Three-body Hydrogen Bond Defects Contribute Significantly to the DielectricProperties of the Liquid Water-Vapor Interface

Sucheol Shin and Adam P. Willard∗

Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Dated: October 1, 2018)

In this Letter, we present a simple model of aqueous interfacial molecular structure and we usethis model to isolate the effects of hydrogen bonding on the dielectric properties of the liquid water-vapor interface. By comparing this model to the results of atomistic simulation we show that theanisotropic distribution of molecular orientations at the interface can be understood by consideringthe behavior of a single water molecule interacting with the average interfacial density field viaan empirical hydrogen bonding potential. We illustrate that the depth dependence of this orien-tational anisotropy is determined by the geometric constraints of hydrogen bonding and we showthat the primary features of simulated orientational distributions can be reproduced by assumingan idealized, perfectly tetrahedral hydrogen bonding geometry. We also demonstrate that non-idealhydrogen bond geometries are required to produce interfacial variations in the average orientationalpolarization and polarizability. We find that these interfacial properties contain significant con-tributions from a specific type of geometrically distorted three-body hydrogen bond defect that ispreferentially stabilized at the interface. Our findings thus reveal that the dielectric properties ofthe liquid water-vapor interface are determined by collective molecular interactions that are uniqueto the interfacial environment.

The dielectric properties of liquid water are determinedin large part by the orientational fluctuations of dipo-lar water molecules [1–4]. Near a liquid water-vapor in-terface these orientational fluctuations are anisotropic,leading to dielectric properties that differ significantlyfrom their bulk values [5–7]. These differences are funda-mental to interface-selective chemical and physical pro-cesses [8–11], but they are generally difficult to accessexperimentally. In this Letter, we study these differenceswith a statistical mechanical model of interfacial hydro-gen bonding. This model provides the ability to relatethe microscopic characteristics of hydrogen bonding tothe emergent properties of the liquid water interface. Bycomparing this model to the results of atomistic simula-tion we isolate the specific role of hydrogen bonding indetermining interfacial molecular structure. We demon-strate that water’s interfacial molecular structure is de-termined by the interplay between the anisotropic fea-tures of the interfacial density field and the moleculargeometry of hydrogen bonding interactions. While manydetails of interfacial molecular structure reflect an ide-alized tetrahedral hydrogen bonding geometry, we showthat only distorted non-tetrahedral hydrogen bond struc-tures can contribute to changes in interfacial dielectricproperties. We identify a specific type of non-tetrahedralhydrogen bond defect that is preferentially stabilized atthe interface and show that it contributes significantly tothe unique dielectric properties of the interfacial environ-ment.

Our microscopic understanding of interfacial molec-ular structure derives primarily from a combination ofsurface-sensitive experiments, such as vibrational sumfrequency generation [12–15], and atomistic simula-tion [16–19]. This combination has revealed that theinterfacial environment contains depth dependent molec-

ular populations that vary in their orientational align-ments. The details of this depth-dependent orientationalmolecular structure cannot be determined from existingexperimental data alone and thus we rely on molecularsimulation to supplement our molecular level understand-ing. The molecular structure predicted through the useof standard classical force fields, such as SPC/E [20] andTIP5P [21], are widely used in the study of water inter-faces because they are both computationally efficient andhave been shown to be consistent with available experi-mental data [22, 23].

Even at microscopic length scales the position of aliquid-vapor interface exhibits capillary wave-like spatialundulations [24, 25]. These undulations contribute tothe properties of the interface but they also serve to blurout molecular scale details. The microscopic details ofthe actual phase boundary, i.e., the intrinsic interface isdifficult to isolate experimentally. In theoretical studies,however, the intrinsic interface and its spatial deforma-tions can be treated separately. This Letter focuses onthe intrinsic interface, defining all molecular coordinatesrelative to the time varying position of the instantaneousliquid phase boundary [19]. We identify the position ofthe phase boundary following the procedure described inRef. 19. Our model is thus most valid over microscopiclength scales, where the effects of surface roughening arenegligible, however, since the deformations in the liquid-vapor phase boundary are well characterized by capillarywave theory [26–29], their effects are trivial to reincorpo-rate to enable comparison to experiment.

Atomistic simulations have revealed that the molec-ular structure of the intrinsic water-vapor interface isanisotropic extending about 1 nm into the bulk liquid[19, 30]. This anisotropic molecular structure can be de-composed into two separate components: (1) an inter-

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facial density field, ρ(r), where r denotes the positionmeasured relative to the instantaneous interface, and (2)a position-dependent probability distribution for molec-ular orientations, P (~κ|r), where ~κ uniquely specifies therotational configuration of a water molecule. The in-terplay between these two components is mediated by acombination of molecular packing effects and collectivehydrogen bonding interactions. Classical density func-tional theory can be used to compute ρ(r), but it doesnot provide explicit information about P (~κ|r) [28, 31].Here we present a theoretical model for computing P (~κ|r)from ρ(r), noting that the complete intrinsic interfacialmolecular structure can therefore be derived by combin-ing this model with an existing approach for computingρ(r).

Our model utilizes a mean field approach by computingP (~κ|r) based on the orientational preferences of a singleprobe molecule immersed in the anisotropic mean den-sity field of the intrinsic interface, ρ(r). As illustrated inFig. 1(a), the probe molecule is modeled as a point par-ticle with four tetrahedrally coordinated hydrogen bondvectors, denoted b1, b2, b3, and b4. The length of thesevectors are chosen to correspond to that of a hydrogenbond, so that each vector indicates the preferred posi-tion of a hydrogen bond partner. To mimic the hydrogenbonding properties of water, each bond vector is assigneda directionality, with b1 and b2 acting as hydrogen bonddonors and b3 and b4 acting as hydrogen bond acceptors.The absolute orientations of these tetrahedral hydrogenbond vectors are specified by ~κ.

The probe molecule interacts with the interfacial den-sity field via an empirical hydrogen bonding potential,E(~κ, r, nk), which specifies the potential energy of aprobe molecule with position r, orientation ~κ, and hy-drogen bonding configuration nk. We define nk ≡(n1, n2, n3, n4), where ni is a binary variable that indi-cates the hydrogen bonding state of the ith bond vector.Specifically, ni = 1 if the probe molecule has formed ahydrogen bond along bi, and ni = 0 if it has not. Thereare many possible ways to define this empirical hydrogenbonding potential. We begin by considering the simpleform,

E(~κ, r, nk) =

4∑

i=1

εwni(r,bi) , (1)

where εw is the energy associated with forming a hy-drogen bond. We treat the ni’s as independent randomvariables that are distributed according to,

ni(r,bi) =

1, with probability PHB(ri),0, with probability 1− PHB(ri),

(2)

where ri = r + bi denotes the terminal position of theith bond vector and PHB(ri) specifies the probability forsuccessful hydrogen bonding at position ri.

In the context of this model, the probability for amolecule at position r to adopt an orientation, ~κ, canthus be expressed as,

P (~κ|r) =⟨e−βE(~κ,r,nk)

⟩b/Z(r), (3)

where 〈· · · 〉b denotes an average over all possible hy-drogen bonding states (i.e., variations in the ni’s), 1/βis the Boltzmann constant, kB , times temperature T ,and Z(r) =

∫d~κ⟨e−βE(~κ,r,nk)

⟩b

is the orientationalpartition function for the probe molecule at position r.By evaluating the average explicitly, the numerator ofEq. (S.12) can be written as,

⟨e−βE(~κ,r,nk)

⟩b

=

4∏

i=1

[1 + PHB

(ri)(e

−βεw − 1)],

(4)and this equation, when combined with Eq. (S.12) pro-vides a general analytical framework for computing theorientational molecular structure of an interface with agiven density profile, ρ(r).

(a) (b)

(c)

−2

(a)

Vapor Water

probe molecule

2 0 2 4 6

a / A

interfacial water density

0 4 6

b1

b3

b4

b2

n

aAa1 a3 aprobe

1

0.5

0

0.5

1

0 2 4 6 8 10

cos

a / A

0

0.5

10.5

−0.5

−1

1

0

0

0.5

1

cos O

H

P(cos

O

H |a)

1

0.5

0

0.5

1

0 2 4 6 8 10

cos

a / A

0

0.5

1

0 4

0.5

−0.5

−1

1

0

2 6 8 100

0.5

1

cos O

H

aA

P(cos

O

H |a)

FIG. 1. (a) Schematic depiction of the mean-field model show-ing a probe molecule with tetrahedrally coordinated bondvectors (white for donor, blue for acceptor) within the liq-uid (blue shaded region) at a distance aprobe from the posi-tion of the instantaneous interface (solid blue line). A plotof the interfacial density profile, ρ(a), obtained from the MDsimulation with TIP5P water [32], is shown with dotted linesindicating the termination points of bond vectors b1 and b3.Panels (b) and (c) contain plots of the orientational distri-bution function, P (cos θOH|a) (see Eq. (5)), as indicated byshading, computed from atomistic simulation and from therigid tetrahedral model respectively.

We simplify this general theoretical model by assumingthat PHB(ri) ∝ ρ(ri), where the value of the proportion-ality constant is chosen to match bulk hydrogen bondingstatistics. Furthermore, we assume that the intrinsic in-terface is laterally isotropic and planar over molecularlength scales. According to this assumptions, functionsthat depend on r to be expressed in terms of a, whichdenotes the scalar distance from the instantaneous in-terface measured perpendicular to the interfacial plane.

3

For instance, with this assumption the function PHB(r)is simplified to PHB(ai) ∝ ρ(ai), where ai is given byai = a − bi · n, and n is the unit vector normal to theplane of the interface (see Fig. 1(a)).

We parameterize our model by comparing to the re-sults of atomistic simulations [32]. We perform this com-parison using the reduced orientational distribution func-tion,

P (cos θOH|a) =

∫d~κP (~κ|a)

[1

2

2∑

i=1

δ(cos θi − cos θOH)

],

(5)where the summation is taken over the two donor bondvectors, cos θi = bi · n/|bi|, and δ(x) is the Diracdelta function. Specifically, we determine free parame-ter, εw, by minimizing the Kullback-Leibler divergencefor P (cos θOH|a) computed from our model and thatfrom atomistic simulation [32]. For the TIP5P forcefield [21] this parameterization yields εw = −3.47 kJ/molat T = 298 K. We find that different water models yieldsimilar value of εw [32]. We note that this value includesthe effects of environmental stabilization on broken hy-drogen bonds and is therefore significantly lower in mag-nitude than absolute hydrogen bond energies [33, 34].This demonstrates that our parameterize hydrogen bondstrength is similar to estimates based on X-ray absorp-tion measurements [35, 36], which highlights that thiseffective bonding energy determines the relevant energyscale for fluctuations in the structure of aqueous hydro-gen bond networks.

The effect of molecular dipole orientations on interfa-cial polarization and polarizability can be computed fromP (~κ|a). We specify orientational polarization in terms ofthe average dipole field,

〈µn(a)〉 = µw

∫d~κP (~κ|a) [µ(~κ) · n] , (6)

where µ(~κ) is the unit dipole vector of tagged molecule inparticular orientation (i.e., µ = (b1 +b2)/|b1 +b2|) andµw is the dipole moment of an individual water molecule.Similarly, the orientational polarizability can be relatedto the fluctuations in the dipole field [2],

〈(δµn(a))2〉 = µ2w

∫d~κP (~κ|a) [µ(~κ) · n]

2−〈µn(a)〉2. (7)

We derive physical insight into the role of hydrogen bond-ing in water’s interfacial molecular structure by compar-ing the functions in Eqs. (5)-(7) to the same quantitiescomputed from atomistic simulations.

We first consider an idealized variation of our modelin which the hydrogen bond vectors are of fixed length,dHB = 2.8 A (i.e., the equilibrium hydrogen bond dis-tance [34]), and rigidly arranged with an ideal tetrahedralgeometry. We refer to this model as the rigid tetrahedralmodel. In Fig. 1(b)-(c), we show that P (cos θOH|a) com-puted from the rigid tetrahedral model is similar to that

computed from simulations with TIP5P water. This sim-ilarity illustrates that the seemingly complicated depthdependent orientational patterns in P (cos θOH|a) havesimple physical origins. Namely, these patterns are de-termined by the constraints imposed on tetrahedral co-ordination by the anisotropic interfacial density field.

As the atomistic simulation data in Fig. 2 illustrates,both 〈µn(a)〉 and 〈(δµn(a))2〉 vary significantly fromtheir bulk values at the liquid vapor interface. However,these variations are not captured by the rigid tetrahedralmodel, despite the ability of this model to capture the pri-mary features of P (cos θOH|a). In fact, the rigid tetrahe-dral model predicts that 〈µn(a)〉 and 〈(δµn(a))2〉 are in-dependent of depth (dashed linesd in Fig. 2). This depth-independent behavior is a mathematical consequence ofmodeling hydrogen bonds as perfectly tetrahedral andenergetically symmetric for donor and acceptor bonds[37]. Interfacial variations in orientational polarizationand polarizability must therefore arise through a combi-nation of (1) distortions in tetrahedral coordination ge-ometry and (2) asymmetry in donor/acceptor hydrogenbond energies. We can study the specific influence ofthese effects on 〈µn(a)〉 and 〈(δµn(a))2〉 by explicitly in-cluding their effects in the empirical hydrogen bondinginteraction of our mean field model.

0.04

0.02

0

0.02

0.04

0 2 4 6 8 10

hµni/

µw

aA

0.15

0.2

0.25

0.3

0.35

0 2 4 6 8 10 (

µn)2↵ /µ

2 waA

MDRigid-Tet

3-Body-Fluct

FIG. 2. Interfacial mean dipole orientation, 〈µn(a)〉, anddipole fluctuations, 〈(δµn(a))2〉, computed from molecular dy-namics (MD) simulation and different variations of our meanfield model. Solid red lines correspond to atomistic simula-tion data [32]. Dashed (Rigid-Tet) and solid (3-Body-Fluct)blue lines correspond to the rigid tetrahedral model and thethree-body fluctuation model (see Eq. (8)), respectively.

We modify our empirical description of hydrogen bond-ing based on insight generated from analyzing the micro-scopic hydrogen bonding properties from atomistic sim-ulation. We quantify the hydrogen bond geometry ofindividual molecules by specifying the inter-bond angle,ψ, between pairs of hydrogen bonds. As Fig. 3(a) illus-trates, there are six such angles for a molecule with fourunique hydrogen bond partners. The probability distri-bution for ψ, P (ψ), computed from atomistic simulationis plotted in Fig. 3(b). This plot highlights that the in-terfacial environment features highly distorted non-idealhydrogen bond geometries that depend on the direction-ality (i.e., donor or acceptor) of the two adjacent hy-drogen bonds. Specifically interfacial inter-bond angles

4

(a)

(d)

1.21

0.80.60.40.2

00.2

0 10 20 30 40 50 60

u(

)/E

HB

[deg]

DonorAcceptor 1.2

10.80.60.40.2

00.2

0 30 60 90 120 150 180

v(

)/E

HB

[deg]

D-DA-AD-A

0

0.01

0.02

0.03

0 30 60 90 120 150 180

P(

)

[deg]

D-DA-AD-A

(c)

(b)

b1 b2

b3b4

14 23

1 2

34

FIG. 3. (a) A schematic illustration of the angles used toquantify hydrogen bond geometries. (b) Probability dis-tributions for inter-bond angles, ψ, generated from atom-istic simulation, computed separately for donor-donor (D-D),acceptor-acceptor (A-A), and donor-acceptor (D-A) pairs ofbonds [32]. Solid and dashed lines correspond to statisticsgenerated within the bulk liquid and at the interface (i.e.,|a| < .1 A) respectively. (c) Average direct interaction en-ergy, u(φ), expressed in units of the average bulk hydrogenbond energy, EHB = 20.9 kJ/mol, between a tagged moleculeand individual hydrogen bond partners, computed separatelyfor donor and acceptor bonds. (d) Average direct interac-tion energy, v(ψ), between two hydrogen bond partners of atagged molecule, as indicated by the dotted arcs in panel (a),computed separately for the case of two donor partners, twoacceptor partners, and one acceptor and one donor.

are narrowed relative to that of the bulk, and this nar-rowing is especially significant between bonds with likedirectionality. Furthermore, as illustrated in Fig. 4(a),we observe a significant increase in the relative fractionof highly distorted hydrogen bond configurations withinthe first 2 A of the interfacial region.

We quantify the energetic properties of these highlydistorted hydrogen bonds by analyzing atomistic simula-tion data. As Fig. 3(c) illustrates, the average direct in-teraction energy between a tagged molecule and one of itshydrogen bond partners is significantly weakened whenthe bond is distorted away from its preferred tetrahedralgeometry. Despite this weakening, however, we observethat highly distorted hydrogen bond configurations canbe stabilized by favorable interactions between hydrogenbond partners. As Fig. 3(d) illustrates, these interactionsbecome particularly favorable when ψ ≈ 60 deg, wherethe hydrogen bond partners are separated by approxi-mately dHB, and thus well situated to form a hydrogenbond. The resulting structure, a triangular three-bodyhydrogen bond defect, is unfavorable in the bulk liquid,however, at the interface this defect structure is stabilizedby an increased availability of broken hydrogen bonds[38], which due to the presence of the liquid phase bound-

ary, cannot all be satisfied without significant distortionsin hydrogen bond geometries.

Three-body interactions have been found to be an im-portant element of water’s molecular structure both inthe bulk and at the interface [39–42]. The three-bodydefects that we have identified here are structurally differ-ent from those that have been found to facilitate the ori-entational relaxation dynamics within the bulk liquid [43]and are uniquely stabilized at the interface. We quantifythe stabilizing effect of the interface by computing the rel-ative free energy for squeezed triangular hydrogen bonddefects, β∆Fsqz(a) = − ln [Psqz(a)/(1− Psqz(a))], wherePsqz(a) denotes the probability to observe a moleculewith position a that is part of such a defect [32]. AsFig. 4(b) illustrates, these defects are more stable at theinterface than in the bulk liquid by about 2 kBT .

(a) (b)

1.5

2

2.5

3

3.5

4

4.5

0 2 4 6 8 10

Fsq

z(a)

aA

0

0.1

0.2

0.3

0 2 4 6 8 10

0

0.3

0.6

0.9

0 2 4 6

P(

<60

deg

)

aA

squeezed triangular H-bond defect

D-D

A-A

D-A

FIG. 4. (a) Probability of observing ψ < 60 deg at varied in-terfacial depth from the atomistic simulation and our model(inset). Labels indicating the types of bond pairs are given inthe same colors of lines. (b) Interfacial profile of free energychange associated with the distortion into a squeezed trian-gular hydrogen bond defect. Plot shown above is for the caseincluding two donor bonds.

We incorporate the effects of these three-body inter-actions into our model by including angle dependence inthe empirical hydrogen bond potential. We accomplishthis by allowing the angles of the hydrogen bonds to fluc-tuate, subject to the following expression,

E(~κ, a, nk) =

4∑

i=1

uα(φi)ni

+∑

i<j

[vαγ(ψij)− λαγ(ai, aj , ψij)]ninj ,(8)

where φi denotes the angle of deviation of bi from itsideal tetrahedral orientation and ψij is the angle betweenbi and bj (see Fig. 3(a)). In this expression the directhydrogen bond energy, uα(φ), is described separately fordonor and acceptor bonds, as denoted by the subscriptα. Similarly, the effects of three-body interactions aredescribed by vαγ(ψ), which depends on the directional-ity of the bonds involved in ψ, denoted by α and γ. Thefunction λαγ(ai, aj , ψij) is designed to attenuate the ef-fects of three-body interactions based on the availabilityof broken hydrogen bonds at ai or aj . Here, we assign

5

uα and vαγ to reflect the atomistic simulation data plot-ted in Fig. 3(c) and Fig. 3(d), respectively. We refer tothis variation of our model as the three-body fluctuationmodel. The details of this model variation are describedin the Supplemental Material [32].

The three-body fluctuation model exhibits interface-specific non-ideal hydrogen bond structure that is simi-lar to that observed in atomistic simulation. As the insetof Fig. 4(a) illustrates, this includes an interfacial en-hancement of donor-donor and acceptor-acceptor anglesof ψ < 60 deg, similar to that observed in atomistic sim-ulation. We find that including the effects of three-bodyhydrogen bond defects significantly improves the abilityof the model to accurately describe interfacial polariza-tion and polarizability, as illustrated in Fig. 2. These ef-fects do not, however, completely account for the interfa-cial variations in 〈(δµn(a))2〉, which suggests that interfa-cial polarizability also includes contributions from othermicroscopic effects, such as higher-order many-body ef-fects.

The characteristics of interfacial molecular structureas derived from molecular dynamics simulations dependsomewhat on the identity of the water force field. Weillustrate some of these differences in the SupplementalMaterial [32]. We observe that different force fields ex-hibit similar trends in dipolar polarization and polariz-ability, but that they differ in their quantitative char-acteristics. These differences reflect subtle variations inforce field structure that are not easily captured withinthe constraints of our mean field model. We discussthis issue in more detail within the Supplemental Ma-terial [32].

To conclude, we summarize two fundamental aspectsof interfacial molecular structure that have been revealedby our mean field model of the liquid water-vapor inter-face. First, we observed that the depth dependent vari-ations in molecular orientations at the interface are de-termined by the molecular geometry of hydrogen bond-ing and how it conforms to the anisotropic features ofthe interfacial density field. We demonstrated that idealbulk-like hydrogen bond geometry was sufficient to ex-plain most of the interfacial variations in molecular ori-entations. Second, our model revealed that interfacialvariations in molecular orientational polarization and po-larizability arise due to distorted non-tetrahedral hydro-gen bond structures. We showed that the interface fea-tures squeezed triangular hydrogen bond defects thatcontribute significantly to determining interfacial dielec-tric properties.

We thank David Chandler, Liang Shi, and Rick C.Remsing for the useful discussions. This work was sup-ported by the National Science Foundation under CHE-1654415 and also partially (SS) by the Kwanjeong Edu-cational Foundation in Korea.

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102, 14171 (2005).[37] For the perfectly tetrahedral coordination,

∑4i=1 bi = 0.

Because of the symmetry, any pair of distinct i and j canrepresent the donor bonds such that 〈µ〉 = µw〈µ〉 ∝ 〈bi+bj〉 = 0 for any orientational distribution. The dipole vec-tor also can be decomposed such that µ =

∑3k=1 µek ek

where ek’s are orthonormal vectors in three dimensions.Choosing e1 = n, we have 〈µn〉 = 〈µ · n〉 = 〈µ〉 · n = 0.Here n could be other two unit vectors and there is nopreference along specific direction. That is, since |µ|2 =∑3k=1 µ

2ek

= µ2w, we expect 〈µ2

n〉 = µ2w/3.

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Supplemental Material: “Three-body hydrogen bond defects contribute significantlyto the dielectric properties of the liquid water-vapor interface”

INTERFACIAL STRUCTURES OF DIFFERENT CLASSICAL WATER FORCE FIELDS

To compare the interfacial structures of classical water force fields, we simulated the liquid-vapor interfaces ofSPC/E [S1], TIP4P/2005 [S2], and TIP5P [S3] waters in NVT ensembles. Each system has 1944 water moleculesin a slab geometry with dimensions of 5.0 nm × 5.0 nm × 3.0 nm. The system was equilibrated at T = 298 K andParticle Mesh Ewald was used to handle the long-range part of electrostatic interactions. The SHAKE and SETTLEalgorithms were used to constrain the geometry of water. The simulations were performed using the LAMMPS package[S4] for SPC/E and TIP4P and the GROMACS package [S5] for TIP5P. Following the same procedure in Ref. S6,instantaneous liquid interface was constructed for each configuration of the generated statistics. Density profile, ρ(a),and orientational distribution, P (cos θ1, cos θ2|a), were computed according to Eqs. (7) and (11) in Ref. S6. FigureS1 shows the density profiles along with the reduced orientational distributions as defined in Eq. (5) of the main text.Both ρ(a) and P (cos θOH|a) exhibit the qualitatively same structural characteristics for three different force fields ofwater.

The orientational polarization, 〈µn(a)〉, and polarizability, 〈(δµn(a))2〉, were computed according to Eqs. (6) and (7)of the main text, and their interfacial profiles are plotted in Fig. S2. The plots show that these interfacial propertiesalso share the qualitatively same feature among three different force fields of water. It is notable that while thereexists some quantitative difference in the scale of the polarization, the polarizability shows the almost identical trendacross the force fields.

OPTIMIZATION FOR THE HYDROGEN BOND ENERGY PARAMETER

To determine the optimal model parameter, εw, we compute the Kullback-Leibler divergence [S7],

Γ(εw) =

∫da

∫d(cos θOH)Pref(cos θOH|a) ln

[Pref(cos θOH|a)

P (cos θOH|a, εw)

], (S.1)

where Pref(cos θOH|a) and P (cos θOH|a, εw) are the reduced orientational distributions obtained from atomistic sim-ulation and our mean-field model, respectively. This quantity measures how far the probability distribution of ourmodel deviates from the reference given εw. As indicated above, it becomes a function of εw and thus we choose theparameter that minimizes the fitness function,

ε∗w = arg minεw

Γ(εw), (S.2)

as the effective hydrogen bond energy.

2

1

0.5

0

0.5

1

1

0.5

0

0.5

1

1

0.5

0

0.5

1

0 2 4 6 8 10

cos O

Hco

s O

Hco

s O

H

aA

0

0.5

1

1.5

0

0.5

1

1.5

2 0 2 4 6 8 10

(a

)/

b

aA

SPC/ETIP4PTIP5P

(a)

(b)

FIG. S1. (a) Density profiles computed from the atomistic simulations of SPC/E, TIP4P, and TIP5P. They are normalized bythe bulk density, ρb. (b) Orientational distributions, P (cos θOH|a), for SPC/E (top), TIP4P (middle), and TIP5P (bottom).Color shading indicates the probability density.

3

0.2

0.1

0

0.1

0.2

0.3

0 2 4 6 8 10

hµni/

µw

aA

SPC/ETIP4PTIP5P

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 2 4 6 8 10

(µ

n)2↵ /µ

2 w

aA

SPC/ETIP4PTIP5P

FIG. S2. Interfacial polarization, 〈µn(a)〉, and polarizability, 〈(δµn(a))2〉, computed from the atomistic simulations of SPC/E,TIP4P, and TIP5P.

FLUCTUATIONS IN HYDROGEN BOND GEOMETRY

We quantify the distortions in hydrogen bond geometry and the associated energetics by analyzing the atomisticsimulation results as follows. Notably, here relatively simple algorithms for quantifying various aspects of molecular

geometry translate into complicated mathematical expressions. Let v(i)k = r

(i)k − r

(i)O , where r

(i)O is the position of the

oxygen of the ith water molecule and r(i)k is the position of the kth bonding site on it (k = 1, 2 indicate the hydrogens

and k = 3, 4 indicate the lone pairs for TIP5P water). Then v(i)k represents the direction of ideal hydrogen bonding

coordination through the kth site of the ith molecule. For the jth molecule neighboring the ith, its deviation anglefrom the ideal coordination to the ith molecule is given by

φ(i)j = min

k

cos−1

v

(i)k · b

(i)j∣∣∣v(i)

k

∣∣∣∣∣∣b(i)j

∣∣∣

, (S.3)

where b(i)j = r

(j)O − r

(i)O represents the hydrogen bond vector of the ith molecule to the jth. The corresponding index,

y(i)j = arg min

k

cos−1

v

(i)k · b

(i)j∣∣∣v(i)

k

∣∣∣∣∣∣b(i)j

∣∣∣

, (S.4)

indicates which ideal boding direction b(i)j is distorted from and thus whether it belongs to donor or acceptor hydrogen

bond. The inter-bond angle between the jth and kth molecules with respect to the ith is given by

ψ(i)jk = cos−1

b

(i)j · b

(i)k∣∣∣b(i)

j

∣∣∣∣∣∣b(i)k

∣∣∣

. (S.5)

Then the probability distribution of inter-bond angle at given distance a is computed as,

Pαγ(ψ|a) =

⟨∑

i

∑

j 6=i

∑

k 6=i>j

δ(ψ(i)jk − ψ)δ(a(i) − a)Θhyd

(b(i)j

)Θhyd

(b(i)k

)Φα

(y(i)j

)Φγ

(y(i)k

)⟩

sinψ

⟨∑

i

∑

j 6=i

∑

k 6=i>j

δ(a(i) − a)Θhyd

(b(i)j

)Θhyd

(b(i)k

)Φα

(y(i)j

)Φγ

(y(i)k

)⟩ , (S.6)

where a(i) is the interfacial depth of the ith molecule,

Θhyd(r) = H(|r| − 2.4 A)H(3.2 A− |r|) (S.7)

4

selects the molecules only in the first hydration shell of the ith molecule using the Heaviside step function, H(x), and

Φα (x) =

H (2.5− x) , if α is Donor,H (x− 2.5) , if α is Acceptor,

(S.8)

selects the neighboring molecule of specific bond type, α. Here the geometric factor, sinψ, corrects the bias comingfrom the variation of solid angle. Fig. 3(b) of the main text shows the plots of Pαγ(ψ|a) with a = 10 A and a = 0 Afor the bulk and interface respectively (0.1 A was used for the binning width of histogram). For the plots in Fig. 4(a)

of the main text, the distribution is integrated such that Pαγ(ψ < 60|a) =∫ 60

0Pαγ(ψ|a) sinψdψ.

Computing Psqz(a) needs to specify the certain type of defect among the configurations of ψ < 60 deg. There is theother type of defect than the squeezed triangular one, which is known as the intermediate of the water reorientation

[S8]. This type of defect has bifurcated hydrogen bonds through one site of the molecule such that y(i)j = y

(i)k . By

excluding such cases, we can compute the probability to observe a squeezed configuration of two donor bonds as,

Psqz,DD(a) =

⟨∑

i

∑

j 6=i

∑

k 6=i>j

H(

60 − ψ(i)jk

)δ(a(i) − a)Θhyd

(b(i)j

)Θhyd

(b(i)k

)δ(y(i)j y

(i)k − 2

)⟩

⟨∑

i

∑

j 6=i

∑

k 6=i>j

δ(a(i) − a)Θhyd

(b(i)j

)Θhyd

(b(i)k

)ΦD

(y(i)j

)ΦD

(y(i)k

)⟩ . (S.9)

The average direct interaction energy for a hydrogen bond pair is computed in the bulk phase as a function of thedeviation angle, φ, such that

uα(φ) =

⟨∑

i

∑

j 6=i

Uij δ(φ(i)j − φ)H

(a(i) − ab

)Θhyd

(b(i)j

)Φα

(y(i)j

)⟩

⟨∑

i

∑

j 6=i

δ(φ(i)j − φ)H

(a(i) − ab

)Θhyd

(b(i)j

)Φα

(y(i)j

)⟩ , (S.10)

where Uij is the pair potential energy between the ith and jth molecules and ab = 10 A. The plots are given in Fig.3(c) of the main text, normalized by the average bulk hydrogen bond energy where we used EHB = 9.0 kBT (see thenext section below for more details on this). Similarly, the average direct interaction energy between two neighborsof a tagged molecule is computed as,

vαγ(ψ) =

⟨∑

i

∑

j 6=i

∑

k 6=i>j

Ujk δ(ψ(i)jk − ψ)H

(a(i) − ab

)Θhyd

(b(i)j

)Θhyd

(b(i)k

)Φα

(y(i)j

)Φγ

(y(i)k

)⟩

⟨∑

i

∑

j 6=i

∑

k 6=i>j

δ(ψ(i)jk − ψ)H

(a(i) − ab

)Θhyd

(b(i)j

)Θhyd

(b(i)k

)Φα

(y(i)j

)Φγ

(y(i)k

)⟩ . (S.11)

IMPLEMENTATION OF THE THREE-BODY FLUCTUATION MODEL

Following the notations used in the previous section, let v1, v2, v3, v4 be the unit vectors of ideal hydrogenbonding directions through the hydrogens and lone pairs of a probe water molecule of given orientation ~κ. Wesample the hydrogen bond vectors, b1,b2,b3,b4, each of which is within a certain solid angle around vi such thatbi · vi = |bi| cosφi where the value for cosφi is drawn from the uniform random distribution of [cos 70, 1]. Eachhydrogen bond vector is assigned a directionality of either donor or acceptor based on the proximity to the bondingsites (see Eq. (S.4)). The six inter-bond angles, ψij , are calculated based on Eq. (S.5). If the bond vectors are tooclose with one another, i.e. ψij < 44, they are not taken into account for computing P (~κ|a) since there is almost nostatistics below that in the atomistic simulation. Hence P (~κ|a) is computed as,

P (~κ|a) =

∫ 4∏

i=1

dbiδ(|bi| − dHB)H

(bi|bi|· vi − cos 70

)∏

j>i

H

(cos 44 − bi · bj

|bi||bj |

)⟨e−βE(~κ,a,nk)

⟩b

Z(a), (S.12)

5

1.2

1

0.8

0.6

0.4

0.2

0

0.2

0 10 20 30 40 50 60 70 80 90

[deg]

uD,max = +0.2

uA,max = 3.7

u↵(

)/ w

FIG. S3. uα(φ) compared with uα(φ) in the same scale. Solid lines are the simulation data of uα(φ)/EHB, as originally shownin Fig. 3(c) of the main text, and circles indicate the rescaled energy functions that are optimized for the accurate trend of〈µn(a)〉. Gray and blue color correspond to α = D and α = A, respectively. The blue dashed line corresponds to uA(φ|a = 1 A),i.e., the average direct interaction energy for a hydrogen bond pair at a = 1 A.

where E(~κ, a, nk) follows the Eq. (8) of the main text. Here we impose the same constraint on the length of hydrogenbond vectors as that in the rigid tetrahedral model. Implementing the fluctuations in |bi| provokes more details aboutthe energetics, uα and vαγ , such as their dependence on both lengths and angles of the hydrogen bond vectors, whichwe have not detailed so far in this model.

The energy functions, uα(φ) and vαγ(ψ), are rescaled from the atomistic simulation data of uα(φ) and vαγ(ψ).Additionally, we parametrize uα(φ) by tuning the maximum value of uα(φ) such that

uα(φ) =

[uα(φ)− uα(0)

] u∗α − uα(0)

uα,max − uα(0)+ uα(0)

|εw|EHB

, if φ ≤ φc,

0, if φ > φc,

(S.13)

where φc = 72, uα,max = maxφ<φc uα(φ), and u∗α is the parameter that sets the new maximum (in the originalscale of uα). Here the factor of |εw|/EHB rescales the functions in units of the effective hydrogen bond energy of ourmodel. We observed that the behavior of 〈µn(a)〉 is largely sensitive to uα(φ), and thus we optimized the parameters,u∗α and εw, for the result expected from atomistic simulations. For the results presented in Fig. 2 of the main text,we used εw = −5.0 kBT , u∗D = +0.2 kBT , and u∗A = −3.7 kBT . We found that the optimized uA(φ) is quite differentfrom uA(φ) but more like the corresponding energy computed near the interface (see Fig. S3). For vαγ(ψ), we simplytake the values from the atomistic simulation data and rescale them in units of εw, that is,

vαγ(ψ) = vαγ(ψ)|εw|EHB

. (S.14)

As given in Eq. (8) of the main text, vαγ(ψ) is combined with the auxiliary function, λ(ai, aj , ψij), in order toaccounts for the interface-specific stability of hydrogen bond defects. This auxiliary function represents the energeticcost for the defects to pay based on the hydrogen bonding status of the ith or jth hydrogen bond partner. Assumingthat this penalty is imposed on the one that donates hydrogen, we describe this function in terms of the averagenumber and energy of hydrogen bonds through donor sites, denoted by ND(a) and ED(a) respectively. Specifically itis given by,

λαγ(ai, aj , ψ) =

ND(aαγ)

2ED(aαγ)

|εw|EHB

, if ψ ≤ ψαγ ,

[1− vαγ(ψ)− vαγ(ψαγ)

vαγ(ψc)− vαγ(ψαγ)

]λαγ(ai, aj , ψαγ), if ψαγ < ψ < ψc,

0, if ψ ≥ ψc,

(S.15)

6

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 108.6

8.4

8.2

8

ND(a

)/2

ED(a

)/kBT

aA

1.2

1

0.8

0.6

0.4

0.2

0

0.2

45 60 75 90 105 120

[vA

A(

)

AA(

)]/

w

[deg]

AA = 0aAA = 0 AaAA = 1 AaAA = 2 A

(a) (b)ND(a)/2

ED(a)

FIG. S4. (a) Interfacial profiles of average number and energy of hydrogen bonds through donor sites, rendered in orangeand blue lines respectively. (b) Illustration of a three-body interaction term implemented in our model. Solid lines show thedifferent energetic preferences for highly distorted configurations depending on the interfacial depth of hydrogen bond partner.Dashed line corresponds to vAA without the attenuation by λAA.

where ψαγ = arg minψ vαγ(ψ), ψc = arg maxψ vAA(ψ), and

aαγ(ai, aj) =

minai, aj, if α = γ,

ΦD(yi) ai + ΦD(yj) aj , if α 6= γ.(S.16)

Here we let the penalty taken by the hydrogen bond partner located closer to the interface, but we make the exceptionfor donor-acceptor bond pairs based on the typical structure of cyclic water trimer [S9]. ND(a) and ED(a) are computedfrom atomistic simulation as,

ND(a) =

⟨∑

i

∑

j 6=i

δ(a(i) − a)H(

30 − φ(i)j)H(

3.5 A−∣∣∣b(i)j

∣∣∣)

ΦD

(y(i)j

)⟩

⟨∑

i

δ(a(i) − a

)⟩ , (S.17)

and

ED(a) =

⟨∑

i

∑

j 6=i

Uij δ(a(i) − a)H

(30 − φ(i)j

)H(

3.5 A−∣∣∣b(i)j

∣∣∣)

ΦD

(y(i)j

)⟩

⟨∑

i

∑

j 6=i

δ(a(i) − a)H(

30 − φ(i)j)H(

3.5 A−∣∣∣b(i)j

∣∣∣)

ΦD

(y(i)j

)⟩ , (S.18)

where the definition of a good hydrogen bond follows the one by Luzar and Chandler [S10]. As illustrated in Fig. S4,they change dramatically within the first 2 A of the interfacial region such that the effect of three-body interactionalso becomes significant in that region. Here we take the value of EHB from ED(a) by setting EHB = |ED(ab)| =8.45 kBT = 20.9 kJ/mol at T = 298 K.

In order to evaluate P (~κ|a), we obtain an approximate analytic expression for⟨e−βE(~κ,a,nk)

⟩b

in Eq. (S.12). Herewe made the same assumption as that of the rigid tetrahedral model, such that 〈ninj〉 ≈ 〈ni〉〈nj〉 for i 6= j [S11].

7

(a)

(b)1

0.5

0

0.5

1

0 2 4 6 8 10

cos O

HaA

0

0.5

1

1

0.5

0

0.5

1

0 2 4 6 8 10

cos O

H

aA

0

0.5

1

P(cos

O

H |a)

P(cos

O

H |a)

FIG. S5. Orientational distributions, P (cos θOH|a), computed from (a) the three-body fluctuation model and (b) the rigidtetrahedral model for the TIP5P force field. Color shading indicates the probability density.

Within this approximation, we can write

⟨e−βE(~κ,a,nk)

⟩b

=

4∏

i=1

〈ni〉be−βuα(φi)4∏

j>i

e−βvαγ(ψij) +

4∑

k=1

[1− 〈nk〉b]

4∏

i 6=k

〈ni〉be−βuα(φi)4∏

j 6=k>i

e−βvαγ(ψij)

+

4∑

i=1

4∑

j>i

〈ni〉b〈nj〉b [1− 〈nk〉b] [1− 〈nl〉b] e−β[uα(φi)+uα(φj)+vαγ(ψij)]

+

4∑

k=1

〈nk〉be−βuα(φk)4∏

i 6=k

[1− 〈ni〉b] +

4∏

i=1

[1− 〈ni〉b] , (S.19)

where 〈ni〉b = PHB(ai) = ρ(ai)/2ρb and the dummy indices, k and l, in the third term are the numbers among1, 2, 3, 4 such that i 6= j 6= k 6= l. The resulting reduced probability distribution, P (cos θOH|a), is given in Fig. S5.Although its qualitative feature is still the same as the result from the rigid tetrahedral model, its details are closerto that from the atomistic simulation of TIP5P water (especially at a < 3 A).

APPLICATION OF MEAN-FIELD MODELS TO SPC/E FORCE FIELD

Despite sharing similar bulk hydrogen bonding structures, different classical water models can yield non-trivialdifferences in interfacial structure. This is highlighted to some extent in Figs. S1 and S2. As Fig. S2 illustrates, differentwater models exhibit similar trends in 〈µn(a)〉 and 〈(δµn(a))2〉, but they differ in their quantitative characteristics.To evaluate the ability of our mean field model to capture these differences have also applied our model to the SPC/Eforce field. To do this, we have followed the same procedure described herein but using the data from moleculardynamics simulations of SPC/E water rather than TIP5P. For the SPC/E-parameterized mean field model we havefound similarly good agreement in reproducing P (cos θOH|a), as illustrated in Fig. S6, but as Fig. S7 illustrates, theagreement to 〈µn(a)〉 and 〈(δµn(a))2〉, is not as strong for SPC/E as it is for TIP5P.

We understand that the difference in the ability of our model to reproduce dipolar polarization/polarizabilitybetween SPC/E and TIP5P arises due to the differences in the geometric tendencies inherent to these force fields.

8

(a)

(b)1

0.5

0

0.5

1

0 2 4 6 8 10

cos O

H

aA

0

0.5

1

1.5

1

0.5

0

0.5

1

0 2 4 6 8 10

cos O

H

aA

0

0.5

1

1.5

P(co

sO

H |a)

P(co

sO

H |a)

FIG. S6. P (cos θOH|a) computed from (a) the atomistic simulation with SPC/E water and (b) the rigid tetrahedral modeloptimized for the SPC/E force field (ε∗w = −1.8 kBT = −4.46 kJ/mol).

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2 4 6 8 10

hµni/

µw

aA

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 2 4 6 8 10

(µ

n)2↵ /µ

2 w

aA

MDRigid-Tet

3-Body-Fluct

FIG. S7. Interfacial polarization, 〈µn(a)〉, and polarizability, 〈(δµn(a))2〉, computed from the three-body fluctuation modelparametrized for the SPC/E force field, in comparison to the molecular dynamics simulation results.

The TIP5P force field is built upon a tetrahedral charge scaffold, so non-ideal hydrogen bond structures are morenaturally described in terms of their deviations from this scaffold. On the other hand, SPC/E is built upon a triangularcharge scaffold (albeit with a tetrahedral bond angle) so non-ideal hydrogen bond structures are less well representedin term of deviation from a tetrahedral scaffold. We speculate that we could improve quantitative accuracy of ourmodel in reproducing SPC/E results by modifying the details of the underlying geometry of our mean field model.

∗ awillard@mit.edu[S1] H. J. C. Berendsen, J. R. Grigera, and T. P. Straatsma, J. Phys. Chem. 91, 6269 (1987).[S2] J. L. Abascal and C. Vega, J. Chem. Phys 123, 234505 (2005).[S3] M. W. Mahoney and W. L. Jorgensen, J. Chem. Phys. 112, 8910 (2000).[S4] S. Plimpton, J. Comp. Phys. 117, 1 (1995).[S5] S. Pronk, S. Pall, R. Schulz, P. Larsson, P. Bjelkmar, R. Apostolov, M. R. Shirts, J. C. Smith, P. M. Kasson, D. van der

Spoel, et al., Bioinformatics 29, 845 (2013).[S6] A. P. Willard and D. Chandler, J. Phys. Chem. B 114, 1954 (2010).

9

[S7] S. Kullback and R. A. Leibler, Ann. Math. Stat. 22, 79 (1951).[S8] D. Laage and J. T. Hynes, Science 311, 832 (2006).[S9] F. N. Keutsch, J. D. Cruzan, and R. J. Saykally, Chem. Rev. 103, 2533 (2003).

[S10] A. Luzar and D. Chandler, Phys. Rev. Lett. 76, 928 (1996).[S11] This is definitely a rough approximation since it neglects the density correlation between hydrogen bond partners even

in the squeezed configurations. However, more accurate treatment for the density correlation provokes again the distancedependence of the energy functions which we have not detailed so far in this model.