Determining the hydration free energies of selected small molecules with MP2 and local MP2 through adaptive force matching

Abstract

Force fields for seven small solute molecules, ethanol, 2-methyl-1-propanol, 2-butanol, cyclohexene, tetrahydropyran, 1,4-dioxane, and 1,4-butanediol, in dilute aqueous solutions were created with the adaptive force matching (AFM) method using MP2 or local MP2 as reference. The force fields provide a way to predict the hydration free energies (HFEs) of these molecules with only electronic structure calculations as reference. For six of the seven molecules, the predicted HFEs are in very good agreement with experiments. For 1,4-butanediol, the model created by force matching LMP2 provides a HFE that is too positive. Further investigation suggests that LMP2 may not be sufficiently accurate for computing HFEs for alcohols with AFM. Other properties, such as enthalpy of hydration, diffusion constants, and vibrational spectra, are also computed with the force field developed. The force fields developed by AFM provide a bridge for computing ensemble properties of the reference electronic structure method. With MP2 and LMP2 as reference methods, the computed properties of the small molecular solutes are found to be in good agreement with experiments.

I. INTRODUCTION

Hydration free energy (HFE) is an important property related to many areas of science that directly influence the human life, such as pharmaceutical chemistry and environmental chemistry. A desirable drug needs to have adequate solubility, which corresponds to a sufficiently negative HFE.¹ The HFE is proportional to the logarithm of Henry’s law constant.² Henry’s law constant of volatility is the ratio of the equilibrium vapor pressure and concentration of the solute.³ Thus, partitioning and distribution of environmentally important molecules in the atmosphere is determined by their HFEs.

The large chemical diversity leads to a very high number of possible molecules of interest to a human being; measuring the HFE for each of them experimentally is tedious and time-consuming. Although seemingly straightforward, the measurement of the gas–liquid partition coefficients for HFE determination has many technical difficulties, especially for substances with very low solubility. Adsorptions on walls of apparatus, the long time needed to reach equilibrium, a tendency for supersaturation, losses in mass balances, and low concentrations and pressures that approach the detection limits of instruments all hinder the accurate determination of HFEs.⁴ With all these difficulties, it is not surprising to see experimental values differ quite significantly for certain compounds in measurements conducted by different groups.^5,6

It is desirable to have the capacity to predict HFEs from the first principles.^7,8 A lot of research has been conducted in recent years for HFE predictions. Successful methods have been reported using implicit solvent based approaches, such as SMD.⁹ With explicit water models, the 3D-RISM method is capable of providing predictions of HFEs.^2,10 Pure force field (FF) based methods have also been developed.^11–14 For example, HFEs for a library of compounds can be fitted during force field parameterization and the force field could then make predictions on test molecules that bear similarity to the training molecules in the library.¹⁵

In this work, we present a force field based method that relies only on electronic structure information. The force fields are fitted to electronic structure references with the adaptive force matching (AFM) method developed previously by our group.^16–19 Although we focus on the ability of the force fields to predict HFEs, the force field can be used to model dilute aqueous solutions of the solute and investigate other properties, such as enthalpy of hydration, radial distribution functions (RDFs), and diffusion constants. We note that the fitting procedure of the force fields is in no way biased to reproduce HFEs. HFEs are highlighted only because of its significance and the good availability of experimental data for validation. Contrary to force field models that fit directly to some experimental properties, the AFM procedure only fits to electronic structure gradient calculations. When a force field is fitted to selected properties, the confidence in predicting properties not being fitted would suffer. By not fitting to any properties, good predictions truly reflect the accuracy in representing the underlying potential energy surface. Prediction of accurate HFEs thus also lends confidence to the accuracy of other properties, where no experimental values are available as validation.

Many force field development procedures fit the property of the neat liquid of the solute and rely on combination rules to model its interactions with water in a solution.^11–14 The accuracy of such a prediction is thus dependent on the appropriateness of the combination rules. Leveraging the ability of AFM to fit a large number of parameters robustly, no combination rules are used with our approach. All solute–solvent interactions are fitted directly to condensed phase references that include many-body effects. Such a procedure allows simple energy expressions to capture many-body effects implicitly. In addition to eliminating combination rules, exponential repulsion terms are used with our potential instead of the popular Lennard-Jones form. Steric hindrance mostly arises from exchange repulsion as a result of the Pauli exclusion effect, and an exponential term is thus much more physical.

This paper presents force field models for simulations of dilute aqueous solutions of ethanol, 2-methyl-1-propanol, 2-butanol, cyclohexene, tetrahydropyran, 1,4-dioxane, and 1,4-butanediol. These molecules are of environmental and pharmaceutical significance. For smaller molecules with up to five heavy atoms, we will use the MP2 method to provide reference gradients. For larger molecules, the local MP2 (LMP2) method will be utilized to provide reference forces. It is worth noting that the QM region in a typical AFM fit includes not only the solute but also some solvent molecules. The number of solvent molecules to be modeled with QM is generally more than the number of molecules in the first solvation shell. Thus, an MP2 computation for AFM is significantly more expensive than an MP2 computation on the solute alone. In addition, MP2 gradient evaluations are more computationally demanding than MP2 energies due to the need to solve the coupled-perturbed Hartree–Fock equations.²⁰ The use of LMP2 increases the size of molecules that can be fitted with AFM with post-Hartree–Fock methods.

The HFE of ethanol has been predicted in a previous study by Rogers and Wang using MP2 as reference following a similar AFM based procedure.¹⁹ The force field to be developed for ethanol in this work is slightly different due to simplifications to the energy expression used. The simplified procedure is easier to generalize to other solute molecules.

In this work, it is shown that the HFEs based on MP2 and LMP2 reference are in very good agreement with experiments for the majority of the molecules investigated. The only molecule where agreement is not satisfactory is 1,4-butanediol. Investigation of an LMP2 based model for ethanol indicates that LMP2 seems to underestimate the absolute HFE of simple alcohols when used with AFM. Other properties, such as enthalpy of hydration and diffusion constants of the solutes, are also reported and compared to corresponding experimental values where available.

This paper is organized in five sections. After the Introduction section, specific details of AFM in the creation of the force field for the solutes are described in Sec. II. The computational details for the property determinations are presented in Sec. III. Performances of our force fields are evaluated in Sec. IV. Summary and conclusions are provided in Sec. V.

II. SPECIFIC DETAILS FOR THE DEVELOPMENT OF SOLUTE–WATER MODELS WITH AFM

In this work, only the solute–water interactions will be fitted with AFM. The water model will be taken as the BLYPSP-4F model.²¹ Without using combination rules, the solute–water terms do not depend on the water–water terms, and it has been shown that replacing the water models for solute models developed by AFM has minimal effects on thermodynamical properties, such as HFE.²² The BLYPSP-4F model is a four-site model that was developed by AFM relying on coupled cluster quality reference forces obtained with the DFT-SP method.^23,24 The model gives excellent properties for water, such as heat of vaporization, density, diffusion constants, surface tension, and dielectric constant.^21,25,26 Figure 1 summarizes the atom typing used for ethanol, 2-butanol, 2-methyl-1-propanol, cyclohexene, tetrahydropyran, 1,4-dioxane, and 1,4-butanediol. With AFM, each solute model was fitted independently of others. Thus, the atom types are specific to each molecule. For example, C1 of cyclohexene will carry different parameters from C1 in tetrahydropyran.

FIG. 1.

Atom type definitions for the seven molecules investigated. Note that atoms with the same type but in different molecules have different parameters.

The basic principle of atom typing is to assign each symmetry unique atom a different atom type. For example, the hydroxyl α-C and β-C in 1,4-butanediol will have different atom types as there are no symmetry operations that make the two carbons equivalent. The only exception to this rule is for tetrahydropyran, where the β-C and γ-C of the ether oxygen are modeled using the same atom type. For tetrahydropyran, assigning a different type for γ-C will also be appropriate. However, we assume the difference between β-C and γ-C of ether oxygen is minute. Modeling them with the same atom type would lead to no loss of accuracy. We do anticipate that exploring the use of identical parameters for atoms with similar chemical environments would be a valuable future direction for AFM based force fields.

Dispersion is weaker than other types of interactions, and it is better determined with symmetry adapted perturbation theory (SAPT) before the AFM iterations.¹⁹ SAPT has the unique ability to separate intermolecular interactions into different components with high accuracy.^27,28 In our force fields, the dispersion parameters are only placed between heavy atoms and assume the following form:

$$E\left(r_{ij}\right)=\frac{C_6}{r_0^6+r_{ij}^6},$$ (1)

where

$$r_0=0.6R_{ij}^{vdW}.$$ (2)

In Eq. (2), the R^vdW parameter of each atom was taken from the van der Waals radius of the element used previously by Tkatchenko.²⁹ The use of r₀ in Eq. (1) will damp short-range contribution to dispersion. This form was proposed in our previous work³⁰ for the development of a force field for alanine peptide and was found to provide stronger damping and better agreement with reference PBE-D3 forces when compared to the Tang–Toennies form.³¹

To fit dispersion, each solute is first divided into fragments (Fig. S1). Dispersion parameters were determined by fitting Eq. (1) to SAPT E2 dispersion energies between two separate solute fragments or between a fragment and water. To extract conformations, one or two solute fragments are placed in a box of BLYPSP-4F water and simulated for 4 ns at 298 K and 1 bar. The fragments assume the Hartree–Fock/aug-cc-pVDZ geometry and are modeled with the OPLS-AA parameters for the simulation. Since the BLYPSP-4F water does not have Lennard-Jones σ and ε, σ and ε of TIP4P water were used to derive the cross terms for the solute–water interactions. A similar procedure is also used for obtaining cross terms for the first generation of AFM (vide infra).

Care was taken not to select dimers that are too close. At short range, exchange dispersion and higher order dispersion may be important. To select a dimer, a random water or solute fragment is selected at a distance between 6.0 Å and 12.0 Å from another fragment with approximately uniform distribution. When measuring distances, the distance of the nearest atoms of the two fragments (or water) is used. A total of ∼100 dimer conformations were selected for each of the fragment–fragment or fragment–water dimers.

The SAPT calculations were performed with the aug-cc-pVDZ dimer centered basis set (DCBS) on the fragment–water and fragment–fragment dimers. The fitted C6 parameters are reported in Table S1.

After the dispersion parameters are determined, AFM iterations are performed to fit other parameters. A standard AFM procedure iterates through three steps: the sampling step, the QM/MM step, and the force-match step.

The objective of the sampling step is to obtain the most representative conformations for the thermodynamic condition of interest. This would be a dilute solution at ambient temperature and pressure. The initial guess force field for the solute molecules will be OPLS-AA. In the first generation of AFM, the solute is solvated by BLYPSP-4F water. Since BLYPSP-4F water uses exponential repulsion, the water is assumed to have the Lennard-Jones parameters of TIP4P to derive cross terms with the solute model. After the first iteration, the force field created by AFM will be used in this step and combination rules will no longer be needed.

MD simulations are performed at 298 K and 328 K with one solute in a box of 343 water molecules. The slightly elevated temperature allows better sampling of the repulsive region of the potential. After a brief 200 ps equilibration with velocity rescaling, sampling is accomplished with 10 ns MD in the NPT ensemble using the Nose–Hoover thermostat^32,33 and Parrinello–Rahman barostat^34,35 with relaxation times of 1 ps and 5 ps, respectively. 100 configurations are extracted from the final 9 ns of trajectory for each temperature in the sampling step for QM/MM.

In the QM/MM step, QM/MM conformations are constructed using the configurations from the sampling step. The QM part of the QM/MM contains a center region and a boundary region. Only the forces in the center region are to be fitted. The boundary region is treated quantum-mechanically, although their forces are not fitted due to their proximity to the MM charges. For each conformation from the sampling step, the QM/MM conformation is constructed with the following procedure:

• 1.The solute will be in the QM center region. Any water molecule with an atom within 3 Å from the solute will be defined as a first hydration shell water molecule and included in the QM part.
• 2.From the first hydration shell water molecules, five are chosen randomly to be included in the center region. Any water within 2.6 Å of these five water molecules will be included in the QM part.
• 3.Any water with at least one atom within 9 Å of the solute but not in the QM part will be kept as MM particles. Water molecules further away will be ignored.
• 4.Water molecules in the QM part but not in the center region will be in the boundary region.

With this protocol, the number of QM water molecules for the QM/MM calculation varies from an average of 25 for ethanol to an average of 29 for 1,4-butandiol. However, only five QM water molecules are used for fitting the force fields in each case. A representative conformation is displayed in Fig. 2 showing the condensed phase-like environment surrounding the solute and water molecules in the center region.

FIG. 2.

A snapshot showing the QM/MM setup for the electronic structure calculations. One ethanol and five water molecules are the central region (CPK) of the QM part. The water molecules in the boundary region (yellow) are treated quantum mechanically but are not fitted in force matching. The QM part is surrounded by MM water (lines) modeled as point charges.

The forces for the fitting are computed with density fitting MP2^36,37 for ethanol, 2-butanol, and 2-methyl-1-propanol. Density fitting LMP2^38,39 is used for cyclohexene, tetrahydropyran, 1,4-dioxane, and 1,4-butanediol. All the electronic structure calculations were performed with the Molpro code version 2015.1.^40,41 For MP2, the solute is modeled with the aug-cc-pVTZ basis set and the water in the center region is modeled with the aug-cc-pVTZ(-df) basis set. In the aug-cc-pVTZ(-df) basis set, the f orbitals were removed from water oxygen and the d orbitals were removed from the hydrogen. Our previous work has shown that this slightly reduced basis set is adequate for modeling water.^16,19 To further reduce the basis set, the water in the boundary region is modeled with aug-cc-pVDZ for the MP2 calculations. For LMP2, the aug-cc-pVTZ basis set is used for all heavy atoms in the QM part and the cc-pVTZ basis set for all hydrogens. The omission of the diffuse functions on hydrogen atoms reduces the basis set linear dependency.

In the force-matching step, the parameters are obtained with a two-step fitting procedure. As described previously, dispersion parameters have been determined by fitting to SAPT energies before the AFM iterations. With the dispersion parameters fixed, the parameters responsible for modeling intermolecular interactions are fitted in the first step to reproduce total forces and torques of the molecules in the center region of the QM part. The charge neutrality constraint is enforced with a weight of 1000 as done in our previous work on AFM.²¹

The intermolecular interactions will be modeled with

$$U_{ij}^{non-bonded}=\frac{q_i{q}_j}{r_{ij}}+A_{ij}\text{exp}\left(-{\alpha }_{ij}{r}_{ij}\right)+\frac{C_{6ij}}{r_0^6+r_{ij}^6},$$ (3)

where r_ij are atomic distances between atom i on the solute and atom j on water; q_i are partial charges on solute atoms, which are determined from charge products with BLYPSP-4F water in the center region of the QM part.²⁶ An exponential term is placed between water oxygen and every heavy atom of the solute to model short range repulsion. In addition, repulsion terms are also placed between each pair of atoms with opposite charges so that each water hydrogen has a repulsion term with every negatively charged atom on the solute and each water oxygen has a repulsion term with every positively charged solute atom. The last term of Eq. (3) is the short-range damped dispersion shown in Eq. (1).

After the intermolecular parameters have been determined, intramolecular parameters of the solute are fitted in a second fitting step, where the root mean square error (RMSE) of atomic forces is minimized. The intramolecular interactions are modeled with harmonic bond, harmonic angle, and cosine dihedral terms. In addition, any pair of atoms separated by more than two bonds will have short-range non-bonded interactions. No special scaling is used for 1–4 interactions.

The bonded terms can be summarized as

$$U^{bonded}=\sum \frac{k_b}{2}{\left(r-r_e\right)}^2+\sum \frac{k_{\theta }}{2}{\left(\theta {-\theta }_e\right)}^2+\sum V_D\left[1+\text{cos}\left(m\omega -\delta \right)\right],$$ (4)

where r, θ, and ω are the bond length, angle, and torsional angle, respectively. The first sum in Eq. (4) includes any pairs of atoms connected by a covalent bond and the second sum includes all angles formed by two covalent bonds sharing one atom. For the bond and angle terms, the equilibrium bond length r_e and equilibrium angle θ_e along with the harmonic constants, k_b and k_θ are the parameters to be fitted.

It can be shown that for any single bond connecting two sp³ hybridized carbons, only one torsional degree of freedom (DOF) exists.³⁰ Thus, only one torsional term will be used. The list of torsional terms used for each solute is summarized in Table S2 and can also be seen from Tables S3–S9. It is worth noting that more torsional terms than the number of DOFs are used for double bonds to achieve better planarity for double bonded atoms. For the torsional term, m is 3 and the equilibrium angle δ is fixed to zero if the two hinge atoms are connected by a single bond. For double bonds, the torsion term has a multiplicity m of 2 and a δ of 180°.

The intramolecular non-bonded interactions are modeled with the same formula as shown in Eq. (3) except that i and j are intramolecular atomic pairs separated by at least three covalent bonds. The dispersion parameters and partial charges have been determined by the SAPT fit and the intermolecular fit, respectively. Thus, only the short-range repulsion terms will be fitted in this step. Due to the limited flexibility of intramolecular atomic pairs, the atom pair for intramolecular short-range nonbonded interactions tends to visit only a narrow range of distances leading to additional challenges in fitting these parameters. We thus decided to fix α in the intramolecular repulsion term to 3.6 Å⁻¹. This value was determined previously by fitting the SAPT exchange repulsion between similar atoms.³⁰

III. COMPUTATION DETAILS FOR PROPERTY DETERMINATIONS

To validate each model, the HFE, enthalpy of hydration, radial distribution functions, diffusion constants, and vibrational spectra are computed. All MD simulations were performed with GROMACS version 2016.3.⁴²

For computing HFE, the Bennet acceptance ratio (BAR) method^43,44 as implemented in Gromacs was used. Although Gromacs is able to automatically account for gas phase contribution to HFE by setting the couple-intramol keyword to no, such implementation is not compatible with using different tabulated pair potentials for different intramolecular short-range non-bonded interactions. The HFE in this work was obtained by subtracting the alchemical free energy in the gas phase from the free energy in the solution phase with couple-intramol set to yes.

A total of 21 λ windows were used to alchemically decouple the solute in each phase. After the Coulombic interactions were decoupled in 11 λ windows, the short-range non-bonded interactions were decoupled in 10 λ windows. The soft-core potential⁴⁵ was used when decoupling short-range non-bonded interactions to avoid numerical instability when the solute and water overlap. We note that unlike the Lennard-Jones form, the core of our short-range non-bonded potential has no singularity. Thus, the computed HFE does not depend sensitively on the choice of soft-core parameters. In this work, the soft-core parameters α and σ were chosen to be 1 nm and 0.3 nm, respectively.^46,47

For the BAR simulations, each alchemical step was simulated in the NPT ensemble for 5 ns after a 0.5 ns equilibration. The equations of motion were integrated with a 0.5 fs time step with stochastic molecular dynamics. The Parrinello–Rahman barostat was employed with a relaxation constant of 5 ps for pressure control. Short-range non-bonded interactions were truncated at 10 Å. For 1,4-butanediol, the HFE was measured also with 20 ns alchemical windows. The result agrees with that obtained using 5 ns alchemical windows within 1 kJ/mol, suggesting that the 5 ns simulation time is adequate. The electrostatics were handled with the PME method.⁴⁸ Long-range corrections for van der Waals interactions are employed for both energy and pressure.

The GAFF⁴⁹ and OPLS-AA¹¹ HFEs were computed with the TIP4P water model with couple-intramol set to no. All other simulation parameters are identical to those used for the AFM models. For GAFF, it has been shown that the semi-empirical AM1-BCC charges could produce better agreement with experiments.⁵⁰ However, for these molecules, the restrained electrostatic potential (RESP) charges were used as it was found to provide better agreement with experiments when compared to the GAFF/AM1-BCC HFEs reported in the literature.

The heat of vaporization was computed at 298.15 K with the following formula:

$$\mathrm{\Delta }{H}_{solution}={⟨E⟩}_{sol}-\left({⟨E⟩}_{\mathit{\text{w}}ater}+{⟨E⟩}_{solute}\right)+P{⟨V⟩}_{sol}-P{⟨V⟩}_{\mathit{\text{w}}ater}-RT,$$ (5)

following the same procedure as described previously by Rogers and Wang.¹⁹ Each trajectory for computing the total energy, E, and volume, V, was simulated for 50 ns.

The diffusion constants were simulated with ten independent, 3 ns trajectories in the NVT ensemble. The temperature was maintained at 298 K with the Nose–Hoover thermostat with a relatively long 5 ps relaxation constant. The box contains one solute and 784 water molecules and was fixed at the average volume at 1 bar for each solution. The diffusion constants were obtained by fitting the block averaged mean squared displacements (MSDs) in the range from 20 ps to 40 ps using the GROMACS “msd” tool.

The power spectra were computed with the Fourier transform of the velocity autocorrelation function for the solute. Each spectrum is based on 1 ns of NVE simulation with one solute in a cubic box of 343 water molecules at an average temperature of 298 K. A relatively small 0.25 fs time step was used to ensure accuracy for the high frequency modes.

IV. RESULTS AND DISCUSSION

All the fitted parameters of the seven molecules, ethanol, 2-butanol, 2-methyl-1-propanol, cyclohexene, tetrahydropyran, 1,4-dioxane, and 1,4-butanediol, in dilute aqueous solutions are summarized in Tables S3–S9 of the supplementary material.

The HFEs of the seven solutes are reported in Table I. AFM lists the predictions based on the models developed in this work. While OPLS-AA/TIP4P and GAFF/RESP/TIP4P values were computed by us according to the description in Sec. III, the GAFF/AM1-BCC/TIP3P values were taken from Matos et al.⁵¹ and were computed with the GAFF model with AM1-BCC charges in TIP3P water. For some solute molecules, several experimental measurements were reported. The experimental numbers from different studies could differ by 4 kJ/mol or more, leading to difficulties for validating the performance of the force field models.

TABLE I.

HFEs of the seven solutes studied in this work. The HFEs of the AFM models, OPLS-AA/TIP4P, and GAFF/RESP/TIP4P were computed as described in Sec. III. The GAFF/AM1-BCC/TIP3P and experimental numbers were obtained from the literature. All values are in kJ/mol.

		OPLS-AA/	GAFF/RESP/	GAFF/AM1-BCC/
Solute	AFM	TIP4P	TIP4P	TIP3P50	Experiment
Ethanol	−19.65 ± 0.14	−20.00 ± 0.14	−19.27 ± 0.12	−14.18	−19.61,67 −20.02,4 −20.83,68
					−20.96,69 −21.21,70 −21.4471
2-butanol	−14.13 ± 0.12	−21.68 ± 0.14	−19.04 ± 0.19		−15.02,52 −19.32,3 −19.6169
2-methyl-1-propanol	−15.25 ± 0.21	−25.23 ± 0.10	−22.31 ± 0.22		−14.62,52 −15.62,53 −16.91,54 −18.8255
Cyclohexene	9.69 ± 0.29	6.06 ± 0.13	7.52 ± 0.16	4.94	6.8372
Tetrahydropyran	−12.48 ± 0.20	−5.62 ± 0.16	−6.35 ± 0.27	−7.57	−13 × 1072
1,4-dioxane	−22.20 ± 0.20	−10.82 ± 0.13	−18.71 ± 0.24	−17.87	−21.2572
1,4-butanediol	−28.71 ± 0.10	−49.59 ± 0.48	−48.45 ± 0.33		<−36.21,4 −41.53,3 −45.3273

For the AFM ethanol model, the HFE of −19.65 kJ/mol compares favorably to the −20.45 kJ/mol HFE obtained previously by Rogers also with MP2 gradients and AFM.¹⁹ The Rogers ethanol model used a slightly different placement of interaction sites that were optimized specifically for the molecule.¹⁹ In this work, the placements of the interaction sites follow the simple rule outlined in Sec. II that is universal for all solutes. The predicted HFE in this work compares favorably to the experimental values in the range of −19.61 kJ/mol to −21.44 kJ/mol. Both OPLS-AA and GAFF/RESP/TIP4P performed well for ethanol, whereas the GAFF/AM1-BCC/TIP3P value is too positive.

For 2-butanol and 2-methyl-1-propanol fitted also with MP2 reference forces, the AFM values of −14.13 kJ/mol and −15.25 kJ/mol compare favorably to the experimental reference from Butler et al. and Kim et al. The Butler estimate for 2-butanol is −15.02 kJ/mol,⁵² and the Kim estimate for 2-methyl-1-propanol is −15.62 kJ/mol.⁵³ The Butler estimate for 2-methyl-1-propanol is more positive by 1 kJ/mol. However, other experiments have reported more negative HFEs for these solutes, which are about 3 kJ/mol–5 kJ/mol lower than the AFM values.^54,55 Thus the performance of the AFM predictions is difficult to judge but is not worse than the uncertainty in different experimental measurements published to date. For 2-butanol, the OPLS-AA and GAFF/RESP/TIP4P potentials provide similar estimates and also in good agreement with experiments. For 2-methyl-1-propanol, both the OPLS-AA and GAFF/RESP/TIP4P predictions are outside the range of the experimental estimates by a small error of 4 kJ/mol–6 kJ/mol.

For cyclohexene, the HFE prediction based on AFM and LMP2 is 9.69 kJ/mol, which is more positive than the experimental reference of 6.83 kJ/mol by 3 kJ/mol. We note this is smaller than the chemical accuracy typically chosen to be 1 kcal/mol. Compared to the experimental reference, OPLS-AA and both GAFF HFEs are in good agreement.

For tetrahydropyran and 1,4-dioxane, the LMP2 based AFM force fields gave HFE predictions of −12.48 kJ/mol and −22.20 kJ/mol, respectively, comparing favorably with experimental values of −13.10 kJ/mol and −21.25 kJ/mol. For these two molecules, OPLS-AA is performing worse, underestimating experimental values by a factor of two in each case. Both GAFF based HFEs performed similarly to OPLS-AA for tetrahydropyran but gave better predictions for 1,4-dioxane.

For each molecule, if one considers a model that predicts HFE within 1 kcal/mol (∼4 kJ/mol) from any one of the experimental references to be a good prediction. For all six molecules, the AFM predictions are good. OPLS-AA gave good predictions for three of these, and GAFF/RESP/TIP4P gave good predictions for five of the six.

For 1,4-butanediol, however, the LMP2 based AFM prediction is −28.71 kJ/mol, quite far away from the closest experimental estimate of −36.21 kJ/mol or lower. Several other experimental estimates gave even lower HFE for 1,4-butanediol. The poor prediction for 1,4-butanediol is especially puzzling considering the good HFE for ethanol. 1,4-butanediol has a structure similar to two ethanol molecules linked at the methyl group, and its HFE is approximately twice that of ethanol. The ethanol force field was developed with MP2 as reference. MP2 gradient computations are costly both in central processing unit (CPU) time and disk space. 1,4-butanediol with the associated larger hydration shell makes MP2 force computations challenging even with density fitting. To judge the performance of LMP2 as the reference method, we decided to create a force field for ethanol with the LMP2 reference.

The development of the LMP2 ethanol model followed exactly the same procedure as the MP2 model, except with LMP2 gradients computed with the basis set used for the larger molecules as reference. The parameters for the LMP2 ethanol model are reported in Table S10. The HFE of the LMP2 based AFM model is −10.05 kJ/mol, which is more positive than the MP2 based value by about 9 kJ/mol. Since only the reference method was changed in our development of the LMP2 ethanol model, this result suggests a possible deficiency of the LMP2 as reference for alcohols. Figure 3 shows two scans of the ethanol–water potential energy surface close to the hydroxyl group of ethanol. In both scans, the LMP2 ethanol–water interaction curve is too shallow by 3 kJ/mol to 5 kJ/mol at the minimum and also exhibit a stronger short range repulsion. The stronger short range repulsion would lead to a larger hydration cavity to fit the solute. The creation of cavities in water is associated with a cavitation penalty leading to weaker hydration. We thus conclude the difference between LMP2 and MP2 contributes to the overly positive HFE for ethanol and 1,4-butanediol. If LMP2 underestimates the absolute HFE of each hydroxyl group by 9 kJ/mol, the 1,4-butanediol HFE would be underestimated by about 18 kJ/mol, which is very close to the observed difference from some of the more negative experimental HFEs. It is worth mentioning that there is more than one way to localize canonical orbitals in LMP2 computations. For example, Molpro also supports the pair natural orbital (PNO) based local MP2 approach. Another method for adding in missing correlation in LMP2 is through the explicitly correlated LMP2-F12 approach.⁵⁶ A more thorough study of other localization schemes is valuable and will be deferred to subsequent work.

FIG. 3.

Potential energy scans of the ethanol–water dimer around the ethanol hydroxyl group. The monomer geometries were fixed during the scans. The DF-LMP2 scans were performed with the aug-cc-pVTZ basis set for heavy atoms and cc-pVTZ for hydrogens, while the MP2 scans used the aug-cc-pVTZ basis set for all atoms. The interaction energy was computed by the difference between the dimer energy and the monomer energy.

Table II reports the enthalpies of hydration at 298.15 K for the seven solutes and the corresponding experimental values, where available. Unlike HFEs, where multiple experimental values are reported for some compounds, the experimental enthalpies of hydration are less abundant. We were only able to find one experimental enthalpy of hydration for these molecules except for 1,4-dioxane, and we could find no experimental values for tetrahydropyran. Thus, the prediction from the AFM model based on LMP2 cannot be validated for this molecule. The agreement for 1,4-butanediol is the worst, which is not surprising considering the model also predicts an overly positive HFE. For 2-butanol and 2-methyl-1-propanol, the AFM model underestimates the absolute enthalpy of hydration by almost 9 kJ/mol. However, considering that the several experimental HFEs for these molecules differ by 4 kJ/mol, part of the discrepancy may have been caused by experimental uncertainty. For all the other molecules, the AFM enthalpies of hydration are generally slightly more positive than the experimental values. However, the error is either comparable to or smaller than 1 kcal/mol. It is worth noting that quantum nuclear effects have not been considered in the simulations.^57–59 It is possible that proper consideration of the quantum nuclear effect could affect the simulated enthalpy of hydration. Thus, a definitive assessment on the performance of the models for the enthalpies of hydration is premature at this stage.

TABLE II.

Enthalpies of hydration of the solutes at 298 K for the AFM based models. The corresponding experimental values are listed where available. All values are in kJ/mol.

Solute	AFM	Experiment
Ethanol	−48.0 ± 0.6	−50.674
2-butanol	−53.9 ± 0.5	−62.774
2-methyl-1-propanol	−53.0 ± 0.7	−60.274
Cyclohexene	−22.2 ± 0.6	−27.374
Tetrahydropyran	−50.0 ± 0.5	…
1,4-dioxane	−42.5 ± 0.6	−48.4,74 −47.975
1,4-butanediol	−64.3 ± 0.7	−89.674

Table III reports the diffusion constants of the models and available experimental values for comparison. For cyclohexene, we borrowed the diffusion constant of cyclohexane as we are unable to locate the cyclohexene diffusion constant from the literature. It can be seen that the cyclohexene diffusion constant computed with the AFM model is in close agreement with the experimental value for cyclohexane. The diffusion constants predicted for the other six molecules are also in close agreement with experiments. Rogers has shown¹⁹ that for other small solutes, AFM based models gave very good diffusion constants when compared to experiments despite the omission of finite size corrections⁶⁰ and quantum nuclear corrections.⁵⁹ Our data thus suggest diffusion constants of small molecules can be quite reliably predicted with models developed by AFM when the AFM based BLYPSP-4F model is used for the solvent.

TABLE III.

Diffusion coefficients of seven aqueous solutes at infinite dilution at 298 K. Error bars are standard error of the mean. All values are in 10⁻⁵ cm²/s.

Solute	AFM	Experiment
Ethanol	1.15 ± 0.03	1.24,74 1.2276
2-butanol	0.90 ± 0.03	1.0276
2-methyl-1-propanol	0.87 ± 0.04	0.9576
Cyclohexene	0.79 ± 0.04	0.84a
Tetrahydropyran	0.82 ± 0.03	0.9777
1,4-dioxane	0.91 ± 0.03	1.1177
1,4-butanediol	0.98 ± 0.03	0.9178

^aThis is the diffusion coefficient of cyclohexane in water at 293 K.⁷⁴

Figure 4 reports the power spectra of each solute in dilute aqueous solutions. The computed spectra were scaled by 0.9598 and 1.012 for the high and low frequency regions, respectively, as proposed by Merrick for MP2/aug-cc-pVTZ.⁶¹ We note that the intensity of simulated spectra has no significance for our point charge based potentials, as such potentials do not have the correct form to capture the dipole derivative surface.^62,63 Focusing on peak locations, the spectra from the simulation are in fairly good agreement with experiments with occasional blueshift in the computed spectra. Two leading sources could be responsible for the blueshifts. One is the missing of higher order terms in the intramolecular energy expressions for bonded interactions, and the other is the neglect of quantum nuclear effects in classical molecular dynamics. Although the reference force calculation has anharmonicity, it will not be captured by the AFM potentials as the bond and angle terms used in this work are harmonic. Explicit treatment of nuclear quantum effects is known to redshift the simulated peaks for high frequency stretching modes.^64,65

FIG. 4.

Power spectra of the solutes in aqueous solutions. Peak locations were scaled by the scaling factors proposed by Merrick for MP2/aug-cc-pVTZ. Experimental spectra are shown in black lines. Peak intensities have been scaled arbitrarily to aid viewing.

The radial distribution functions (RDFs) for each solute are also computed. The results are only shown in the supplementary material since there are no available experimental references for validation.

V. SUMMARY AND CONCLUSION

The AFM method has been used to fit pairwise additive potentials for modeling the aqueous solutions of seven solutes. For molecules with up to five heavy atoms, the MP2 method was used to provide reference forces. LMP2 was used for larger molecules. With AFM, the MP2 or local MP2 computations were performed on QM/MM clusters designed to mimic a condensed phase environment. This procedure allows pairwise additive potentials to capture many-body effects implicitly. In addition, the conformations for the reference calculations were obtained with MD performed under the thermodynamic condition of interest.

Comparisons with experiments are complicated by the fact that a range of experimental values have been reported in the literature for some of the molecules. The HFEs predicted for ethanol, 2-butanol, 2-methyl-1-propanol, tetrahydropyran, and 1,4-dioxane are within 1 kJ/mol to the closest reported experimental value. For cyclohexene, the predicted HFE based on AFM and local MP2 is about 3 kJ/mol from the experimental reference, which is within the typically accepted chemical accuracy of 1 kcal/mol.

For 1,4-butanediol, the AFM based model seriously underestimates the HFE by 8 kJ/mol to 17 kJ/mol depending on which experimental reference to use. 1,4-butanediol was fit with LMP2 because of the high computational demand for modeling the solute along its hydration water with MP2. When LMP2 is used to model ethanol, the HFE is underestimated by about 9 kJ/mol. A potential energy surface scan shows that LMP2 underestimates the binding energy of ethanol–water dimer and is too repulsive at short range. It is thus highly likely that LMP2 predicts too weak hydration for alcohols when used as reference for AFM. We note that the localization algorithm in LMP2 is not unique. The deficiency only applies to the default pair atomic orbital (PAO) LMP2 as implemented in Molpro. The improved PNO-LMP2, especially with F12 correction, might perform much better.⁶⁶

The AFM developed models provide better agreement with experimental HFE when compared to OPLS-AA and previously reported GAFF/AM1-BCC/TIP3P values. The GAFF/RESP/TIP4P also performs fairly well for these molecules.

Once a force field is developed with AFM, it can be used to model other ensemble properties of the solute. For all solutes, where experimental reference is available, it was shown that our AFM based model provides good description for the enthalpy of hydration except for 1,4-butanediol. The AFM models also provide excellent agreement with experimental diffusion constants. With only harmonic bond and angle terms, the AFM vibrational peaks determined from a classical molecular dynamics simulation tend to be blue-shifted but overall in reasonable agreement with experiments.

A significant strength of AFM developed models is that they are completely based on electronic structure computations without fitting to any experiments. With sufficiently high-quality QM/MM calculations, AFM based models could potentially provide high quality descriptions of free energy and other ensemble properties of small molecules, although more work needs to be done to fully study the contribution from quantum nuclear effects for some properties of interest.

SUPPLEMENTARY MATERIAL

See the supplementary material for fragments used for SAPT calculations, the summary of the parameters of the seven solute molecules, LMP2 parameters for ethanol, and radial distribution functions for the seven solutes in dilute aqueous solutions.

DATA AVAILABILITY

The force field models developed in this work are available at http://wanglab.uark.edu/Models. Other data, such as results from the MP2 and LMP2 gradient calculations, are available from the corresponding author upon request.