General Many-Body Framework for Data-Driven Potentials with Arbitrary Quantum Mechanical Accuracy: Water as a Case Study

The Near-Sightedness of Many-Body Interactions in Anharmonic Vibrational Couplings

Couplings between vibrational motions are driven by electronic interactions, and these couplings carry special significance in vibrational energy transfer, multidimensional spectroscopy experiments, and simulations of vibrational spectra. In this investigation, the many-body contributions to these couplings are analyzed computationally in the context of clathrate-like alkali metal cation hydrates, including Cs+(H2O)20, Rb+(H2O)20, and K+(H2O)20, using both analytic and quantum-chemistry potential energy surfaces. Although the harmonic spectra and one-dimensional anharmonic spectra depend strongly on these many-body interactions, the mode-pair couplings were, perhaps surprisingly, found to be dominated by one-body effects, even in cases of couplings to low-frequency modes that involved the motion of multiple water molecules. The origin of this effect was traced mainly to geometric distortion within water monomers and cancellation of many-body effects in differential couplings, and the effect was also shown to be agnostic to the identity of the ion. These outcomes provide new understanding of vibrational couplings and suggest the possibility of improved computational methods for the simulation of infrared and Raman spectra.

Many-body interactions and deep neural network potentials for water

We present a detailed assessment of deep neural network potentials developed within the Deep Potential Molecular Dynamics (DeePMD) framework and trained on the MB-pol data-driven many-body potential energy function. Specific focus is directed at the ability of DeePMD-based potentials to correctly reproduce the accuracy of MB-pol across various water systems. Analyses of bulk and interfacial properties as well as many-body interactions characteristic of water elucidate inherent limitations in the transferability and predictive accuracy of DeePMD-based potentials. These limitations can be traced back to an incomplete implementation of the "nearsightedness of electronic matter" principle, which may be common throughout machine learning potentials that do not include a proper representation of self-consistently determined long-range electric fields. These findings provide further support for the "short-blanket dilemma" faced by DeePMD-based potentials, highlighting the challenges in achieving a balance between computational efficiency and a rigorous, physics-based representation of the properties of water. Finally, we believe that our study contributes to the ongoing discourse on the development and application of machine learning models in simulating water systems, offering insights that could guide future improvements in the field.

Balance between Physical Interpretability and Energetic Predictability in Widely Used Dispersion-Corrected Density Functionals

We assess the performance of different dispersion models for several popular density functionals across a diverse set of noncovalent systems, ranging from the benzene dimer to molecular crystals. By analyzing the interaction energies and their individual components, we demonstrate that there exists variability across different systems for empirical dispersion models, which are calibrated for reproducing the interaction energies of specific systems. Thus, parameter fitting may undermine the underlying physics, as dispersion models rely on error compensation among the different components of the interaction energy. Energy decomposition analyses reveal that, the accuracy of revPBE-D3 for some aqueous systems originates from significant compensation between dispersion and charge transfer energies. However, revPBE-D3 is less accurate in describing systems where error compensation is incomplete, such as the benzene dimer. Such cases highlight the propensity for unpredictable behavior in various dispersion-corrected density functionals across a wide range of molecular systems, akin to the behavior of force fields. On the other hand, we find that SCAN-rVV10, a targeted-dispersion approach, affords significant reductions in errors associated with the lattice energies of molecular crystals, while it has limited accuracy in reproducing structural properties. Given the ubiquitous nature of noncovalent interactions and the key role of density functional theory in computational sciences, the future development of dispersion models should prioritize the faithful description of the dispersion energy, a shift that promises greater accuracy in capturing the underlying physics across diverse molecular and extended systems.

Quantifying the Relevance of Long-Range Forces for Crystal Nucleation in Water

Understanding nucleation from aqueous solutions is of fundamental importance in a multitude of fields, ranging from materials science to biophysics. The complex solvent-mediated interactions in aqueous solutions hamper the development of a simple physical picture, elucidating the roles of different interactions in nucleation processes. In this work, we make use of three complementary techniques to disentangle the role played by short- and long-range interactions in solvent-mediated nucleation. Specifically, the first approach we utilize is the local molecular field (LMF) theory to renormalize long-range Coulomb electrostatics. Second, we use well-tempered metadynamics to speed up rare events governed by short-range interactions. Third, the deep learning-based State Predictive Information Bottleneck approach is employed in analyzing the reaction coordinate of the nucleation processes obtained from the LMF treatment coupled with well-tempered metadynamics. We find that the two-step nucleation mechanism can largely be captured by the short-range interactions, while the long-range interactions further contribute to the stability of the primary crystal state under ambient conditions. Furthermore, by analyzing the reaction coordinate obtained from the combined LMF-metadynamics treatment, we discern the fluctuations on different time scales, highlighting the need for long-range interactions when accounting for metastability.

Pitfalls in the n-mode representation of vibrational potentials

Simulations of anharmonic vibrational motion rely on computationally expedient representations of the governing potential energy surface. The n-mode representation (n-MR)-effectively a many-body expansion in the space of molecular vibrations-is a general and efficient approach that is often used for this purpose in vibrational self-consistent field (VSCF) calculations and correlated analogues thereof. In the present analysis, a lack of convergence in many VSCF calculations is shown to originate from negative and unbound potentials at truncated orders of the n-MR expansion. For cases of strong anharmonic coupling between modes, the n-MR can both dip below the true global minimum of the potential surface and lead to effective single-mode potentials in VSCF that do not correspond to bound vibrational problems, even for bound total potentials. The present analysis serves mainly as a pathology report of this issue. Furthermore, this insight into the origin of VSCF non-convergence provides a simple, albeit ad hoc, route to correct the problem by "painting in" the full representation of groups of modes that exhibit these negative potentials at little additional computational cost. Somewhat surprisingly, this approach also reasonably approximates the results of the next-higher n-MR order and identifies groups of modes with particularly strong coupling. The method is shown to identify and correct problematic triples of modes-and restore SCF convergence-in two-mode representations of challenging test systems, including the water dimer and trimer, as well as protonated tropine.

Consistent density functional theory-based description of ion hydration through density-corrected many-body representations

Delocalization error constrains the accuracy of density functional theory in describing molecular interactions in ion-water systems. Using Na+ and Cl- in water as model systems, we calculate the effects of delocalization error in the SCAN functional for describing ion-water and water-water interactions in hydrated ions, and demonstrate that density-corrected SCAN (DC-SCAN) predicts n-body and interaction energies with an accuracy approaching coupled cluster theory. The performance of DC-SCAN is size-consistent, maintaining an accurate description of molecular interactions well beyond the first solvation shell. Molecular dynamics simulations at ambient conditions with many-body MB-SCAN(DC) potentials, derived from the many-body expansion, predict the solvation structure of Na+ and Cl- in quantitative agreement with reference data, while simultaneously reproducing the structure of liquid water. Beyond rationalizing the accuracy of density-corrected models of ion hydration, our findings suggest that our unified density-corrected MB formalism holds great promise for efficient DFT-based simulations of condensed-phase systems with chemical accuracy.

Simple and Efficient Route toward Improved Energetics within the Framework of Density-Corrected Density Functional Theory

The crucial step in density-corrected Hartree-Fock density functional theory (DC(HF)-DFT) is to decide whether the density produced by the density functional for a specific calculation is erroneous and, hence, should be replaced by, in this case, the HF density. We introduce an indicator, based on the difference in noninteracting kinetic energies between DFT and HF calculations, to determine when the HF density is the better option. Our kinetic energy indicator directly compares the self-consistent density of the analyzed functional with the HF density, is size-intensive, reliable, and most importantly highly efficient. Moreover, we present a procedure that makes best use of the computed quantities necessary for DC(HF)-DFT by additionally evaluating a related hybrid functional and, in that way, not only "corrects" the density but also the functional itself; we call that procedure corrected Hartree-Fock density functional theory (C(HF)-DFT).

Accelerating Anharmonic Spectroscopy Simulations via Local-Mode, Multilevel Methods

Ab initio computer simulations of anharmonic vibrational spectra provide nuanced insight into the vibrational behavior of molecules and complexes. The computational bottleneck in such simulations, particularly for ab initio potentials, is often the generation of mode-coupling potentials. Focusing specifically on two-mode couplings in this analysis, the combination of a local-mode representation and multilevel methods is demonstrated to be particularly symbiotic. In this approach, a low-level quantum chemistry method is employed to predict the pairwise couplings that should be included at the target level of theory in vibrational self-consistent field (and similar) calculations. Pairs that are excluded by this approach are "recycled" at the low level of theory. Furthermore, because this low-level pre-screening will eventually become the computational bottleneck for sufficiently large chemical systems, distance-based truncation is applied to these low-level predictions without substantive loss of accuracy. This combination is demonstrated to yield sub-wavenumber fidelity with reference vibrational transitions when including only a small fraction of target-level couplings; the overhead of predicting these couplings, particularly when employing distance-based, local-mode cutoffs, is a trivial added cost. This combined approach is assessed on a series of test cases, including ethylene, hexatriene, and the alanine dipeptide. Vibrational self-consistent field (VSCF) spectra were obtained with an RI-MP2/cc-pVTZ potential for the dipeptide, at approximately a 5-fold reduction in computational cost. Considerable optimism for increased accelerations for larger systems and higher-order couplings is also justified, based on this investigation.

Realistic phase diagram of water from "first principles" data-driven quantum simulations

Since the experimental characterization of the low-pressure region of water's phase diagram in the early 1900s, scientists have been on a quest to understand the thermodynamic stability of ice polymorphs on the molecular level. In this study, we demonstrate that combining the MB-pol data-driven many-body potential for water, which was rigorously derived from "first principles" and exhibits chemical accuracy, with advanced enhanced-sampling algorithms, which correctly describe the quantum nature of molecular motion and thermodynamic equilibria, enables computer simulations of water's phase diagram with an unprecedented level of realism. Besides providing fundamental insights into how enthalpic, entropic, and nuclear quantum effects shape the free-energy landscape of water, we demonstrate that recent progress in "first principles" data-driven simulations, which rigorously encode many-body molecular interactions, has opened the door to realistic computational studies of complex molecular systems, bridging the gap between experiments and simulations.

Are Neural Network Potentials Trained on Liquid States Transferable to Crystal Nucleation? A Test on Ice Nucleation in the mW Water Model

Neural network potentials (NNPs) are increasingly being used to study processes that happen on long time scales. A typical example is crystal nucleation, which rate is controlled by the occurrence of a rare fluctuation, i.e., the appearance of the critical nucleus. Because the properties of this nucleus are far from those of the bulk crystal, it is yet unclear whether NN potentials trained on equilibrium liquid states can accurately describe nucleation processes. So far, nucleation studies on NNPs have been limited to ab initio models whose nucleation properties are unknown, preventing an accurate comparison. Here we train a NN potential on the mW model of water─a classical three-body potential whose nucleation time scale is accessible in standard simulations. We show that a NNP trained only on a small number of liquid state points can reproduce with great accuracy the nucleation rates and free energy barriers of the original model, computed from both spontaneous and biased trajectories, strongly supporting the use of NNPs to study nucleation events.

A highly accurate full-dimensional ab initio potential surface for the rearrangement of methylhydroxycarbene (H3C-C-OH)

We report here a full-dimensional machine learning global potential surface (PES) for the rearrangement of methylhydroxycarbene (H₃C-C-OH, 1t). The PES is trained with the fundamental invariant neural network (FI-NN) method on 91 564 ab initio energies calculated at the UCCSD(T)-F12a/cc-pVTZ level of theory, covering three possible product channels. FI-NN PES has the correct symmetry properties with respect to permutation of four identical hydrogen atoms and is suitable for dynamics studies of the 1t rearrangement. The averaged root mean square error (RMSE) is 11.4 meV. Six important reaction pathways, as well as the energies and vibrational frequencies at the stationary geometries on these pathways are accurately preproduced by our FI-NN PES. To demonstrate the capacity of the PES, we calculated the rate coefficient of hydrogen migration in -CH₃ (path A) and hydrogen migration of -OH (path B) with instanton theory on this PES. Our calculations predicted the half-life of 1t to be 95 min, which is excellent in agreement with experimental observations.

A neural network potential with self-trained atomic fingerprints: A test with the mW water potential

We present a neural network (NN) potential based on a new set of atomic fingerprints built upon two- and three-body contributions that probe distances and local orientational order, respectively. Compared with the existing NN potentials, the atomic fingerprints depend on a small set of tunable parameters that are trained together with the NN weights. In addition to simplifying the selection of the atomic fingerprints, this strategy can also considerably increase the overall accuracy of the network representation. To tackle the simultaneous training of the atomic fingerprint parameters and NN weights, we adopt an annealing protocol that progressively cycles the learning rate, significantly improving the accuracy of the NN potential. We test the performance of the network potential against the mW model of water, which is a classical three-body potential that well captures the anomalies of the liquid phase. Trained on just three state points, the NN potential is able to reproduce the mW model in a very wide range of densities and temperatures, from negative pressures to several GPa, capturing the transition from an open random tetrahedral network to a dense interpenetrated network. The NN potential also reproduces very well properties for which it was not explicitly trained, such as dynamical properties and the structure of the stable crystalline phases of mW.

Local Molecular Field Theory for Coulomb Interactions in Aqueous Solutions

Coulomb interactions play a crucial role in a wide array of processes in aqueous solutions but present conceptual and computational challenges to both theory and simulations. We review recent developments in an approach addressing these challenges─local molecular field (LMF) theory. LMF theory exploits an exact and physically suggestive separation of intermolecular Coulomb interactions into strong short-range and uniformly slowly varying long-range components. This allows us to accurately determine the averaged effects of the long-range components on the short-range structure using effective single particle fields and analytical corrections, greatly reducing the need for complex lattice summation techniques used in most standard approaches. The simplest use of these ideas in aqueous solutions leads to the short solvent (SS) model, where both solvent-solvent and solute-solvent Coulomb interactions have only short-range components. Here we use the SS model to give a simple description of pairing of nucleobases and biologically relevant ions in water.

Δ-Machine Learned Potential Energy Surfaces and Force Fields

There has been great progress in developing machine-learned potential energy surfaces (PESs) for molecules and clusters with more than 10 atoms. Unfortunately, this number of atoms generally limits the level of electronic structure theory to less than the "gold standard" CCSD(T) level. Indeed, for the well-known MD17 dataset for molecules with 9-20 atoms, all of the energies and forces were obtained with DFT calculations (PBE). This Perspective is focused on a Δ-machine learning method that we recently proposed and applied to bring DFT-based PESs to close to CCSD(T) accuracy. This is demonstrated for hydronium, N-methylacetamide, acetyl acetone, and ethanol. For 15-atom tropolone, it appears that special approaches (e.g., molecular tailoring, local CCSD(T)) are needed to obtain the CCSD(T) energies. A new aspect of this approach is the extension of Δ-machine learning to force fields. The approach is based on many-body corrections to polarizable force field potentials. This is examined in detail using the TTM2.1 water potential. The corrections make use of our recent CCSD(T) datasets for 2-b, 3-b, and 4-b interactions for water. These datasets were used to develop a new fully ab initio potential for water, termed q-AQUA.

Accurate Modeling of Bromide and Iodide Hydration with Data-Driven Many-Body Potentials

Ion-water interactions play a central role in determining the properties of aqueous systems in a wide range of environments. However, a quantitative understanding of how the hydration properties of ions evolve from small aqueous clusters to bulk solutions and interfaces remains elusive. Here, we introduce the second generation of data-driven many-body energy (MB-nrg) potential energy functions (PEFs) representing bromide-water and iodide-water interactions. The MB-nrg PEFs use permutationally invariant polynomials to reproduce two-body and three-body energies calculated at the coupled cluster level of theory, and implicitly represent all higher-body energies using classical many-body polarization. A systematic analysis of the hydration structure of small Br^-(H₂O)_n and I^-(H₂O)_n clusters demonstrates that the MB-nrg PEFs predict interaction energies in quantitative agreement with "gold standard" coupled cluster reference values. Importantly, when used in molecular dynamics simulations carried out in the isothermal-isobaric ensemble for single bromide and iodide ions in liquid water, the MB-nrg PEFs predict extended X-ray absorption fine structure (EXAFS) spectra that accurately reproduce the experimental spectra, which thus allows for characterizing the hydration structure of the two ions with a high level of confidence.

Adaptive Subsystem Density Functional Theory

Subsystem density functional theory (DFT) is emerging as a powerful electronic structure method for large-scale simulations of molecular condensed phases and interfaces. Key to its computational efficiency is the use of approximate nonadditive noninteracting kinetic energy functionals. Unfortunately, currently available nonadditive functionals lead to inaccurate results when the subsystems interact strongly such as when they engage in chemical reactions. This work disrupts the status quo by devising a workflow that extends subsystem DFT's applicability also to strongly interacting subsystems. This is achieved by implementing a fully automated adaptive definition of subsystems which is realized during geometry optimizations or ab initio molecular dynamics simulations. The new method prescribes subsystem merging and splitting events redistributing the resources (both for work and data) in an efficient way making use of modern parallelization strategies and object-oriented programming. We showcase the method with examples probing from moderate-to-strong inter-subsystem interactions, opening the door to using subsystem DFT for modeling chemical reactions in molecular condensed phases with a black box computational tool.

Data-Driven Many-Body Potential Energy Functions for Generic Molecules: Linear Alkanes as a Proof-of-Concept Application

We present a generalization of the many-body energy (MB-nrg) theoretical/computational framework that enables the development of data-driven potential energy functions (PEFs) for generic covalently bonded molecules, with arbitrary quantum mechanical accuracy. The "nearsightedness of electronic matter" is exploited to define monomers as "natural building blocks" on the basis of their distinct chemical identity. The energy of generic molecules is then expressed as a sum of individual many-body energies of incrementally larger subsystems. The MB-nrg PEFs represent the low-order n-body energies, with n = 1-4, using permutationally invariant polynomials derived from electronic structure data carried out at an arbitrary quantum mechanical level of theory, while all higher-order n-body terms (n > 4) are represented by a classical many-body polarization term. As a proof-of-concept application of the general MB-nrg framework, we present MB-nrg PEFs for linear alkanes. The MB-nrg PEFs are shown to accurately reproduce reference energies, harmonic frequencies, and potential energy scans of alkanes, independently of their length. Since, by construction, the MB-nrg framework introduced here can be applied to generic covalently bonded molecules, we envision future computer simulations of complex molecular systems using data-driven MB-nrg PEFs, with arbitrary quantum mechanical accuracy.

How Good Is the Density-Corrected SCAN Functional for Neutral and Ionic Aqueous Systems, and What Is So Right about the Hartree-Fock Density?

Density functional theory (DFT) is the most widely used electronic structure method, due to its simplicity and cost effectiveness. The accuracy of a DFT calculation depends not only on the choice of the density functional approximation (DFA) adopted but also on the electron density produced by the DFA. SCAN is a modern functional that satisfies all known constraints for meta-GGA functionals. The density-driven errors, defined as energy errors arising from errors of the self-consistent DFA electron density, can hinder SCAN from achieving chemical accuracy in some systems, including water. Density-corrected DFT (DC-DFT) can alleviate this shortcoming by adopting a more accurate electron density which, in most applications, is the electron density obtained at the Hartree-Fock level of theory due to its relatively low computational cost. In this work, we present extensive calculations aimed at determining the accuracy of the DC-SCAN functional for various aqueous systems. DC-SCAN (SCAN@HF) shows remarkable consistency in reproducing reference data obtained at the coupled cluster level of theory, with minimal loss of accuracy. Density-driven errors in the description of ionic aqueous clusters are thoroughly investigated. By comparison with the orbital-optimized CCD density in the water dimer, we find that the self-consistent SCAN density transfers a spurious fraction of an electron across the hydrogen bond to the hydrogen atom (H^*, covalently bound to the donor oxygen atom) from the acceptor (O_A) and donor (O_D) oxygen atoms, while HF makes a much smaller spurious transfer in the opposite direction, consistent with DC-SCAN (SCAN@HF) reduction of SCAN overbinding due to delocalization error. While LDA seems to be the conventional extreme of density delocalization error, and HF the conventional extreme of (usually much smaller) density localization error, these two densities do not quite yield the conventional range of density-driven error in energy differences. Finally, comparisons of the DC-SCAN results with those obtained with the Fermi-Löwdin orbital self-interaction correction (FLOSIC) method show that DC-SCAN represents a more accurate approach to reducing density-driven errors in SCAN calculations of ionic aqueous clusters. While the HF density is superior to that of SCAN for noncompact water clusters, the opposite is true for the compact water molecule with exactly 10 electrons.

The MD17 datasets from the perspective of datasets for gas-phase “small” molecule potentials

There has been great progress in developing methods for machine-learned potential energy surfaces. There have also been important assessments of these methods by comparing so-called learning curves on datasets of electronic energies and forces, notably the MD17 database. The dataset for each molecule in this database generally consists of tens of thousands of energies and forces obtained from DFT direct dynamics at 500 K. We contrast the datasets from this database for three “small” molecules, ethanol, malonaldehyde, and glycine, with datasets we have generated with specific targets for the potential energy surfaces (PESs) in mind: a rigorous calculation of the zero-point energy and wavefunction, the tunneling splitting in malonaldehyde, and, in the case of glycine, a description of all eight low-lying conformers. We found that the MD17 datasets are too limited for these targets. We also examine recent datasets for several PESs that describe small-molecule but complex chemical reactions. Finally, we introduce a new database, “QM-22,” which contains datasets of molecules ranging from 4 to 15 atoms that extend to high energies and a large span of configurations.

Toward High-level Machine Learning Potential for Water Based on Quantum Fragmentation and Neural Networks

Accurate and efficient simulation of liquids, such as water and salt solutions, using high-level wave function theories is still a formidable task for computational chemists owing to the high computational costs. In this study, we develop a deep machine learning potential based on fragment-based second-order Møller-Plesset perturbation theory (DP-MP2) for water through neural networks. We show that the DP-MP2 potential predicts the structural, dynamical, and thermodynamic properties of liquid water in better agreement with the experimental data than previous studies based on density functional theory (DFT). The nuclear quantum effects (NQEs) on the properties of liquid water are also examined, which are noticeable in affecting the structural and dynamical properties of liquid water under ambient conditions. This work provides a general framework for quantitative predictions of the properties of condensed-phase systems with the accuracy of high-level wave function theory while achieving significant computational savings compared to ab initio simulations.

Assessing the Interplay between Functional-Driven and Density-Driven Errors in DFT Models of Water

We investigate the interplay between functional-driven and density-driven errors in different density functional approximations within density functional theory (DFT) and the implications of these errors for simulations of water with DFT-based data-driven potentials. Specifically, we quantify density-driven errors in two widely used dispersion-corrected functionals derived within the generalized gradient approximation (GGA), namely BLYP-D3 and revPBE-D3, and two modern meta-GGA functionals, namely strongly constrained and appropriately normed (SCAN) and B97M-rV. The effects of functional-driven and density-driven errors on the interaction energies are first assessed for the water clusters of the BEGDB dataset. Further insights into the nature of functional-driven errors are gained from applying the absolutely localized molecular orbital energy decomposition analysis (ALMO-EDA) to the interaction energies, which demonstrates that functional-driven errors are strongly correlated with the nature of the interactions. We discuss cases where density-corrected DFT (DC-DFT) models display higher accuracy than the original DFT models and cases where reducing the density-driven errors leads to larger deviations from the reference energies due to the presence of large functional-driven errors. Finally, molecular dynamics simulations are performed with data-driven many-body potentials derived from DFT and DC-DFT data to determine the effect that minimizing density-driven errors has on the description of liquid water. Besides rationalizing the performance of widely used DFT models of water, we believe that our findings unveil fundamental relations between the shortcomings of some common DFT approximations and the requirements for accurate descriptions of molecular interactions, which will aid the development of a consistent, DFT-based framework for the development of data-driven and machine-learned potentials for simulations of condensed-phase systems.

Elevating density functional theory to chemical accuracy for water simulations through a density-corrected many-body formalism

Density functional theory (DFT) has been extensively used to model the properties of water. Albeit maintaining a good balance between accuracy and efficiency, no density functional has so far achieved the degree of accuracy necessary to correctly predict the properties of water across the entire phase diagram. Here, we present density-corrected SCAN (DC-SCAN) calculations for water which, minimizing density-driven errors, elevate the accuracy of the SCAN functional to that of "gold standard" coupled-cluster theory. Building upon the accuracy of DC-SCAN within a many-body formalism, we introduce a data-driven many-body potential energy function, MB-SCAN(DC), that quantitatively reproduces coupled cluster reference values for interaction, binding, and individual many-body energies of water clusters. Importantly, molecular dynamics simulations carried out with MB-SCAN(DC) also reproduce the properties of liquid water, which thus demonstrates that MB-SCAN(DC) is effectively the first DFT-based model that correctly describes water from the gas to the liquid phase.