H bond network in water

Croucher ASI, Hong Kong, Dec. 7 2005

Roberto Car, Princeton University

Many special properties of water are associated to its H-bond network

The H-bond network imposes characteristic correlations on the positions and relative orientations of neighboring molecules

Here we focus on the effect of these correlations on the static and dynamic dielectric properties of water within ab-initio Molecular Dynamics

H-bond network

Water molecules

Maximally Localized Wannier (Boys) Orbitals in H2O: a polar molecule

bond (green) and lone (pink) pairs

A water dimer: two water molecules form a H-bond

Hydrogen Bonds

water dimer

Local tetrahedral H-bond order :

Donors (D) and Acceptors (A)

Pauling ice rule: 2D + 2A

Proton disorder

The static dielectric constant of water is unusually large: why?

Reproduced from L. Pauling, General Chemistry (1970)

In condensed phases the “effective” dipole moment of a water molecule is enhanced (from 1.8D to ~ 3D) but this is not sufficient.

Alignment of independent dipoles:

2

cos3 B

N EP N

k T

4

1P

E

In water and/or ice this gives (with 3 )D 30

To explain the large dielectric constant one needs a more refined model including the effect of the H-bonds

Short-range dipole-dipole correlations are consequence of the Pauling ice

rules

Anti-parallel correlations between neighboring dipoles would necessitate H-bond breaking and are suppressed. Thus the local dipole is enhanced

First-principles Molecular Dynamics

I II

M RR

from electronic ground-state within DFT

In presence of en electric field: electric enthalpy functional

P E E

Use Maximally Localized Wannier Functions (MLWF defined according to Marzari and Vanderbilt) at each time step

P is calculated from the displacements of the ions and the MLWF centers

Distribution of “dipoles” in liquid water

from Silvestrelli and Parrinello, PRL (1999): molecular “dipoles” are defined using MLWFs

First-principles water has a large dielectric constant

At the temperature of the simulation (T~325K) the experimental dielectric constant is = 70 , the calculated value is = 79 5

41

P

E

jiji ji

ji

dd

dd

rNrC

,,)(

1)(

Dipole-dipole correlations:

Short range correlations are present even in absence of electric field

A simple lattice model to model H-bonds correlations

• We take a cubic (diamond) model of ice (standard structure of ice is hexagonal)

• We assume that Pauling ice rules are exactly satisfied: configurations in absence of an applied field are only distinguished by entropy not by energy

• We apply a finite electric field and include the dipolar coupling to the field for each molecule characterized by an average dipole moment (a parameter of the model)

EpWN

ii

1

Tk

W

accBeP ,1min

Dipole configurations satisfying the Pauling ice rules in presence of an external electric field can be generated by a Monte Carlo (MC) procedure

Bernal-Fowler-Pauling (BFP) model of ice in electric field

The polarization in the lattice model has values close to those of first-principles water, if the molecular dipoles in the lattice model are equal to the average dipole of first-principles water ( ~ 3D)

jiji ji

ji

dd

dd

rNrC

,,)(

1)(

Dipole-dipole correlations in ab-initio water and in the BFP model of ice show strikingly similar behavior

The best fit corresponds to = 2.64 D

Interestingly this is very close to the value ( ~ 2.60 D) predicted for ice by Coulson and Eisenberg (Proc. Roy. Soc. 1966)

Fitting the BFP model to experiment at one temperature reproduces well the observed temperature behavior

Some historical perspective:

Onsager and Kirkwood applied the concept of Lorentz field (local field) to understand the dielectric properties of polar liquids

All these treatments describe a polar liquid as a collection of dipolar molecules

Onsager treats a molecule as a cavity in a statistical continuum of uniform dielectric constant equal to that of the liquid in the bulk. In this treatment environmental effects are only due to long range Coulomb effects, the effect of the H-bonds enters at most as an enhancement of the dipole moment of a molecule in the liquid phase compared to the gas phase

In Kirkwood’s treatment the cavity includes a molecule and its shell of neighbors. In this way he models the effect of hindered relative rotations of neighboring molecules due to H-bonds. This is a different kind of environmental effect directly related to the chemistry of the bonds

Kirkwood’s theory is a phenomenological model that depends on 2 parameters: the bond angle among water molecules and the dipole moment of a molecule in the liquid phase. Using a tetrahedral bond angle and a reasonable estimate for the enhanced dipole he gets a value of 55 for the dielectric constant of water (experiment: 79). The unmodified Onsager theory leads to a value of 31.

Applications to lattice models of liquid water by Pople (in the 1950’s) led him to conclude that the dielectric constant of water is a strong indication for the existence of an H-bond network in the liquid

There is no way of estimating the 2 parameters in the context of the theory. Besides, the concept of a molecular dipole is ill-defined in condensed phase. Furthermore the precise value of the local field acting on the molecules is difficult to estimate. These difficulties are solved in our first-principle treatment which models water as a collection of nuclei and electrons

Issues of interest

• dielectric response in supercritical water

• proton disordered and proton ordered forms of ice

Do H-bonds play a role also in the dynamical response of

water?

We focus here on the Infrared Response (IR) of water

From M. Sharma, R. Resta, and R.C., PRL 2005

Dynamic response of water to an electric field: IR spectroscopy

Within linear response theory the infrared absorption coefficient derives from the fluctuations of the cell dipole moment M = i i

We focus on the modes at ~ 185 cm-1 which are associated to hindered translations of the water molecules

Translations of a rigid dipole do not couple to uniform electric fields. Hence the origin of the IR feature at ~185 cm-1 must be electronic. It has been attributed (Madden and Impey, CPL 1986) to an induced molecular dipole, a consequence of the dynamic polarizability of the water molecule (induced intramolecular dipole)

Rigid translations of the central molecule are hindered by the H-bonds that a molecule forms with its neighbors, which define a (distorted) local tetrahedral cage

Ab-initio MD simulations do not support this interpretation

Spectrum including intramolecular (i=j) correlations only

The origin of the 185 cm-1 feature must be intermolecular!

Pasquarello and Resta (PRB 2003) suggested that the IR activity of the translational modes in water originates from dynamic charge transfer between neighboring H-bonded molecules, i.e. the analogue, for rigid molecular translations in a condensed environment, of dynamical Born effective charges in ionic crystals (induced intermolecular dipole)

Correlations of the distances from the center of mass of the central molecule of the centers of the 4 neighboring MLWF that are bonded to the central molecule

Our analysis is fully consistent with the intermolecular dipole model

The strongest coupling is in the plane perpendicular to the z axis

Strong environmental effects due to H-bonds (local chemistry)

This should be contrasted with the classical treatment of Madden and Impey in which the effect of the environment only enters through the value of the local field (long range Coulomb correlations) at the molecular dipole.

Covalent charge fluctuations in water?Intermolecular induced dipoles are a manifestation of electron charge fluctuations between neighboring molecules (a quantum effect)

This should be contrasted with the standard interpretation of the translational IR activity as due to intramolecular charge fluctuations (a “classical” effect)

In principle the issue could be resolved experimentally because the two effects have different selection rules

Modes at 60 cm-1: why are they absent or very weak in the

experimental spectra?

These are bending modes of the H-bond network for which intermolecular dipole fluctuations are absent to a first approximation

ConclusionsClassical simulations consider water as a collection of dipolar molecules

Ab-initio simulations consider water as an assembly of classical nuclei and quantum electrons: this allows us to determine from first principles the response of water to both static and dynamic fields

The resulting picture agrees with earlier models for the static dielectric constant and leads to an entirely new picture for the dynamic response, showing that even in the latter case the environmental effect of the H bonds plays a crucial role.

Hydrophobic interactions: another H-bond effect

• Non-polar solutes like hydrocarbons in water experience a solvent mediated attractive interaction: the hydrophobic interaction

• Thermodynamic properties (e.g. T dependence) suggest that the effect (for small hydrophobic solutes) is controlled by entropy, i.e. few or none H-bonds are broken but the network rearranges around the solute in a way that reduces the entropy (more order)

Two methane (CH4) molecules in water solution attract each other by solvent mediated forces (J. Luen Li et al. 2005)

hydrocarbon-water accessible area and transfer energy

[1] Sharp et. al. Science, 252, 106 (1991)

Experimentally, the hydrocarbon-water transfer energy is proportional to the“accessible surface area”: G = A

The same surface tension parameter =47 cal mol-1 Å-2 [1]

is valid for all paraffin series (methane, ethane, …, decane)

Association of two methanes in water (surface tension

model)free energy per accessible surface area: E= A

water radius: 1.4 Å methane radius: 1.95 Å 0=3.35 Å methane contact distance d =3.9 Å =47 cal mol-1 Å-2 [1]

[1] Sharp et. al., Science, 252, 106 (1991)

0 047 2 (2 ) 2.7E d kcal mol-1

The free energy change from two “well-separated” methanes to two methanes in contact is E = 20 r

W

0

r CH4

Ab-initio MD calculation of the Potential of Mean Force between a methane pair

The strength of the hydrophobic interaction is in rough agreement with the surface tension model

There is only a shallow solvent-separated minimum

Diffusion of H2O near methane is substantially reduced

time [ps]

dis

pla

cem

en

t2 [

Å2]

H2O

CH4-CH4

H2OH2O near the methane pairaverage over all 63 H2O in a unit cellcenter of the methane pair

H2O near the methane pair

Number of waters in the first solvation shell

plateau indicates a more stable local structure

Clathrate structures?

Conclusions

• Ab-initio simulations reveal features of the H-bond network

• Two crucial aspects of H-bonds that emerge from the studies presented here are:

(a) their directional character

(b) the possibility of many nearly degenerate configurations leading to important entropic effects

Acknowledgement

Collaborators:

Manu Sharma and Raffaele Resta (Trieste) – dielectric properties;

Je-Luen Li, Chao Tang (NEC), Ned Wingreen – hydrophobic effect;

Support from NSF and from ONR is gratefully acknowledged