AMOEBA+ Classical Potential for Modeling Molecular Interactions

Abstract

Classical potentials based on isotropic and additive atomic charges have been widely used to model molecules in computers for the past few decades. The crude approximations in the underlying physics are hindering both their accuracy and transferability across chemical and physical environments. Here we present a new classical potential, AMOEBA+, to capture essential intermolecular forces, including permanent electrostatics, repulsion, dispersion, many-body polarization, short-range charge penetration and charge transfer, by extending the polarizable multipole-based AMOEBA (Atomic Multipole Optimized Energetics for Biomolecular Applications) model. For a set of common organic molecules, we show that AMOEBA+ with general parameters can reproduce both quantum mechanical interactions and energy decompositions according to the Symmetry-Adapted Perturbation Theory (SAPT). Additionally, a new water model developed based on the AMOEBA+ framework captures various liquid phase properties in molecular dynamics simulations while remains consistent with SAPT energy decompositions, utilizing both ab initio data and experimental liquid properties. Our results demonstrate that it is possible to improve the physical basis of classical force fields to advance their accuracy and general applicability.

Graphical Abstract

I. Introduction

Classical molecular dynamics (MD) simulations are widely used to study the physical properties of chemical and biological systems. One of the essential ingredients in the MD simulations is the accuracy of the underlying classical potentials, or force fields (FFs), which include the functional forms that describe the intra- and intermolecular potential energy surface (PES) and the parameters associated with different chemistry. The traditional FFs, such as AMBER (Assisted Model Building with Energy Refinement),^1–4 CHARMM (Chemistry at HARvard Macromolecular Mechanics),^5–7 GROMOS (Groningen Molecular Simulation),^8–9 and OPLS (Optimized Potential for Liquid Simulations),^10–12 employ fixed atomic charges to model the electrostatic interactions via pairwise additive Coulombic interactions. While they have been widely used in complex systems due to their efficiency in simulations, there has been increasing effort to improve the underlying physics, particularly the many-body polarization that can vary significantly depending on chemical and physical environments.^13–15 For example, it is well known that a water molecule in isolation has a dipole moment of 1.85 Debye which increases by about 50% to 2.4 ~ 3.0 Debye in the liquid phase due to polarization at room temperature.^16–18 Mainly three approaches have been adopted to explicitly capture many-body polarization in FFs, including the Drude oscillators,^19–21 fluctuating charges^{6, 22} and induced dipole schemes.^{13, 16, 23–24} Polarizable AMBER,²⁵ OPLS,²⁶ GROMOS,²⁷ CHARMM,²⁸ NEMO (Non-empirical Molecular Orbital),^29–30 and PFF (Polarizable Force Field for proteins)³¹ employ one of the three approaches. Another area of improvement focuses on the deficiency of spherical atomic charge representation of permanent electrostatics. Atomic multipole expansion was adopted to capture the anisotropic electrostatic potentials around atoms, examples include AMOEBA,¹⁶ NEMO,³² EFP (Effective Fragment Potential),^33–34 and SIBFA (Sum of Interactions Between Fragments Ab initio computed).^35–37 More sophisticated approaches are used by Gaussian-based GEM (Gaussian Electrostatic Model)^38–39 and fragment-based electronic structure method X-Pol.^{24, 40} We have been developing the AMOEBA potential that employs atomic multipole expansion up to quadrupoles to represent the permanent charge distributions and inducible atomic dipoles to account for the polarization response.^{16, 41} It has been applied to water,^{16, 42} ions,^43–45 small organic molecules²³ and complex proteins⁴⁶ and nucleic acids,⁴¹ with extensive applications to protein-ligand and ion binding.^47–52

While computationally expensive, ab initio quantum mechanical (QM) energy decomposition analysis (EDA) such as SAPT⁵³ provide detailed components of interaction energy including electrostatics, induction, exchange-repulsion, and dispersion. However, classical force fields have not systematically utilized such information due to the intrinsic limitations of the underlying physical approximations. Our studies over the past few years suggest that it is possible to model after SAPT in with relatively simple classical terms. For example, electrostatic interactions between molecules are well represented by multipole expansion at long distance. At short distances where the electron clouds overlap, the electrostatic potentials (ESP) of atoms deviate from the pure Coulomb form (1/r) due to electron shielding. This is known as the charge penetration (CP) effect.⁵⁴ While the CP correction is short-ranged, it can be significant at equilibrium geometry of typical molecular complexes. Empirical damping formula for treating CP have been proposed by several research groups.^55–60 We have demonstrated, combining CP correction with point multipoles^58,60 leads to electrostatic interactions closely matching those from SAPT decomposition even at short distances while the added computational cost is minimal. This is a major step towards eliminating error cancellation between what are referred to as electrostatic and van der Waals interactions in classical FFs. A prominent example of the CP effect is the benzene-benzene system.⁶¹ The electrostatic interaction of a stacked benzene is attractive according to ab initio energy decomposition analysis (EDA).^62–63 However, classical point electrostatic models will yield repulsive interactions when C and H atoms are stacked on top of the same C and H. Thus to achieve reasonable benzene stacking energy, the vdW repulsion has to be reduced artificially, which however leads to transferability issues with other molecules. Induction is another fundamental force and often separated in concept into charge transfer (CT) and polarization. In contrast to CP, the definition of CT effect has been far from clear, despite the fact that this concept has been proposed for more than 60 years.⁶⁴ Several quantum chemical EDA methods attempted to separate the CT and polarization from the total induction energy. However, the resulting CT energy ranges from −38.37 to −1.49 kJ·mol⁻¹ for the hydrogen-bonding water dimer depending on the methods used (see paper by Mao et al⁶⁵ and references therein). Stone has pointed out that the EDAs based on the natural bond orbital to separate CT and polarization are intrinsically problematic.⁶⁶ An empirical scheme has been proposed to separate charge transfer from SAPT induction energy.⁶⁷ To overcome the uncertain in CT energy, we have employed an effective approach where we first determine the polarization model against the many-body MP2 energy for a wide range of molecular clusters in various intermolecular distances.⁶⁸ CT energy is then obtained by subtracting the model polarization energy from the SAPT induction energy.

In this paper, we integrate our progress over the past few years and present a new intermolecular potential, AMOEBA+. By incorporating explicit CP and CT terms, as well as improved polarization and van der Waals functions, AMOEBA+ potential is able to accurately describe intermolecular interactions and their components with general parameters. In addition, a water model based on the AMOEBA+ framework is demonstrated. The water model performs well in simulating liquid phase properties while the individual intermolecular force components remain consistent with SAPT energy decomposition.

II. Methodology

2.1. AMOEBA+ potential energy terms

The total potential energy of AMOEBA+ model can be expressed as the sum of the bonded and non-bonded energy terms

$$E_{total}=E_{bonded}+E_{non-bonded}$$ (1)

where E_bonded and E_non-bonded can be expressed as

$$E_{bonded}=E_{bond}+E_{angle}+E_{b-a}+E_{oop}+E_{torsion}$$ (2)

$$E_{non-bonded}=E_{electrostatics}^{CP-corrected}+E_{polarization}+E_{charge-transfer}+E_{VDW}$$ (3)

The detailed functional forms for bonded terms, including bond, angle, bond-angle coupling, out-of-plane, and torsion of the AMOEBA model have been retained in AMOEBA+.^{14, 23} As to AMOEBA+ water model, the bonded terms including bond stretching, angle bending, and the Urey-Bradley, as with the current AMOEBA water models.^{16, 69} The main improvements of AMOEBA+ over AMOEBA FF on functional form lie in the non-bonded interactions, as will be described in detail next.

Electrostatics.

The current AMOEBA model employs the point atomic multipoles truncated at quadrupoles to compute the electrostatic interactions. In AMOEBA the atomic multipole moments are usually derived with distributed multipole analysis (DMA) approach⁷⁰ and then optimized against high-level ab initio electrostatic potential (ESP). Although the atomic charge is not a quantum observable, various population analysis schemes exist to derive atomic multipoles that facilitate our chemical understanding and FF development. Atom-in-molecule (AIM) methods, such as Hirshfeld,⁷¹ Hirshfeld-I^72–73 and iterative stockholder analysis (ISA),^74–76 underwent much improvement in the past decades. Several attempts to develop molecular mechanical potentials using these approaches also appeared recently.^{72, 77–78} In this work, both the DMA and ISA based multipoles were explored to examine their performance on simulating water properties. The charge penetration effect was incorporated into AMOEBA+ using the scheme by Gordon and coworkers,^{55, 60} where damping functions were applied to the full multipole-multipole interaction. As an illustration, the CP-corrected charge-charge interaction between sites i and j is expressed as

$$E_{elst}^{chg-chg}=\frac{Z_i{Z}_j}{r}+\frac{Z_i{q}_j}{r}{f}_{damp}(r)+\frac{Z_j{q}_i}{r}{f}_{damp}(r)+\frac{q_i{q}_j}{r}{f}_{damp}^{overlap}(r)$$ (4)

where the Z_i and Z_j are nuclei charges and q_i and q_j are electron charges. Thus $Z_i+q_i=c_i^{DMA}$ will be the actual net charge on atom i. When the damping is removed (f = 1), the above equation reduces to the familiar Coulombic form ($c_i^{DMA}{c}_j^{DMA}/r$) in common FFs. In AMOEBA+, damping is systematically applied to monopole, dipole and quadrupole. The electrostatic energy of AMOEBA+ is then the sum of multipole-multipole interaction energy and the CP correction. A compact form of the CP-corrected electrostatic energy between sites i and j can be written as

$$E_{elst}^{CP-corr}=\frac{Z_i{Z}_j}{r}+Z_i{T}_{ij}^{damp}{M}_j+Z_j{T}_{ji}^{damp}{M}_i+M_i^t{T}_{ij}^{overlap}{M}_j$$ (5)

Within the particle mesh Ewald (PME), the CP was incorporated as “modifications” in real-space, i.e. the original (non-Ewald) pairwise multipole interactions were subtracted while the CP-corrected terms described above are added. The effect of CP damping is short-ranged and dies off at typical real-space Ewald cutoffs. For example, the magnitude of CP correction is on the order of 10⁻⁶ and 10⁻⁸ kcal·mol⁻¹ when two water molecules are 6 and 7 Å apart, respectively.

Polarization.

AMOEBA potential utilizes the interactive atomic dipole induction scheme to represent the many-body polarization effect. A point dipole is induced at each polarizable site (atom) by the total electric field felt by that site:

$${\mu }_i^{ind}={\alpha }_i\left({\sum }_j{T}_{ij}^{damped}{M}_j+{\sum }_j{T}_{ij}^{damped}{\mu }_j^{ind}\right)$$ (6)

where T_ij is the multipole-multipole interaction matrix for direct induction and $T_{ij}^{11}$ only includes the dipole field related terms for mutual induction. Thole damping scheme⁷⁹ is used in the AMOEBA model to ensure the finite nature of the intermolecular polarization and proper anisotropic molecular response. This scheme is rather successful in reproducing molecular polarizability tensors for a broad range of organic molecules using element-based isotropic atomic polarizabilities.^79–80 AMOEBA uses the same damping function for the direct and mutual induction. The damping function corresponding to the T_α matrix is

$$f_{Thole}(r)=1-{\text{e}}^{-a{u}^3(r)}$$ (7)

where $u(r)=r_{ij}/{\left({\alpha }_i{\alpha }_j\right)}^{\frac{1}{6}}$ is the polarizability-normalized distance between sites i and j; and r_ij is the actual distance; α is the atomic polarizability. The current AMOEBA model uses the same damping factor a for both the direct and mutual parts.¹⁶ In our recent work, we systematically examined the damping functional form and the damping factor by explicitly comparing against the MP2 many-body interaction energy for a range of molecular clusters at various orientations and separtions.⁶⁸ We found that a better distance dependence behavior of the 3-body energy (E_3B) can be achieved by modifying the functional form for damping the permanent field to

$$f_{MB}^{direct}(r)=1-e^{-a{u}^{\frac{3}{2}}(r)}$$ (8)

This new damping function for the permanent field was adopted in the AMOEBA+ potential. The damping functional form and parameter for mutual induction remain the same as AMOEBA. More details of the modified polarization model can be found in a previous publication.⁶⁸

Charge transfer.

The CT refers to the stabilization energy between atoms at near covalent distances. In our model, we are not explicitly considering the transfer of electrons between them. In our model, CT energy is computed as the difference between SAPT2+⁸¹ induction energy and the polarization energy as described above: $E_{Induction}^{SAPT2+}-E_{Polarization}^{AMOEBA+}$. SIBFA has an elaborated many-body charger transfer term using triple overlap function of the damped distributed point multipoles.⁸² Simpler pairwise additive functions are also used to model CT energy especially in water, including discrete charge transfer approach^83–85 and electron density overlap approximation method³⁸ where the monomer density is approximated by exponential functions. We noted the CT with exponential summation only between hydrogen and oxygen atoms in water⁸⁵ may have a deficiency in treating the case where two heavy atoms closely contact. Here we employ a pairwise exponential function to describe this interaction between any two atoms belonging to two molecules:

$$E_{charge-transfer}=-{\sum }_{ij}{a}_{ij}\text{exp}\left(-b_{ij}{r}_{ij}\right)$$ (9)

where a_ij is related to the magnitude of the energy and b_ij controls the distance dependence behavior. Combining rules were used for two hetero-atoms as

$$a_{ij}=\sqrt{a_i{a}_j}$$ (10)

$$b_{ij}=\frac{1}{2}\left(b_i+b_j\right)$$ (11)

In practice, CT interaction is very short-ranged and a 6 Å cutoff is used for CT energy and gradient. Polynomial functions are also added to switch off the CT energy and forces near the cutoff distance. The many-body effect of CT can be important for certain systems such as high-valence ions, which will be treated separately in our future work.

Van der Waals.

AMOEBA+ model retains the same vdW functional form used in the current AMOEBA, the Halgren’s buffered-14–7 potential⁹²

$$E_{vdW}={\epsilon }_{ij}{\left(\frac{1+\delta }{{\sigma }_{ij}+\gamma }\right)}^7\left(\frac{1+\gamma }{{\sigma }_{ij}^7+\gamma }-2\right)$$ (12)

where ε_ij is the potential well depth, ${\sigma }_{ij}=r_{ij}/r_{ij}^0$ with r_ij as the separation between i and j. We use fixed values of δ = 0.07 and γ = 0.12 in AMOEBA+. In the current AMOEBA FF, CUBIC-MEAN (Eq.13) and HHG (Eq. 14) combining rules are used for $r_{ij}^0$ and ε_ij, respectively.

$$r_{ij}^0=\frac{{\left(r_{ii}^0\right)}^3+{\left(r_{jj}^0\right)}^3}{{\left(r_{ii}^0\right)}^2+{\left(r_{jj}^0\right)}^2}$$ (13)

$${\epsilon }_{ij}=\frac{4{\epsilon }_{ii}{\epsilon }_{jj}}{{\left(\sqrt{{\epsilon }_{ii}}+\sqrt{{\epsilon }_{jj}}\right)}^2}$$ (14)

In this work, we also examined the performance of the Waldman-Hagler (W-H) rule (Eq. 15) for ε_ij.

$${\epsilon }_{ij}=2\sqrt{{\epsilon }_{ii}{\epsilon }_{jj}}\frac{{\left(r_{ii}^0{r}_{jj}^0\right)}^3}{{\left(r_{ii}^0\right)}^6+{\left(r_{jj}^0\right)}^6}$$ (15)

2.2. Code implementation on CPU and GPU platforms

The AMOEBA+ model was implemented on the CPU platform based on the Tinker 8.2 source code⁸⁶ and CUDA platform supported in Tinker-OpenMM toolkit.^87–88 It mostly involves adding the new CP and CT energy and forces and the new vdW mixing rule to the existing AMOEBA routines, and modifying the damping for permanent (direct) polarization. Specifically, CT related subroutines in Tinker-CPU and CUDA kernel functions in Tinker-OpenMM were implemented in the same manner as the pairwise vdW. An individual cutoff distance (default 6 Å) and a switching function were used in the calculation of CT energy and forces. CP was incorporated as pairwise corrections to the multipole-multipole interactions in the real space of Ewald sum. The reciprocal space Ewald summation code was not affected. The AMOEBA+ polarization model, where the damping functions of direct polarization were changed, was implemented as a modification to the direct polarization energy, field and force. The code related to mutual induction was not affected. On a single CPU, the total computational cost of AMOEBA+ energy and gradient evaluation, comparing to AMOEBA, is increased slightly by ~ 6%, which includes the cost of added CP and CT, as well as modified (direct) polarization energy and gradient. The detailed computational cost of both energy and gradient by individual energy components is also provided in Table 1. The canonical Tinker CPU code serves as the reference code for developing new algorithms and FFs. It is also critical to implement computationally efficient and high-performance simulation packages that support the use of new potentials with more complex potential energy functions. It has been shown the GPU implementation of AMOEBA-based MD engine⁸⁸ offers a roughly 200-fold acceleration compared to a single CPU core. The preliminary implementation of AMOEBA+ on GPU can achieve ~20 ns/day on RTX2080 for a DHFR system (23,555 atoms) using a 2-fs time step. In another implementation, Tinker-HP⁸⁹ takes advantage of the massive parallelization of MPI and provides excellent scalability of performance over a large number of CPU cores for systems of ~ 1 million atoms. Preliminary implementation of AMOEBA+ GPU code in Tinker-OpenMM has been completed and utilized in this work. Implementation of AMOEBA+ in Tinker-HP, for CPU and CPU-GPU hybrid platform, is in progress. The Tinker-CPU code is available at TinkerTools GitHub site as AMOEBA+ branch.

Table 1.

Evaluation of computational cost of AMOEBA+ model.^a,b Total wall time (in second) of 500 energy and gradient calculations for each potential energy component was evaluated by TIMER program in Tinker.

	AMOEBA	AMOEBA+	Difference	Difference %	Change in AMOEBA+
Eelst	9.7	11.2	+1.5	+15.2%	Added charge penetration correction
Epol	40.9	41.9	+1.1	+2.6%	Split the direct and mutual induction
Ect	/	0.9	+0.9	/	Added new charge transfer term
Efot	53.1	56.4	+3.3	+6.3%	All of the above

^aCPU architecture: Intel (R) Xeon (R) CPU E5–2680 0 @ 2.70GHz.

^bImportant settings: The 36 ų simulation box contains 1600 water molecules; cutoffs for vdW and Ewald real space and charge transfer were 12, 7 and 6 Å, respectively; neighbor-list method was used; induced dipoles were converged to 10E-4. 500 repetitions were run to evaluate the energy and force.

2.3. Param eterization of organic molecules based on SAPT

All SAPT2+ data, Cartesian coordinate structure and parameter files are also included on the SI. As described in our previous work, the S108×7 dimer set was systematically constructed and the interaction energy was decomposed using SAPT2+ method. Briefly, the S108×7 set contains 108 organic dimers, each with seven intermolecular separations, namely, at 0.70, 0.80, 0.90, 0.95, 1.00, 1.05, and 1.10 times of the equilibrium distances. All the 38 molecules in S108 set are listed in Figure 1.

Figure 1.

Molecular structures of the 38 molecules in S108 dataset. These molecules cover nine chemical elements: hydrogen (white), oxygen (red), carbon (gray), nitrogen (blue), sulfur (yellow), phosphor (orange) and halogen (green). Both the neutral molecules and charged molecules/fragments (e.g., 04, 19, 20, 23, 31 and 32) are included due to their relevance in biology as amino acid side chain analogs or fragments of nucleic acids.

The interaction energy components of SAPT2+ can be expressed as

$$E_{SAPT2+}=E_{electrostatics}+E_{induction}+E_{exchange-repulsion}+E_{dispersion}$$ (16)

We first parametrized AMOEBA+ model (Eq. 3) from QM calculations of the monomer and energy components by SAPT2+. The detailed one-to-one mappings are

$$\begin{gathered} E_{electrostatics}^{AMOEBA+}\to E_{electrostatics}^{SAPT2+} \\ {E}_{polarization}^{AMOEBA+}+E_{charge-transfer}^{AMOEBA+}\to E_{induction}^{SAPT2+} \\ {E}_{vanderWaals}^{AMOEBA+}\to E_{exchange-repulsion}^{SAPT2+}+E_{dispersion}^{SAPT2+} \end{gathered}$$ (17)

The atomic multipoles were derived for the molecules in S108 dataset in a previous work using a systematic procedure,⁴¹ where the DMA approach was used to derive the initial multipoles which were then optimized to high-level ESP. The CP parameters of the organic molecules have been derived for S101 dataset⁴² and here we directly expanded to S108 set without further optimization. We modified the direct damping factor from the original 0.75 in our previous publication⁶⁸ to 0.70 in this work. It was found that this change leads to better agreement in many-body energy of water clusters with MP2 values and only has a subtle influence on organic compounds. After subtracting polarization energy from total induction energy of SAPT2+, we parametrized CT model with the remaining energy as the target. In the parametrization of the vdW model, we experimented both the HHG and W-H combining rules for ε_ij. The cost function in the optimization of the vdW model was designed as the root mean square deviation (RMSE) of the model predicted vdW energy and that from SAPT2+. We fixed the shape parameters γ and δ to their original values due to a better dispersion behavior beyond the equilibrium distances, which influence the bulk properties much more than small clusters.^90–92 To prevent overfitting and ensure the physical sense of vdW parameters, we slightly constrained the parameters in reasonable ranges by using a regularization term in the cost function.

2.4. Param etrization of AMOEBA+ w ater model based on ab initio and experimental data

As an application of AMOEBA+ potential, the AMOEBA+ water model has been parameterized for liquid simulations using ForceBalance (FB).^{3, 93–94} FB uses a series of thermodynamic fluctuation equations to obtain the parametric derivatives of condensed phase properties from MD simulations. Thus it allows one to use both experimental properties and ab initio data in parameter optimization. FB has been used to develop new FFs, such as the iAMOEBA⁹⁵ and uAMOEBA,⁹⁶ and to revise the parameters of existing FFs, such as AMBER for proteins,³ AMOEBA water,⁶⁹ and TIP3P/TIP4P waters.⁹⁴ In the optimization, the FF parameters were updated by FB iteratively until satisfied gas phase and liquid properties were obtained (or cost function minimized). The initial parameters of the bonded terms were taken from AMOEBA water03.prm.¹⁶ Table 2 lists the targeting data in the parametrization of AMOEBA+ water model. The objective (cost) function was designed as the sum of weighted mean square errors from gas and liquid targets. Weights were applied among different fitting targets and physical properties within a target (Table 2). In this work, the optimization was carried out using the trust-radius Newton-Raphson algorithm with an adaptive trust radius. The algorithm requires the first and second derivatives of the objective function in the parameter space.

Table 2.

Target ab initio QM data and experimental measurements for temperatures ranging from 265 to 369 K (12 data points in total). All of these data can be found in the “studies/015_amoeba_tinker” directory in ForceBalance program, except for the “Hydrogen-bonding dimer” set, of which the interaction energy was decomposed using SAPT2+ for dimers in 10 intermolecular separations.

System and property		Data point	Data type	Weighta
Clusters	Hydrogen-bonding dimer	10	SAPT2+ interaction energy	1.0
	Smith dimer	10	CCSD(T) binding energy	1.0
	Trimer to hexamer	11	CCSD(T) binding energy	1.0
	Octamer to 20-mer	18	MP2 binding energy	0.5
Liquid	Density	12	Experiment	1.0	3.0b
	Enthalpy of vaporization	12	Experiment		2.0b
	Thermal expansion coefficient	12	Experiment		1.0b
	Static dielectric constant	12	Experiment		1.0b
	Isobaric heat capacity	12	Experiment		0.1b
	Isothermal compressibility	12	Experiment		1.0b

^aWeights applied in ForceBalance optimization among different targets and properties within the target. Here each cluster system is a target, and liquid is another target.

^bWeights applied to the six liquid properties. Weights were also applied to properties at different temperatures (not shown).

We tested both DMA and ISA-based multipoles in developing water models. POLTYPE tool⁹⁷ was used to derive the initial DMA multipoles. We detail the computational procedure for deriving ISA multipoles in SI. In FB optimization, ANALYZE program of Tinker was used to compute the energy of water clusters and liquid boxes from MD simulations. DYNAMIC_OMM program was used to perform liquid phase NPT simulations. The box dimension was ~26×26×26 ų containing 590 water molecules. The vdW cutoff was set to 10 A in FB run with a long-range correction. For simulations to correct the size effect on self-diffusion constant, 12 Å was used in the larger water boxes. Long-range electrostatic interaction was treated using particle mesh Ewald (PME) with 7 Å real space cutoff, which was also used as the cutoff for calculating CP correction. RESPA integrator,^98–99 BUSSI thermostat,¹⁰⁰ Monte Carlo barostat, and 0.5 fs integrating time step were used in the NPT simulations. We also note that a larger time step (2.0 fs) using RESPA is very stable but will lead to slightly different properties. For example, with 2-fs MD the water density will be about 1.0 kg·m⁻³ lower at 298 K than that of 0.5 fs. In each FB iteration, 250 ps of equilibration NPT simulation was first performed to equilibrate the water box. Then 5 ns of production run was performed to calculate the physical properties and gradients w.r.t. the parameters, which are listed in Table 5.

Table 5.

Initial and optimized parameters of AMOEBA+ water model.

Term	Parameter	Unit	Initial	Optimal
Multipole	O monopole	e	−0.382800	−0.558246
	O dipole Z	E·bohr	0.054770	−0.144923
	O quadrupole XX	e·bohr2	0.698660	0.451599
	O quadrupole YY	e·bohr2	−0.604710	−0.280108
	O quadrupole ZZ	e·bohr2	−0.093950	−0.171491
	H monopole	E	0.191400	0.279123
	H dipole Z	e·bohr	−0.200970	−0.230060
	H quadrupole XX	e·bohr2	0.038810	0.215207
	H quadrupole YY	e·bohr2	0.022140	−0.029761
	H quadrupole ZZ	e·bohr2	−0.060950	0.185446
	H quadrupole XZ	e·bohr2	0.000000	0.191100
CP	O damping factor	none	4.1615	4.0483
	H damping factor	none	3.2632	3.2748
CT	O parameter aCT	103 kcal-mol−1	3.4761	3.2003
	O parameter bCT	Å−1	3.6034	3.7188
	H parameter aCT	103 kcal-mol−1	3.7994	2.9436
	H parameter bCT	Å−1	4.8850	4.7135
Polarization	O polarizability	Å	0.837	0.948
	H polarizability	Å	0.496	0.416
	direct damping factor	none	0.70	0.70
	mutual damping factor	none	0.39	0.39
vdW	O vdW diameter	Å	3.813189	3.808992
	O vdW epsilon	kcal-mol−1	0.084785	0.061361
	H vdW diameter	Å	3.339858	3.340781
	H vdW epsilon	kcal-mol−1	0.002449	0.004571
	H vdW reduction	none	0.980000	0.983604
Bonded	O-H bond length	Å	0.96	0.94
	Bond force constant	kcal-mol−1· Å−2	556.85	556.85
	H-O-H angle	Degree	108.50	108.81
	Angle force constant	kcal-mol−1·rad−2	48.70	48.70
	U-B H-H length	Å	−7.60	−7.60
	U-B force constant	kcal-mol−1·Å−2	1.53	1.53

III. Results and discussion

We previously obtained SAPT2+ intermolecular energy components for the S108×7 dataset.⁵⁸ The parameters of CP, CT, and vdW obtained in this study are summarized in Table S1–S2. In total AMOEBA+ has 18 atom classes for CP and CT, 28 atom classes for vdW, and nine element-based atomic polarizability parameters. Thus a very limited number of parameters (18 for CP, 36 for CT, 9 for polarizability and 56 for vdW) are used in AMOEBA+. With optimized parameters, AMOEBA+ model can accurately capture three energy components. We then developed an AMOEBA+ model for water, where both the gas phase cluster energy and various liquid properties are included in parametrization and validation processes.

3.1. Interm olecular interactions for organic molecules

The three non-bonded interactions of AMOEBA+ (electrostatics, CT and vdW) were parameterized separately using the SAPT2+ S108 database as the training set. Polarization parameters were derived separately using the MP2 interaction energy with many-body expansion.⁶⁸ As explained above, the sum of AMOEBA+ polarization and CT energy is compared to SAPT induction energy. The performance of AMOEBA+ for energy components are summarized in Table 3 and the correlations between AMOEBA+ vs. SAPT2+ are shown in Figure 2. To validate the transferability, these parameters were further tested on additional SAPT2+ data published by others. In total there are 707 data points in the training set and 309 in the testing set.

Table 3.

Statistics evaluation of the AMOEBA+ model on each energy component and total interactions for S108×7 dimer set compared to SAPT2+ data. All energy values are in kcal·mol⁻¹.

Data points	Statisticsa	Eelst	Eind	EvdWb		Einterc
short distances (0.70~0.80×req)	MUE	1.56	0.96	2.82	(3.06)	3.15
	MSE	0.25	−0.15	0.26	(−0.06)	0.36
	RMSE	2.32	1.54	4.52	(4.87)	4.89
medium distances (0.90~1.10×req)	MUE	0.39	0.33	0.88	(1.04)	0.68
	MSE	0.06	0.27	−0.33	(−0.58)	0.00
	RMSE	0.61	0.48	1.34	(1.70)	1.10
all distances (0.70~1.10×req)	MUE	0.73	0.51	1.43	(1.62)	1.39
	MSE	0.12	0.15	−0.16	(−0.43)	0.11
	RMSE	1.34	0.92	2.67	(2.97)	2.78

^aMUE: mean unsigned error; MSE: mean signed error; RMSE: root mean square error.

^bValues from W-H and HHG combining rule (in parentheses) for ε_ij are provided.

^cSum of three components where the E_vdw was calculated with W-H combining rule for ε_ij.

Figure 2.

Correlation plot of the intermolecular energy and three decomposed components for AMOEBA+ model against SAPT2+. (a) Electrostatics energy; (b) Induction energy, where sums of E_CT and E_polarization are served as the AMOEBA+ values, and (d) Van der Waals energy, where the AMOEBA+ values are calculated from parameters with W-H combining rule, and SAPT2+ values are sums of E_dispersion and E_{exchange-repuision}^. Different intermolecular distances are labeled with three different colors (orange, green and blue). The black solid line shows the perfect correlation.

3.1.1. S108×7 database (training set)

Electrostatics.

The multipole parameters were previously derived based on the DMA-fitting procedure.⁵⁸ The CP model this work used was previously developed and parametrized on S101 dataset.⁶⁰ In this work, we directly applied CP on the S108 dataset without re-optimizing the parameters. Using transferable, expanded element-based parameters, the CP-corrected point multipoles are able to reproduce SAPT2+ electrostatic energy for 108 homo- and heterodimers at 7 intermolecular distances, with an RMSE of 1.34 kcal·mol⁻¹ (Table 3). Excellent agreement was found for the medium-distance regions (0.90~1.10 panel in Table 3 and blue dots in Figure 2), with an RMSE of 0.61 kcal·mol⁻¹. It is worth to mention that for several organic molecules we studied (not shown here), the CP parameters derived from DMA-based multipoles can also be directly applied on the ISA multipoles without adjusting the CP parameters, which indicates the robustness and transferability of the CP model and parameters.

Charge transfer.

Considering the simplicity and isotropic nature of the current CT potential, a possible concern is its capability of capturing the CT energy of different configurations across various chemical environments. We examined the robustness of the model on water and various organic dimers in S108 data set. Figure 3 shows water dimers of very different configurations at various intermolecular separations. It is encouraging that the absolute errors w.r.t. the QM target energy, $E_{Ind.}^{SAPT2+}-E_{Pol.}^{AMOEBA+}$, are within 0.5 kcal·mol⁻¹ in the equilibrium or even shorter distances. Figure 3a shows the excellent agreement between the exponential function used by AMOEBA+ and SAPT2+ for the classic hydrogen-bonding dimer. AMOEBA+ CT energy is also consistent with that given by the ALMO (absolutely localized molecular orbital)^101–102 EDA method. For the “Smith Dimer” configurations (Figure 3b), the two obvious but still small errors appear for the 5^th and 6^th dimers, which have cyclic structures with the oxygen of one molecule pointing to the hydrogen of the other, suggesting that CT is directional. For the OO-faced dimers (Figure 3c and d), the deviations are at the most 0.5 kcal·mol⁻¹ in all intermolecular distances. Besides water molecules, we also examined other special cases for several organic dimers in different orientations (Figure S1). Overall, we can conclude that CT model is robust across these molecules and configurations. We combined the AMOEBA+ CT and polarization energy when comparing with the induction energy of SAPT2+. The agreement is remarkably well even for the short-distance dimers (Table 3 and Figure 2). For all seven intermolecular distances, AMOEBA+ predicted the induction energy of SAPT2+ with an RMSE of 0.92 kcal·mol⁻¹.

Figure 3.

Comparison of QM and AMOEBA+ CT energy for water dimers of different configurations and intermolecular distances. The target CT energy is defined as $E_{Ind}^{SAPT2+}-E_{Pol}^{AMOEBA+}$. (a) hydrogen-bonding water dimer at 10 intermolecular distances, where “CT.(ALMO)” is the energy predicted by ALMO EDA approach at HF/6–311++G(2d,2p) level; (b) Smith dimers at various orientations; (c) and (d) oxygen-oxygen faced dimers at seven intermolecular distances.

Van der Waals.

Here we mainly explore the difference between W-H and HHG combining rules of ε in the buffered-14–7 potential by fitting two sets of vdW parameters to the SAPT2+ repulsion and dispersion energy. Our results indicate that the W-H is slightly better than the HHG rule combining rule for ε_ij, with a reduction of 0.36 and 0.30 kcal·mol⁻¹ in RMSE for the medium distances and all seven intermolecular distances, respectively (Table 3). We note that W-H is a more complicated functional form that incorporates the effects of atom radius into ε_ij.

Interestingly, when examining the total interaction energy of AMOEBA+, the errors in individual components also partially cancel. AMOEBA+ displays an RMSE of 1.10 kcal·mol⁻¹ for the medium-range distances and the most substantial deviations are in the short distances. The RMSE for all distance points is 2.78 and it will be reduced to 2.55 kcal·mol⁻¹ if very repulsive total energy values (> 10.0 kcal·mol⁻¹) are excluded.

3.1.2. Validation of AMOEBA+ on additional data set

We have further performed validation of AMOEBA+ using SAPT2+(3) energy decomposition data published by others.^103–107 Molecular dimers in the S59 data set are the same in composition as our training set but only the structures with intermolecular separations not present in S108 are included. Majority of the dimers in the other three sets, S14, Ionic and X6, include heterodimers not present in S108. Overall, AMOEBA+ performs similarly well on the training and testing sets, with an RMSE of 0.97 kcal·mol⁻¹ for total interaction energy of SAPT2+(3) and < 0.7 kcal·mol⁻¹ for the individual energy components. The errors on the neutral sets (S14, S59 and X6) are well within 1.0 kcal·mol⁻¹ and those on the charged molecules (Ionic set) are higher (1.2~2.0 kcal·mol⁻¹) as expected.

This set of initial AMOEBA+ parameters for organic molecules were derived purely based on the S108 SAPT2+ data. Due to various limitations in the ab initio methods and approximations in classical models, the resulted potential is unlikely to provide chemical accuracy when applied to condensed-phase simulations. Further optimization using the condensed-phase properties is needed, which requires extensive molecular simulations. As an example, we will next demonstrate the refinement and application of AMOEBA+ potential to water, where the parameters were further optimized based on selected liquid phase properties.

3.2. AMOEBA+ water model for gas and liquid-phase

Using water as an example, we illustrate an approach where we integrate experimental measurements with ab initio data in training AMOEBA+ parameters. In addition, to systematically evaluate different multipole methods (DMA vs ISA) and combining rules for ε in vdW potential (HHG vs W-H), we optimized four sets of water parameters, i.e., DMA/W-H, ISA/W-H, DMA/HHG and ISA/HHG combinations. For clarity, hereafter we will report the results from the “DMA/W-H” model and refer it to AMOEBA+ in the remaining sections. The discussion will also be made on the other three models, for which we provided the detailed results in SI.

Parameters of AMOEBA+ water model are shown in Table 5. The initial non-bonded parameters were derived within the S108 dataset, as described above, and the initial bonded parameters were taken from the AMOEBA03 water model.¹⁶ The multipoles in S108 were first derived using DMA at MP2/6–311G** level of theory. The dipole and quadrupole parameters were further optimized by fitting to the ESP at MP2/aug-cc-pvtz level.⁵⁸ This procedure resulted in a partial charge of −0.38 e on oxygen. The atomic multipoles in AMOEBA03 were generated via DMA at the MP2/aug-cc-pvtz with the experimental monomer geometry, with a partial charge of −0.52 e on oxygen.¹⁶ Interestingly, the partial charge on oxygen for AMOEBA+ model, after FB optimization, converges to that of the AMOEBA03. Please note that the atomic multipoles cannot be uniquely determined. DMA and ISA methods could lead to dramatically different atomic multipole values and yet both produce accurate electrostatic potential around the molecules, at least at the far distances (the deficiency in the short distances is addressed by CP correction). As will be shown next, both the initial and optimized electrostatic parameters give electrostatic interaction energy consist with SAPT decomposition. In addition, the vdW parameters changed slightly from those initial values derived by fitting to SAPT repulsion-dispersion energy, so did the valence parameters. This again shows that, to achieve the desired chemical accuracy in condensed-phase, we need to incorporate experimental measurements in classic potential parameter optimization. To understand the sensitivity of liquid properties to these parameters, we analyzed the final numerical gradient values of the six thermodynamic properties w.r.t. the each parameter (Table S17 and S18) and found that 1) almost all of the properties are sensitive to the vdW and multipole parameters; 2) static dielectric constant is strongly affected by atomic dipole moments, which is not surprising as the static dielectric constant is calculated from the cell dipole fluctuation (see the following section); 3) isothermal compressibility is sensitive to the equilibrium OH bond length and 4) charge penetration parameters have a strong effect on liquid density.

3.2.1. Gas phase cluster properties

Monomer and Equilibrium Dimer.

Multipole moments are the most basic properties of a water molecule and are largely responsible for interesting water properties across multiple phases.¹⁰⁸ In Table 6, AMOEBA+ molecular multipole moments calculated at the model-optimized geometry and an “ideal” water geometry determined by rotation-vibration spectra of water vapor¹⁰⁹ are compared with experimental values and those from a previous AMOEBA14 model. With the model-optimized geometry, AMOEBA+ gives a total dipole moment of 1.778 Debye, lower than the ab initio and experimental values, mostly due to the larger HOH angle. The molecular dipole moment increases to 1.918 Debye when the “ideal” geometry is used. AMOEBA+ has larger quadrupole moments but smaller polarizability comparing to the AMOEBA14. In Table 7, the dimer properties at equilibrium geometry are given. The dimer dissociation energy of AMOEBA+ agrees better with the ab initio value than AMOEBA14, which is also true for other separation distances as shown in Figure 5a. The total dipole moment of the water dimer calculated with AMOEBA+ based on the MP2-optimized geometry agrees well with both experimental and ab initio values. It is noted that the water models employing DMA multipoles better predict the monomer and equilibrium dimer properties than those using ISA multipoles, using either HHG or W-H vdW mixing rules (Table S10 and S11).

Table 6.

Monomer properties of AMOEBA+ compared to AMOEBA14 and experimental values. AMOEBA+ values were given for both model-optimized and “ideal” geometry, with the latter given in parenthesis.

Property		AMOEBA14a	AMOEBA+b	ab initioc	Expt.
Dipole (Debye)	dz	1.808	1.778 (1.918)	1.840	1.855d
Quadrupole	Qxx	2.626	3.314 (3.193)	2.57	2.63e
(Debye A)	Qyy	−2.178	−2.628 (−2.679)	−2.42	−2.50e
	Qzz	−0.045	−0.684 (−0.514)	−0.14	−0.13e
Polarizability	αxx	1.767	1.583 (1.589)	1.47	1.53f
(Å2·s4·kg−1)	αyy	1.308	1.209 (1.213)	1.38	1.42f
	αzz	1.420	1.308 (1.329)	1.42	1.47f

^aReference⁶⁹;

^bIn parentheses are the values calculated with the “ideal” geometry, where O-H bond length is 0.9572 Å, and angle ∠HOH is 104.52 degree;

^cReference 110;

^dReference 111;

^eReference 112;

^fReference 113.

Table 7.

Dimer properties at equilibrium geometry predicted by AMOEBA+ compared to AMOEBA14 and experimental values. D_e (in kcal·mol⁻¹) is the dissociation energy with monomer optimized. μ_tot (in Debye) is the total dipole moment. Other geometrical properties are illustrated in Figure 4. Optimized dimer geometry was used for AMOEBA+ unless explicitly noted.

Property	AMOEBA14a	AMOEBA+	ab initio	Expt.
De	4.64	4.80	4.98b	5.44±0.7d
r0–0	2.908	2.892	2.907b	2.976e
α	4.61	14.38	5.5b	−1±10e
θ	64.9	79.7	56.9b	57±10e
μtot	2.20	1.79 (2.85f)	2.76c	2.643e

^aReference 69;

^bReference 114;

^cReference 115;

^dReference 116;

^eReference 117;

^fValue calculated using AMOEBA+ model with MP2-optimized geometry.

Figure 5.

Intermolecular interaction energy of hydrogen-bonding dimer at 10 separation distances. (a) Energy components from SAPT2+ and AMOEBA+ model. (b) Total interaction energy calculated by two QM methods compared to AMOEBA+ and AMOEBA14 water models. The orange shadows indicate the equilibrium distance region of the water dimer.

Energy Decomposition Analysis.

The primary goal of AMOEBA+ potential over the current AMOEBA is to improve the physical representation of individual energy components, reduce error cancellation, and improve the transferability. Figure 5 shows the comparison of the hydrogen-bonding water dimer interaction energy and its components from AMOEBA+ and SAPT2+, using the final (liquid) optimized AMOEBA+ parameters. Here we defined the intermolecular interaction energy at the cluster geometry without relaxing the monomers. Interaction energy at CCSD(T)/aug-cc-pv5z level of theory was also calculated and provided here. Even with liquid-optimization, the electrostatics, induction and vdW energies are still well reproduced in all distances w.r.t the SAPT2+ EDA. The errors are noticeable mostly at very close contact. At O⋯O distance of 2.37 Å, the errors are +0.4, +0.9, and +3.0 kcal·mol⁻¹ for the three components, respectively. The total dimer interaction energy values from two ab initio methods, SAPT2+ and CCSD(T), show nontrivial disagreements at short distances (O⋯O distances < 2.77 Å), as shown in Figure 5b. AMOEBA+ total interaction energy closely follows those of SAPT2+ and CCSD(T) at the equilibrium distance and beyond. It is also clear that AMOEBA14 underestimates the dimer interaction energy for all distances. The deviation between AMOEBA+ and SAPT2+ vdW energy at the very short distance (e.g. 2.37 Å) may imply the uncertainty of exchange-repulsion by SAPT2+, which is also less repulsive than CCSD(T). Additional validation on the Smith05 dimer, which possesses a cyclic geometry with non-typical O⋯H interactions, and the hydrogen-bonding dimer with various flap angles, shows good transferability of AMOEBA+ water model in predicting interaction energy components (Figure S2).

Binding Energy of Water Clusters.

Binding energy (BE) of a molecular cluster is defined as the energy required to separate each monomer from the cluster, with the cluster and monomer optimized to their own minimum energy. The AMOEBA+ BEs of clusters ranging from dimer to 20-mers are compared with high-level QM results as well as AMOEBA14 in Table 8 and Table 9. Compared to AMOEBA14 model, AMOEBA+ offers better BEs for eight out of ten Smith dimers. RMSE of AMOEBA+ water dimer BE with respect to CCSD(T) is reduced from 0.62 kcal·mol⁻¹ of AMOEBA14 to 0.28. The Smith dimer set is usually thought as a test for “anisotropy” of a water model as different H-bond configurations are present. Both AMOEBA+ and AMOEBA14 perform similarly well on clusters from trimer to octamers. However, AMOEBA+ does noticeably better than AMOEBA14 for almost all larger clusters from 11-mer to 20-mers. Overall, an improvement of ~0.6 kcal·mol⁻¹ in terms of RMSE is achieved by AMOEBA+ over AMOEBA14 model (Table 9). It is interesting that both DMA-WH and DMA-HHG models predict the BEs of Smith dimers better than ISA-WH and ISA-HHG models (Table S14). Considering the greater monopole and smaller quadrupole in the ISA-based multipoles, this could suggest ISA multipoles lack in anisotropy in comparison with DMA. Nonetheless, all four water models, regardless of the multipole methods and vdW combining rules, all give better BEs than AMOEBA14 for large clusters (Table S15).

Table 8.

The binding energy of Smith dimers predicted by AMOEBA+ and AMOEBA14 models compared to QM data at CCSD(T)/CBS. Deviations (RMSE) from CCSD(T) data are reported for two water models. Values in bold indicate better agreement to CCSD(T) reference. All energy values are in kcal·mol⁻¹.

(H2O)2	CCSD(T)a	AMOEBA+	AMOEBA14b
Smith01	−4.97	−4.96	−4.65
Smith02	−4.45	−4.11	−4.22
Smith03	−4.42	−4.00	−4.19
Smith04	−4.25	−4.75	−3.54
Smith05	−4.00	−4.08	−3.06
Smith06	−3.96	−3.90	−2.92
Smith07	−3.26	−3.69	−2.49
Smith08	−1.30	−1.39	−1.02
Smith09	−3.05	−3.22	−2.37
Smith10	−2.18	−2.34	−1.96
RMSE		0.28	0.62

^aReference 126;

^bReference 69.

Table 9.

The binding energy of trimer to 20-mer water clusters predicted by AMOEBA+ and AMOEBA14 models compared to QM data, where the BEs of small clusters (n=3~6) were calculated at CCSD(T)/CBS and the BEs of large clusters (n>6) were calculated at MP2/CBS. Deviations from QM data are reported for two water models. Values in bold indicate better agreement to QM reference. All energy values are in kcal·mol⁻¹.

(H2O)n	Geometry	QM	AMOEBA+	AMOEBA14g
n=3a	Cyclic	−15.74	−16.07	−15.38
n=4a	Cyclic	−27.40	−28.26	−27.43
n=5a	Cyclic	−35.93	−36.41	−35.78
n=6b	Prism	−45.92	−46.03	−45.18
	Cage	−45.67	−46.10	−45.83
	Bag	−44.30	−44.65	−44.52
	Cyclic chair	−44.12	−44.88	−43.53
	Book1	−45.20	−45.87	−45.08
	Book2	−44.90	−45.25	−45.06
	Cyclic boat1	−43.13	−43.54	−42.99
	Cyclic boat2	−43.07	−43.51	−43.07
n=8c	S4	−72.70	−73.56	−72.22
	D2d	−72.70	−73.74	−72.24
n=11d	434	−105.72	−101.65	−101.11
	515	−105.18	−101.54	−100.99
	551	−104.92	−101.23	−100.58
	443	−104.76	−101.55	−101.17
	4412	−103.97	−100.94	−100.33
n=16e	Boat-a	−170.80	−162.50	−160.45
	Boat-b	−170.63	−162.09	−160.30
	Anti-boat	−170.54	−161.77	−160.30
	ABAB	−171.05	−163.59	−161.20
	AABB	−170.51	−163.19	−160.89
n=17e	Sphere	−182.54	−172.24	−171.53
	5525	−181.83	−171.00	−170.42
n=20f	Dodecahedron	−200.10	−193.58	−193.81
	Fused Cubes	−212.10	−208.65	−205.77
	Face Sharing Prisms	−215.20	−205.31	−204.41
	Edge Sharing Prisms	−218.10	−208.53	−207.06
RMSE			5.46	6.33

^aReference 16;

^bReference 118.

^cReference 119;

^dReference 120;

^eReference 121;

^fReference 122;

^gReference 69.

3.2.2. Liquid properties

Thermodynamic properties of water calculated by AMOEBA+ model include density, enthalpy of vaporization, thermal expansion coefficient, isothermal compressibility, static dielectric constant, and isobaric heat capacity. Structural (radial distribution function) and dynamic properties (self-diffusion constant) that were not included as the parametrization targets but only used as validation are also presented.

Density.

One of the anomalous properties of water is its maximum density at 4 °C. The density-temperature profile simulated by AMOEBA+ model shows excellent agreement with experiment with small deviations up to 0.6% at very high temperature (369 K). Compared to the AMOEBA14 model, AMOEBA+ shows better agreement with experiment in the near-density-maximum temperatures and similar agreement in the ambient temperature (numerical data shown in SI). For example, the density predicted by AMOEBA+ at 277 K is 1000.0 kg·m⁻³, in excellent agreement with the experimental value (1000.0 kg·m⁻³), and AMOEBA14 gives 1001.5 kg·m⁻³. At ambient temperature (298 K), the average density by AMOEBA+ is 998.3 kg·m⁻³ and AMOEBA14 gives 997.9 kg·m⁻³. We note that a larger time step (2 fs) results in an average density of 997.7 kg·m⁻³ by AMOEBA+, compared to the experimental density of 997.0 kg·m⁻³. Other three models parametrized in this work (DMA/HHG, ISA/W-H, ISA/HHG) show similar performance (Table S3).

Enthalpy of Vaporization.

The enthalpy of vaporization is computed from the potential energy difference between liquid and gas phases, assuming water vapor to be an ideal gas

$$\Delta H_{vap}=-\Delta E+\Delta PV=-E_{liq}+E_{gas}+RT$$ (18)

The potential energy of the gas as obtained from the stochastic dynamics simulation of a monomer at specific temperatures using a time step of 0.1 fs. The AMOEBA+ ΔH_vap at 298 K is 44.5 kJ·mol⁻¹, in agreement with the experimental value (44.0 kJ·mol⁻¹). The ΔH_vap values by AMOEBA+ and AMOEBA14 are within 0.1 kJ·mol⁻¹ and both are slightly larger (+0.8~0.9 kJ·mol⁻¹) at low temperature (265 K) than the experimental value. While the absolute value of ΔH_vap reflects the magnitude of interaction energy of water molecules and have a strong influence on the self-diffusion constant of water (see later discussion), the slope of the ΔH_vap-T curve is also important as it defines the heat capacity.

Thermal Expansion Coefficient.

The thermal expansion coefficient measures the volume change of water in response to the temperature. It was calculated using fluctuation formula.

$$\alpha =\frac{1}{V}{\left(\frac{\partial V}{\partial T}\right)}_{p,N}=\frac{1}{k_B{T}^2}\frac{\langle HV\rangle -\langle H\rangle \langle V\rangle }{\langle V\rangle }$$ (19)

AMOEBA+ α values are consistent with the experimental data at temperatures below 300 K but begin to deviate as the temperature rises (Figure 8). The transition of α from negative to positive follows the same trend as the experiment.

Figure 8.

Liquid water thermal expansion coefficient at temperatures ranging from 265 to 369 K and atmospheric pressure (1 atm) calculated with AMOEBA+ model. These values are obtained from a small box simulation with Nose-Hoover integrator. Experimental data are taken from reference¹²³.

Isothermal Compressibility.

The isothermal compressibility measures the volume change of water in response to the pressure. It was calculated from the fluctuation equation

$${\kappa }_T=-\frac{1}{V}{\left(\frac{\partial V}{\partial p}\right)}_{T,N}=\frac{1}{k_{B}T}\frac{\langle V^2\rangle -{\langle V\rangle }^2}{\langle V\rangle }$$ (20)

To compare with experimental κ_T at selected temperatures, we ran the MD simulations using the AMOEBA+ water final parameters. Here we used on a small water box (216 molecules) and the Nose-Hoover integrator in Tinker (CPU) code to calculate the κ_T values, which again agree well with the experimental measurements at low temperatures but are overestimated above 290 K.

Static Dielectric Constant.

The static dielectric constant (£0) is primarily determined by the electrostatics and polarization component of the water model (Table S16). The ε₀ is calculated from the fluctuation of the total dipole moment of the simulation box:

$${\epsilon }_0=1+\frac{4\pi }{3k_{B}T\langle V\rangle }\left(\langle M^2\rangle -\langle M\rangle \cdot \langle M\rangle \right)$$ (21)

where 〈M〉 is the ensemble is average of the total box dipole moment; V is the volume of the simulation box and T is the temperature. The simulated ε₀ at all temperatures matches reasonably well with the experimental measurements, with the uncertainties increase quickly for low temperatures due to slow dynamics (Figure 10). The ε₀ for AMOEBA+ is 80.4 at 298 K and under 1 atmosphere, whereas the experimental measurement is 78.4.

Figure 10.

Liquid water static dielectric constant at temperatures ranging from 265 K to 369 K and under atmospheric pressure (1 atm) calculated with AMOEBA+ model. Experimental data are taken from reference 124.

Isobaric Heat Capacity.

The Isobaric heat capacity is related to liquid enthalpy fluctuation via

$$C_p=\frac{1}{N{k}_{B}T}\left(\langle H_{liq}^2\rangle -{\langle H_{liq}\rangle }^2\right)$$ (22)

In the above equation, N is the number of molecules in the simulation box, k_B is the Boltzmann constant and T is temperature. It is noted that classical approximation of the intra- and intermolecular vibration overestimates the heat capacity compared with a quantum oscillator model. Thus the temperature dependent inter- and intra-molecular corrections are usually necessary when compare simulations with experiments.¹²⁵ Another approach for calculating C_p independent of fluctuation is to use the differentiation equation¹²⁶

$$C_p={\left(\frac{\partial \langle H_{liq}\rangle }{\partial T}\right)}_P={\left(\frac{\partial \langle U_{liq}\rangle }{\partial T}\right)}_P+\frac{9}{2}R$$ (23)

Where the first term is the differentiation of liquid potential energy w.r.t. the temperature, and the second term is the kinetic energy contribution from nine degrees of freedom. Using Eq. 23, we simply calculated C_p of a certain temperature from the ΔH_vap values of two neighboring temperatures. At 298.15 K, the C_p calculated by the differential equation, after addition of quantum corrections (in the range of −2.5 to −1.6 cal·mol⁻¹·K⁻¹),¹²⁵ is ~19.4 cal·mol⁻¹·K⁻¹, in closer agreement with the experimental value (18.0 cal·mol⁻¹·K⁻¹).

Radial Distribution Function.

The experimental radial distribution functions (RDFs) are not included in our parametrization thus this comparison serves as a test of our model. RDFs in the experiment can be derived from X-ray scattering¹²⁷ or neutron diffraction¹²⁸ techniques and reflect the structural features of liquid water. An ideal water model is expected to reproduce the RDF peak positions and the intensity of the experimental RDFs since many of water properties are encoded in its structure.¹²⁹ Figure 12 provides RDFs of O⋯O, O⋯H and H⋯H pairs from AMOBA+ simulation at 298 K and under 1 atmosphere. The first peak of g_oo(r) from AMOEBA+ resides at 2.75 Å (Figure 12a), in between the values derived from neutron diffraction (2.73 Å)¹²⁸ and X-ray scattering (2.80 Å) experiments.¹²⁷ The intensity of the first and second peaks as well as the position of the second peak of AMOEBA+ g_oo(r) agrees slightly better with the neutron diffraction than the X-ray scattering data. The higher first peak of g_oo(r) by AMOEBA^{16, 69, 95} and AMOEBA+ water models may be a result of the Buffered-14–7 vdW functional form. A recent study shows that the exponential Buckingham potential has advantage over Lennard-Jones on describing RDFs of water by explicit fitting using ForceBalance.¹³⁰ Nonetheless, our previous work also showed that the Buckingham potential has difficulty to capture SAPT2+ exchange-repulsion energy.¹³¹ Features beyond the first peaks are also consistent with the neutron diffraction curves, which is essential since the tetrahedral structure of water is characterized by the second and third peaks.¹²⁸ The AMOEBA+ predicts the first and second peaks of g_OH(r) at 1.84 and 3.25 A respectively, vs. the experimental data (1.77 and 3.33 Å). The positions of the first and second peaks of g_HH(r) (2.36 and 3.85 Å) predicted by AMOEBA+ are in excellent agreement with the neutron scattering measurement (2.34 and 3.84 Å). As seen from Figure 12c, the height of the second peak from AMOEBA+ overlaps with the experiment.

Figure 12.

AMOEBA+ predicted RDFs at 298.15 K and 1-atmosphere compared to experimental measurement. (a) O⋯O pair, (b) O⋯H pair and (c) H⋯H pair. RDFs between intramolecular pairs were not calculated and the experimental data below 1.65 Å (b) and 1.80 Å (c) are not shown on the plot.

Self-Diffusion Constant.

The self-diffusion constant (D₀) was not used as a parameterization target thus this comparison serves as another validation. It reflects the dynamic feature of water, which is crucial for the kinetics processes such as chemical reactions in solution. It was evaluated with the Einstein equation

$$D_0=\underset{t\to \infty }{\text{lim}}\frac{d}{dt}\langle {\left|r(t)-r\left(t_0\right)\right|}^2\rangle$$ (24)

It has been shown that the D₀ could be underestimated in a limited size of water box in simulations. To correct the size-effect, we computed the D₀ with the cubic boxes of 18 Å, 40 Å, 60 Å and 90 Å, where N_water is 216, 2210, 7500 and 25150, respectively. The time length of NPT production MD simulations are 6 ns, 5 ns, 4 ns and 2 ns, respectively. The size-independent D₀ was obtained by a linear fit to $D~\frac{1}{L}$ and extrapolating to $\frac{1}{L}=0$. As seen from Figure 13, AMOEBA+ water D₀ matches very well with the experimental data. AMOEBA+ D₀ at room temperature and under 1 atmosphere is 2.23 x10⁵ cm²·s⁻¹ while the corresponding experimental value is 2.30×10⁵ cm²·s⁻¹. This agreement is better than widely used non-polarizable water models, such as SPC¹⁴¹ and TIPnP (n=3,4 and 5)¹³² as well as polarizable models, such as AMOEBA models^{16, 69} and AMOEBA variants.^95–96 As discussed above, W-H vdW combining rule performs slightly better than HHG on capturing the vdW component of SAPT2+ energy for the organic S108 sets (Table 3). It is seen here that seemingly small differences in potential energy can have noticeable impacts on the dynamic and thermodynamic property of liquid water. Due to less repulsive energy that HHG rule provides, HHG-based water models underestimate D₀ of water at a series of temperatures. For example, the D₀ with the HHG combining rule, with either DMA or ISA based multipoles, is systematically slower than experimental value (298 K) by 8.0~12.6% (Figure S4 and S5). In contrast, models employing W-H rule are able to predict appropriate D₀ for various temperatures.

Figure 13.

Size-corrected water self-diffusion constant as a function of temperatures under 1-atmosphere pressure compared to experiment. Detailed extrapolation over different system sizes is provided in Figure S3–S6.

Summary of AMOEBA+ and AM OEBA water models.

The first AMOEBA water model was published in 2003 and the parametrization relied particularly on high-level ab initio data of small water clusters, and limited liquid properties (density and heat of vaporization) at only room temperature.¹⁶ It was encouraging that various bulk properties can be reasonably reproduced by AMOEBA03 water model even beyond the ambient conditions.⁴² AMOEBA14 model was a parameter optimization effort explicitly utilizing a wide range of liquid properties at various temperatures, with every single parameter allowed to be optimized. Nevertheless, the gas phase properties were sacrificed somewhat during the optimization. More importantly, the physical meaning of the parameter, e.g. atomic charges, after optimization becomes unclear. When mixing this water model with other molecules, the “electrostatic” interactions may no longer be truly electrostatic and errors summing over different energy components may be amplified instead of canceled. To address this challenge, we have developed a new potential energy function that allows us to systematically examine and model the individual components of intermolecular forces, by explicitly accounting for CP, CT and many-body dependence of polarization effects using relatively simple, computationally tractable empirical functions. The resulting AMOEBA+ water model shows comparable performance on liquid properties and improvements on gas phase cluster properties over the AMOEBA14 model. Most importantly, the AMOEBA+ water potential also provides interaction energy components that are consistent with SAPT method. By extending this approach from water to other chemical species, we will be able to arrive at a general and transferable force field for molecular simulations.

IV. Conclusion

Classical potentials based on isotropic and additive atomic interactions have been widely used over the past few decades to simulate small molecules to large proteins in computers. While computationally attractive, it is well understood that severe approximations are made in the underlying physics. The inaccuracy can be hidden due to error-cancellation in over-parameterization or insufficient sampling in molecular simulations. To fundamentally advance the accuracy and transferability of classical potentials, we propose a new classical potential, AMOEBA+, to model the essential intermolecular forces, including permanent electrostatics, repulsion and dispersion, as well as many-body polarization, short-range charge penetration and charge transfer, by extending the polarizable atomic multipole-based AMOEBA model. The key improvements are the inclusion of short-range charge penetration and charge transfer effects and modification of direct polarization damping and vdW combining rules. We adopted a unique approach to separate the CT energy out of SAPT induction energy by first determining the many-body polarization energy independent of ab initio EDA such as SAPT. The AMOEBA+ framework was then successfully demonstrated on a set of common organic molecules, where we showed that a classical potential with limited, element-based parameters reproduced SAPT2+ intermolecular energies and their components for homo- and heterodimers at various configurations and separations. The accuracy and transferability of AMOEBA+ potential were further validated on the testing database generated by other researchers. Direct application ab initio potential to condensed-phase is limited by the accuracy of quantum mechanical methods, as well as approximations and simplifications made in classical potentials. Thus the most effective approach to extend ab initio potentials to condensed-phase is to directly incorporate accurate liquid thermodynamic properties that are widely available. Following this strategy, we have derived a new water model based on the AMOEBA+ framework, by incorporating both ab initio data and experimental thermodynamic properties. The optimization against liquid properties lead to subtle changes in the parameters, however, the agreement with initial gas-phase SAPT2+ EDA remains excellent for a relatively simple classical potential. To develop a general AMOEBA+ FF beyond water, we will extend this approach to other molecular species in the future, where the AMOEBA+ parameters derived in this study will be further refined by using additional cluster QM data and well established experimental properties. This will require extensive molecular simulations that can be achieved by using high-performance Tinker simulation platforms, including Tinker-HP⁸⁹ and Tinker-OpenMM.⁸⁸ We believe that the new classical mechanics model and approach presented in this work has the potential to fundamentally advance the general applicability of classical force fields.

Figure 4.

Illustration of geometrical parameters for equilibrium water dimer. Angle θ is the angle between the extended O⋯O vector and the bisector of the ∠HOH angle of the hydrogen-bond acceptor molecule.

Figure 6.

Liquid water density at temperatures ranging from 265 K to 369 K and under atmospheric pressure (1 atm) calculated with AMOEBA+ model. The standard deviations for the simulated density are in 0.15~0.47 kg·m⁻³ and not shown on the plot. Experimental data are taken from reference¹²³.

Figure 7.

Liquid water enthalpy of vaporization at temperatures ranging from 265 to 369 K and under atmospheric pressure (1 atm) calculated with AMOEBA+ model. The standard deviations for the simulated ΔH_vap are in 0.03~0.06 kJ·mol⁻¹ and not shown on the plot. Experimental data are taken from reference¹²⁴.

Figure 9.

Liquid water isothermal compressibility at temperatures ranging from 265K to 369K and under atmospheric pressure (1 atm) calculated with AMOEBA+ model. Experimental data are taken from reference 123.

Figure 11.

Liquid water heat capacity at temperatures ranging from 265 to 369 K and under atmospheric pressure (1 atm) calculated with AMOEBA+ model. C_p values were calculated using both the fluctuation formula and differential equation. Quantum corrections were included in both approaches. The standard deviations for the C_p calculated by fluctuation formula are in 0.6~1.4 cal·mol⁻¹·K⁻¹ and not shown in the plot. Experimental data are taken from reference 124.

Table 4.

Validation of the accuracy and transferability of AMOEBA+ on additional SAPT data. RMSEs (in kcal·mol⁻¹) of AMOEBA+ against SAPT2+(3) are reported for individual energy components.

Database	Relative intermolecular distances	Data points	RMSEe
			Eele	Eind	EydW	Einter
S14a	1.2, 1.5, 2.0	42	0.65	0.18	0.21	0.82
S59b	1.25, 1.5, 2.0	177	0.24	0.14	0.21	0.44
Ionicc	1.0, 1.05, 1.1, 1.25, 1.5, 2.0	54	1.21	1.63	1.69	2.00
X6d	1.0, 1.05, 1.1, 1.25, 1.5, 2.0	36	0.35	0.05	0.40	0.53
Overall	1.0~2.0	309	0.60	0.69	0.74	0.97

^aSubset of S22×5;¹⁰³

^bSubset of S66×8 with 1.2 and above distances;¹⁰⁵

^cSubset of “Ionic” with 1.0 and above distances;¹⁰⁶

^dSubset of X40 with 1.0 and above distances;¹⁰⁴

^eSAPT2+(3) energy decomposition data are taken from reference 107.