heterodyne detection, whereby the emitted signal is mixed
with a strong local oscillator pulse, with electric field
denoted ELO. The difference signal resulting from the
sample (the difference between the mixed local oscillator/
echo signal and isolated local oscillator) is then given by
[52]
Z1
? exp½iðxLO? x3? x2þ x1Þt?:
The 2DIR spectrum is calculated from the real part of the
Fourier transform of the above signal over the s and s0
dimensions (with frequencies x1and x3); this is equivalent
to summing the individual one-sided Fourier transforms of
the rephasing and non-rephasing signals, as described pre-
viously [51]. The ‘‘dynamic line width,’’ C(T), is defined to
be the FWHM of a one-dimensional slice through the max-
imum of the two-dimensional 0-1 resonance, along the x1
axis for a particular waiting time T [43–45]. The ‘‘nodal
slope’’ is a complementary metric to the dynamic line
width, and provides an alternative way to quantify the
inherentlytwo-dimensional
the 2DIR spectrum [50]. The nodal slope is defined to be
the slope of the node separating the positive-going 0-1
and negative-going 1-2 resonances in the two-dimensional
spectrum. As correlation is lost as a function of T, between
the frequencies in the s and t periods, the 0-1 and 1-2
resonances become more circular, and the slope of the
nodal line goes to zero. Thus, the nodal slope is a measure
of the residual frequency correlation, and thus a measure of
the dynamics.
Iðs;T;s0Þ ¼ ?2Im
?1
dtE?
LOðt ? s0ÞPð3Þð~ks;t;s;TÞ
ð8Þ
informationcontained in
2.3. Calculation of frequencies and transition dipoles
To calculate the line shapes and non-linear response
functions, one needs time-dependent trajectories of transi-
tion frequencies and dipoles. As we have done before
[25,26,30,51,56], we will utilize the empirical frequency cor-
relation (EFC) method in order to calculate the required
quantities. Briefly, the EFC technique begins by sampling
solute/solvent clusters from an MD simulation, and using
ab initio techniques to calculate the frequencies and transi-
tion dipoles of the solute in the presence of the solvent. An
empirical correlation is then derived between the ab initio
calculated frequencies and transition dipoles, and the exter-
nal electric field generated by the point charges on the sur-
rounding water molecules. This correlation thus provides a
computationally efficient way to estimate the instantaneous
transition frequencies and dipoles of a solute during a sub-
sequent MD simulation. Complete details of the procedure
are available elsewhere [25,26,51].
We first apply the EFC method to the SPC/E water
model, as has been done in previous work [25,51]. The
resulting relations between the SPC/E generated electric
field, and the OH (OD) transition frequencies are given
in Table 2. Note that the expressions for the transition fre-
quencies are identical to those reported in Refs. [25,51], but
differ from those reported in Ref. [26], since the latter were
Table 1
Summary of relaxation factors for the non-linear response functions [68]
Response functions Factor
R1, R2, R4, R5, R7
R3, R6, R8
e?ðt3þ2t2þt1Þ=2T1
e?ð3t3þ2t2þt1Þ=2T1
J.R. Schmidt et al. / Chemical Physics 341 (2007) 143–157
147