On independence of the solvation of interaction sites of a water molecule M. Předota 1, A. Ben-Naim 2, I. Nezbeda 1,3 1 Institute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic, Prague, Czech Republic; 2 Department of Physical Chemistry, Hebrew University, Jerusalem, Israel; 3 Department of Physics, J. E. Purkyně University, Ústí n. Lab., Czech Republic; On independence of the solvation of interaction sites of a water molecule M. Předota 1, A. Ben-Naim 2, I. Nezbeda 1,3 1 Institute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic, Prague, Czech Republic; 2 Department of Physical Chemistry, Hebrew University, Jerusalem, Israel; 3 Department of Physics, J. E. Purkyně University, Ústí n. Lab., Czech Republic;

On independence of the solvation of interaction sites of a water molecule M. Předota 1, A. Ben-Naim 2, I. Nezbeda 1,3 1 Institute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic, Prague, Czech Republic; 2 Department of Physical Chemistry, Hebrew University, Jerusalem, Israel; 3 Department of Physics, J. E. Purkyně University, Ústí n. Lab., Czech Republic; On independence of the solvation of interaction sites of a water molecule M. Předota 1, A. Ben-Naim 2, I. Nezbeda 1,3 1 Institute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic, Prague, Czech Republic; 2 Department of Physical Chemistry, Hebrew University, Jerusalem, Israel; 3 Department of Physics, J. E. Purkyně University, Ústí n. Lab., Czech Republic;

On independence of the solvation of interaction sites of a water molecule M. Předota 1, A. Ben-Naim 2, I. Nezbeda 1,3 1 Institute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic, Prague, Czech Republic 2 Department of Physical Chemistry, Hebrew University, Jerusalem, Israel 3 Department of Physics, J. E. Purkyně University, Ústí n. Lab., Czech Republic Institute of Chemical Process Fundamentals

Aim Support simplifying assumptions used in analytic theories of aqueous systems Justify previously used speculative approximations for the calculation of the solvation Helmholtz free energy of a water molecule † Lend support to the first order thermodynamic perturbation theory of Wertheim ‡ Examine correlations in the bonding of the individual sites of a water molecule using two qualitatively different extended primitive models Implication:  A W =  A core +N sites  A site, where  A W solvation free energy of a water molecule  A core solvation free energy of the core (typically LJ sphere)  A site solvation free energy of an interaction site  A site =-log  exp[-  B site ]  W+core Calculation of solvation free energy reduces to the calculation of the average energy of the individual interaction sites † A. Ben-Naim, Solvation thermodynamics (Plenum Press, New York, 1987), A. Ben-Naim, Statistical thermodynamics for chemists and biochemists, (Kluwer-Plenum, New York, 1992) ‡ M. S. Wertheim, J. Stat. Phys. 42, 459 (1986) Assumption: Interaction sites of a molecule act independently

Extended primitive models of water Short-range model of water, interactions on the simplest level † Hard core and like site repulsions as hard sphere repulsion Hydrogen bonding resulting from unlike site attraction as square-well attraction Geometry of models EPM4 = 0.7,  EPM4 = 0.8 |OM| = 0.15,  OO = 1.0 EPM5 = 0.4,  EPM4 = 0.8 |OM| = |OM| = 0.5,  OO = 1.0 † I. Nezbeda, J. Mol. Liq , 317 (1997)

Number of sites Core + 3 off-center sitesCore + 4 off-center sites Parent model TIP4PST2 Geometry Planar, tetrahedral angle HOH,Off-center sites arranged M site on bisectortetrahedrally on a sphere Role of H and M sites Single M site plays the role of doublyFull symmetry of H and M sites degenerated bonding siteDirectionality of hydrogen bonds Combination of M site attraction and H sitedictated by the arrangement of repulsion essential for hydrogen bondingsites Sites can form multiple bonds Maximum 1 bond per site EPM4 and EPM5 primitive models of water EPM5 EPM4 M H H O M M H O H

6 different molecules obtained by removal (turning off) of some of the interaction sites of EMP5 water molecule Other combinations symmetrical by exchanging all H and M sites Labeled by active sites Solute molecules descending from EPM5 water molecule hard sphere H HH HHM HHMM =EPM5 HM

6 different molecules obtained by removal (turning off) of some of the interaction sites of EMP4 water molecule No symmetry of H and M sites Labeled by active sites Solute molecules descending from EPM4 water molecule hard sphere H HH HHM =EPM4 M HM

Bonding Both models prevent double bonding between two water molecules Each site of EPM5 can form no more than one bond Molecule can create maximum of 4 bonds H site of EPM4 can form up to 2 bonds and M site up to 3 bonds Since M site plays the role of a degenerated (geometrically collapsed) double site, it ordinarily forms 2 bonds Molecule can forms up to 5 bonds, 6 bonds maximum If the sites acted independently, the probability of the number of bonds of the solute to be n would be binomial where N sites is the number of sites of the solute and p is half of solute’s average energy

Definition of angular distribution of molecules around the solute  defined as the angle between the OH vector of the solute molecule and the projection of the solute-solvent OO vector onto the reference plane of the solute In-plane molecules Lying close to the HOH plane of the solute Bonded mostly to H sites of the solute Energies Total internal energy E is given by the water-water interaction, E WW, and by the solute water interaction, E WS, which are given directly by the number of corresponding bonds E = E WW + E SW = - N WW - N SW Splitting the total energy E into the energy of water molecules, E W, and the energy of the solute, E S E = E W + E S E W = E WW + 1/2E SW ; E S = 1/2E SW Definitions

Simulation method Monte Carlo simulation of N W =215 water molecules and a single solute – molecule originating from water molecule when some of its interaction sites are removed (turned off) Packing fraction  =(  /6)(N/V)  W 3 ; N=N W + 1  EPM4 =0.35,  EPM5 =0.3 Temperature  =1/kT  EPM4 =6,  EPM5 =5 5  10 5 equilibration cycles, 18  10 6 productive cycles Preferential sampling f(r SW )=(1+D)/(r SW 2 +D); f(L/2)=0.1 Properties observed Average energy of water molecule (solvent) Average energy of solute molecule Average number of bonds of each site of the solute Probability distribution of the solute to form n bonds Angular distribution of water molecules around the solute All (i.e. both bonded and nonbonded) and only bonded to the solute studied separately Solute-solvent pair correlation function

Average energy and number of bonds of EPM5 Probability distributions of different solutes to form n bonds with the solvent molecules, and average energies for the EPM5 solvent. P th (n) is the the binomial distribution with p=0.9225, and P sim (n) is the simulation result; E S is the average energy of the solute, E S /N sites is the average energy per site of the solute, and E W is the average energy of solvent per water molecule

Average energy and number of bonds of EPM4 The probabilities of creation of n bonds from simulation are given separately for the individual sites, P H sim (n) and P M sim (n), and for the entire solute, P sim (n). The theoretical prediction P th (n) is given by the binomial distribution with p=0.775 nHMHHHMHHM 00.25— — — Average0.76— —0.06— —0.38— —0.55— —0.02— 0.01 Average—1.53— — ——— ———— — —— ———00.02 =-2E S 0.76      E S /N sites 1.52      E W /N W 1.54     0.01 nHMHHHMHHM 00.25— — — Average0.76— —0.06— —0.38— —0.55— —0.02— 0.01 Average—1.53— — ——— ———— — —— ———00.02 =-2E S 0.76      E S /N sites 1.52      E W /N W 1.54     0.01

Probabilities of a creation of n bonds for the H site and M site in different solutes descending from the EPM4 water molecule The probabilities follow binomial distribution with p=0.775 Proved that M site acts as degenerated double site

Angular distribution of bonded molecules around different solutes EPM5 The peaks are independent of the presence of other sites EPM4 Little correlations of the peaks

Angular distribution of EPM5 water molecules (bonded and nonbonded) around different solutes Complex behavior resulting from the combination of additive distribution of bonded molecules and nonadditive distribution of nonbonded molecules Combination of water-like and hard-sphere-like structure

Solute-solvent pair correlation function EPM5 Turning on sites changes the PCF from hard-sphere like to water- like Molecules cannot approach close each other because of site-site repulsions EPM4 Turning on sites forces molecules to approach each other closer Behavior originates from the position of M site closer to the central of molecules

Conclusions Independence of bonding of individual sites of water molecule proved for both EPM4 and EPM5 models For EPM5 independence exactly, for EPM4 it does not hold exactly but correlations are very small Fully justified previously used speculative approximations for the calculation of the solvation Helmholtz free energy of a water molecule Support to the first order thermodynamic perturbation theory of Wertheim Assumption of independence of bonding justified for practical applications Reduction of the calculation of average quantities over up to quadruplet distribution function to calculations of averages over pair distributions only Drastic simplification which we hope will render the development of an analytical theory of water (and aqueous systems in general) feasible Studied not only fully interacting water molecules (considered as a solute) but also a series of other solutes made from the water molecule by turning off some of its interaction sites Additional information on the behaviour of water M. Předota, A. Ben-Naim, I. Nezbeda, J. Chem. Phys. 118, (2003) Reference: