Absolute ion hydration free energy scale and the surface potential of water via quantum simulation

Significance

Establishing an absolute free energy scale for single-ion hydration has been an elusive goal of condensed-phase chemistry for nearly a century. In this paper, a zero for the free energy scale is determined by employing quasi-chemical theory and quantum mechanical simulations of the sodium ion in water. The simulation results produce an accurate estimate of the effective surface potential of water by comparison of the computed “bulk” hydration free energy with experimental values for the “real” hydration free energy. This work provides a firm quantitative footing for studies of interfacial specific ion structure, thermodynamics, and kinetics in fields as diverse as colloid science, electrochemistry, biophysics, and energy storage materials.

Abstract

With a goal of determining an absolute free energy scale for ion hydration, quasi-chemical theory and ab initio quantum mechanical simulations are employed to obtain an accurate value for the bulk hydration free energy of the Na⁺ ion. The free energy is partitioned into three parts: 1) the inner-shell or chemical contribution that includes direct interactions of the ion with nearby waters, 2) the packing free energy that is the work to produce a cavity of size $\lambda$ in water, and 3) the long-range contribution that involves all interactions outside the inner shell. The interfacial potential contribution to the free energy resides in the long-range term. By averaging cation and anion data for that contribution, cumulant terms of all odd orders in the electrostatic potential are removed. The computed total is then the bulk hydration free energy. Comparison with the experimentally derived real hydration free energy produces an effective surface potential of water in the range −0.4 to −0.5 V. The result is consistent with a variety of experiments concerning acid–base chemistry, ion distributions near hydrophobic interfaces, and electric fields near the surface of water droplets.

Ion solvation phenomena are central to diverse condensed-phase physical, chemical, biological, materials, and energy-related systems (1–6). The physical behaviors of solutions that include ions of the same charge but with different size, shape, and/or chemical makeup can vary dramatically. Well after the pioneering work on protein precipitation by Hofmeister, the study of these specific-ion effects has seen a major resurgence in the last two decades due to the recognized relevance to problems spanning fields from fundamental colloid science and electrochemistry to biological channel proteins and battery materials (7).

Progress has been made due to a set of experimental and theoretical/computational tools that have begun to unravel the fundamental interactions leading to observed behavior. These tools include spectroscopic probes that interrogate specific ions at interfaces (8, 9), molecular theories that partition free energies into distinct physical parts (10–12), and quantum simulation methods that include polarization and charge transfer effects not contained in simple classical models (13–15).

Tied to the aim of gaining fundamental insights into specific ion effects is the long-standing goal of establishing a single-ion free energy scale for hydration (16, 17). While ion pair hydration thermodynamic quantities are directly measurable, estimation of the single-ion values typically requires theoretical input (18–21). The need for theoretical input is in turn linked to the Gibbs–Guggenheim principle (22–24) that states that the electrostatic potential shifts across chemically dissimilar interfaces are not measurable. The single-ion surface potential term cancels when composing the ion pair enthalpy or free energy (25, 26). Further motivations for developing quantitative theories of single-ion free energies include gaining a better understanding of ion–solvent interactions, ion exchange in resins, ion pairing and its specificity, binding in or on proteins, distributions near interfaces, ion transfer reactions, and electrode kinetics (16). In addition, establishing a single-ion hydration free energy scale facilitates the development of more accurate force fields employed in molecular simulations (27, 28).

We have suggested previously that there may be physical and chemical consequences to solvent-induced potential gradients through an interface even though the shifts are not directly measurable (29). The consequences include nonuniform ion density profiles (8, 30), altered acid–base chemistry (31–34), and electric fields probed by spectroscopic measurements at droplet surfaces (35). Experimental evidence is accumulating that suggests the effective surface potential of water (passing from the gas phase or a hydrophobic medium into water) is approximately −0.4 V, in agreement with earlier estimates (29, 36, 37).

For the sake of the discussion below, consider the case of a single spherical monovalent ion located near the center of a large water droplet. We assume the droplet is large enough to obtain a stable value for the average electrostatic potential in the region between rapidly varying zones near the ion and near the distant liquid–vapor interface, but not so large as to involve extraneous ions from the autoionization of water (or from other sources).

The excess chemical potential for the ion $X$ is given by the Widom formula (10)

$${\mu }_X^{ex}=-kT\ln {⟨\exp \left[-{\epsilon }_X/kT\right]⟩}_0,$$ [1]

where ${\epsilon }_X$ is the total interaction energy of the ion with all solvent molecules and the zero subscript indicates the ion and waters are uncoupled during sampling. Since the Coulomb potential is long ranged, any inhomogeneous charge density regions in the droplet contribute to the chemical potential value.

The traditional expression for the excess chemical potential of an ion is of the form (38)

$${\mu }_X^{ex}={\mu }_{X,che}^{ex}+q\varphi ,$$ [2]

where ${\mu }_{X,che}^{ex}$ is a “chemical” value deep in the liquid and $\varphi$ is a contribution from surface effects. It is clear by comparing Eqs. 1 and 2 that Eq. 2 involves some kind of partitioning of the electrostatic potential. The questions addressed in this paper are, what is the physical content of $\varphi$, and what is its numerical value?

There are multiple choices for defining the generic interfacial potential $\varphi$ above. First, following refs. 18 and 39 we can compute the (planar) liquid–vapor surface potential by integrating the Poisson equation through the interface,

$${\varphi }_{sp}=4\pi {\int }_{-\infty }^{\infty }z\rho \left(z\right)dz,$$ [3]

where $z$ is the coordinate perpendicular to the interface and $\rho \left(z\right)$ is the charge density (due to electron and nuclear distributions). It is known that, for realistic charge distributions, this potential is large and positive (40, 41), with a value of about 4 V for the water liquid–vapor interface. The large magnitude arises from the internal molecular structure with electrons spatially distributed around point-like nuclei (producing the quadrupolar Bethe potential). The associated intrinsic chemical term ${\mu }_{X,int}^{ex}$ (42) includes all local interactions of the ion with the solvent, including a large approximately canceling quadrupole term. Thus, while the sum produces the “real” excess chemical potential ${\mu }_X^{ex}$,

$${\mu }_X^{ex}={\mu }_{X,int}^{ex}+q{\varphi }_{sp},$$ [4]

the two contributing terms are of very large magnitude. Another proposed partitioning of the surface potentials involves the dipolar surface potential for the liquid–vapor interface, as opposed to the full ${\varphi }_{sp}$ discussed above (43). This definition leads to an alternative “intrinsic” chemical potential.

We have previously termed the potential contribution due to a varying charge distribution near a molecular-sized cavity the local potential ${\varphi }_{lp}$ (29, 44, 45). The total electrostatic cavity potential from the two inhomogeneous water domains is then the net potential

$${\varphi }_{np}={\varphi }_{lp}+{\varphi }_{sp}.$$ [5]

The resulting excess chemical potential is

$${\mu }_X^{ex}={\mu }_{X,b}^{ex}+q{\varphi }_{np},$$ [6]

where ${\mu }_{X,b}^{ex}$ is the bulk hydration free energy in the absence of the net interfacial potential (figure 1 of ref. 29). The net potential ${\varphi }_{np}$ should be defined in such a way that it is independent of ion identity. It is thus associated with longer-ranged electrostatic effects outside the first solvation shell. Ref. 46 displays a stable net potential beyond a cavity size of about 4 Å. The quasi-chemical theory discussed below provides a natural definition based on length-scale partitioning of the free energy into near and far-field contributions (47).

The real hydration free energy ${\mu }_X^{ex}$ on the left side of Eq. 6 can be obtained indirectly from experiment using a combination of conventional ion hydration free energies (referenced to the proton) and cluster free energies of formation for an array of ions (48–50) (the cluster-pair approximation). We previously obtained a revised estimate, shifted by $+1.7$ kcal/mol from the value obtained in ref. 48, for the hydration free energy of the proton (36, 37) after isolating and eliminating an extrathermodynamic assumption. That assumption results from replacing a difference with a sum (equations 16–19 in ref. 36) in the basic free energy expressions. We note that ref. 37 utilizes a correlation method and standard thermodynamic expressions to extract the single-ion quantities without further extrathermodynamic assumptions. Thus, the results do not depend appreciably on the ion–water cluster size used in the analysis.

In the present paper, the target quantity is the bulk free energy ${\mu }_{X,b}^{ex}$ for the Na⁺ ion (right side of Eq. 6). This bulk value has been shown (36) to correspond to the Marcus value (51) that includes no interfacial potential contribution. Toward this end we utilize quasi-chemical theory and quantum mechanical simulation methodology to partition the free energy into spatially resolved and manageable parts. Employing the difference $\left({\mu }_X^{ex}-{\mu }_{X,b}^{ex}\right)/q={\varphi }_{np}$ via Eq. 6, we are able to estimate the effective surface potential of water (the net potential) in the absence of other interfering ions. In other words, we obtain the solvent contribution to the net potential. The computational results and experimental values are summarized in the last five lines of Table 1 (below). The resulting estimate of the effective surface potential of water (−0.4 V) implies an electric field near the interface of magnitude $10^7$ V/cm assuming an interfacial width of approximately 5 Å.

Table 1.

Contributions to the bulk hydration free energy from the quantum mechanical simulations

Contribution	Calculation	Value, kcal/mol
Inner shell	Cavity growth	−49.40 $\pm$ 0.18
Packing	Cavity growth	14.46 $\pm$ 0.26
Long-range parts
1) ${⟨ϵ⟩}_{0,cav}$ (Na+)	Mean bind energy	−93.14 $\pm$ 0.36
2) ${⟨ϵ⟩}_{cav}$ (Na+)	Mean bind energy	−129.75 $\pm$ 0.41
3) ${⟨ϵ⟩}_{0,cav}$ (${\mathrm{F}}^{-}$)	Mean bind energy	70.42 $\pm$ 0.33
4) ${⟨ϵ⟩}_{0,m}=\left(\left(1\right)+\left(3\right)\right)/2$	Mean bind	−11.36 $\pm$ 0.24
	energy (bulk)
5) $\delta \overline{ϵ}/2$	[(1)− (2)]/2	18.31 $\pm$ 0.27
6) ${\mu }_{LR}^{ex}$	$\left(4\right)-\left(5\right)$	−29.67 $\pm$ 0.36
7) $\frac{1}{2}\beta {⟨\delta {ϵ}^2⟩}_{0,cav}$	Fluctuation	16.02 $\pm$ 0.57
8) $\frac{1}{2}\beta {⟨\delta {ϵ}^2⟩}_{cav}$	Fluctuation	18.35 $\pm$ 0.65
9) Total fluctuation	$\left(\left(8\right)-\left(7\right)\right)/2$	1.17 $\pm$ 0.43
Total long-range bulk FE	$\left(6\right)+\left(9\right)$	−28.5 $\pm$ 0.56
Corrections
Finite-size corr	$L=16.04$ Å	−29.36
Quantum corr	CCSD(T) (5W)	$\approx$2.1 $\pm$ 0.09
Dispersion corr	SAPT2 (6W)	−0.8 $\pm$ 0.01
Total bulk hydration FE	Sum	−92.8 $\pm$ 0.64
Total bulk hydration (FE (corr)	Sum	−90.7 $\pm$ 0.65
Experiment (real)		−101.5 (37, 48)
Experiment (bulk)		−91.5 (51)
Computed net potential		[−0.38, −0.47] V

This work builds upon previous efforts aimed at ab initio real ion hydration free energy calculations (14, 52–54), but differs by focusing on the bulk value. This avoids difficulties related to the zero of the energy scale and finite-size effects for long-range electrostatics in periodic boundary simulations. Once a value is established for one ion, free energies for other ions can be obtained from thermodynamic measurements. Our results resolve a recurrent debate concerning the surface electrostatic contribution to single-ion hydration free energies.

We first outline the necessary theory for analysis of the ab initio molecular dynamics (AIMD) simulations at the density functional theory (DFT) level. We then present the results with relevant discussion and finish with our conclusions. The computational methods are summarized following the conclusions.

Theory

We employ quasi-chemical theory (QCT) (6, 10, 11) to compute the hydration free energy of the Na⁺ ion. QCT provides a means to partition the free energy spatially into three physical parts. The three components correspond to the processes of 1) forming a cavity somewhat larger than the ion in water (packing), 2) inserting the ion into the cavity (long-ranged or outer shell), and 3) allowing the water solvent back into contact with the ion (inner shell).

In the limit of a purely repulsive hard sphere cavity-forming potential, the hydration free energy can then be expressed (in the order given above) as (11)

$${\mu }_X^{ex}=-kT\ln p_0\left(\lambda \right)-kT\ln {⟨\exp \left[-{\epsilon }_X/kT\right]⟩}_{\lambda }+kT\ln x_0\left(\lambda \right),$$ [7]

where $p_0\left(\lambda \right)$ is the probability to observe no solvent molecules within the cavity of size $\lambda$ (with the ion not included in the sampling), the second term is the free energy for inserting the ion into the system while the cavity is present, and $x_0\left(\lambda \right)$ is the probability to observe no solvent molecules within the cavity of size $\lambda$ with the ion included.

The first and third terms of Eq. 7 can alternatively be viewed as the reversible work to produce a cavity of size $\lambda$ in pure water and minus the work to create a cavity of size $\lambda$ with the ion at the center, respectively. Operationally, the growth process can be enacted using a smooth yet strongly repulsive potential (47); the limit of a purely repulsive wall yields Eq. 7. Since in the middle term of Eq. 7 the waters are constrained to locations some distance from the central ion, useful approximations can be used for its estimation (see below). This “mechanical” picture of the hydration process yields physical insights into the driving forces through analysis of the separate terms. Note that the final result is independent of the details of the potential used to create the cavities.

Through an insightful exact reformulation of the solvation free energy derived in ref. 11, Eq. 7 is the same as

$${\mu }_X^{ex}=-kT\ln K_n^{\left(0\right)}{\rho }_W^n+kT\ln p\left(n\right)+{\mu }_{X{W}_n}^{ex}-n{\mu }_W^{ex}$$ [8]

where $K_n^{\left(0\right)}$ is the equilibrium constant for forming the ion/$n$-water cluster in the gas phase (where $n$ is typically chosen as the most probable hydration number in the condensed phase), ${\rho }_W$ is the bulk liquid density of water, $p\left(n\right)$ is the probability to observe that coordination number, ${\mu }_{X{W}_n}^{ex}$ is the free energy to insert the ion/water cluster into water, and ${\mu }_W^{ex}$ is the hydration free energy of water in water (−6.3 kcal/mol) (55).

In previous work (36), we found that, for small $n$ values consistent with the hydration number in the first shell (typically 4 to 6), there is little or no surface potential contribution from that first shell. Thus, any interfacial potential contribution arises from the third term on the right-hand side of Eq. 8. This yields the physical insight that the solvent contribution to the interfacial potential can usefully be defined as the net potential at the center of a cavity in the 4- to 6-Å size range.

Classical simulations (46) have shown that the cavity potential, plotted as a function of cavity size, stabilizes at a consistent value for this size range. In addition, an empirical observation from ref. 47 suggests there is a common length scale for all monovalent monatomic ions at which the inner-shell term equals the bulk hydration free energy (at roughly 6 Å in classical simulations). At this length scale, the sum of the cavity formation free energy and a Born-like long-ranged electrostatic free energy contribution is zero, leaving the inner-shell term as an accurate estimate of the bulk hydration free energy.

Here we utilize a soft-cavity version of Eq. 7 and AIMD simulations to compute each QCT contribution to the free energy for the hydration of the Na⁺ ion. For the long-range term, we compute an average of the free energies for a cation (Na⁺) and an anion (${\mathrm{F}}^{-}$) to eliminate the interfacial potential contribution. Thus, we obtain the bulk hydration free energy in Eq. 6. This allows for a fully quantum mechanical estimate of the net potential ${\varphi }_{np}$. See Materials and Methods for further details.

Results and Discussion

We first display the solvation structure through the ion–water radial distribution functions (rdfs) obtained from classical and quantum simulations (Fig. 1). The classical simulations utilized the simple point charge–extended (SPC/E) water model and the Horinek et al. (56) ion–water force field. As discussed in Materials and Methods, the dispersion interaction between the ion and the waters was not included in the quantum simulations. The inner-shell hydration structure is clearly softened in the quantum simulation relative to the classical results. The integrated hydration numbers vs. distance from the ion are relatively consistent between the classical and quantum models, however. Up to the first minimum at 3.2 Å, the coordination number from the quantum simulation is 5.3 vs. the experimental results ranging from 5.3 to 5.5 (57, 58). The classical simulation produces a value of 5.8.

Fig. 1.

The ${\mathrm{N}\mathrm{a}}^{+}$-O radial distribution functions and resulting coordination numbers.

The first maximum location of the rdf in the quantum simulation occurs at 2.42 Å, consistent with other AIMD simulations (57) and close to a recent experimental estimate of 2.38 Å (58) and previous experimental results in the range 2.40 to 2.43 Å (57). A recent simulation that included D3 dispersion interactions between the ion and water produced a first maximum at 2.56 Å (58), which is well outside the range of experimental results. (In a single simulation employing the D3 dispersion correction for the ion–water and water–water interactions, we observed a peak maximum at 2.53 Å.) As discussed in Materials and Methods, the D3 interactions for sodium–water are likely overestimated, and thus we address those perturbatively with correlated electron quantum chemical calculations (see below). We conclude that the solvation structure is reproduced to sufficient accuracy by the quantum simulations.

Next, we present numerical results related to computing the inner-shell and packing contributions to the QCT free energy. The numerical processes to compute these terms involve growing physical cavities of radius $\lambda$ around the ion (producing minus the inner-shell contribution) and in pure water (the packing contribution), respectively. The cavities are produced using the same methodology as in our previous classical simulations in ref. 47.

Numerical results are presented in Fig. 2 by exhibiting the cumulative work computed from the growth process enacted during the classical and quantum simulations. The quantum mechanical results are −49.40 kcal/mol for the inner-shell term and 14.46 kcal/mol for the packing term (Table 1).

Fig. 2.

The cavity formation free energies as a function of coupling parameter $\gamma$. The open symbols are for the classical force field, and the solid symbols are for the DFT calculations. The circles are for the packing (PK) contributions, indicating that the SPC/E model gives a lower cavity formation free energy, which agrees with the observations by Galib et al. (59). The squares are for the inner-shell (IS) contributions, indicating that the SPC/E water model produces a larger cavity formation free energy around the ion.

We now turn to the long-ranged contribution that carries the interfacial potential portion of the free energy. In a study that analyzed ion hydration in water droplets (60), the electrostatic potential profile outside a central cavity shows that at least 1,000 water molecule systems are required to obtain a stable potential profile in the bulk-like region of the liquid. For the vast majority of AIMD quantum simulations of water, systems of much smaller size are modeled. Thus, there may be an indeterminate potential shift within the periodic simulation box due to the lack of convergence of the potential to a constant value moving away from the cavity or ion.

In addition, there is a second factor in the quantum model that produces a potential shift addressed previously in classical simulations (for which the shift is of much smaller magnitude than in the quantum simulations) (28). Namely, the Ewald potential integrates to zero over the simulation box. As mentioned above, previous work (41) has revealed an enormous potential jump across the water liquid–vapor interface due to the distribution of matter in quantum mechanical systems. When considering a 4-Å cavity at the center of a modest-sized box (here $L=16$ Å for an $N=128$ water system), the potential at the cavity center is roughly −4 V. Then a simple calculation shows that the average value outside the cavity must be of magnitude +0.3 V to cancel the cavity value. This implies a shift of −0.3 V that should be included in estimating a real single-ion hydration free energy, and it is of the same magnitude as the interfacial potential quantity that we seek. Thus, it cannot be neglected.

Rather than pursuing the daunting computational challenge of modeling very large systems (at least 1,000 water molecules) over long times (multiple nanoseconds) to obtain accurate estimates of the required shifts, here we eliminate the interfacial potential contribution by estimating the bulk single-ion hydration free energy.

Fig. 3 displays the log of the interaction energy distributions for the long-ranged term computed both for sampling with the ion coupled to the water molecules (left distribution) and then uncoupled (right distribution). We note that the two distributions are accurately Gaussian with comparable widths.

Fig. 3.

Logarithms of the distributions of ion–water interaction energies obtained during sampling with and without the ion present at the cavity center. The blue line is for the uncoupled sampling, where there is no ion–water interaction, while the black line is for the coupled sampling case. The blue dashed and black dashed-dotted lines are the Gaussian fits to the data. The average interaction energies for the uncoupled sampling are −93.14 kcal/mol with a standard derivation of 4.56 kcal/mol and a fluctuation term in the free energy of 16.02 kcal/mol; for the coupled sampling, the results are −129.75 kcal/mol and $\sigma$ = 4.88 kcal/mol, with a corresponding fluctuation term of 18.35 kcal/mol.

We focus on the two mean quantities < ε >_0,cav and < ε >_cav. Since the dispersion part of the ion–water interactions has been removed (Materials and Methods), the interaction energy takes the form

$$\epsilon =q\varphi +{\epsilon }_{ind},$$ [9]

where ${\epsilon }_{ind}$ is the induction part of the interaction. The potential $\varphi$ here is the cavity potential from the unperturbed molecular charge distributions.

It is expected that the electric field at the center of an approximately spherical cavity is very small. Thus, for the long-range term, the major contribution to the induction energy is likely to arise from induced polarization of the water molecules and not of the ion. It is also plausible that the polarization of the water molecules is symmetric in the ion charge. Below we test these hypotheses.

To estimate the long-ranged contribution free of interfacial potential effects, we consider the (Gaussian) approximation

$$\begin{array}{ll}\hfill {\mu }_{LR}^{ex}& \approx \frac{1}{2}\left(<\epsilon >0_{,cav}+<\epsilon {>}_{cav}\right)\hfill \\ \hfill & \hspace{1em}+\frac{1}{4kT}\left(<\delta {\epsilon }^2{>}_{cav}-<\delta {\epsilon }^2{>}_{0,cav}\right),\hfill \end{array}$$ [10]

where the sampling includes the cavity. For a purely Gaussian process, the second term is zero. Hence, that term can serve as an indicator of the accuracy of the approximation. In classical simulations, it is apparent from figure 7 of ref. 47 that the long-range interactions are accurately Gaussian for cations with a cavity size of roughly 4 Å. The computed quantum free energy contribution from the second-order difference in Eq. 10 is of low magnitude (1 kcal/mol). Thus, the above approximation is of good accuracy, which is not surprising since the waters are restricted to be greater than 4 Å from the ion.

Since we have available the difference of the two means $\delta \overline{\epsilon }$, the first-order expression above can be rewritten as

$${\mu }_{LR}^{ex}\approx <\epsilon {>}_{0,cav}-\delta \overline{\epsilon }/2.$$ [11]

The final step is to approximate $<\epsilon {>}_{0,cav}$ as the average of the results computed for the Na⁺ and ${\mathrm{F}}^{-}$ ions. The first electrostatic term in Eq. 9 cancels during the averaging process, along with all higher-order odd terms in the free energy cumulant expansion.

The accuracy of this approach relies on the assumption that the induction contribution satisfies the above hypotheses. To test the approximation, we computed the induction contribution to the binding energy of the ions with the nearest six waters (outside the cavity). Indeed, we obtained induction energy results of −4.38 kcal/mol for the Na⁺ ion and −4.36 kcal/mol for the ${\mathrm{F}}^{-}$ ion. The results for the average interaction energies for the two ions are listed in Table 1. (Note the large positive result for the ${\mathrm{F}}^{-}$ ion caused by the large negative cavity potential discussed above.)

The average binding energy for the two ions is −11.36 kcal/mol, while half the shift between the two means in Fig. 3 is 18.31 kcal/mol. Thus, the approximation for the long-range term is −29.67 kcal/mol (Table 1). Finally, the finite-size self-energy correction (61) for this system size is −29.36 kcal/mol. Taking all of the results together produces a bulk hydration free energy of −92.8 kcal/mol in the absence of ion–water dispersion interactions. A previous QCT estimate at the DFT (B3LYP) level without dispersion corrections is −93.3 kcal/mol (6, 62). The corresponding Marcus value for the bulk hydration free energy is −91.5 kcal/mol (51).

To complete our analysis of the bulk hydration free energy of the Na⁺ ion, we compute 1) quantum chemical corrections to the hydration free energy obtained from high-level quantum chemical calculations (CCSD) mainly due to induction and dispersion and 2) the dispersion portion of the correction.

To estimate the quantum correction, the interaction energies for the ion with five waters were computed at the accurate CCSD(T) level. The DFT level of theory was found to overbind the ion to the water molecules by 1.9 kcal/mol on average. The full correction was estimated by integrating out to longer range, assuming that induction effects dominate the interactions. The total energy correction was then estimated to be 4.2 kcal/mol, producing an approximate free energy correction of roughly 2.1 kcal/mol (63). This is likely an overestimate since it assumes pair interactions outside the first shell with no screening. The shift is positive due to the known overpolarization in DFT vs. higher-level quantum calculations (64). In addition, there is a finite-size dielectric correction that is expected to be of small magnitude (61).

For the dispersion part of the correction, a first-order approximation is quite accurate (12). To estimate this portion of the above 2.1-kcal/mol correction, configurations of the closest six waters were drawn from a fully coupled AIMD simulation (with no cavity). The dispersion contribution to the interaction was computed at the SAPT2 level, producing a result of −0.64 kcal/mol. An estimate (via integration assuming a $g\left(r\right)$ value of 1) of the remainder of the interaction due to more distant waters leads to a total dispersion free energy contribution of −0.8 kcal/mol. This implies that the induction part of the correction is 2.9 kcal/mol.

The resulting corrected total bulk free energy estimate is −92.8 + 2.1 = −90.7 kcal/mol (Table 1). We previously obtained a revised experimental estimate of the real hydration free energy of the proton (37) that is shifted $+1.7$ kcal/mol relative to the value reported in Tissandier et al. (48). Shifting the value for the Na⁺ ion by this amount produces a real hydration free energy of −101.5 kcal/mol for the left side of Eq. 6 (that includes the net potential contribution).

Thus the present quantum simulations and analysis produce an estimate of the effective interfacial potential in the range of −8.7 to −10.8 kcal/mol-e or −0.38 to −0.47 V, consistent with our previous estimates arising from alternative methods (29, 36, 37).

We close the discussion of the computational results with an independent but approximate estimate of the bulk hydration free energy. In classical simulations discussed in ref. 47, we observed a near-linear behavior of the inner-shell free energy as a function of increasing cavity size (beyond a nonlinear regime at small length scales). Later, we suggested that the quasi-linearity resulted from the addition of two nonlinear terms over the chosen range (figure 1 of ref. 65). Nonetheless, it is clear from figures 5 and 6 from ref. 47 that linear extrapolation of the inner-shell term out to the 6- to 8-Å length scale is appropriate.

We performed a similar analysis using the quantum mechanical (QM) data generated in our CP2K simulations. The QM simulations were performed on a system consisting of the ion and 128 waters. The smaller system size vs. the classical results (512 waters) necessitates a downward shift of the QM results by roughly 4.1 kcal/mol (also with a change of slope), as determined by classical simulations of the size dependence of the inner-shell term. Interestingly, the inner-shell and long-range size dependences have opposite signs (and similar magnitudes), so the total free energy is largely independent of system size, as noted earlier (61). There has been extensive work on finite-size corrections and convergence to the thermodynamic limit (20, 66).

Fig. 4 shows that the resulting QM curve intersects the computed bulk free energy at roughly 5.6 Å. This length scale is somewhat less than that observed for the classical simulations (6.15 Å). This result is not surprising since, in a simple continuum model, the expected crossing-point radius should be proportional to the inverse of the cube root of the liquid surface tension. Recent ab initio simulations (67) indicate the quantum models exhibit a significantly higher surface tension than both experiment and the classical models. Thus, the magnitude of the slope of the inner-shell free energy vs. cavity size is larger for the QM simulations. In fact, a simple scaling calculation for the cavity size at which the cavity formation and long-ranged (Born-type) free energies balance predicts a crossing point within 0.1 Å of the observed length scale.

Fig. 4.

Linear extrapolation of the inner-shell contributions to the computed bulk free energies in classical and quantum simulations. The open squares (blue) are for the classical simulations and the solid squares (black) are for the quantum simulations with $N=128$ waters. The downward-shifted dashed blue line results from classical simulations that include one ion and 512 water molecules. The predicted shift in the quantum data is obtained from the size dependence of the classical data. The dashed (blue) horizontal line is the bulk free energy computed classically (−98.3 kcal/mol), and the solid (black) horizontal line is for the quantum calculation (−92.8 kcal/mol). The $N=512$ classical inner-shell contribution crosses the computed bulk free energy at 6.17 Å while that from quantum simulation crosses the exact bulk free energy at 5.60 Å. Our previous result for the crossing point is shown as a dashed vertical line at 6.15 Å.

This extrapolation procedure provides an independent estimate of the bulk hydration free energy, very close to −92.8 kcal/mol, that does not include any interfacial potential effects. It is not as accurate as the direct estimate above, but provides further support to the prediction of an effective surface potential of water close to −0.4 V.

Conclusions

A previous quasi-chemical estimate for the Na⁺ ion bulk hydration free energy (−93.3 kcal/mol, B3LYP-DFT) (6, 62), which omits interfacial potential effects and ion–water dispersion interactions, is very close to our corresponding quantum mechanical result (−92.8 kcal/mol). The quasi-chemical calculation in ref. 62 employed density functional theory methods for the inner-shell term and a dielectric model for the long-range term. Thus, our result provides confirmation of the accuracy of the QCT since we employ a consistent quantum mechanical description for the interactions at all length scales.

The computed effective surface potential of water, obtained from Eq. 6, is close to −0.4 V. This is consistent with previous estimates that involved cluster simulations employing polarizable classical models (68). The value is also consistent with our previous modified cluster-pair approximation result (37) and several standard models of single-ion hydration free energies that are free of interfacial potential effects (29, 69). Further related work consistent with the present results is found in other calculations of surface-potential–free hydration free energies (70, 71). Finally, the quantum mechanical result obtained here fully justifies the association of the Marcus experimental values (51) with the bulk hydration free energies.

The potential limitations of the present results are 1) finite-size effects in the simulations, 2) the lack of a full quantum treatment for the nuclear motions, 3) the pairwise assumption employed to estimate the induction/dispersion correction, 4) the accuracy of the underlying DFT functional, and 5) convergence of the simulations. The near equality of the computed free energy result and the tabulated Marcus value (neither of which contains an interfacial potential contribution) suggests that the resulting errors are not of large magnitude.

The results indicate a long-ranged electrostatic potential shift acting on ions near the water liquid–vapor surface over and above the strong inner-shell interactions that can display non-Gaussian behavior. Such uniform shifts (independent of the ion size) were seen previously in classical simulations of ions in bulk water (47, 69). Experimental evidence related to ion distributions near hydrophobic surfaces (8, 30) and acid–base shifts of several ${\mathrm{p}\mathrm{K}}_a$ units relative to bulk water (31, 32) provides support to these conclusions. The observation in ref. 30 that the anion in the tetraphenyl-arsonium/tetraphenyl-borate (TATB) hydrophobic ion pair prefers the interface while spectroscopy indicates the anion is more strongly hydrated in bulk water (72) gives a clear indication of an electrostatic driving force at work near the interface. It is also interesting to note that, in recent spectroscopic experiments (35), electric fields of magnitude $10^7$ V/cm were estimated from Stark shifts of fluorescent probes near droplet surfaces, very close to the effective field observed in the present simulations. Ref. 35 ascribed the field to a buildup of anions near the surface, but the above results imply there is a significant contribution from water itself.

Materials and Methods

In the present work we set the cavity size at $\lambda$ = 4.1 Å. All classical results were obtained just as in ref. 47. For the cavity potential, the purely repulsive Weeks–Chandler–Andersen (WCA) potential was implemented within the FIST module of the ab initio molecular dynamics package CP2K 2.6.1 (73). With the QuickStep module, we performed all of the DFT-based simulations for systems consisting of 128 waters and 1 ${\mathrm{N}\mathrm{a}}^{+}$ ion fixed at the center of a cubic box. The box size was determined by $L={\left(\frac{128}{\rho }+\frac{4\pi }{3}{r}_c^3\right)}^{\frac{1}{3}}$, where the water number density is $\rho =33.3285$ (nm)⁻³, $r_c=4.1\hspace{0.17em}\gamma$ (Å), and $\gamma$ was varied from 0 to 1 during the thermodynamic integration.

The initial configurations were generated by classical molecular dynamics simulations using the GROMACS 4.5.5 package (74), in which waters were modeled with the SPC/E force field (75) and the ${\mathrm{N}\mathrm{a}}^{+}$ ion was modeled with Lennard-Jones parameters taken from ref. 56 (set one). With 1 fs as the time step, all classical simulations were run for 2 ns in the canonical (NVT) ensemble after 1 ns of equilibration in the isothermal–isobaric (NPT) ensemble. We used the same classical simulation protocols as in our earlier work (47). The temperature was set at 300 K for the classical simulations.

For quantum simulations, we employed dual basis sets of Gaussian-type orbitals of shorter-range double zeta bases (DZVP-MOLOPT-SR-GTH) and plane waves (with 400-Ry cutoff) (76). Atomic cores were modeled with the Goedecker–Teter–Hutter (GTH) pseudopotentials (77). The revised Perdew–Burke–Ernzerhof (revPBE) (78, 79) functionals were used for all atoms in the system, while the D3 dispersion corrections (80, 81) were used only for water–water interactions. The Ewald method was applied for the electrostatic interactions under periodic boundary conditions and the Nosé–Hoover thermostat chain (82) of length 3 was coupled to each molecule to maintain a temperature of 330 K for all of the NVT ensemble simulations. This higher temperature mimics the excess kinetic energy of the protons in water due to quantum zero-point fluctuations. The theoretical rationale for employing a higher temperature to model nuclear quantum effects can be found in ref. 83, p. 104. Similar techniques have been used previously in simulations of water (84), and the full path integral simulations have been compared to higher-temperature classical models in ref. 85. Due to the requirement of many simulations along the thermodynamic integration paths to grow in the nanoscale cavities, we were not able to employ the path integral formalism for incorporating nuclear quantum effects. The time step was taken as 0.5 fs and the configurations were saved every 10 fs. For the long-range contributions, two 30-ps simulations (coupled and uncoupled) were implemented with the first 10 ps to equilibrate the system. For the 23-step integration performed for the packing contribution and the 27-step integration for the inner-shell contribution, we performed 15-ps simulations with the first 5 ps for equilibration.

During preliminary simulations for the long-ranged contribution at 400 K (to speed up the equilibration), it was observed that the D3 dispersion energy between ${\mathrm{N}\mathrm{a}}^{+}$ and waters is more negative than that between ${\mathrm{F}}^{-}$ and waters, in contrast to the fact that the ${\mathrm{F}}^{-}$ ion is significantly more polarizable than the ${\mathrm{N}\mathrm{a}}^{+}$ ion. In a dispersion calculation for the optimized geometry of a system consisting of one ion and 128 waters (without WCA cavity potentials), we obtained a ${\mathrm{N}\mathrm{a}}^{+}$–water dispersion energy of −10.6 kcal/mol and a ${\mathrm{F}}^{-}$–water dispersion energy of −4.5 kcal/mol. With inclusion of a WCA cavity potential with $\lambda =4.1$, the ${\mathrm{N}\mathrm{a}}^{+}$–water dispersion energy is −11.2 kcal/mol and the ${\mathrm{F}}^{-}$–water dispersion energy is −1.5 kcal/mol. In the simulations of the present work, we thus removed the ion–water dispersion interactions and used the D3 correction only for the water–water interactions. As discussed in Results and Discussion, this leads to structural results that agree significantly better with experiment.

To calculate the induction and dispersion interaction energies between the ion and the closest six waters in the long-ranged contributions, we employed the SAPT2 (86) module with the basis set aug-cc-pvdz (87) in the PSI4 package (88). The calculations that compare the induction interactions with the closest six waters outside the cavity for the Na⁺ ion and the ${\mathrm{F}}^{-}$ ion included 101 configurations, while the calculations of the dispersion interactions with six waters at close contact included 590 configurations. We also calculated the interaction energy between the ${\mathrm{N}\mathrm{a}}^{+}$ ion and five waters (101 configurations) at the CCSD(T) (89) and DFT levels with the same basis set as above using the PSI4 package.

Data Availability.

All study data are included in this article.