Neural Network Corrections to Intermolecular Interaction Terms of a Molecular Force Field Capture Nuclear Quantum Effects in Calculations of Liquid Thermodynamic Properties

Abstract

We incorporate nuclear quantum effects (NQE) in condensed matter simulations by introducing short-range neural network ($\text{NN}$) corrections to the ab initio fitted molecular force field ARROW. Force field $\text{NN}$ corrections are fitted to average interaction energies and forces of molecular dimers, which are simulated using the Path Integral Molecular Dynamics (PIMD) technique with restrained centroid positions. The $\text{NN}$-corrected force field allows reproduction of the NQE for computed liquid water and methane properties such as density, radial distribution function (RDF), heat of evaporation (HVAP), and solvation free energy. Accounting for NQE through molecular force field corrections circumvents the need for explicit computationally expensive PIMD simulations in accurate calculations of the properties of chemical and biological systems. The accuracy and locality of pairwise $\text{NN}$ NQE corrections indicate that this approach could be applicable to complex heterogeneous systems, such as proteins.

Graphical Abstract

INTRODUCTION

All-atom molecular simulations are readily accessible and provide a very detailed description of chemical and biological systems, allowing computation of quantities of interest such as ligand binding free energies and rates of chemical reactions. The practical usefulness of molecular simulations is often limited by the insufficient accuracy of model Hamiltonians describing complex atomic interactions. One of the key ingredients toward achieving this accuracy is including nuclear quantum effects (NQE).^1–4 The traditional path integral approach has a significant computational cost compared with standard molecular dynamics. As a result, various methods have been developed, for example, in refs 5–9, to reduce the overhead of modeling NQEs to classical molecular dynamics (MD). In this work, we present a method to encapsulate some of the NQE in an effective intermolecular Hamiltonian term, encoded by a neural network, thus enabling more accurate predictions without the Path Integral Molecular Dynamics (PIMD) overhead. The required local corrections to the intermolecular interaction energies can be learned from PIMD calculations of dimers only. Combined with an accurate ab initio force field,^10,11 the effective hybrid Hamiltonian produces accurate predictions of condensed matter properties from a classical MD simulation.

Recently, there has been a resurgence of models that are derived from ab initio quantum computations, using analytical expressions,¹⁰ neural networks,^12,13 or both.¹¹ Molecular force fields (FF) derived from ab initio quantum chemical data describe molecular system potential energy surfaces (PES) with very high fidelity.^10,14–18 However, to reliably compute condensed matter properties in statistical simulations with accurate FFs, NQE must be taken into account^1–3 explicitly. (Most popular molecular force fields account for NQE implicitly as their parameters are fitted to reproduce experimental observables such as liquid density, heat of evaporation (HVAP), and solvation free energies.) For example, the errors of computed hydration free energy values for common neutral solutes are reduced from 0.8 to 0.22 kcal/mol when the PIMD technique was used in calculations with the ARROW FF.^3,10

NQE are usually modeled via the Path Integral Molecular Dynamics (PIMD) technique, which is significantly computationally more expensive than classical MD. The PIMD method models the molecular system via multiple replicas, with each atom represented by a necklace of beads connected by harmonic springs (ring-polymer). While up to 128 replicas are needed for convergence of NQE for some thermodynamic properties such as heat capacity, 8 replicas are usually sufficient to capture most of NQE for properties dependent mostly on intermolecular energies, such as solvation and binding free energies.^3,10,19,20 At room temperature, the dynamics of molecular systems are close to the classical limit, and beads representing atoms in PIMD simulations are localized in small volumes of $\mathrm{0.1-0.2}\hspace{0.17em}\text{Å}$ across (corresponding to quantum uncertainty in atom positions). We expect that approximating the statistics of the full bead necklaces by those of the centroid may provide an accurate description of the statistical properties of ring-polymers and NQE effects.

Many methods have been developed to encompass NQE while preserving the classical description of molecular dynamics. One of these is the centroid molecular dynamics (CMD) method devised by Voth and his co-workers.^21,22 In this approach, the dynamics of a molecular system are represented by the motion of atom centroids in the effective mean-field potential created by ring-polymer beads.⁵ Computing the effective centroid forces in each point of a CMD trajectory is expensive, and thus several approximate techniques have been developed to speed up the calculations.^23,24 Another approach to account for NQE in classical molecular dynamics simulations involves the Generalized Langevin Equation (GLE)-based frequency-dependent thermostats,^25,26 perturbed path-integral approximations,²⁷ ring-polymer contraction,⁵ and multiple time stepping.²⁸

Machine learning techniques are increasingly used to describe molecular potential surfaces.^29–35 Neural networks are able to accurately fit very complex multidimensional functions and are thus uniquely suited to describe the intricate dependence of molecular energy on atomic coordinates. $\text{NN}$ potentials were used successfully to derive coarse-grained potentials^36,37 that described the free energy of molecular systems in the space of reduced coordinates. A derivation of an effective centroid energy functional can also be considered as a coarse-graining of ring-polymer PIMD simulations. Coarse-graining theory of PIMD simulations has been recently reported in refs 38,39. Derivations of effective centroid potentials for chemical systems using analytical functions and neural network based on atom-centered descriptors have been reported previously in^8,9,24,39,40

Here, we focus on an accurate description of intermolecular interactions that play a dominant role in determining properties of interest in biochemistry such as ligand binding free energies and solvation free energies. We expect the PIMD intermolecular interactions to be numerically close to intermolecular energies evaluated in centroid coordinates, with corrections rapidly decaying with increasing atom–atom distances. To fit NQE corrections to intermolecular energies, we used a neural network ($\text{NN}$) potential correction defined with descriptors that describe the relative geometries of intermolecular atomic contacts.

Because of the localized nature of NQE, we hypothesize that corrections to intermolecular energies and forces, trained on PIMD simulations of molecular dimers, can accurately model NQE. Atom-centered machine learning methods achieved a good transferability of quantum corrections to centroid forces across phases⁹ or learning them using very short distance cutoffs.⁴⁰ Intermolecular interactions are relatively small perturbations of total energies of molecular systems; therefore, we expect that NQE in intermolecular interactions can be adequately described with even more local descriptors than those used for total potential energies and forces.

Below we describe a derivation of intermolecular $\text{NN}$ NQE corrections to the ARROW FF for liquid water and methane. Water was chosen because of its obvious importance and the large amount of literature on its NQE and their modeling. However, in water, many of the NQE offset and partially cancel each other.^1,41 We therefore also model methane because its NQE are fairly straightforward and also because many of its NQE come from the change of behavior of bonded interactions, which our technique does not explicitly embrace. We were curious if an effective intermolecular Hamiltonian can reproduce the energetic effects of an intramolecular perturbation. As NQE in both systems are significant as was shown in the previous studies,^3,15 the two liquids provide a good proof of principle.

The parameters of ARROW FF are fitted exclusively to high-level ab initio quantum chemical data for monomers, dimers, and multimers of small molecules. We previously reported³ that the ARROW FF allowed accurate calculations of solvation free energies of neutral molecules, provided NQE are taken into account by PIMD calculations using at least 8 replicas. We now show that $\text{NN}$ corrections to the ARROW FF accurately reproduce the NQE in calculations of the structural and thermodynamic properties of liquid water and methane. $\text{NN}$ corrections allow the construction of a molecular FF from high-level quantum chemical data, which is inclusive of NQE and does not require PIMD calculations for accurate predictions of properties of interest. While the ARROW FF is polarizable and a multipolar force field, we note that it may not capture small effects related to electronic degrees of freedom.^42–44

METHODS

An encapsulation of PIMD corrections in a neural network has been discussed recently,³⁹ and so it is worthwhile to explain why we choose to do this in an alternative manner. First and foremost, we recently published¹¹ a hybrid $\text{NN}$-analytical model of intermolecular interactions, parametrized from first-principles, which achieves chemical accuracy for prediction of properties of diverse molecular systems. The structure of the $\text{NN}$ term and the choices made in designing it are described fully there. In short, the construction is computationally economical, naturally aligns with the pairwise force formulation of our FF model, and can be enabled in small subsets of the chemical interaction space. The current work is an application of the same techniques to encapsulate NQE and its specific purpose is to be consistent with our overall methodology. Focusing on intermolecular interactions was a necessary and conscious choice for Illarionov et al., and in this work, we want to see if the same choice of simulation mechanisms can be applied to encoding PIMD in a classical effective Hamiltonian. A diagram of the interaction model is shown in Figure 1, along with a schematic of the neural network that computes the NQE correction.

Figure 1.

(a) Diagram of the intermolecular interaction fingerprint. $\text{A}$ and $\text{B}$ are the interacting atoms, $\text{X}$ is located at the midpoint between $\text{A}$ and $\text{B}$, and the atom pair symmetry functions (APSF) are atomic symmetry functions centered at $\text{X}$. The location of $\text{X}$ automatically symmetrizes the construction with respect to the $\text{A}\leftrightarrow \text{B}$ permutation. The APSF summations (blue dashed lines) are over neighborhoods of both $\text{A}$ and $\text{B}$. The fingerprint is fed into an ($\text{AB}$) specific neural network to produce an energy correction to the ARROW interaction energy between $\text{A}$ and $\text{B}\hspace{0.33em}(\text{Δ}{E}_{\text{AB}})E_{\text{PIMD}}=E_{\text{FF}}+E_{\text{NN}}$. (b) Diagram of the neural network term for pair interactions between two water molecules. Each pair interaction fingerprint is fed into its corresponding trained neural network (e.g., HH, HO, and OO in this case). The output is then smoothly truncated by a TensorFlow $\lambda$-layer to zero beyond a cutoff distance. All of the individual pair energy contributions are then summed to a final dimer energy correction $\text{Δ}{E}_{\text{NN}}$, which is then added to the energy $E_{\text{FF}}$ output by the analytical force field to produce the total energy.

Spherical Bessel ($\text{SB}$) descriptors were assigned to intermolecular atom pairs. Here, a pair between the oxygen atoms of water ($\text{A}$) and acetaldehyde ($\text{B}$) is shown. $\text{X}$ is the midpoint of the $\text{A-B}$ atom pair, which is the reference point for computed $\text{SB}$ descriptors. $\text{SB}$ descriptors involve a sum over atoms A and B and their immediate surroundings (atoms covalently bound to $\text{A}$ and $\text{B}$ and if $\text{A}$ or $\text{B}$ has only one bond, the next layer of covalent neighbors is included). Feedforward $\text{NN}$ having $\text{SB}$ descriptors as the input layer compute the contribution of atom pairs $\text{A-B}$ to the intermolecular energy NQE corrections.

The structure of the neural network is illustrated in Figure 1b. We constructed the $\text{NN}$ using the Keras TensorFlow functional API that predicts changes in the intermolecular energies and forces of the molecular dimers due to $\text{NQE}$ (${\text{Δ}}_{\text{NQE}}{E}_{\text{intermol}}\left({\overrightarrow{r}}_{\text{dimer}}\right)$ and ${\text{Δ}}_{\text{NQE}}{E}_{\text{intermol}}\left({\overrightarrow{r}}_{\text{dimer}}\right)$). The $\text{NN}$ component of the intermolecular interaction energy is decomposed into a sum of interactions between pairs of atoms ($\text{AB}$) with each atom of the pair residing on a different monomer and having a separation distance of less than $R_{\text{cutoff}}=5\text{Å}$.

$${\text{Δ}}_{\text{NQE}}{E}_{\text{intermol}}\left({\overrightarrow{r}}_{\text{dimer}}\right)\approx E_{\text{NN}}=\sum _{\text{AB}}{E}_{\text{NN}}^{\text{AB}}\text{|}{\overrightarrow{r}}_{\text{A}}-{\overrightarrow{r}}_{\text{B}}\text{|}=r_{\text{AB}}<R_{\text{cutoff}}=5\hspace{0.17em}\text{Å}$$ (1)

The interaction center of the pair is chosen to be at the geometric midpoint $\text{X}$ between $\text{A}$ and $\text{B}$. This symmetrizes the construction to the permutation $\text{A}\leftrightarrow \text{B}$ and permits the use of atomic pair symmetry functions (APSF) describing atom pair geometry centered at $\text{X}$. Figure 1 shows a diagram of the APSF function for the intermolecular atom pair $\text{A-B}$. The specific basis we chose is Spherical Bessel ($\text{SB}$) descriptors.^45,46

$$p_{nl}=\frac{2l+1}{4\pi }\sum _{kj}{g}_{n-l,l}\left(\left|{\overrightarrow{r}}_j-\overrightarrow{X}\right|\right)g_{n-l,l}\left(\left|{\overrightarrow{r}}_k-\overrightarrow{X}\right|\right)P_l(\cos \hspace{0.17em}{\gamma }_{jk})$$ (2)

where $0\le n\le n_{\text{max}}$, $0\le l\le l_{\text{max}}=n_{\text{max}}$.

The atomic set for the convolution ($j$,$k$ indices in (2)) includes atoms covalently bonded to atoms A and B and the next covalent neighbors if $\text{A}$ or $\text{B}$ is a terminal atom. We employ 15 $\text{SB}$ descriptors $\left(n_{\max }=l_{\max }=4\right)+1\text{/}r$ (reverse distance descriptor) for each interatomic contact. This number of descriptors was chosen to create a balance between accurately describing pair geometries and avoiding overtraining for the systems considered (the MAE of the training set is similar to that of the validation set).

The $\text{NN}$ model for the atom pair intramolecular energy contribution was composed of 3 layers: 2 dense layers with 20 nodes each and the tanh activation function and a $\lambda$-layer that smoothly scales the $\text{NN}$ correction to zero at interatomic contacts greater than $4\hspace{0.17em}\text{Å}$.

$$\begin{gathered} \left.E_{\text{NN}}^{\text{AB}}=\sum f_2\left(W_2\left(f_1\left(W_1{\overrightarrow{x}}_{\text{AB}}\right)+{\overrightarrow{b}}_1\right)\right)+{\overrightarrow{b}}_2\right) \\ \times \frac{1}{2}\left(1+\tanh \left(-\alpha \left(r_{\text{AB}}-R_0\right)\right)\right) \end{gathered}$$ (3)

where ${\overrightarrow{x}}_{\text{AB}}$ is a vector of atom pair descriptors $p_{nl}$ eq 2; $W_L$, ${\overrightarrow{b}}_L$, and $f_L$ are the matrix of weights, biasing vector, and activation functions (tanh) for the $L\text{th}$ layer of the $\text{NN}$; $r_{\text{AB}}$ is the atom–atom distance of the atom pair; and $R_0$ is the scaling layer cutoff distance ($4\hspace{0.17em}\text{Å}$) and the distance decay parameter $\alpha =5\hspace{0.17em}{\text{Å}}^{-1}$.

Atomic forces due to $\text{NN}$ corrections were computed using an automatic differentiation utility provided by the Tensor-Flow library and analytical derivatives of $\text{SB}$ descriptors

$$F_{js}^{\text{NN}}=\frac{\partial }{r_{js}}\sum _{\text{AB}}{E}_{\text{NN}}^{\text{AB}}=\sum _{\text{AB}}\frac{\partial p_{nl}^{\text{AB}}}{r_{js}}\frac{\partial }{\partial p_{nl}}{E}_{\text{NN}}^{\text{AB}}$$ (4)

where $r_{js}$ is a Cartesian coordinate $s$ of atom $j$ that is involved in $p_{nl}^{\text{AB}}$ descriptor described by eq 2.

Computing Changes of Dimer Intermolecular Energies and Forces due to NQE.

Ten thousand dimers were sampled from PIMD R8 (8 replicas) trajectories of liquid water in the ARROW FF at 298 K. For liquid methane, 2400 dimers were sampled from PIMD R8 trajectories at 120 K. The dimers were chosen to have an intermolecular separation of less than $4.5\hspace{0.17em}\text{Å}$.

Figure S1 shows the distribution of the closest intermolecular distances for water and methane dimers used in calculations. For each molecular dimer, we performed PIMD R8 calculations where the initial dimer atom coordinates were used as positional restraints for atom centroids. A rigid force constant of 1000 kcal/mol/ ${\text{Å}}^2$ was used for centroid restraints to ensure that the average centroid coordinates were close to the atom coordinates of the reference dimers. We ran 1 ns PIMD simulations for each water dimer and corresponding monomers. 5 ns PIMD simulations were run for methane dimers and monomers. Averaging interaction energies and forces along PIMD trajectories while keeping atom centroids restrained allows us to obtain effective changes of intermolecular interactions due to NQE for the given dimer coordinates.

For each dimer, we computed changes of intermolecular energies and forces due to $\text{NQE}-{\text{Δ}}_{\text{NQE}}{E}_{\text{intermol}}\left({\overrightarrow{r}}_{\text{dimer}}\right)$ and ${\text{Δ}}_{\text{NQE}}{\overrightarrow{F}}_{\text{intermol}}\left({\overrightarrow{r}}_{\text{dimer}}\right)$ with the following procedure:

1. PIMD intermolecular energy and forces of the dimer are computed by subtracting averaged PIMD potential energies and forces (including the bead spring term which arises from the quantum kinetic energy operator) of the dimer and averaged PIMD potential energies and forces of the monomers with restrained centroid coordinates ${\overrightarrow{r}}_{\text{dimer}}$

   $$E_{\text{intermol}}^{\text{PIMD}}\left({\overrightarrow{r}}_{\text{dimer}}\right)={〈E_{\text{dimer}}^{\text{pot}}〉}_{\text{PIMD}}-\left({〈E_{\text{monomer 1}}^{\text{pot}}〉}_{\text{PIMD}}\right.+\left.{〈E_{\text{monomer 2}}^{\text{pot}}〉}_{\text{PIMD}}\right)$$ (5)

   $${\overrightarrow{F}}_{\text{intermol}}^{\text{PIMD}}\left({\overrightarrow{r}}_{\text{dimer}}\right)={〈{\overrightarrow{F}}_{\text{dimer}}〉}_{\text{PIMD}}-\left({〈{\overrightarrow{F}}_{\text{monomer 1}}〉}_{\text{PIMD}}\right.+\left.{〈{\overrightarrow{F}}_{\text{monomer 2}}〉}_{\text{PIMD}}\right)$$ (6)
2. Reference ARROW FF intermolecular energy and forces of the dimer are computed by subtracting FF energies and forces of the dimer and those of monomers at averaged centroid coordinates in PIMD calculations

   $$E_{\text{intermol}}^{\text{FF}}\left({\overrightarrow{r}}_{\text{dimer}}\right)=E_{\text{dimer}}^{\text{FF}}\left({\overrightarrow{r}}_{\text{dimer}}\right)-\left(E_{\text{monomer}\hspace{0.17em}1}^{\text{FF}}\left({\overrightarrow{r}}_{\text{monomer 1}}\right)\right.+\left.E_{\text{monomer}\hspace{0.17em}2}^{\text{FF}}\left({\overrightarrow{r}}_{\text{monomer 2}}\right)\right)$$ (7)

   $${\overrightarrow{F}}_{\text{intermol}}^{\text{FF}}\left({\overrightarrow{r}}_{\text{dimer}}\right)={\overrightarrow{F}}_{\text{dimer}}^{\text{FF}}\left({\overrightarrow{r}}_{\text{dimer}}\right)-\left({\overrightarrow{F}}_{\text{monomer 1}}^{\text{FF}}\left({\overrightarrow{r}}_{\text{monomer 1}}\right)\right.+\left.{\overrightarrow{F}}_{\text{monomer 2}}^{\text{FF}}\left({\overrightarrow{r}}_{\text{monomer 2}}\right)\right)$$ (8)
3. Changes of intermolecular energies and forces due to NQE were then computed as

   $${\text{Δ}}_{\text{NQE}}{E}_{\text{intermol}}\left({\overrightarrow{r}}_{\text{dimer}}\right)=E_{\text{intermol}}^{\text{PIMD}}\left({\overrightarrow{r}}_{\text{dimer}}\right)-E_{\text{intermol}}^{\text{FF}}\left({\overrightarrow{r}}_{\text{dimer}}\right)$$ (9)

   $${\text{Δ}}_{\text{NQE}}{\overrightarrow{F}}_{\text{intermol}}\left({\overrightarrow{r}}_{\text{dimer}}\right)={\overrightarrow{F}}_{\text{intermol}}^{\text{PIMD}}\left({\overrightarrow{r}}_{\text{dimer}}\right)-{\overrightarrow{F}}_{\text{intermol}}^{\text{FF}}\left({\overrightarrow{r}}_{\text{dimer}}\right)$$ (10)

We thus isolated the NQE corrections to intermolecular dimer energies and forces, which were then fitted by a $\text{NN}$.

The dimer data set was split into training and validation sets (80/20%). We did not use a separate “set aside” test set, as the final quality of the model was evaluated on the computed bulk properties of liquid water and methane. $\text{NN}$ training used the weighted mean square error (MSE) loss function $f_{\text{loss}}$ for predicted NQE corrections to intermolecular energies and atomic forces.

$$f_{\text{loss}}=\frac{1}{N_{\text{dimer}}}\times \sum _{\text{dimer}}\left[\left(1-w_{\text{F}}\right)\right.\times {\left(E_{\text{NN}}-{\text{Δ}}_{\text{NQE}}{E}_{\text{intermol}}\left({\overrightarrow{r}}_{\text{dimer}}\right)\right)}^2\left.+w_{\text{F}}\times \sum _{ij}{\left(F_{ij}^{\text{NN}}-{\text{Δ}}_{\text{NQE}}{F}_{\text{intermol}}{\left({\overrightarrow{r}}_{\text{dimer}}\right)}_{ij}\right)}^2\right]$$ (11)

Here, $w_{\text{F}}$ is the relative weight of intermolecular forces in the loss function. In this work, $w_{\text{F}}=0.05$, see Figure S7 in the Supporting Information (SI) for more details on $w_{\text{F}}$ choice.

The Adam stochastic optimizer was used with a batch size of 256 and a patience value of 300, in the optimization of the loss function $f_{\text{loss}}$ via the back-propagation algorithm.

MD Simulations Details and Liquid Properties Calculations.

To perform condensed matter simulations, we implemented calculations of $\text{NN}$ NQE corrections to the ARROW FF in our ARBALEST program.^3,10,18,47 $\text{NN}$ corrections to energies and forces are computed for each water dimer within a $4.5\hspace{0.17em}\text{Å}$ cutoff. $\text{SB}$ descriptor computations were found to be the most expensive step of calculations of $\text{NN}$ corrections and were implemented to run on NVIDIA GPUs using CUDA libraries. The intermolecular energies and forces were also computed on the GPUs, while the intramolecular terms and integration of the equations of motion were processed on the CPUs.

Boxes of water or methane of $\text{32}\times \text{32}\times \text{32}\hspace{0.17em}{\text{Å}}^3$ sizes were simulated. The Particle-Mesh Ewald (PME) algorithm⁴⁸ was used to compute the electrostatic interactions. A grid size mesh of $1\hspace{0.17em}\text{Å}$ and a fifth power spline interpolation order was used to compute the inverse PME sum. Nine $\text{Å}$ cutoffs were used to compute the direct PME sum as well as the exchange and dispersion interactions. Bulk corrections were applied to account for long-distance dispersion interactions. Solvation free energies were computed by decoupling the interactions between the solute and the solvent molecules with the Bennett acceptance ratio (BAR)⁴⁹ method. Electrostatic, exchange-repulsion, and dispersion interactions were switched off simultaneously using a lambda-dependent scaling (a scaling factor power of 2 was used) and a soft-coring algorithm (maximal soft-coring radius $=1.5\hspace{0.17em}\text{Å}$, soft-coring radius scaling factor power = 1). Eleven equally spaced $\lambda$ points were used to decouple the solute–solvent intermolecular interactions.

Energy minimization (1000 steps) using the steepest descent algorithm was initially performed on the simulated systems. This was followed by a 50 ps equilibration and a 1 ns production run in the isothermal–isobaric ensemble (NPT). The temperature was maintained at 298 K (for water) and 120 K (for methane) using a Langevin thermostat⁵⁰ (relaxation time = 0.1 ps). The pressure was maintained at 1 atm with a Berendsen barostat⁵¹ (relaxation time = 0.5 ps). A Multiple Time Step (MTS) algorithm⁵² was used to integrate the equations of motion both for the classical and path-integral runs. Bonded and PIMD^53,54 bead interactions were integrated with a time step of 0.125 fs, while intermolecular interactions were integrated with a time step of 2 fs. The simulation protocol was the same for the ARROW and ARROW $\text{NN}$ calculations.

RESULTS AND DISCUSSION

First, we found that NQE induce significant alterations in intermolecular interactions within water dimers, especially when the dimers are in close proximity. These alterations manifest as substantial changes in energies, reaching up to 1 kcal/mol, and forces, with deviations of up to 7 kcal/mol/ ${\text{Å}}^2$ (see Figure 2a,b). As the distance between the molecules increases, the influence of the NQE rapidly diminish.

Figure 2.

NQE (PIMD-MD) in intermolecular energies eq 9 (a) and forces eq 10 (b) of water dimers computed by PIMD R8 (green) and corrected by $\text{NN}$ (red) as a function of the closest distance between water monomers (the validation set). $\text{NN}$ corrections to dimer intermolecular energies (c) and forces (d) vs NQE changes. NQE corrections of intermolecular interactions of water dimers rapidly decay with intermolecular distance and are accurately reproduced by $\text{NN}$ based on $\text{SB}$ atom pair descriptors.

$\text{NN}$ corrections are detailed in the Methods section are able to accurately fit changes of intermolecular interactions due to NQE, as evidenced by the data presented in Figure 2a–d. The MAE of the $\text{NN}$ fitted to NQE corrections to intermolecular energies was around 0.03 kcal/mol for the training set and 0.04 kcal/mol for the validation set. The convergence of the MAE values for the intermolecular energy fit for one of the $\text{NN}$ optimization runs is depicted in Figure S2 in the Supporting Information.

For the $\text{NN}$ fit to forces, the MAE was consistent at 0.11 kcal/mol/ ${\text{Å}}^2$ for both the training and the validation sets. Notably, when the $\text{NN}$ fitting process exclusively utilized energies $(w_{\text{F}}=0.0)$, the MAE for the $\text{NN}$-corrected intermolecular forces increased by approximately 3-fold, reaching 0.3 kcal/mol/ ${\text{Å}}^2$ (see Figure S7 in the SI).

The outcomes of computations of bulk properties of water with the ARROW FF, enhanced by $\text{NN}$ corrections that account for NQE in intermolecular interactions, are displayed in Figure 3. PIMD calculations of water with the ARROW FF reveal minor variations of water density that are non-monotonic with the number of PIMD replicas utilized (Figure 3a). NQE $\text{NN}$ corrections to the FF, derived from dimer PIMD calculations utilizing 8 and 32 replicas, yield results that closely align with the water density derived from bulk PIMD simulations with the corresponding number of replicas (1.029 g/mL compared to 1.026 g/mL for R8, and 1.023 vs 1.020 g/mL for R32).

Figure 3.

Water properties at $T=298\hspace{0.17em}\text{K}$, $P=1\hspace{0.17em}\text{atm}$ computed with ARROW FF (MD (blue), PIMD with 4, 8, 16, and 32 replicas (lime to dark green)) and with ARROW FF + NN PIMD corrections fit to PIMD R8 (salmon) and PIMD R32 (red) dimer calculations (a) Density. (b) O–O RDF. (c) Evaporation enthalpy (HVAP). (d) Solvation free energy. Computed O–O RDF, HVAP, and $\text{Δ}G$ solvation changes due to NQE are accurately reproduced by $\text{NN}$ NQE corrections to the ARROW FF.

It is worth emphasizing that the ARROW FF was fitted to quantum chemical data and its parameters were not fine-tuned to replicate water density, unlike the typical approach in widely adopted force fields. In this particular study, we focus on the $\text{NN}$ description of NQE in intermolecular interactions.

In a recent study,⁵⁵ we effectively utilized $\text{NN}$ to improve the fitting of quantum chemical data. Consequently, the resultant ARROW $\text{NN}$ force field demonstrates a heightened ability to accurately reproduce the experimental water density and various other properties of liquids, surpassing the capabilities of the original ARROW FF.

In Figure 3b, the water oxygen–oxygen RDF is depicted. The RDF is calculated using the ARROW force field (FF) with MD and PIMD simulations employing 8 replicas and 32 Replicas. Additionally, the RDF was obtained using the ARROW FF with NQE $\text{NN}$ corrections. NQE $\text{NN}$ corrections were fitted to a combination of energies and forces $(w_{\text{F}}=0.05)$. The dependence of the MAE of NQE corrections of energies and forces on the parameter $w_{\text{F}}$ is shown in Figure S7. The $\text{NN}$ corrections successfully capture the alterations in the oxygen–oxygen RDF stemming from NQE. Additional RDF’s for the O–H and H–H pairs in water are shown in Figures S3–S4 in the SI. O–H and H–H RDFs are not reproduced as well as the O–O RDF, with the RDF obtained in direct PIMD calculations showing much lower and wider peaks than in MD simulations with $\text{NN}$ NQE corrections. Here, the limitations of our approach are apparent, as quantum delocalization of hydrogen atoms does not appear directly in MD simulations with $\text{NN}$ NQE corrections, but is only reflected indirectly in the statistics of atom centroids.

In Figure 3c,d, the computed results for NQE in the water enthalpy of evaporation (HVAP) and hydration free energy under standard conditions ($T=298\hspace{0.17em}\text{K}$ and $P=1\hspace{0.17em}\text{atm}$) are presented. The computations were carried out using various setups: the ARROW FF in conjunction with MD, PIMD simulations employing 4, 8, 16, and 32 replicas, and the ARROW FF augmented with NQE $\text{NN}$ corrections fitted to dimer PIMD calculations using 8 and 32 replicas. Fairly large NQE changes in HVAP ∼ 1.05 kcal/mol and in solvation free energy 0.7 kcal/mol of this water model are correctly reproduced by MD $\text{NN}$-corrected FF. The RMS deviations of the computed HVAP and solvation free energy values computed with $\text{NN}$ NQE corrections obtained in multiple independent $\text{NN}$ fits were about 0.03 and 0.06 kcal/mol, respectively.

Remarkably, the calculations using the augmented ARROW FF with $\text{NN}$ corrections (fitted using PIMD R32 calculations of water dimers) closely reproduce the PIMD R32 ARROW FF results with an accuracy of approximately 0.03 kcal/mol. This deviation is notably smaller than the difference of 0.08 kcal/mol between HVAP values computed by using PIMD R8 and R32 (10.88 and 10.96 kcal/mol, respectively). Therefore, it can be inferred that simulations utilizing $\text{NN}$ corrections derived from water dimer PIMD calculations with a higher number of replicas (such as 32) are expected to yield greater accuracy compared with direct PIMD calculations involving a lower number of replicas (such as 8).

To explore the performance of $\text{NN}$ NQE interactions for other water FFs, we fittted $\text{NN}$ corrections running restrained centroid PIMD calculations for the same 10 000 set of water dimers using a simple flexible TIP3P-like water model. Some computational results using fitted $\text{NN}$ corrections for this water model are presented in Figure S8 in the SI. The accuracy with which NQE $\text{NN}$ corrections for the TIP3P-like water model reproduce results of direct bulk PIMD calculations is similar to the results obtained for ARROW FF water. We can conclude from this fact that the accuracy of $\text{NN}$ NQE corrections is likely to be weakly dependent on the underlying FF model.

In order to validate the applicability of our method across various systems, we extended our computations to liquid methane at a temperature of 120 K. Our earlier study³ demonstrated pronounced NQE in this particular system, leading to approximately a 5% alteration in equilibrium density, HVAP, and solvation free energies.

As shown in Figure 4a–d, the results underscore that the NQE corrections to methane dimer intermolecular energies and forces are approximately 5 times smaller in magnitude compared with their counterparts in water dimers. The corrections amount to a maximum of 0.15 kcal/mol for energies and 1.5 kcal/mol/ ${\text{Å}}^2$ for forces. Furthermore, achieving convergence in the average intermolecular PIMD energies and forces required extended simulation times for methane dimers. Specifically, 5 ns long trajectories were necessary for methane, whereas only 1 ns was sufficient for water dimers to attain convergence.

Figure 4.

NQE (PIMD-MD) changes in intermolecular energies (a) and forces (b) of methane dimers computed by PIMD R8 (green) and corrected by $\text{NN}$ (red) as a function of the closest distance between methane monomers (validation set). NN NQE corrections to intermolecular energies (c) and forces (d) of methane dimers vs NQE changes.

An illustration of the computational outcomes of the NQE on the properties of liquid methane at 120 K is shown in Figure 5. As seen in Figure 5a, using PIMD R8, there is a noticeable decrease in the density of liquid methane due to NQE, going from 0.437 to 0.416 g/mL. As the number of PIMD replicas increases to 16 and 32, the density further reduces to 0.413 and 0.411 g/mL, respectively, though with a diminishing rate of change. Methane density computed with the ARROW FF, enhanced with NQE corrections based on PIMD R8 dimer data, aligns well with the densities deduced from PIMD for matching number of replicas (0.417 vs 0.416 g/mL for R8 and 0.413 vs 0.410 for R32). Figure 5b demonstrates the effects of NQE on methane’s C–C RDF. Additional RDFs for the C–H and H–H pairs in methane are shown in Figures S5–S6 in the SI. As in the case of the water model, NQE effects on RDF involving hydrogen atoms are not fully reproduced in MD calculations with $\text{NN}$ NQE corrections, while heavy atom RDFs are fairly accurate.

Figure 5.

Liquid methane properties at $T=120\hspace{0.17em}\text{K}$ and $P=1\hspace{0.17em}\text{atm}$ computed with ARROW FF (MD) (blue), PIMD 4, 8, 16, and 32 replicas (lime-dark green), and with ARROW FF + NN NQE corrections fitted to dimer PIMD calculations with 8 (salmon) and 32 (red) replicas. (a) Density. (b) C–C RDF NN. (c) evaporation enthalpy (HVAP). (d) Solvation free energy. Ca. 5–6% changes in density, HVAP, and $\text{Δ}G$ solvation due to NQE are correctly reproduced by MD simulations with ARROW FF + NN correction.

As can be seen in Figure 5c,d for liquid methane, HVAP and solvation free energy changes due to NQE are reproduced by calculations with ARROW FF augmented by $\text{NN}$ NQE corrections. Average HVAP and solvation free energy values computed over multiple independently fitted $\text{NN}$ corrections give a better accuracy of reproducing results of direct PIMD calculations, while results given by individual $\text{NN}$ differ with RMS of ∼0.02 kcal/mol for HVAP and ∼0.03 kcal/mol for solvation free energies (shown by error bars in Figure 5a,c,d).

Calculations of $\text{NN}$ NQE based on atom pair $\text{SB}$ descriptors have been implemented in the ARBALEST program^10,18,47 to run on GPUs using CUDA and TensorFlow⁵⁶ libraries. With the chosen parameters of $\text{SB}$ descriptors, calculations with the ARROW FF augmented by $\text{NN}$ NQE corrections were about 3 times slower than calculations with the ARROW FF without $\text{NN}$. Specifically, the computation speed for $32\times 32\times 32\hspace{0.17em}{\text{Å}}^3$ of a water or methane box using a single GTX 1080-TI GPU was ∼1 ns/h for the ARROW FF and ∼0.35 ns/h for the ARROW FF with $\text{NN}$ NQE. This makes the method more computationally efficient than doing direct PIMD calculations with this FF.

CONCLUSIONS

We showed that $\text{NN}$ adjustments to the intermolecular energies and forces to the QM parametrized force field ARROW can accurately replicate NQE when calculating bulk properties of liquid water and methane in classical (1-bead) MD simulations. These $\text{NN}$ corrections employ Spherical Bessel descriptors linked to atomic pairs between interacting molecules and are trained on averaged intermolecular energies and centroid forces obtained in PIMD (RPMD) calculations of molecular dimers.

The method we have discussed shares several similarities with the techniques presented in refs 9,38–40, which employed the atom-centered descriptors and different types of neural network models to fit effective forces acting on centroids in CMD and RPMD calculations. It was shown that effective $\text{NN}$ potentials reproduced NQE effects for properties such as RDF, diffusion constants, pressure, and vibrational spectra. The authors of ref 40 emphasized the local nature of NQE corrections that allowed them to reduce the size of water clusters used in $\text{NN}$ training to 7 water molecules, while ∼30 water molecule clusters were used in ref 39. In contrast to the cited works, we fitted $\text{NN}$ corrections only to intermolecular energies and forces of molecular dimers, resulting in a more localized description of NQE effects. We also focus on NQE for properties that depend mostly on intermolecular interactions such as solvation and binding free energies that are focal points in drug development and material design. Given that these corrections provide a faithful representation of NQE for the analyzed properties, the technique could be suitably extended to complex heterogeneous systems like proteins. Here, $\text{NN}$ corrections can be individually fitted for the different interacting chemical species.

The procedure we described has several significant limitations. First, it is only applicable at or around a certain temperature, and the correction Hamiltonian will need to be retrained if a different temperature is desired. Second, as we employ intermolecular corrections only, the NQE inherently produced by intramolecular degrees of freedom, such as heat capacity, will not be properly reproduced. We note, however, that some intramolecular effects, such as increased fluctuation of bond lengths, are picked up by the effective intermolecular correction Hamiltonian. Third, centroid approximations are known to have difficulties at low temperatures,⁵⁷ and this correction inherits the deficiency. Fourth, as the statistics of centroid fluctuations are an approximation of the fluctuations of full path-integral replicas, the method is subject to the inaccuracies of this approximation. Fifth, the momentum distribution in the model will still remain Maxwell–Boltzmann, so properties dependent on proper quantum momentum could not be accurately computed. Nonetheless, given the limitations stated above, the method does offer a way to get, after some training cost, very accurate NQE corrections for observables of interest (Figures 3 and 5) at temperatures of importance to many biological and chemical applications at MD computational expense. Additionally, $\text{NN}$ corrections can be trained on dimer conformations only, which enables the tractability of training for diverse and heterogeneous systems.

Recently, we extended the ARROW FF with $\text{NN}$ corrections targeting intermolecular interactions and employing the same atom pair-centered Spherical Bessel descriptors described above.⁵⁵ $\text{NN}$ Corrections to the ARROW FF provided a significant improvement in the description of strong intermolecular interactions involving charged species and allowed very accurate calculations of ion solvation energetics and ligand binding free energies. By systematic refining of the ARROW $\text{NN}$ to incorporate NQE with the approach described in this work, we anticipate a quantitative description of complex chemical phenomena, such as protein–ligand binding and protonation equilibrium in classical MD simulations, that so far were not sufficiently reliable due to the deficiencies of the force fields used.

ABBREVIATIONS

MD

molecular dynamics

PIMD

path integral molecular dynamics

NQE

nuclear quantum effects

FF

force field

NN

neural network

RDF

radial distribution function

HVAP

enthalpy of vaporization