^{1}and Philippe H. Hünenberger

^{1,a)}

### Abstract

The raw single-ion solvation free energies computed from atomistic (explicit-solvent) simulations are extremely sensitive to the boundary conditions and treatment of electrostatic interactions used during these simulations. However, as shown recently [M. A. Kastenholz and P. H. Hünenberger, J. Chem. Phys.124, 224501 (2006)10.1529/biophysj.106.083667; M. M. Reif and P. H. Hünenberger, J. Chem. Phys.134, 144103 (2010)], the application of appropriate correction terms permits to obtain methodology-independent results. The corrected values are then exclusively characteristic of the underlying molecular model including in particular the ion–solvent van der Waals interaction parameters, determining the effective ion size and the magnitude of its dispersion interactions. In the present study, the comparison of calculated (corrected) hydration free energies with experimental data (along with the consideration of ionic polarizabilities) is used to calibrate new sets of ion-solvent van der Waals (Lennard-Jones) interaction parameters for the alkali (Li^{+}, Na^{+}, K^{+}, Rb^{+}, Cs^{+}) and halide (F^{−}, Cl^{−}, Br^{−}, I^{−}) ions along with either the SPC or the SPC/E water models. The experimental dataset is defined by conventional single-ion hydration free energies [Tissandier *et al.*, J. Phys. Chem. A102, 7787 (1998)10.1021/jp982638r; Fawcett, J. Phys. Chem. B103, 11181] along with three plausible choices for the (experimentally elusive) value of the absolute (intrinsic) hydration free energy of the proton, namely, , −1075 or −1050 kJ mol^{−1}, resulting in three sets L, M, and H for the SPC water model and three sets L_{ E }, M_{ E }, and H_{ E } for the SPC/E water model (alternative sets can easily be interpolated to intermediate values). The residual sensitivity of the calculated (corrected) hydration free energies on the volume-pressure boundary conditions and on the effective ionic radius entering into the calculation of the correction terms is also evaluated and found to be very limited. Ultimately, it is expected that comparison with other experimental ionic properties (e.g., derivative single-ion solvation properties, as well as data concerning ionic crystals, melts, solutions at finite concentrations, or nonaqueous solutions) will permit to validate one specific set and thus, the associated value (atomistic consistency assumption). Preliminary results (first-peak positions in the ion-water radial distribution functions, partial molar volumes of ionic salts in water, and structural properties of ionic crystals) support a value of close to −1100 kJ·mol^{−1}.

Financial support from the Swiss National Science Foundation (Project No. NF20-109261) is gratefully acknowledged. The authors also wish to warmly thank Mika Kastenholz for setting the stage to the calculation of methodology-independent single-ion solvation free energies and for his help in the initial steps of the present study.

I. INTRODUCTION

II. COMPUTATIONAL DETAILS

A. Molecular dynamics simulations

B. Electrostatic interactions

C. Volume–pressure boundary conditions

D. Structural solvation properties

E. Free energy calculations

F. Free energy correction terms

G. Effective ionic radii

H. Ion–solvent Lennard-Jones interaction parameters

I. Simulations of ionic crystals

III. RESULTS AND DISCUSSION

A. Structural solvation properties

B. Available parameter sets for the alkali and halide ions

C. Sensitivity of the correction terms to the ionic radius

D. Influence of the volume–pressure boundary conditions

E. Reoptimized ion-solvent Lennard-Jones interaction parameters

F. Initial assessment of the reoptimized parameters

IV. CONCLUSIONS

### Key Topics

- Free energy
- 102.0
- Sodium
- 52.0
- Solvents
- 45.0
- Electrostatics
- 44.0
- Boundary value problems
- 22.0

## Figures

Structural solvation properties around a monovalent ion evaluated from explicit-solvent MD simulations using different electrostatic schemes. The curves correspond to 1 ns simulations at 298.15 K and constant volume (solvent density 997 kg m^{−3}). The ion–solvent Lennard-Jones interaction parameters are those of set St for Na^{+} (Table I), together with the SPC water model (Ref. 73), the ion bearing either a positive charge (Na^{+}; a, c, e) or a negative charge (Na^{−}; b, d, f). The properties displayed (Sec. II D) are the ion-dipole radial distribution function [Eq. (1)], the ion–dipole orientation correlation function [Eq. (2)], and the radial polarization *P*(*r*) [Eq. (3)]. The dashed curves in (e) and (f) represent the Born polarization [Eq. (5) with ]. The successive curves, corresponding to different electrostatic schemes (LS, BA, BM, or CM; Sec. II B), are shifted upward on the vertical axis by 0.10 in (a) and (b) or 0.05 in (c) and (e), or downward by 0.05 in (d) and (f). In the CT schemes (BA, BM, and CM), the cutoff distance was set to *R* _{ C } = 1.2 nm (indicated by an arrow). In the Barker–Watts reaction-field schemes (BA and BM), the reaction-field permittivity was set to .

Structural solvation properties around a monovalent ion evaluated from explicit-solvent MD simulations using different electrostatic schemes. The curves correspond to 1 ns simulations at 298.15 K and constant volume (solvent density 997 kg m^{−3}). The ion–solvent Lennard-Jones interaction parameters are those of set St for Na^{+} (Table I), together with the SPC water model (Ref. 73), the ion bearing either a positive charge (Na^{+}; a, c, e) or a negative charge (Na^{−}; b, d, f). The properties displayed (Sec. II D) are the ion-dipole radial distribution function [Eq. (1)], the ion–dipole orientation correlation function [Eq. (2)], and the radial polarization *P*(*r*) [Eq. (3)]. The dashed curves in (e) and (f) represent the Born polarization [Eq. (5) with ]. The successive curves, corresponding to different electrostatic schemes (LS, BA, BM, or CM; Sec. II B), are shifted upward on the vertical axis by 0.10 in (a) and (b) or 0.05 in (c) and (e), or downward by 0.05 in (d) and (f). In the CT schemes (BA, BM, and CM), the cutoff distance was set to *R* _{ C } = 1.2 nm (indicated by an arrow). In the Barker–Watts reaction-field schemes (BA and BM), the reaction-field permittivity was set to .

Illustration of the ionic sizes corresponding to the three reoptimized ion–water Lennard-Jones interaction parameter sets. The drawings refer to the ion–solvent Lennard-Jones parameters of sets L, M, and H (Table I), optimized for , −1075, and −1050 kJ mol^{−1}, respectively, together with the SPC water model (Ref. 73). The effective ionic radius (Sec. II G; minimum of the Lennard-Jones curve for the ion–solvent interactions; Tables II and III) and the calculated standard intrinsic hydration free energy of the ion (Table VIII) are indicated.

Illustration of the ionic sizes corresponding to the three reoptimized ion–water Lennard-Jones interaction parameter sets. The drawings refer to the ion–solvent Lennard-Jones parameters of sets L, M, and H (Table I), optimized for , −1075, and −1050 kJ mol^{−1}, respectively, together with the SPC water model (Ref. 73). The effective ionic radius (Sec. II G; minimum of the Lennard-Jones curve for the ion–solvent interactions; Tables II and III) and the calculated standard intrinsic hydration free energy of the ion (Table VIII) are indicated.

Comparison between experimental and simulated properties for the three reoptimized Lennard-Jones interaction parameter sets L, M, and H (Table I) at 298.15 K. The properties include (a) the first peak positions in the ion-water (oxygen) radial distribution functions (Tables II and III); (b) the salt partial molar volumes [density-corrected standard-state variant (Ref. 3)] in water; (c,d) the salt crystal lattice parameters *L* _{ u } and energies *E* _{ l }, considering ion–ion Lennard-Jones interaction parameters derived by application of the geometric-mean combination rule (Ref. 87); (e,f) corresponding lattice parameters and energies, considering parameters derived by application of the Waldman–Hagler combination rule (Ref. 86). More details on these calculations can be found in supplementary material S2 (Ref. 161).

Comparison between experimental and simulated properties for the three reoptimized Lennard-Jones interaction parameter sets L, M, and H (Table I) at 298.15 K. The properties include (a) the first peak positions in the ion-water (oxygen) radial distribution functions (Tables II and III); (b) the salt partial molar volumes [density-corrected standard-state variant (Ref. 3)] in water; (c,d) the salt crystal lattice parameters *L* _{ u } and energies *E* _{ l }, considering ion–ion Lennard-Jones interaction parameters derived by application of the geometric-mean combination rule (Ref. 87); (e,f) corresponding lattice parameters and energies, considering parameters derived by application of the Waldman–Hagler combination rule (Ref. 86). More details on these calculations can be found in supplementary material S2 (Ref. 161).

Equilibrated configurations of the LiF, LiCl, LiBr, and LiI salts after 0.8 ns simulation at *P*° = 1 bar and *T* ^{−} = 298.15 K, using the ion–ion Lennard-Jones interaction parameters deduced from the ion–water interaction parameters of set L based on the geometric-mean (Ref. 87) (GM), Lorentz–Berthelot (Refs. 289 and 290) (LB) or Waldman–Hagler (Ref. 86) (WH) combination rules. The Li^{+} cation is depicted in white and the *X* ^{−} anion (F^{−}, Cl^{−}, Br^{−} or I^{−}) is depicted in black. Structures differing from a rocksalt lattice (indicative of a phase transition) are framed. Note that the images are scaled by the edge length (and thus not representative of the real box sizes) and that the surface patterns in the rocksalt crystal structures are an artifact due to the arbitrary location of the clipping plane (these patterns are not seen for LiF due to more limited thermal fluctuations of the atoms around their lattice positions).

Equilibrated configurations of the LiF, LiCl, LiBr, and LiI salts after 0.8 ns simulation at *P*° = 1 bar and *T* ^{−} = 298.15 K, using the ion–ion Lennard-Jones interaction parameters deduced from the ion–water interaction parameters of set L based on the geometric-mean (Ref. 87) (GM), Lorentz–Berthelot (Refs. 289 and 290) (LB) or Waldman–Hagler (Ref. 86) (WH) combination rules. The Li^{+} cation is depicted in white and the *X* ^{−} anion (F^{−}, Cl^{−}, Br^{−} or I^{−}) is depicted in black. Structures differing from a rocksalt lattice (indicative of a phase transition) are framed. Note that the images are scaled by the edge length (and thus not representative of the real box sizes) and that the surface patterns in the rocksalt crystal structures are an artifact due to the arbitrary location of the clipping plane (these patterns are not seen for LiF due to more limited thermal fluctuations of the atoms around their lattice positions).

Properties that could in principle be used for the (in)validation of a specific ion–water (and ion–ion) Lennard-Jones interaction parameter set. It is assumed that the set has been optimized against intrinsic single-ion hydration free energies for a given value of . The comparison against other experimental properties aims at (in)validating this specific value. This approach can be viewed as a form of extra-thermodynamic assumption, termed here the “atomistic consistency assumption”.

Properties that could in principle be used for the (in)validation of a specific ion–water (and ion–ion) Lennard-Jones interaction parameter set. It is assumed that the set has been optimized against intrinsic single-ion hydration free energies for a given value of . The comparison against other experimental properties aims at (in)validating this specific value. This approach can be viewed as a form of extra-thermodynamic assumption, termed here the “atomistic consistency assumption”.

## Tables

Overview of the 14 main sets of ion–solvent (water) Lennard-Jones interaction parameters currently available for the alkali and halide ions (force fields with implicit polarizability). The six new sets derived in the present article are also included (last entry). The sets are listed in chronological order of publication. The reported information includes the authors of the set and the corresponding original references, the water models considered and the corresponding original references, and the combination rule compatible with this set (if specified in the original reference; GM: geometric-mean (Ref. 87) combination rule; LB: Lorentz–Berthelot (Refs. 289 and 290) combination rule; WH: Waldman–Hagler (Ref. 86) combination rule). Parameters for sets Aq, St, We, and Re can be found in Tables II and III.

Overview of the 14 main sets of ion–solvent (water) Lennard-Jones interaction parameters currently available for the alkali and halide ions (force fields with implicit polarizability). The six new sets derived in the present article are also included (last entry). The sets are listed in chronological order of publication. The reported information includes the authors of the set and the corresponding original references, the water models considered and the corresponding original references, and the combination rule compatible with this set (if specified in the original reference; GM: geometric-mean (Ref. 87) combination rule; LB: Lorentz–Berthelot (Refs. 289 and 290) combination rule; WH: Waldman–Hagler (Ref. 86) combination rule). Parameters for sets Aq, St, We, and Re can be found in Tables II and III.

Ion–solvent Lennard-Jones interaction parameters and effective ionic radii for the alkali ions based on different parameter sets and radius definitions. The parameter sets considered are (Table I): St, Aq, We, and Re (L, M, or H) for the SPC water model, as well as Re (L_{ E }, M_{ E }, or H_{ E }) for the SPC/E water model. The Lennard-Jones interaction parameters (*C* _{6} and *C* _{12}) represent the coefficients for the interaction between the ion and the oxygen atom of the water model (note that the SPC and SPC/E water models have the same water–water Lennard-Jones interaction parameters). The different radius definitions are provided in Sec. II G. Values characterizing set We, but derived from simulations involving the SPC/E water model, are given between parentheses. The experimental ranges were obtained from neutron diffraction, x-ray diffraction, and x-ray absorption fine structure measurements (Refs. 287 and 288).

Ion–solvent Lennard-Jones interaction parameters and effective ionic radii for the alkali ions based on different parameter sets and radius definitions. The parameter sets considered are (Table I): St, Aq, We, and Re (L, M, or H) for the SPC water model, as well as Re (L_{ E }, M_{ E }, or H_{ E }) for the SPC/E water model. The Lennard-Jones interaction parameters (*C* _{6} and *C* _{12}) represent the coefficients for the interaction between the ion and the oxygen atom of the water model (note that the SPC and SPC/E water models have the same water–water Lennard-Jones interaction parameters). The different radius definitions are provided in Sec. II G. Values characterizing set We, but derived from simulations involving the SPC/E water model, are given between parentheses. The experimental ranges were obtained from neutron diffraction, x-ray diffraction, and x-ray absorption fine structure measurements (Refs. 287 and 288).

Ion–solvent Lennard-Jones interaction parameters and effective ionic radii for the halide ions based on different parameter sets and radius definitions. See legend of Table II.

Ion–solvent Lennard-Jones interaction parameters and effective ionic radii for the halide ions based on different parameter sets and radius definitions. See legend of Table II.

Standard intrinsic hydration free energies of the alkali and halide ions calculated from MD simulations based on three different ion–solvent Lennard-Jones interaction parameter sets, along with 17 different combinations of electrostatic schemes and simulation parameters. The results correspond to free energy calculations (including error estimates; Sec. II E) at 298.15 K and constant volume (NVT; Sec. II C; solvent density 997 kg m^{−3}). The ion–solvent Lennard-Jones interaction parameters are those of sets St, Aq, and We (Table I) together with the SPC water model (Ref. 73). The different electrostatic schemes considered are LS, BA, BM, and CM (Sec. II B), along with different parameters (number of water molecules *N* _{ S }; cutoff radius *R* _{ C } for the three latter schemes). The evaluation of the correction term (Sec. II F) relied on (Sec. II G) in all cases. Error estimates for the 17 individual solvation free energies are indicated (±). The corresponding average values are also reported, along with the [bias-corrected (Ref. 280)] standard deviations and the error estimates . Since the We, Na^{+}, and Cl^{−} ions were actually optimized (Ref. 98) for the SPC/E water model (Ref. 76), solvation free energies calculated for these ions using the SPC/E water model are also reported for comparison (values given between parentheses for the LS scheme with *N* _{ S } = 1024 only). More details concerning these calculations can be found in supplementary material S2 (Ref. 161).

Standard intrinsic hydration free energies of the alkali and halide ions calculated from MD simulations based on three different ion–solvent Lennard-Jones interaction parameter sets, along with 17 different combinations of electrostatic schemes and simulation parameters. The results correspond to free energy calculations (including error estimates; Sec. II E) at 298.15 K and constant volume (NVT; Sec. II C; solvent density 997 kg m^{−3}). The ion–solvent Lennard-Jones interaction parameters are those of sets St, Aq, and We (Table I) together with the SPC water model (Ref. 73). The different electrostatic schemes considered are LS, BA, BM, and CM (Sec. II B), along with different parameters (number of water molecules *N* _{ S }; cutoff radius *R* _{ C } for the three latter schemes). The evaluation of the correction term (Sec. II F) relied on (Sec. II G) in all cases. Error estimates for the 17 individual solvation free energies are indicated (±). The corresponding average values are also reported, along with the [bias-corrected (Ref. 280)] standard deviations and the error estimates . Since the We, Na^{+}, and Cl^{−} ions were actually optimized (Ref. 98) for the SPC/E water model (Ref. 76), solvation free energies calculated for these ions using the SPC/E water model are also reported for comparison (values given between parentheses for the LS scheme with *N* _{ S } = 1024 only). More details concerning these calculations can be found in supplementary material S2 (Ref. 161).

Conventional hydration free energies of the alkali and halide ions calculated from MD simulations based on different ion–solvent Lennard-Jones interaction parameter sets or derived from experiment. The simulation results correspond to free energy calculations at 298.15 K and either constant volume (NVT; Sec. II C; solvent density 997 kg m^{−3}) or constant pressure (NPT_{s}; Sec. II C; 1 bar), and are derived from the results (and associated error estimates ) reported in Tables IV and VIII. The ion–solvent Lennard-Jones interaction parameters considered are those of sets St, Aq, We, L, M, and H together with the SPC water model (Ref. 73), or L_{ E }, M_{ E }, and H_{ E } together with the SPC/E water model (Ref. 76) (Table I). For the three former sets, the simulations correspond to NVT conditions and the conventional values were derived using kJ mol^{−1}. For the six latter sets, the simulations correspond to NPT_{s} conditions and the conventional values were derived using the values for which each set was optimized. Values between parentheses for set We are based on simulations using SPC/E water model (Ref. 76) rather than the SPC water model (Ref. 73). The experimental data is taken from five literature sources by Rosseinsky (Ref. 252) (Ro65), Gomer and Tryson (Ref. 253) (Go77), Marcus (Ref. 248) (Ma85), Tissandier *et al.* (Ref. 218) (Ti98) and Fawcett (Ref. 219) (Fa99).

Conventional hydration free energies of the alkali and halide ions calculated from MD simulations based on different ion–solvent Lennard-Jones interaction parameter sets or derived from experiment. The simulation results correspond to free energy calculations at 298.15 K and either constant volume (NVT; Sec. II C; solvent density 997 kg m^{−3}) or constant pressure (NPT_{s}; Sec. II C; 1 bar), and are derived from the results (and associated error estimates ) reported in Tables IV and VIII. The ion–solvent Lennard-Jones interaction parameters considered are those of sets St, Aq, We, L, M, and H together with the SPC water model (Ref. 73), or L_{ E }, M_{ E }, and H_{ E } together with the SPC/E water model (Ref. 76) (Table I). For the three former sets, the simulations correspond to NVT conditions and the conventional values were derived using kJ mol^{−1}. For the six latter sets, the simulations correspond to NPT_{s} conditions and the conventional values were derived using the values for which each set was optimized. Values between parentheses for set We are based on simulations using SPC/E water model (Ref. 76) rather than the SPC water model (Ref. 73). The experimental data is taken from five literature sources by Rosseinsky (Ref. 252) (Ro65), Gomer and Tryson (Ref. 253) (Go77), Marcus (Ref. 248) (Ma85), Tissandier *et al.* (Ref. 218) (Ti98) and Fawcett (Ref. 219) (Fa99).

Sensitivity of the calculated (average) standard intrinsic hydration free energies (and associated spread and estimated error) to the definition of the effective ionic radius entering into the evaluation of the correction terms. The reported results correspond to average values and associated [bias-corrected (Ref. 280)] standard deviations and error estimates , evaluated as in Table IV, but considering four alternative choices concerning *R* _{ I }, namely, *R* _{ I } = 0.8*R* _{mean}, *R* _{mean}, or (Sec. II G). Absolute values (, , ) are reported for the reference choice (identical to the entries of Table IV), while changes relative to this reference values (, , ; alternative value minus reference value) are reported for the other choices. More details on these calculations can be found in supplementary material S2 (Ref. 161).

Sensitivity of the calculated (average) standard intrinsic hydration free energies (and associated spread and estimated error) to the definition of the effective ionic radius entering into the evaluation of the correction terms. The reported results correspond to average values and associated [bias-corrected (Ref. 280)] standard deviations and error estimates , evaluated as in Table IV, but considering four alternative choices concerning *R* _{ I }, namely, *R* _{ I } = 0.8*R* _{mean}, *R* _{mean}, or (Sec. II G). Absolute values (, , ) are reported for the reference choice (identical to the entries of Table IV), while changes relative to this reference values (, , ; alternative value minus reference value) are reported for the other choices. More details on these calculations can be found in supplementary material S2 (Ref. 161).

Sensitivity of the calculated standard intrinsic hydration free energies to the volume–pressure boundary conditions employed in explicit-solvent MD simulations. The reported results correspond to hydration free energies (including error estimates; Sec. II E) calculated at 298.15 K under various volume–pressure boundary conditions (Sec. II C). The ion–solvent Lennard-Jones interaction parameters are those of set St for Na^{+} (Table I) together with the SPC water model (Ref. 73), the ion bearing either a positive charge (Na^{+}) or a negative charge (Na^{−}). Two different electrostatic schemes (LS or BM, the latter with *R* _{ C } = 1.0 or 1.2 nm) and system sizes (*N* _{ S } = 512 or 1024) are considered. The cavity formation terms were computed using the BM scheme (*R* _{ C } = 1.4 nm) with *N* _{ S } = 1024 only (the same values were used for the LS scheme and for *N* _{ S } = 512). The average and standard deviation of the box edge *L* were computed from 1 ns simulations at zero ionic charge. The correction term was calculated using and the reported average value 〈*L*〉 of *L*. More details on these calculations can be found in supplementary material S2 (Ref. 161).

Sensitivity of the calculated standard intrinsic hydration free energies to the volume–pressure boundary conditions employed in explicit-solvent MD simulations. The reported results correspond to hydration free energies (including error estimates; Sec. II E) calculated at 298.15 K under various volume–pressure boundary conditions (Sec. II C). The ion–solvent Lennard-Jones interaction parameters are those of set St for Na^{+} (Table I) together with the SPC water model (Ref. 73), the ion bearing either a positive charge (Na^{+}) or a negative charge (Na^{−}). Two different electrostatic schemes (LS or BM, the latter with *R* _{ C } = 1.0 or 1.2 nm) and system sizes (*N* _{ S } = 512 or 1024) are considered. The cavity formation terms were computed using the BM scheme (*R* _{ C } = 1.4 nm) with *N* _{ S } = 1024 only (the same values were used for the LS scheme and for *N* _{ S } = 512). The average and standard deviation of the box edge *L* were computed from 1 ns simulations at zero ionic charge. The correction term was calculated using and the reported average value 〈*L*〉 of *L*. More details on these calculations can be found in supplementary material S2 (Ref. 161).

Standard intrinsic hydration free energies of the alkali and halide ions calculated from MD simulations using reoptimized ion–solvent Lennard-Jones interaction parameter sets. The reported results correspond to hydration free energies (including error estimates; Sec. II E) calculated at 298.15 K and constant pressure (NPT_{s} conditions; Sec. II C; 1 bar). The ion–solvent Lennard-Jones interaction parameters are those of sets L, M, and H together with the SPC water model (Ref. 73), or L_{ E }, M_{ E }, and H_{ E } together with the SPC/E water model (Ref. 76) (Table I). The simulations were performed using the LS scheme with *N* _{ S } = 1024. The evaluation of the correction terms (Sec. II F) relied on . More details on these calculations can be found in supplementary material S2 (Ref. 161).

Standard intrinsic hydration free energies of the alkali and halide ions calculated from MD simulations using reoptimized ion–solvent Lennard-Jones interaction parameter sets. The reported results correspond to hydration free energies (including error estimates; Sec. II E) calculated at 298.15 K and constant pressure (NPT_{s} conditions; Sec. II C; 1 bar). The ion–solvent Lennard-Jones interaction parameters are those of sets L, M, and H together with the SPC water model (Ref. 73), or L_{ E }, M_{ E }, and H_{ E } together with the SPC/E water model (Ref. 76) (Table I). The simulations were performed using the LS scheme with *N* _{ S } = 1024. The evaluation of the correction terms (Sec. II F) relied on . More details on these calculations can be found in supplementary material S2 (Ref. 161).

Differences between predicted and computed values (Sec. III E) for the M and M_{ E } sets (predicted minus computed). The predicted values derive from postulating a linear relationship between and at constant *C* _{6}, and anchoring this line based on the computed values for sets L or L_{ E } and H or H_{ E }. Interpolation at the value of the corresponding M or M_{ E } set provides the desired prediction.

Differences between predicted and computed values (Sec. III E) for the M and M_{ E } sets (predicted minus computed). The predicted values derive from postulating a linear relationship between and at constant *C* _{6}, and anchoring this line based on the computed values for sets L or L_{ E } and H or H_{ E }. Interpolation at the value of the corresponding M or M_{ E } set provides the desired prediction.

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