A deep potential model with long-range electrostatic interactions

Linfeng Zhang; Han Wang; Maria Carolina Muniz; Athanassios Z. Panagiotopoulos; Roberto Car; Weinan E

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A deep potential model with long-range electrostatic interactions

Linfeng Zhang, Han Wang, Maria Carolina Muniz, Athanassios Z. Panagiotopoulos, Roberto Car, Weinan E

arXiv:2112.13327v2Dec 26, 2021physics.chem-phphysics.comp-ph

TL;DR

Short-range DP models omit explicit long-range electrostatics, limiting their treatment of Coulomb-tail properties. The paper introduces DPLR, which adds Wannier-based Gaussian electrostatics to DP; it recovers missing long-range effects and extrapolates learned energy surfaces to larger systems. The approach is smooth and analytically differentiable, but its dynamics capability is limited by an assumption about the modeled trajectories.

Problem
Most DP-like models omit explicit long-range electrostatic interactions, limiting properties governed by the Coulombic tail, including long-wavelength polar-crystal behavior.
Method
DPLR combines standard short-range DP interactions with electrostatics from spherical Gaussian charges centered on ions and maximally localized Wannier distributions.
Results
DPLR recovers missing long-range effects, correctly extrapolates quantum-mechanical energy surfaces learned on smaller systems to larger systems, and reproduces size-dependent NaCl phonon behavior with partial LO-TO splitting recovery.
Takeaways & Limitations
The model provides a smooth, symmetry-preserving potential with analytical forces and virial for systems requiring explicit long-range electrostatics.
Takeaways & Limitations
The model's dynamics capability is limited by an assumption about the modeled trajectories.

Abstract

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Machine learning models for the potential energy of multi-atomic systems, such as the deep potential (DP) model, make possible molecular simulations with the accuracy of quantum mechanical density functional theory, at a cost only moderately higher than that of empirical force fields. However, the majority of these models lack explicit long-range interactions and fail to describe properties that derive from the Coulombic tail of the forces. To overcome this limitation we extend the DP model by approximating the long-range electrostatic interaction between ions (nuclei+core electrons) and valence electrons with that of distributions of spherical Gaussian charges located at ionic and electronic sites. The latter are rigorously defined in terms of the centers of the maximally localized Wannier distributions, whose dependence on the local atomic environment is modeled accurately by a deep neural network. In the deep potential long-range (DPLR) model, the electrostatic energy of the Gaussian charge system is added to short-range interactions that are represented as in the standard DP model. The resulting potential energy surface is smooth and possesses analytical forces and virial. Missing effects in the standard DP scheme are recovered, improving on accuracy and predictive power. By including long-range electrostatics, DPLR correctly extrapolates to large systems the potential energy surface learned from quantum mechanical calculations on smaller systems. We illustrate the approach with three examples, the potential energy profile of the water dimer, the free energy of interaction of a water molecule with a liquid water slab, and the phonon dispersion curves of the NaCl crystal.

I. INTRODUCTION

Local short-range ML potentials are efficient and often accurate, but omit explicit Coulomb interactions needed for several nonlocal properties. DPLR combines deep potential with Wannier-based long-range electrostatics to recover these effects while retaining smooth, conservative, efficient dynamics.

Motivation: Most ML potentials decompose the PES into atomic terms within a cutoff, giving linear scaling but neglecting long-range Coulomb forces between charged electrons and nuclei.This approximation can fail for cluster, interfacial, vapor-phase, and polar-crystal properties.
Motivation: Point-charge extensions add long-range electrostatics, but atomic charges are ambiguous because overlapping charge densities permit different partitioning results.Some charge-based models also lack rigorous definitions or cannot handle changing chemical bonds.
DPLR approach: DPLR splits the PES into standard short-range interactions and explicit long-range electrostatics represented by spherical Gaussian charges at ionic and Wannier electronic sites.The Wannier-center dependence on local environments is learned with deep neural networks.
DPLR approach: Wannier-centered distributions rigorously describe molecular dipoles and dipolar fluctuations, while conserving integer total charge and improving higher-moment behavior over atom-centered distributions.Atom-centered errors begin at the dipole level and are only conditionally convergent with system size.
DPLR approach: The resulting DPLR potential is physically meaningful, symmetry preserving, smooth, and provides analytical forces and virial while remaining computationally efficient.The model supports polarization fluctuations in finite and extended systems and response to externally applied fields.
Applications: For NaCl phonons, DPLR captures size-dependent dispersion and partially recovers longitudinal-transverse optical-mode splitting, unlike DP's size-independent modes.The recovered non-analytic contribution arises from long-range dipole-dipole interactions.

II. THEORY AND METHOD

The framework focuses on periodically repeated supercells, with finite systems treated through sufficiently large cells and neutrality required under periodic boundaries.

The method models extended systems using a periodically repeated supercell.
Finite systems can be represented this way when the supercell is large enough to neglect interactions between periodic images.
Periodic boundary conditions require the system to be electrically neutral.
The scheme could be extended to other choices of boundary conditions.

A. Electrostatic energy

DPLR represents ions and electronic sites with spherical Gaussian charges centered on ionic coordinates and Wannier centroids, then combines their electrostatic energy with short-range DP interactions. The approximation is smooth and efficient, while residual errors arise mainly from neglected long-range multipolar effects and limitations in uniquely assigning Wannier centers.

Electronic sites: Wannier centroids average maximally localized Wannier centers assigned to the same atom, providing an efficient representation of molecular electronic charge.For water, four Wannier centers are associated with each oxygen atom and combined into one centroid.
Scope and limitations: Unique Wannier-center assignment fails during electron transfer between atoms, requiring a significant generalization of the method.
Electrostatic representation: DPLR approximates electrostatics with spherical Gaussian charge distributions at ionic and Wannier-centroid sites.The ionic and electronic Gaussians integrate to their respective charges and share an adjustable spread parameter.
Model construction: The Gaussian electrostatic energy is combined with a short-range DP contribution that captures additional many-body effects within the cutoff radius.The DP component handles residual short-range interactions, leaving approximation errors associated with distances beyond the cutoff.
Efficient evaluation: The Gaussian electrostatic energy is evaluated efficiently in Fourier space because the smooth distributions yield rapidly convergent sums.The implementation uses particle-particle-particle-mesh methods for the applications discussed.

B. Deep Potential Long-Range model

DPLR combines a short-range DP energy with Gaussian-charge electrostatics whose Wannier-centroid dependence is environment-dependent, while retaining analytical forces and virial. Its energy conservation depends on accurate evaluation of the electrostatic sum.

Energy construction: DPLR defines the potential energy as a short-range contribution plus Gaussian-charge electrostatic energy.The electrostatic term is introduced separately from the short-range DP-like component.
Wannier-centroid dependence: Wannier centroids are assigned bijectively to nearby ions and predicted from each atom’s local environment within the DP cutoff.Their implicit coordinate dependence must be included when differentiating the energy.
Forces and virial: DPLR forces contain standard DP, electrostatic, Wannier-centroid, and environmental-dependence contributions.The first term is the standard DP force; subsequent terms account for electrostatics and the environment-dependent Wannier centroids.
Forces and virial: The virial combines short-range and electrostatic terms, including a correction for the nonlinear dependence of Wannier centroids on the cell tensor.This correction is needed because Wannier-centroid positions do not generally change linearly under cell deformation.

III. DEEP POTENTIAL LONG-RANGE MODEL FOR WATER

The water demonstration uses PBE-labeled DP, DW, and DPLR models to isolate whether DPLR captures long-range electrostatic effects missing from standard DP. The authors state that the functional choice is not critical for this demonstration and can be replaced during labeling.

Model construction: The water models are constructed using the PBE functional approximation, producing DP, DW, and DPLR models.The models are based on PBE density-functional-theory data.
Functional scope: PBE overestimates water hydrogen-bond strength and consequently misdescribes relative ice and water density at ambient pressure.This is identified as a limitation of the chosen exchange-correlation functional.
Purpose and scope: The study is designed to demonstrate that DPLR captures long-range electrostatic effects missing from the DP model.The authors explicitly distinguish this demonstration from constructing a state-of-the-art water model.
Functional scope: More accurate exchange-correlation functionals could be substituted for PBE in the labeling steps.The paper states that the functional choice is not critical to the intended demonstration.

A. Training data and DP model

The DP-GEN workflow expands a water training set by sampling bulk, surface, and low-density configurations, selecting uncertain structures for DFT labeling, and iteratively refining DP models. The resulting dataset contains 583 configurations, with the final cycle showing satisfactory convergence.

Concurrent learning: DP-GEN iteratively enlarges the DFT-labeled training dataset while refining a representative ensemble of DP models.The ensemble samples thermodynamic space through deep-potential molecular-dynamics trajectories.
Configuration sampling: The exploration covers bulk, slab, and low-density water configurations across temperatures from 200 to 400 K.Bulk sampling uses NPT trajectories, while slab and low-density sampling uses NVT trajectories.
Data selection: Configurations with force-deviation values between 0.15 and 0.30 eV/Å are selected for single-shot DFT labeling.The ensemble’s maximal force deviation is monitored along trajectories to classify explored configurations.
Convergence: Less than 0.1% of configurations retain gradient deviations larger than 0.15 eV/Å at the end of each training stage.This reduction is reported as evidence of satisfactory convergence of the DP-GEN cycle.
Training dataset: 583 configurations comprise the final training data: 135 initial configurations plus 448 selected during DP-GEN.The selected configurations include 323 bulk, 94 surface, and 31 low-density structures.

B. DW model

The DW model predicts Wannier-centroid displacements from local atomic environments, enabling DPLR’s Gaussian-charge representation. For water, β ≈ 0.4 Å^-1 minimizes the generalization gap, whereas overly localized charges increase errors and numerical difficulty.

DW model: The DW model predicts each Wannier-centroid position relative to its uniquely associated oxygen atom using a 6 Å local-environment cutoff.The same cutoff is used for the DP model.
Numerical considerations: The intramolecular Coulomb force between the oxygen ion and its associated Wannier centroid is about 10^4 eV/Å, roughly four orders above the average atomic force.This makes resolving labeled forces numerically difficult.
Spread-parameter selection: For β ≥ 0.5 Å^-1, DPLR errors increase because highly localized Gaussian charges create numerical difficulties.The spread parameter is the inverse width of the Gaussian charge distribution.
Spread-parameter selection: β ≈ 0.4 Å^-1 gives DPLR training and test errors of 1.3 and 1.5 meV/H2O, respectively, with minimal generalization gap.The paper uses β = 0.4 Å^-1 thereafter and relates this spread to the physical extent of the water Wannier-centroid distribution.
Long-range contribution: Extending standard DP’s cutoff from 6 Å to 8 Å only modestly reduces errors, unlike explicit long-range electrostatics.The reported test error changes from 2.14 to 2.08 meV/H2O, while bulk and surface training errors also decrease only slightly.

IV. APPLICATION TO TWO WATER SYSTEMS

DPLR improves the description of two water-system interactions by recovering long-range electrostatic effects absent from standard DP. It accurately captures the water-dimer Coulomb tail and describes water absorption into a liquid slab.

DPLR describes long-range electrostatic interactions among water-molecule dipoles that are absent in the DP model.
A. Water dimer: Both DP and DPLR describe short-distance water-dimer energies equally well, while intermediate-distance accuracy improves with larger cutoffs.
A. Water dimer: Beyond the cutoff radius, DP misses the water-dimer 1/d^3_O tail from dipole-dipole interactions, whereas DPLR recovers it accurately.
A. Water dimer: DPLR is concluded to be superior with both tested cutoff choices, including for water-dimer configurations excluded from training.

B. Free energy profile of a molecule at varying distance from a liquid slab

The slab calculation evaluates water-molecule absorption free energy as a function of molecule–slab distance at 300 K. DPLR remains close to DFT, whereas DP shows a smaller near-slab free-energy gain and much larger uncertainty.

Free-energy calculation: The free-energy profile is obtained by varying the molecule–slab distance from 17 Å to 9 Å while maintaining equilibrium at 300 K.
Free-energy calculation: Near 9 Å, the tagged molecule forms hydrogen bonds with neighboring slab molecules, suggesting incorporation into the slab.
DPLR versus DFT: DPLR reproduces DFT energies with an average error of ∼1 meV/H2O, supporting the approximation used to estimate the DFT free-energy deviation.
DPLR versus DP: Near the slab, DP predicts a smaller free-energy gain than DPLR, consistent with missing attractive dipolar interactions.
DPLR versus DP: Near the slab, the standard deviation of independent DP models becomes almost an order of magnitude larger than DPLR uncertainty.

V. PHONONS IN CRYSTALLINE SODIUM CHLORIDE

NaCl phonons expose the importance of long-range electrostatics in polar crystals. DPLR captures size-dependent longitudinal optical behavior and avoids the large-system extrapolation failure of DP.

Polar phonons: In polar NaCl, long-range electrostatics add a non-analytic contribution to the dynamical matrix needed for LO-TO splitting.
2 × 2 × 2 supercell: DP and DPLR analytical phonon curves coincide and closely match DFT on the 2 × 2 × 2 training supercell.
LO-TO splitting: Adding the non-analytic dynamical-matrix contribution restores agreement between calculated and experimental phonon frequencies.
Supercell-size dependence: DP modes are essentially supercell-size independent, whereas DPLR LO modes change strongly with size because of long-range dipolar interactions.
Supercell-size dependence: DPLR recovers more of the correct LO modes in larger supercells, while numerical Fourier interpolation converges slowly near q → 0.
Supercell-size dependence: DPLR retains long-range effects that let it extrapolate the learned potential-energy surface to larger supercells, unlike DP.

VI. EXTERNAL FIELDS

The extended methodology can represent electrostatic response to external fields and support equilibrium or nonequilibrium response calculations. DPLR is relevant when long-range dipolar correlations matter, while DP can suffice for short-range-dominated responses.

An external-field extension couples a time-dependent field to polarization, enabling nonequilibrium molecular dynamics of insulating systems.
The potential-energy surface can use DPLR or DP depending on whether electrostatic size dependence should be included explicitly.
Equilibrium response to external fields can also be studied with molecular dynamics using the Kubo formalism.
DPLR is appropriate for static dielectric properties of water, where long-range correlations among molecular dipoles become important.
For Raman scattering, short-range correlations between molecular polarizabilities make the standard DP model sufficient.

VII. CONCLUSIONS

DPLR extends deep potential models with long-range electrostatics represented through learned Wannier-centroid information and Gaussian charge distributions. The approach models size-dependent electrostatic contributions accurately, while its applicability is bounded by computational cost and assumptions about linear response and fixed atom–Wannier associations.

Method: DPLR adds long-range electrostatic energy from ionic and Wannier-centroid Gaussian charges to the standard DP potential-energy surface.Wannier-centroid environmental dependence is modeled by the DW deep neural network.
Results: The optimal Gaussian spread parameter β is close to the physical value expected from Wannier-centroid spatial delocalization in the water and NaCl examples.This β minimizes the generalization gap in the machine-learning procedure.
Results: Long-range electrostatic contributions whose size variation is ignored by standard DP are approximated accurately in DPLR.The approximation assumes these effects arise from a weak size dependence treatable within linear response.
Limitations: DPLR simulations are more expensive than standard DP, with an expected computational burden increase of approximately a factor of 5.The added cost comes from backward propagation through both DP and DW networks and from Ewald calculations.
Limitations: The linear-response and size-independence assumptions may fail for larger systems exhibiting long-range electrostatic effects beyond the linear-response regime.In such cases, the centroid treatment should be revisited using a self-consistent condition under the long-range electrostatic field.
Limitations: Because Wannier centroids are uniquely associated with specific atoms, they cannot split or recombine, limiting general treatment of electron-transfer reactions.Proton-transfer reactions are allowed, whereas electron-transfer reactions are not generally supported under this assumption.

Appendix A: Modeling the short-range part

The short-range DP model represents each atom through descriptors built from neighboring atoms within cutoff radii. It combines angular-and-radial information for close neighbors with radial information for more distant neighbors before fitting atomic energy contributions with neural networks.

Local environment: The short-range DP model constructs local descriptors from neighboring atoms within specified cutoff radii.Two neighbor sets support descriptors containing angular and radial information.
Descriptor construction: The descriptor preserves local-environment symmetry while encoding angular and radial information.A hybrid descriptor concatenates angular-and-radial and radial-only descriptors.
Energy model: A feed-forward fitting network maps the descriptor to the short-range energy, and the DP networks are trained end-to-end by stochastic gradient descent.The fitting network uses skip connections.
Local environment: Each neighbor contribution uses a smoothly decaying switch function applied to relative-position features.The relative positions are expressed through Cartesian components and inverse-distance-scaled coordinates.
Descriptor construction: The embedding matrix maps scalar neighbor features to higher-dimensional vectors through a feed-forward neural network.The resulting descriptor dimensions depend on the neighbor count and embedding size.
Descriptor construction: Angular-and-radial information is used for close neighbors, whereas radial information alone generally suffices for more distant neighbors.The hybrid descriptor usually has stronger generalization ability than the angular-and-radial descriptor alone.