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Computation of methodology-independent single-ion solvation properties from molecular simulations. III. Correction terms for the solvation free energies, enthalpies, entropies, heat capacities, volumes, compressibilities, and expansivities of solvated ions
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10.1063/1.3567020
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1 Laboratory of Physical Chemistry, ETH Zürich, CH-8093 Zürich, Switzerland
a) Author to whom correspondence should be addressed. Electronic mail: phil@igc.phys.chem.ethz.ch. Tel.: + 41 44 632 5503. Fax: + 41 44 632 1039.
J. Chem. Phys. 134, 144103 (2011)
/content/aip/journal/jcp/134/14/10.1063/1.3567020
http://aip.metastore.ingenta.com/content/aip/journal/jcp/134/14/10.1063/1.3567020

Figures

FIG. 1.

Terms contributing to ΔG cor (correction to the solvation free energy), evaluated for a Na+ ion in water (numerical: circles; empirical: solid lines). The parameters employed for the calculations are listed in Table VI. The type-D correction term vanishes for the chosen parameters (). The standard-state correction is a constant and evaluates to kJ mol−1 for water ( in the density-corrected variant). The simulation parameter varied is the droplet radius S (FBC/CB), the box edge L (PBC/LS), or the cutoff distance R C (PBC/BM, PBC/CM; L = 3.6 nm).

FIG. 2.

Terms contributing to ΔS cor (correction to the solvation entropy), evaluated for a Na+ ion in water (numerical: circles; empirical: solid lines). The parameters employed for the calculations are listed in Table VI. The type-D correction term vanishes for the chosen parameters (). The standard-state correction is a constant and evaluates to J mol−1  K−1 for water ( J mol−1  K−1 in the density-corrected variant). See caption of Figure 1.

FIG. 3.

Terms contributing to ΔC P, cor (correction to the solvation heat capacity), evaluated for a Na+ ion in water (numerical: circles; empirical: solid lines). The parameters employed for the calculations are listed in Table VI. The type-D correction term vanishes for the chosen parameters (). The standard-state correction is a constant and evaluates to J mol−1  K−1 for water ( J mol−1  K−1 in the density-corrected variant). See caption of Figure 1.

FIG. 4.

Terms contributing to ΔV cor (correction to the solvation volume), evaluated for a Na+ ion in water (numerical: circles; empirical: solid lines). The parameters employed for the calculations are listed in Table VI. The type-D correction term vanishes for the chosen parameters (). The standard-state correction is a constant and evaluates to m3  mol−1 for water ( m3  mol−1 in the density-corrected variant). See caption of Figure 1.

FIG. 5.

Terms contributing to ΔK T, cor (correction to the solvation volume-compressibility), evaluated for a Na+ ion in water (numerical: circles; empirical: solid lines). The parameters employed for the calculations are listed in Table VI. The type-D correction term vanishes for the chosen parameters (). The standard-state correction is a constant and evaluates to m3  mol−1  bar−1 for water ( m3  mol−1  bar−1 in the density-corrected variant). See caption of Figure 1.

FIG. 6.

Terms contributing to ΔA P, cor (correction to the solvation volume-expansivity), evaluated for a Na+ ion in water (numerical: circles; empirical: solid lines). The parameters employed for the calculations are listed in Table VI. The type-D correction term vanishes for the chosen parameters (). The standard-state correction is a constant and evaluates to m3  mol−1  K−1 for water ( m3  mol−1  K−1 in the density-corrected variant). See caption of Figure 1.

Tables

Table I.

Fitting coefficients for the empirical evaluation of the correction terms and via Eqs. (42) and (43) or Eqs. (20) and (50)–(53), respectively, in the context of cutoff-based (CT) electrostatic interactions with straight cutoff (SC) truncation or with the Barker–Watts (BW) reaction-field scheme.

Table II.

Error measures εrms, εave and εmax, denoting the root-mean-square, average, and maximum absolute deviations, respectively, between numerical [Eqs. (12)–(15) for ; Eq. (19) for ] and empirical [Eqs. (42) and (43) for ; Eqs. (20) and (50)–(53) for ] correction terms and in the context of cutoff-based (CT) electrostatic interactions with straight cutoff (SC) truncation or with the Barker–Watts (BW) reaction-field scheme. Corresponding error measures concerning the first and second derivatives of these correction terms with respect to , R I , and L are also reported. The calculation of relied on the empirically-evaluated . For , the dataset encompassed 59 800 points for the evaluation of both free energies and derivative properties. For , the dataset encompassed 2142 points for the evaluation of the free energies, and 36 points for the evaluation of the derivative properties. Error measures reported between parentheses correspond to cases where the self-second derivatives were set to zero in the numerical (target) set.

Table III.

First partial derivatives of the correction terms with respect to the solvent permittivity ε (experimental ε S , model , or reaction-field εBW; as indicated), the effective ionic radius R I , the size parameter X (cubic box-edge length L or spherical droplet radius S; as indicated), or the surface properties χ of the pure solvent (surface potential at a planar interface , dependence on the inverse curvature at a convex interface , or corresponding dependence at a concave interface ; as indicated). The equations correspond to the numerical (unprimed entry numbers) or to the empirical (primed: SC; double-primed: BW) formulations of the correction terms , , ΔG B , , , and ΔG D for the FBC/CB, PBC/SC, PBC/BW, and PBC/LS schemes. Note that only nonvanishing derivatives are reported. The quantities N A , ε o , αLS, and q I stand for Avogadro's constant, the dielectric permittivity of vacuum, the LS self-term constant [Eq. (20)], and the ionic charge. The quantity ΔG A and its derivatives have not been expanded in the various expressions [Eq. (16), derivatives provided in this table]. The quantities Q and Z are defined as and Z = N A q I ξ. The quantity noted ξ is the exclusion potential of the solvent model and its dependence on L has not been explicited [Eqs. (21), (40), and (41)]. The function Y CT is defined as Y CT = exp [μCT(2R C )−1 X + νCT], where CT = SC or BW, and the derivatives of μCT and νCT have not been explicited [Eqs. (50)–(53), straightforward derivatives]. The coefficients a i and b i are given in Table I [Eqs. (42) and (43)].

Table IV.

Standard-state correction terms to ionic hydration free energies and derivative quantities calculated by atomistic simulations. The terms (standard variant) and (density-corrected variant) are involved in Eqs. (69) and (81), respectively. The quantity represents their difference, according to Eq. (75). The different terms were evaluated based on a standard water density . The corresponding pressure and temperature derivatives (required for the calculation of and only) were taken from Table VI. The reference pressure, molality, and temperature are set to P° = 1 bar, b° = 1 mol kg−1, and T = 298.15 K. The reference density [Eq. (79)] is set to 997.0 kg m−3.

Table V.

Standard intrinsic hydration free energy of the Na+ cation (Ref. 28) evaluated by explicit-solvent simulations with application of correction terms. The simulations are performed at constant volume and temperature (NVT ensemble; T =298.15 K; density kg m−3). The boundary conditions and electrostatic schemes considered are FBC/CB (droplet of radius S containing N S water molecules), PBC/LS (cubic box of edge L containing N S water molecules), and PBC/CT (cubic box of edge L containing N S water molecules, cutoff distance R C ; for the BA and BM schemes) systems. The correction terms are calculated as detailed in Sec. II B, using the parameters (Ref. 280) R I = 0.2 nm, ε S = 78.4, , , V, V nm, and V nm. The correction terms and ΔG B (as well as the resulting ) are evaluated using the numerical approach. Corresponding values relying on the empirical approach (fitted functions) are reported between parentheses. The empirical estimate also relies on the empirical estimate.

Table VI.

Parameters used for the numerical evaluation of the correction terms in the context of Na+ hydration (Figs. 1–6). The required quantities f are reported, along with their first and second derivatives with respect to temperature (∂ T , ), pressure (∂ P , ), or pressure and temperature (), as estimated for Na+ in water at pressure P° = 1 bar and temperature T = 298.15 K. For the solvent, they include the dielectric permittivity (ε S , ; assumed identical for real and model solvent), the density (ρ S , ; assumed identical for real and model solvent) and the air–liquid interfacial potential () along with its dependence on the inverse of the interface curvature for a concave () or convex () interface. For the ion, the only quantity is the effective ionic radius (R I ). The values of ε S and ρ S (and derivatives) are based on experiment. The values of R I , , and (and derivatives) are based on simulations (SPC water model), or neglected in the absence of available data.

/content/aip/journal/jcp/134/14/10.1063/1.3567020
2011-04-08
2014-04-16

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