Based on the probable importance of the specific outer neighbor molecular-level properties of this liquid to biological structure, function and dynamics, on the accumulating experimental evidence supporting this structural picture of water, and on the failures of currently used models to reproduce faithfully the properties of the real liquid [47,48] or of the ice polymorphs [49], it would seem that any realistic model used in Monte Carlo or molecular dynamics (MD) computational studies of the pure liquid or of aqueous systems must explicitly promote realistic outer neighbor bonding characteristics. Correctly designed water-water bonding outward from a perturbing surface appears essential for a realistic description of the thermodynamics, which controls the equilibrium structures, thus the dynamics, in aqueous systems undergoing a chemical or a biological change. However, the difficulties of incorporating any characteristics beyond the most elementary ones into computational models for chemical or biological hydration studies were recognized many years ago by Berendsen, et al. [50] in their formulation of SPC-type models. The creation of a computationally very simple outer structure model for water was thus the goal described briefly in Ref. [20].

As is the case for currently used water models, any new model must give a strong hydrogen-bond interaction along the O-H···O direction with an O···O potential minimum near 2.8 Å. On the other hand, the H-bond interaction should disappear, much more rapidly than in any current Coulombic modeling method, for angles away from this direction and for O···O distances greater than 2.8 Å. Away from the H-bond regions of the potential, the water-water interaction must be supplanted by a weak O···O van der Waals attraction [19] near 3.5 Å. In all current models, for O···O distances near 3.5 Å, there is a fairly deep "H-bond-type" minimum for the O-H···O configuration and a strong repulsion for the H-O···O configuration. On the other hand, next-nearest-neighbor outer structure from the total interaction should be characterized by a potential giving two O···O minima: open tetrahedral structure with a next-nearest-neighbor O···O minimum at 4.5 Å and a bent H-bond structure with a shallow (or flat) non-H-bonded O···O minimum in the vicinity of 3.5 Å.

An exact mathematical analysis of the proposed outer features in a one-dimensional model [50] has already shown that such a potential provides a sturdy density maximum with the correct pressure effects [20]. Previous water models either show no density maximum at all or a rather fragile one at the wrong temperature with the wrong shape and pressure dependence, moving around with the water model used, run time and sample size [3]. In fact, the fragile nature of the density maximum from the popular SPC and SPC/E models has already been emphasized [47]: "Although a density maximum is detected by monitoring energy-volume correlations as a function of temperature, the large statistical uncertainty in the correlations reduces the significance of this finding." The incorrect representation in this paper of structure, energy and fluctuations was attributed to the absence of a quantum mechanical treatment, not to the absence of realistic outer structure. In more recent work [52] it was found that there is indeed no density maximum for SPC (or TIP3P) water. However, in agreement with the density maximum found [53] near 250 K for 2988 TIP4P molecules, a broad density maximum near 258 K did appear for an ensemble of 512 TIP4P water molecules in the recent work [52]. Significant model dependencies in computational studies of many properties of water, including the density, were discussed in our book [3]. The last two sentences of this book represent our current views precisely: "It is thus difficult to see how any of the earlier water models can provide really trustworthy answers to the myriad of questions concerning the behavior of water near surfaces of chemical or biological importance. In fact, many of the answers found are extremely model dependent, and are therefore ambiguous." It is precisely because of this model dependence that a primary emphasis in our ongoing work has been to start introducing new models that correctly reproduce realistic outer structural characteristics of the ice polymorphs and the liquid. A fairly complicated 3-site model that included realistic outer structure was introduced in Ref. [20].

Are there simpler potential models that might give the desired results? Speaking of potentials at the end of Ref. [20], it was mentioned that to achieve such results, "It is possible that any of the commonly used potentials can be modified using appropriate angular and distance cutoffs". For example, if the Coulomb interactions in SPC or SPC/E [10] are smoothly switched off in the outer structural regions, a modified potential MSPC can be produced,

(1)

 with the Coulomb switching function given by,

Here, r_ab is the switching distance between atom a on molecule i and atom b on molecule j. By adjustment of the parameter a_ab, this water-water potential, at atom-atom distances greater than a chosen switching distance, evolves into a potential with less Coulomb and more of an O···O Lennard-Jones contribution.

Figure 2 illustrates this type of pair potential, as a function of O···O separation, along the non-hydrogen-bond H₂O···OH₂ direction for the currently parameterized MSPC model compared with SPC. This figure should be also compared with the similar Fig. 3 of Ref. [20]. Both of these figures indicate that our new and modified potentials at ~ 3.5  Å for this non-H-bonded orientation substitute a weak minimum for the strongly repulsive Coulomb interactions in ordinary models.

Perhaps a better representation of the differences between MSPC and standard potentials is shown in Fig. 3, which is comparable to Figs. 5 and 6 of Ref. [20]. In this figure, the O···O distance is fixed at two distances, 2.8 Å and 3.5 Å, then one molecule is rotated around its z-axis relative to the other (see Fig. 4). It is seen that for MSPC and other current water models a deep H-bond minimum occurs at 2.8 Å for 0° rotation, the H-bond angle. On the other hand, at 3.5 Å, the H-bond minimum for MSPC is much shallower than it is for the other models. Away from the H-bond angle, a steep potential hill of > 10 kJ/mol, corresponding to an "activation temperature" t >  930° C, has to be surmounted in the standard models for any bent hydrogen bonded structure to form at all. There is thus no way that any of this structure can be formed at ordinary temperatures, so no two-state density maximum can occur using any of the these potentials. On the other hand, at 3.5 Å, the new potential of Ref. [20] and the MSPC potential introduced here,

have been designed to give flatter O···O potential surfaces under this rotation. Instead of the H-bond angle being so strongly favored, the bonds in MSPC can bend without much energy expense to give the desired 3.5 Å non-hydrogen-bonded structure. From MD studies using this modified potential, a sturdy density maximum arises whose temperature again can be varied with parameter adjustment. Without such characteristics, the anomalous heat capacity, the viscosity and the non-Arrhenius behavior, all of which depend on the less rigid dense structure, cannot be reproduced. Particularly troublesome, when based on the standard models, is the very likely incorrect diffusional dynamics near hydrophilic/hydrophobic solutes and other types of surfaces. The flat, more generally accessible regions at O···O distances near 3.5 Å, introduced by these outer bonding characteristics, should also improve five-site potentials such as ST2 and MST-FP, which, because of too small a diffusion coefficient compared with the experimental value, have been considered too "stiffly structured". See the discussion in Sect 6.4 of Ref. [3].

Finally, speaking of flexible/polarizable (FP) models, introduced for the first time in our own papers [2,54-56], experience with such models has taught us that, as far as reproducing the anomalies is concerned, FP effects are of decidedly secondary importance. Thus, while future realistic water models must certainly include flexible intramolecular bonding, which must also be anharmonic [56], together with "instantaneously responsive" polarizability, development of the most simple water-water potentials that lead transparently to the anomalies will not initially require these computational complications. Neither will they require the presence of complicating long-range Coulomb effects, also missing from the potential in Eq. 1.

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