Introduction

Computational predictions of thermodynamic properties of macroscopic systems are challenging. The size of such systems makes sufficient simulations using rigorous electronic structure methods impractical. Empirical force fields can be used, but their parameterization often relies on fitting to specific thermodynamic properties, which can lead to agreement through error cancellation rather than a true representation of the underlying physics. Without a true representation of the physics, the ability of such models to make predictions on properties that are difficult to measure experimentally is compromised. This limitation has motivated efforts to develop force field models based solely on electronic structure information1,2,3,4,5,6,7,8,9,10.

Hydrocarbons are a particularly challenging system for electronic structure-based force field development due to their low heat of vaporization, ΔΗvap. The density of hydrocarbons is strongly correlated with ΔΗvap. Consequently, a minor error of a few kJ/mol in ΔΗvap would lead to large percentage errors in density, and subsequently, many other thermodynamics properties. A good hydrocarbon model thus requires both highly accurate electronic structure references and a robust fitting protocol.

Given the importance of hydrocarbons in various industrial applications, such as solvents11, lubricants12, and refrigerants13, and their role as building blocks of phospholipids14,15, amino-acids, and pharmaceutical compounds16,17,18, developing a rigorous and accurate electronic structure based force field for such molecules is of great significance.

Recent successes using the adaptive force matching (AFM) method2,9,19,20,21,22 have led to highly effective models for hydrocarbons both as a neat liquid23 and as a solute in an aqueous environment24,25. Such models used typical point-charge based energy expressions along with a Buckingham potential for short-range non-bonded interactions.

As hydrocarbons are non-polar, it is interesting to explore if force field models for such non-polar systems can be developed without partial charges. The commonly accepted understanding is that hydrocarbons are primarily bound by dispersion interactions. If this is indeed true, not modeling partial charges is not expected to lead to a deficiency in the model’s ability to predict thermodynamic properties.

To confirm this hypothesis, we will attempt to develop hydrocarbon models using the AFM method without partial charges on hydrogen and carbon atoms. As AFM only fits forces computed by electronic structure methods, such models are not likely to reproduce experimental properties solely through error cancellations.

The Coulombic interaction is long range. Such long-range interactions are more challenging to model and lead to increased computational cost. Although the particle mesh Ewald (PME) method26 offers significant improvements in efficiency over traditional Ewald summation, it remains the dominant computational bottleneck for force evaluations in large systems. A model without using partial charges will eliminate the need for the costly computation of Coulombic interactions. In addition, such a model will allow the use of larger timestep sizes27, further improving simulation efficiency. These efficiency advancements are particularly crucial for simulating complex systems characterized by slow dynamics, like the frigid organic oceans found on distant worlds, including Saturn’s moon Titan28.

From a fitting perspective, while the overall electrostatic interaction is expected to be weak, it is achieved through a cancellation between two strong terms. In the case of hydrocarbons, large attractive Coulombic interactions between hydrogen and carbon cancel strong Coulombic repulsion between like atoms almost entirely. When the carbon atoms are completely surrounded by hydrogens, increasing simultaneously the magnitude of both the positive and negative charges while maintaining neutrality has little effect on the electrostatic potential outside the molecule. Fitting partial charges involving such buried atoms is a known challenge29,30,31.  A model without partial charges, if indeed sufficient, will eliminate such issues in fitting.

Although early studies investigated all-atom hydrocarbon models without partial charges32,33, to the best of our knowledge, all prevailing all-atom force fields uniformly incorporate partial charges to describe these molecules34,35,36,37,38,39,40. The united atom potentials41,42,43,44,45,46,47 have shown success for these systems48,49, which essentially suggest it is possible to reproduce properties of hydrocarbons without modeling partial charges. However, the development of such models all involve fitting directly to thermodynamic properties. Even if not modeling electrostatics could lead to missing physics, direct fitting could cause such problems to be masked.

This study aims to establish first-principles-based force fields for hydrocarbons without using partial charges in two distinct settings: dissolved in water as a solute, and as a pure liquid. Liquid water is a polar medium with a high dielectric constant. If a non-polar molecule has a quadrupole moment, it will interact with the dipole of water, whereas the leading electrostatic term in a neat liquid will be quadrupole-quadrupole interactions. In addition, water could induce dipoles on a solute molecule. Although a force field with fixed partial charges will not be able to explicitly model such induction effects, having partial charges could potentially give the model a better chance of capturing these effects implicitly.

Since creating models without partial charges is more challenging in a polar solvent, we will first discuss fitting of such models for hydrocarbons in aqueous solutions. In particular, models for methane, ethane, cyclopentane, and cyclohexene will be studied in this work. While methane, ethane, and cyclopentane have no dipole moment at their equilibrium conformations, cyclohexene does have a small nonvanishing dipole moment of 0.331 D as measured by a Stark effect spectrometer in the gas phase50.

Hydrated models of hydrocarbons with partial charges

AFM based all atom force fields for hydrocarbons have been developed with partial charges. In particular, AFM based models for methane, ethane and cyclohexene have been published24,25. The performance of such models with partial charges is the most appropriate metric to judge the performance of the corresponding partial-charge free models of primary interest to this study.

The previously published models for hydrated methane and ethane were fit against MP2 reference forces25. We decided not to refit such models. We will fit models for cyclopentane and cyclohexene with partial charges using B3LYP-D3(BJ)51,52,53,54 as the reference method to offer a basis for comparison to the corresponding partial charge free model. B3LYP-D3(BJ) has been shown to provide good accuracy for predicting thermodynamic properties of a range of drug-like molecules in recent studies21.

Water will be modeled with the BLYPSP-4F potential55 developed previously by fitting coupled cluster quality forces with AFM56. The detailed fitting procedure is described in the Methods section, and the parameters and Gromacs input files of the final models are provided in Supplementary Information. We computed HFE, ΔΗhyd, and diffusion constant, D at 298 K and reported them in Table 1 along with related experimental references.

Table 1 Calculated properties for the hydrated hydrocarbon models with partial charges.
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It is clear from Table 1 that the MP2 based methane and ethane models give excellent agreement for all the properties computed. The experimental values from different measurements deviate from each other. For example, the largest58 and smallest60 ΔΗhyd of methane differ by more than 2 kJ/mol. When compared to the closest experimental value, the MP2 based prediction agrees within 0.7 kJ/mol for these molecules. The models also give excellent agreement regarding D. The two experimental D of ethane62,63, differ by 0.3 × 10−5 cm2/s. In this case, the MP2 predicted D is close to the mean of the two experimental values.

The cyclopentane and cyclohexene models were created with the B3LYP-D3(BJ) reference. A slightly larger error is thus expected when compared to those using MP2 reference. For cyclopentane, the HFE is roughly 5 kJ/mol too high, while the ΔΗhyd is 3 kJ/mol too high. This is on par with the commonly accepted chemical accuracy of 1 kcal/mol. For cyclohexene, the agreement for HFE is within 2 kJ/mol, and the experimental enthalpy of hydration agrees within the error bar. It is worth noting that the experimental literature on cyclohexene hydration can be ambiguous. The term “hydration” in this context could refer to the addition of water to an alkene, resulting in an alcohol67. Therefore, careful interpretation is required when evaluating published data on alkene hydration.

Hydrated models of hydrocarbons without partial charges

The models without partial charges were generated using the same set of reference forces used to fit the model with partial charges. Also, the dispersion parameters were reused without refitting. The placement of the exponential repulsion terms remains the same, however, all repulsion parameters were refit without partial charges on any solute atom. Other details of the fitting procedure are provided in the Methods section. Parameters for these models can be found in Supplemental Information along with the corresponding Gromacs input files. The HFE, ΔΗhyd, and D calculated for these models are reported in Table 2 along with the properties of the corresponding models with partial charges.

Table 2 Properties calculated for hydrated models with and without the use of partial charges.
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For most cases, there is a minimal change in properties between models with and without partial charges. The difference in HFE is less than 0.4 kJ/mol for all models except for cyclohexene, and the difference in ΔΗhyd is no more than 2 kJ/mol. The D is also in close agreement.

The HFE of cyclohexene, being the only property that shows a fairly large deviation, is the most interesting. The atom typing of cyclohexene is shown in Fig. 1. Without partial charges, the exponential term between C1 and the water hydrogen, Hw, is fitted to be attractive. While attractive exponential terms are occasionally encountered with AFM, they are typically only observed for interior atoms of a molecule. In such cases, direct steric interaction is not possible, thus fitting such repulsion would lead to numerical issues. In such cases, the standard practice of AFM is to remove attractive exponential terms and refit. The C1 of cyclohexene is the alkene carbon and is accessible to water in the direction normal to the double bond. In the model with partial charges, the C1 atom has a negative charge of -0.334 e, whereas Hw is positively charged. The attractive exponential term is likely a result of the fitting algorithm trying to model the Coulombic attraction. When the attractive exponential term is retained and allowed to fully optimize, the α term becomes 1.584 Å−1, which suggests a very slow decay, consistent with the energy expression trying to capture a long-range Coulombic attraction. The cyclohexene model reported in Table 2 has this attractive exponential term removed according to the standard protocol of AFM.

Fig. 1
figure 1

Atom types used for fitting (a) Methane, (b) Ethane, (c) Cyclopentane, (d) Cyclohexene.

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From a model fitting perspective, it is interesting to examine if it is indeed possible to approximate a Coulombic term with an exponential term. Table 3 summarizes the α parameter and properties of several models fitted for cyclohexene along with the experimental references. The same dispersion parameters and training set is used to fit all these models with the C1-Hw exponential term discarded, fitted, or empirically adjusted to match the experimental HFE. The model with point charges is the same model reported in Table 1. The C1 - Hw exponential term is fitted to be repulsive when partial charges are present.

Table 3 Properties of various cyclohexene models with different α and electrostatics.
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While the model with partial charges produces the best agreement with experiments, the model fitted with the attractive exponential term discarded gives a reasonable ΔΗhyd but overestimates HFE by 4 kJ/mol. Retaining this attractive exponential term with the α parameter fitted leads to a model with a negative HFE and a ΔΗhyd, that is far too low. This outcome suggests that fitting only to forces without a sensible energy expression will lead to a model with poor performance.

Empirically adjusting α to match the experimental HFE results in an overly negative ΔΗhyd, exhibiting characteristic symptoms of overfitting, where the optimized property comes at the expense of poor agreement of other relevant properties.

It is also interesting to note that regardless of how poor the HFE and the ΔΗhyd are, all models in Table 3 give similar D. Due to the relatively long lifetime of hydrogen bonds, the diffusion of a solute in an aqueous environment is strongly affected by the water model. Despite the difference in energy expressions, all cyclohexene models have roughly similar size. This can be seen from the cyclohexene water radial distribution functions (RDF)s shown in the Supplementary Information. As a result of the weak interaction between hydrophobes and water, we anticipate that for such molecules, the diffusion dynamics are not affected much by the specific solute-water interactions but mostly by the size of the molecule.

Cyclohexene only has a small 0.331 D50 dipole moment. Even for such a molecule, our results suggest that the explicit modeling of partial charges becomes important with the mere presence of a double bond.

Neat models of hydrocarbons without partial charges

For saturated hydrocarbons in an aqueous solution, our results indicate that a high-quality model can be obtained without modeling electrostatics explicitly. We anticipate explicit modeling of electrostatics with point charges would be even less important for neat liquids, as dispersion is expected to dominate the cohesive energy of such systems. As the accuracy of DFT for dispersion bound compounds is not well established, we will fit models for the four nonpolar molecules using MP2 as reference with dispersion derived using SAPT68. SAPT is a double perturbation theory; E2 dispersion computed with SAPT is equivalent to the dispersion at the MP4 level68.

We are developing models exclusively without partial charges. The success of our recent partial-charge-free hydrated models suggests that methane, ethane, and cyclopentane are suitable candidates for this approach. Furthermore, our results show good agreement with experiments also for neat cyclohexene using a partial-charge-free model. Therefore, we have decided to focus our efforts and not pursue both types of models.

Models for neat hydrocarbons have been developed previously with AFM using partial charges23. The dispersion was fitted to SAPT using the def2-TZVPD basis set while the reference forces were obtained with the def2-TZVP basis set23. The fitting procedure for the models created in this work is similar to that used previously, and is described in the Methods section.

Density, ΔΗvap, and D for these four molecules were computed and reported in Table 4. The parameters for these models are reported in Supplementary Information along with Gromacs input files for the best performing models.

Table 4 Properties calculated for hydrocarbon models fitted without using partial charges.
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As shown in Table 4, for ethane, cyclopentane, and cyclohexene, the properties of the MP2 based models fitted without partial charges are in excellent agreement with experiments. Notably, the predicted densities deviate by less than 4%, while ΔΗvap are accurate to within 1.5 kJ/mol. The D for these hydrocarbons is correlated with their ΔΗvap, showing faster diffusion for species with weaker binding. The D from the models are in good agreement with experiments in the cases of ethane and cyclopentane. No experimental reference for cyclohexene can be identified. However, we are confident the 2.49 × 10−5 cm2/s estimate is reliable, given the excellent track record of AFM based models in predicting this property.

For methane with the def2-TZVPD basis set, the ΔΗvap is underestimated by only 2 kJ/mol, which is not much worse in an absolute sense when compared to ethane, where the deviation is about 1.5 kJ/mol. However, 2 kJ/mol is 25% of the ΔΗvap of this molecule. It is thus not surprising even such a small deviation leads to a 10% underestimation of the density.

For a molecule that binds so weakly, highly accurate electronic structure calculations are required. While it is costly to resort to a method that account for higher order electron correlation, we explored the effect of using a larger basis set. With the def2-QZVPD basis set, the agreement in both the ΔΗvap and density improves. Using the aug-cc-pVQZ basis set72,73, which has more basis functions than def2-QZVPD, the error in ΔΗvap reduces to 1.3 kJ/mol, whereas the density agrees to roughly 4%, showing a clear improvement as the basis set becomes larger.

We note that aug-cc-pVQZ and def2-QZVPD are both quadruple zeta quality basis sets with def2-QZVPD being optimized to reduce basis set superposition error74. No counter-poise correction75 is used when computing the MP2 forces for AFM. In principle, there is no reason to believe the aug-cc-pVQZ basis set will produce a more accurate dimer binding energy. Table 5 summarizes dimer binding energies computed with the aug-cc-pVQZ basis set at both MP2 and CCSD(T) level and with a complete basis set (CBS) limit estimate obtained with the explicit F12 method76,77. As shown in Table 5, inclusion of higher order electron correlation increases the aug-cc-pVQZ binding energy by about 0.11 kJ/mol. At the same time, the F12 energy is lower than the aug-cc-pVQZ energy by approximately the same amount. This cancellation causes the CBS dimer binding energy at the CCSD(T) level to be in very good agreement with the MP2 binding energy computed with the aug-cc-pVQZ basis set. Thus, the aug-cc-pVQZ basis set might accidentally provide the most accurate dimer surface through the cancellation of a small underbinding due to missing higher order correlation and a minor overbinding due to the incomplete basis set. We note that this study was performed only for one conformation, and the observed energy difference is very small. However, the trend of increasing correlation with higher levels of theory leading to increased binding energy, and the slight overbinding of aug-cc-pVQZ due to BSSE, is consistent with typical expectations for this type of systems.

Table 5 Binding energy of methane dimer.
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We also note that the liquid methane simulations were performed at the experimental boiling temperature of 111 K78. At such a low temperature, quantum nuclear effects79 are expected to play a larger role. Three-body dispersion80,81,82, which is missing from the MP2 based reference data, could also be important. These errors could also contribute to the slightly poorer agreement for the challenging system of liquid methane.

It is also worth mentioning that since methane liquid is a weakly coupled system, intramolecular vibrations are long lived with slow vibrational energy relaxation83. This leads to complications when simulating methane. If care is not taken, a non-equilibrium state could persist for a fairly long duration, with the temperature for the intermolecular degrees of freedom differing from the intramolecular vibrational temperature. In practice, the Langevin thermostat84 is used to ensure proper equilibration of methane for density and ΔΗvap calculations, as described in the Methods section.

Conclusion

This work demonstrates the development of accurate force field models for non-polar molecules, such as hydrocarbons, without relying on partial charges, using the AFM method. When compared to traditional empirical models, AFM models reproduce observed behavior without fitting to experiments, thus ensuring that the underlying physics is properly captured. Fitting models without partial charges improves numerical stability during parameterization. The resulting models are also more efficient, as the computational cost associated with long-range electrostatic interactions is eliminated. AFM models do not use any manual adjustment; they are machine-learned models that utilize physics-based energy expressions rather than neural networks.

Although models for saturated alkanes show good accuracy without using partial charges, in the presence of water, which is a polar solvent, cyclohexene, with only a single double bond, necessitates explicit modeling of electrostatic interactions. Although it is possible to fit an attractive exponential term to reproduce the experimental HFE of cyclohexene directly, such a practice leads to an overfit model that poorly predicted other properties, such as ΔΗhyd.

The determination of partial charges on buried atoms of nonpolar molecules poses numerical challenges due to the near cancelation of electrostatic potentials from oppositely charged atoms. In contrast, models without partial charges are expected to improve fit stability and enable faster simulations of hydrocarbon liquids or their mixtures. Our findings suggest that for modeling liquid non-polar hydrocarbons, a model without partial charges is sufficient. However, when considering aqueous solutions, it is only advisable to model saturated hydrocarbons without partial charges on the solute.

For liquid hydrocarbons, our results show that MP2 with SAPT dispersion provides sufficient accuracy to predict properties such as ΔΗvap, density, and D when fitted to a force field using AFM. For methane, the performance is found to be highly sensitive to the basis set, and the aug-cc-pVQZ-based model yields the best agreement.

Under aqueous conditions, both the B3LYP-D3(BJ) and the MP2-based models exhibit excellent agreement with experiments. Although the B3LYP-D3(BJ) model for cyclopentane showed relatively large deviation, it is still on par with the commonly accepted chemical accuracy of 1 kcal/mol. Furthermore, different cyclohexene models developed using AFM show a minimal dependence of D on the details of hydrocarbon-water interactions, suggesting that cavity size is the primary factor influencing the diffusivity of hydrophobes.

Methods

Software

Gromacs 2019.6 was used for all MD simulations85. Orca 5.0.3 was used for calculation of the QM reference forces and methane dimer binding energies86. Psi4 1.8.2 was used for SAPT calculations87, and Grimme’s D3 program is used to compute the D3(BJ) empirical dispersion contribution53,88. For fitting reference forces to force field parameters, we used CRYOFF v4.2.1, a software developed in house.

Force field development protocol

All force fields were developed with the AFM method. The parameters for the final force fields are reported in the Supplementary Information. The Gromacs input files for the best performing models are also supplied in the Supplementary Information.

In the AFM procedure, the dispersion parameters are determined prior to the iterative determination of other force field parameters. The AFM iterations consist of three steps: the sampling step, the quantum mechanics/molecular mechanics (QM/MM) step, and the fitting step. The dispersion determination step and the three AFM steps for the development of each model are summarized below.

Models for hydrated hydrocarbons

The hydrated force fields for methane and ethane with partial charges were taken from a previous publication25 without refitting. The methane-water and ethane-water dispersion parameters were taken from the models with partial charges25 when developing the hydrated methane and ethane models without partial charges. Dispersion parameters were only fitted for cyclopentane and cyclohexene.

Dispersion

Dispersion was modeled using the short-range-damped form22

$${E_{SRD}}\left( {{r_{ij}}} \right)=\mathop \sum \limits_{n} \frac{{{C_n}}}{{r_{{ij}}^{n}+r_{0}^{n}}}$$
(1)

where \(\:{r}_{0}\) was defined as

$${r_0}=0.6~R_{{ij}}^{{vdw}}$$
(2)

with \(\:{R}_{ij}^{vdw}\) being the sum of van der Waals radius for atoms i and j. The van der Waals radius of each atom was taken from the work of Tkatchenko89. For hydrated hydrocarbons, the sum in Eq. 1 only contained only the term with n being 6.

Since the cyclopentane and cyclohexene models were fit against B3LYP-D3(BJ) as reference, the dispersion parameters were fitted against the corresponding Grimme’s D3 empirical dispersion with Beck-Johnson damping53,54. The training set for the dispersion fit was water-solute dimers extracted from 1.1 ns of MD with 1 solute in 1728 water at 298 K and 1 bar. During the MD simulations, the solute-water interaction was modeled to be identical to that of a OPLS-AA90,91,92 solute in TIP4P93 water and the water-water interaction was modeled with the BLYPSP-4 F force field. The training set contained 560 solute-water dimers. Out of the 560 dimers, 280 conformations (uniform set) were taken from the MD trajectory following a uniform distribution with a nearest atom distance ranging from 5 to 12 Å. Another 280 conformations (random set) were taken from the same distance range but without enforcing a uniform distribution in distance.

Sampling step

The initial guess force field for the solute is OPLS-AA. The sampling for each generation was performed with one solute in 1000 water molecules under NPT condition at 1 bar. Two 1.1 ns trajectory were run, with one trajectory at 298 K and another trajectory at 328 K. 120 conformations were taken from the final 960 ps of each trajectory with one conformation taken every 8 ps. The training set for each generation thus contains 240 conformations.

QM/MM step

The AFM QM region has a fitting zone and a buffer zone; only the forces on the fitting zone atoms are used for force matching. From each configuration, the following protocol was used to generate the QM/MM region:

  1. 1)

    The solute was chosen as the center of the fitting zone. Water molecules within 3 Å of the solute, measuring by nearest atom distance, were classified as first hydration shell waters.

  2. 2)

    Five first hydration shell waters were randomly chosen to be part of the fitting zone. The remaining first hydration shell waters and any other molecule within 2.6 Å of any fitting zone water was classified as buffer zone molecules.

  3. 3)

    All waters within 9 Å of the solute that was not part of the fitting zone nor buffer zone were retained as MM particles, with all molecules further away discarded.

While the MM atoms were modeled as point charges for Coulombic embedding, the QM region was modeled with using B3LYP-D3(BJ) with the aug-cc-pVTZ basis set on heavy atoms but the cc-PVTZ basis set on hydrogens.

FM step

For models with partial charges, each unique atom type (Fig. 1) was assigned a partial charge and interacted with all other charged atoms separated by more than two covalent bonds according to the Coulombic interactions. Note that the same atom type of different molecules uses different parameters.

The repulsion is modeled using an exponential repulsion term,

$${E_{rep}}\left( {{r_{ij}}} \right)={A_{ij}}{e^{ - \alpha {r_{ij}}}}$$
(3)

For hydrated cyclopentane and cyclohexene, repulsion is placed between heavy atoms and atoms with opposite charges.

Intramolecular Coulombic and repulsion is included for all 1–4 pairs and beyond. In addition, harmonic bond and harmonic angle terms are included for all bonds and angles. All angles formed by two bonds sharing one atom are included. The torsional terms are modeled using the cosine function with the form,

$${E_{dih}}\left( \omega \right)={V_D}\left[ {1+\cos \left( {m*\theta - \delta } \right)} \right],$$
(4)

where \(\:m\) = 2 and \(\:\delta\:\) = 180° for double bonds and \(\:m\) = 3 and \(\:\delta\:\) = 0° for single bonds.

A total of 8 generations for each solute were generated, with the last 4 generations being used to produce the final model.

We note that although both cyclopentane and cyclohexene models were fitted, the cyclohexene model did not go through a complete AFM iteration since a prior published training set was already available. Thus for cyclohexene, only B3LYP-D3(BJ) reference forces were recalculated using the conformations from the prior publication24.

Models without partial charges

The models without partial charges were created by refitting the training set for the model with partial charges. The same parameters for dispersion are used. The only difference is thus the atomic partial charge of each solute atom is forced to be zero.

Development of models for pure hydrocarbons in their liquid state

Dispersion

Since MP2 was used as the reference method for AFM, we fit E2 dispersion computed with SAPT on dimers of the molecules. The basis set for the SAPT calculation is def2-TZVPD, except for methane, where the same basis set as shown in Table 4 is used. The dimers were selected from a liquid phase simulation performed at 298 K and 1 bar for cyclopentane and cyclohexene. For methane and ethane, NVT simulations were performed at experimental boiling point and density. All simulations for sampling such dimers were performed using cubic boxes of 1728 molecules with the OPLS-AA force field.

Similar to the solute-water training set, the hydrocarbon dimers also contain a uniform set and a random set. Both were selected from the 5 to 12 Å distance range. 280 dimer conformations are in each of the two sets. As the SAPT dispersion depends weakly on the molecular geometry, before the SAPT calculation, each dimer is optimized for up to 10 steps with restraints to ensure the intermolecular distance and orientation do not change.

Since the cohesive energy of neat hydrocarbons is dominated by dispersion, we include both the n = 6 and n = 8 terms in Eq. 1 when modeling dispersion. L2 regularization is used for the more flexible molecules, cyclopentane and cyclohexene. The λ for the L2 regulation is chosen to be (0.5ssv)2where ssv is the smallest singular value of the singular value decomposition (SVD). In the case of cyclohexene, for some pairs of atoms, we were unable to obtain attractive dispersion for both the n = 6 and n = 8 term. In that case, only the n = 6 will be used for those atom pairs.

Sampling step

For all molecules, the initial guess force field was OPLS-AA. For each generation, the sampling was performed with MD for 1.1 ns with a cubic box containing 343 molecules. The sampling temperature was 100 K for methane, 180 K for ethane and 298 K for cyclopentane and cyclohexene. In addition, a second sampling at 30 K higher was performed for each molecule. For cyclohexene, the MD for both temperatures were performed at 1 bar. For methane and ethane, the samplings were performed under NVT condition using the experimental liquid density94,95 at the boiling temperature. The higher temperature sampling for cyclopentane was performed at the experimental density at 298 K96 as it is above the experimental boiling temperature.

Similar to the hydrated molecules, 120 conformations separated by 8 ps are selected from each MD trajectory, giving a training set of 240 conformations for each molecule at each generation.

The QM step

Since the hydrocarbons are not polar, the MP2 calculations were performed without a MM region. Consequently, there is no need to separate the QM region into a fitting zone and a buffer zone, and all QM forces can be fitted in the FM step.

The following protocol is used to select the QM region:

  1. 1)

    One random molecule was chosen to be the center of the QM region.

  2. 2)

    Five molecules within 3 Å of the first molecule, measuring by nearest atom distance, were chosen to be in the QM region.

All other molecules are discarded. The MP2 calculations were performed with the def2-TZVP basis set unless otherwise mentioned. (See Table 4)

The FM step

The hydrocarbon liquid models use similar energy expressions as the hydrated models except that both 1/r6 and 1/r8 terms are included for dispersion. Exponential repulsion is placed between all pairs of atoms as done with previous work of Nikitin and Wang23. Eight iterations of AFM were run for each molecule with the last 5 generations being used to produce the final force field.

Binding energy calculations

The methane dimer binding energies reported in Table 5 were calculated with density fitting(DF) MP2 and DF-CCSD(T). The density fitting was performed with the corresponding optimized auxiliary basis set where available and the autogenerated (autoaux) auxiliary function of ORCA was used otherwise. The monomer and dimer geometries were optimized with at the MP2 level using the aug-cc-pVQZ basis set. The complete basis set estimate was obtained using the aug-cc-pVQZ basis set with the corresponding geminal F12 basis functions76 with the explicit F12 method.

Property calculations

Simulation details

Unless otherwise discussed, the Parrinello-Rahman barostat97,98 with a relaxation constant of 5 ps is used for pressure control and the Nose-Hoover thermostat99,100 with a relaxation constant of 2 ps is used for temperature regulation. All simulations were performed with a 0.5 fs timestep. For models with partial charges, the Particle Mesh Ewald method26 is used to treat long range electrostatics. The water model for hydrated molecule simulations is BLYPSP-4 F.

The cutoff radius for the van der Waals interaction is 0.9 nm for hydrated molecule simulations and 1.3 nm for simulations of neat hydrocarbon liquid. The use of the larger cutoff is beneficial due to dispersion being the dominating interaction for the latter. Long range corrections to energy and stress were applied for all cases. As mentioned in a prior publication by Nikitin and Wang23, the long-range correction due to 1/r8 is not implemented in Gromacs and had to be applied manually.

Hydration free energy (HFE)

HFE was calculated using the Bennett acceptance ratio (BAR) method101,102 in a cubic box containing the solute and 343 water molecules under NPT conditions. For models with partial charges, the Coulombic interactions were removed in 10 λ steps. The short-range nonbonded interactions were removed in 25 λ steps. For the removal of short-range non-bonded interactions, the soft-core potential as implemented in Gromacs103 was used with the sc-alpha parameter set to 1 and the soft-core radius set to 0.265 nm. For hydrated cyclohexene and cyclopentane models, the couple-intramol parameters were set to ‘yes’ with the BAR simulations performed both in the aqueous solution and in the gas phase. The gas phase simulation was performed under NVT conditions with only the solute molecule in a box the same size as the average box size of the aqueous state simulation.

Enthalpy of hydration (ΔΗ hyd)

ΔΗhyd was calculated with the finite difference method,

$$\Delta {H_{hyd}}={U_{solution}} - \left( {{{\left\langle U \right\rangle }_{water}}+{{\left\langle U \right\rangle }_{solute}}} \right)+P{V_{solution}} - P{V_{water}} - RT,$$
(5)

where \(\left\langle U \right\rangle\) is the mean potential energy measured using 200 ns of MD. While the potential energy of the solution and water was measured under NPT conditions in a cubic box containing 266 water molecules, the solute potential energy was measured from an NVT simulation with a single solute in the gas phase using the Langevin thermostat with inverse friction coefficient of 2 ps−1.

Density and ΔΗ vap for hydrocarbon liquid

All density and ΔΗvap simulations were performed using 343 molecule cubic boxes under NPT conditions. The simulation conditions are summarized in Table 6 to match that of the experimental data. The density was averaged over the final 20 ns of a 25 ns trajectory. The ΔΗvap were computed as

$$\Delta {H_{vap}}={\left\langle U \right\rangle _{gas}} - {\left\langle U \right\rangle _{liq}}+RT$$
(6)

where the liquid phase mean potential energy \({\left\langle U \right\rangle _{liq}}\) was also measured over the final 20 ns of a 25 ns NPT trajectory. The gas phase mean potential energy was computed with 100 ns MD under NVT conditions using the Langevin thermostat with an inverse friction coefficient of 2 ps−1.

Table 6 Simulation conditions for hydrocarbon property calculations.
Full size table

While the NPT simulations for density and ΔΗvap uses the Parrinello-Rahman97,98 barostat and the Nose-Hoover thermostat99,100 for most molecules, such a protocol does not work well for methane, as the weak interaction between methane molecules leads to equilibration issues. The high vibrational energy of the molecule will take a very long time to equilibrate with the translational energy. Thus for methane, the Berendsen barostat104 with a relaxation of 1 ps and the Langevin thermostat with an inverse friction coefficient of 2 ps−1 was used.

Diffusion constant (D)

The D for liquid hydrocarbon was determined from a 2 ns MD using a 500 molecule box. For a hydrocarbon in an aqueous solution, a much longer simulation length is required as there is only one solute molecule to measure mean square displacement. (MSD) Thus, the solute D was averaged over 26 ns. All MSD were measured under NVT conditions, with temperature controlled using a Nose-Hoover thermostat with a 5 ps relaxation time. The box size for the NVT simulation were determined from a NPT simulation at the target temperature and pressure by averaging the final 4 ns of a 5 ns trajectory. The D was determined using the Einstein relationship105 by fitting the MSD from 5 to 15 ps.