Machine Learning Force Fields.

In recent years, the use of machine learning (ML) in computational chemistry has enabled numerous advances previously out of reach due to the computational complexity of traditional electronic-structure methods. One of the most promising applications is the construction of ML-based force fields (FFs), with the aim to narrow the gap between the accuracy of ab initio methods and the efficiency of classical FFs. The key idea is to learn the statistical relation between chemical structure and potential energy without relying on a preconceived notion of fixed chemical bonds or knowledge about the relevant interactions. Such universal ML approximations are in principle only limited by the quality and quantity of the reference data used to train them. This review gives an overview of applications of ML-FFs and the chemical insights that can be obtained from them. The core concepts underlying ML-FFs are described in detail, and a step-by-step guide for constructing and testing them from scratch is given. The text concludes with a discussion of the challenges that remain to be overcome by the next generation of ML-FFs.

Molecular force fields with gradient-domain machine learning (GDML): Comparison and synergies with classical force fields.

Modern machine learning force fields (ML-FF) are able to yield energy and force predictions at the accuracy of high-level ab initio methods, but at a much lower computational cost. On the other hand, classical molecular mechanics force fields (MM-FF) employ fixed functional forms and tend to be less accurate, but considerably faster and transferable between molecules of the same class. In this work, we investigate how both approaches can complement each other. We contrast the ability of ML-FF for reconstructing dynamic and thermodynamic observables to MM-FFs in order to gain a qualitative understanding of the differences between the two approaches. This analysis enables us to modify the generalized AMBER force field by reparametrizing short-range and bonded interactions with more expressive terms to make them more accurate, without sacrificing the key properties that make MM-FFs so successful.

Molecular force fields with gradient-domain machine learning: Construction and application to dynamics of small molecules with coupled cluster forces.

We present the construction of molecular force fields for small molecules (less than 25 atoms) using the recently developed symmetrized gradient-domain machine learning (sGDML) approach [Chmiela et al., Nat. Commun. 9, 3887 (2018) and Chmiela et al., Sci. Adv. 3, e1603015 (2017)]. This approach is able to accurately reconstruct complex high-dimensional potential-energy surfaces from just a few 100s of molecular conformations extracted from ab initio molecular dynamics trajectories. The data efficiency of the sGDML approach implies that atomic forces for these conformations can be computed with high-level wavefunction-based approaches, such as the "gold standard" coupled-cluster theory with single, double and perturbative triple excitations [CCSD(T)]. We demonstrate that the flexible nature of the sGDML model recovers local and non-local electronic interactions (e.g., H-bonding, proton transfer, lone pairs, changes in hybridization states, steric repulsion, and n → π* interactions) without imposing any restriction on the nature of interatomic potentials. The analysis of sGDML molecular dynamics trajectories yields new qualitative insights into dynamics and spectroscopy of small molecules close to spectroscopic accuracy.

Mechanical Vibrations of Atomically Defined Metal Clusters: From Nano- to Molecular-Size Oscillators.

Acoustic vibrations of small nanoparticles are still ruled by continuum mechanics laws down to diameters of a few nanometers. The elastic behavior at lower sizes (<1-2 nm), where nanoparticles become molecular clusters made by few tens to few atoms, is still little explored. The question remains to which extent the transition from small continuous-mass solids to discrete-atom molecular clusters affects their specific low-frequency vibrational modes, whose period is classically expected to linearly scale with diameter. Here, we investigate experimentally by ultrafast time-resolved optical spectroscopy the acoustic response of atomically defined ligand-protected metal clusters Au n(SR) m with a number n of atoms ranging from 10 to 102 (0.5-1.5 nm diameter range). Two periods, corresponding to fundamental breathing- and quadrupolar-like acoustic modes, are detected, with the latter scaling linearly with cluster diameters and the former taking a constant value. Theoretical calculations based on density functional theory (DFT) predict in the case of bare clusters vibrational periods scaling with size down to diatomic molecules. For ligand-protected clusters, they show a pronounced effect of the ligand molecules on the breathing-like mode vibrational period at the origin of its constant value. This deviation from classical elasticity predictions results from mechanical mass-loading effects due to the protecting layer. This study shows that clusters characteristic vibrational frequencies are compatible with extrapolation of continuum mechanics model down to few atoms, which is in agreement with DFT computations.

Vibrational properties and specific heat of core-shell Ag-Au icosahedral nanoparticles.

The vibrational density of states (VDOS) of metal nanoparticles can be a fingerprint of their geometrical structure and determine their low-temperature thermal properties. Theoretical and experimental methods are available nowadays to calculate and measure it over a size range of 1-4 nm. In this work, we present theoretical results regarding the VDOS of Ag-Au icosahedral nanoparticles with a core-shell structure in that size range (147-923 atoms). The results are obtained by changing the size and type of atoms in the core-shell structure. For all sizes investigated, a smooth and monotonic variation of the VDOSs from Ag to Au is obtained by increasing the number of core Au atoms, and vice versa. Nevertheless, the Ag561Au362 nanoparticle, with a Ag core, shows an anomalous enhancement at low frequencies. An analysis of the calculated VDOSs indicates that as a general trend the low-frequency region is mainly due to the shell contribution, whereas at high frequencies the core effect would be dominant. A linear variation with size is obtained for the period of quasi-breathing mode (QBM), in agreement with the behaviour obtained for pure Ag and Au nanoparticles. A non-monotonic variation is obtained for the QBM frequency as a function of the Ag concentration for all nanoparticles investigated. The calculated specific heat at low temperatures of the Ag-Au nanoparticles is smaller (larger) than the corresponding one calculated for the pure Au (Ag) nanoparticles of same size. Nevertheless, the enhancement of VDOS at low frequencies of the Ag561Au362 nanoparticle with a Ag core induced larger values of specific heat than those of the pure Au923 nanoparticle in the temperature range of 5-15 K.

Intercavity polariton slows down dynamics in strongly coupled cavities.

Band engineering stands as an efficient route to induce strongly correlated quantum many-body phenomena. Besides inspiring analogies among diverse physical fields, tuning on demand the group velocity is highly attractive in photonics because it allows unconventional flows of light. Λ-schemes offer a route to control the propagation of light in a lattice-free configurations, enabling exotic phases such as slow-light and allowing for highly optical non-linear systems. Here, we realize room-temperature intercavity Frenkel polaritons excited across two strongly coupled cavities. We demonstrate the formation of a tuneable heavy-polariton, akin to slow light, appearing in the absence of a periodic in-plane potential. Our photonic architecture based on a simple three-level scheme enables the unique spatial segregation of photons and excitons in different cavities and maintains a balanced degree of mixing between them. This unveils a dynamical competition between many-body scattering processes and the underlying polariton nature which leads to an increased fluorescence lifetime. The intercavity polariton features are further revealed under appropriate resonant pumping, where we observe suppression of the polariton fluorescence intensity.

Interpolating Nonadiabatic Molecular Dynamics Hamiltonian with Bidirectional Long Short-Term Memory Networks.

Essential for understanding far-from-equilibrium processes, nonadiabatic (NA) molecular dynamics (MD) requires expensive calculations of the excitation energies and NA couplings. Machine learning (ML) can simplify computation; however, the NA Hamiltonian requires complex ML models due to its intricate relationship to atomic geometry. Working directly in the time domain, we employ bidirectional long short-term memory networks (Bi-LSTM) to interpolate the Hamiltonian. Applying this multiscale approach to three metal-halide perovskite systems, we achieve two orders of magnitude computational savings compared to direct ab initio calculation. Reasonable charge trapping and recombination times are obtained with NA Hamiltonian sampling every half a picosecond. The Bi-LSTM-NAMD method outperforms earlier models and captures both slow and fast time scales. In combination with ML force fields, the methodology extends NAMD simulation times from picoseconds to nanoseconds, comparable to charge carrier lifetimes in many materials. Nanosecond sampling is particularly important in systems containing defects, boundaries, interfaces, etc. that can undergo slow rearrangements.

Accurate global machine learning force fields for molecules with hundreds of atoms.

Global machine learning force fields, with the capacity to capture collective interactions in molecular systems, now scale up to a few dozen atoms due to considerable growth of model complexity with system size. For larger molecules, locality assumptions are introduced, with the consequence that nonlocal interactions are not described. Here, we develop an exact iterative approach to train global symmetric gradient domain machine learning (sGDML) force fields (FFs) for several hundred atoms, without resorting to any potentially uncontrolled approximations. All atomic degrees of freedom remain correlated in the global sGDML FF, allowing the accurate description of complex molecules and materials that present phenomena with far-reaching characteristic correlation lengths. We assess the accuracy and efficiency of sGDML on a newly developed MD22 benchmark dataset containing molecules from 42 to 370 atoms. The robustness of our approach is demonstrated in nanosecond path-integral molecular dynamics simulations for supramolecular complexes in the MD22 dataset.

BIGDML-Towards accurate quantum machine learning force fields for materials.

Machine-learning force fields (MLFF) should be accurate, computationally and data efficient, and applicable to molecules, materials, and interfaces thereof. Currently, MLFFs often introduce tradeoffs that restrict their practical applicability to small subsets of chemical space or require exhaustive datasets for training. Here, we introduce the Bravais-Inspired Gradient-Domain Machine Learning (BIGDML) approach and demonstrate its ability to construct reliable force fields using a training set with just 10-200 geometries for materials including pristine and defect-containing 2D and 3D semiconductors and metals, as well as chemisorbed and physisorbed atomic and molecular adsorbates on surfaces. The BIGDML model employs the full relevant symmetry group for a given material, does not assume artificial atom types or localization of atomic interactions and exhibits high data efficiency and state-of-the-art energy accuracies (errors substantially below 1 meV per atom) for an extended set of materials. Extensive path-integral molecular dynamics carried out with BIGDML models demonstrate the counterintuitive localization of benzene-graphene dynamics induced by nuclear quantum effects and their strong contributions to the hydrogen diffusion coefficient in a Pd crystal for a wide range of temperatures.

On the forbidden graphene's ZO (out-of-plane optic) phononic band-analog vibrational modes in fullerenes.

The study of nanostructures' vibrational properties is at the core of nanoscience research. They are known to represent a fingerprint of the system as well as to hint the underlying nature of chemical bonds. In this work, we focus on addressing how the vibrational density of states (VDOS) of the carbon fullerene family (Cn: n = 20 → 720 atoms) evolves from the molecular to the bulk material (graphene) behavior using density functional theory. We find that the fullerene's VDOS smoothly converges to the graphene characteristic line-shape, with the only noticeable discrepancy in the frequency range of the out-of-plane optic (ZO) phonon band. From a comparison of both systems we obtain as main results that: (1) The pentagonal faces in the fullerenes impede the existence of the analog of the high frequency graphene's ZO phonons, (2) which in the context of phonons could be interpreted as a compression (by 43%) of the ZO phonon band by decreasing its maximum allowed radial-optic vibration frequency. And 3) as a result, the deviation of fullerene's VDOS relative to graphene may hold important thermodynamical implications, such as larger heat capacities compared to graphene at room-temperature. These results provide insights that can be extrapolated to other nanostructures containing pentagonal rings or pentagonal defects.

Dynamical strengthening of covalent and non-covalent molecular interactions by nuclear quantum effects at finite temperature.

Nuclear quantum effects (NQE) tend to generate delocalized molecular dynamics due to the inclusion of the zero point energy and its coupling with the anharmonicities in interatomic interactions. Here, we present evidence that NQE often enhance electronic interactions and, in turn, can result in dynamical molecular stabilization at finite temperature. The underlying physical mechanism promoted by NQE depends on the particular interaction under consideration. First, the effective reduction of interatomic distances between functional groups within a molecule can enhance the n → π* interaction by increasing the overlap between molecular orbitals or by strengthening electrostatic interactions between neighboring charge densities. Second, NQE can localize methyl rotors by temporarily changing molecular bond orders and leading to the emergence of localized transient rotor states. Third, for noncovalent van der Waals interactions the strengthening comes from the increase of the polarizability given the expanded average interatomic distances induced by NQE. The implications of these boosted interactions include counterintuitive hydroxyl-hydroxyl bonding, hindered methyl rotor dynamics, and molecular stiffening which generates smoother free-energy surfaces. Our findings yield new insights into the versatile role of nuclear quantum fluctuations in molecules and materials.

SpookyNet: Learning force fields with electronic degrees of freedom and nonlocal effects.

Machine-learned force fields combine the accuracy of ab initio methods with the efficiency of conventional force fields. However, current machine-learned force fields typically ignore electronic degrees of freedom, such as the total charge or spin state, and assume chemical locality, which is problematic when molecules have inconsistent electronic states, or when nonlocal effects play a significant role. This work introduces SpookyNet, a deep neural network for constructing machine-learned force fields with explicit treatment of electronic degrees of freedom and nonlocality, modeled via self-attention in a transformer architecture. Chemically meaningful inductive biases and analytical corrections built into the network architecture allow it to properly model physical limits. SpookyNet improves upon the current state-of-the-art (or achieves similar performance) on popular quantum chemistry data sets. Notably, it is able to generalize across chemical and conformational space and can leverage the learned chemical insights, e.g. by predicting unknown spin states, thus helping to close a further important remaining gap for today's machine learning models in quantum chemistry.

Towards exact molecular dynamics simulations with machine-learned force fields.

Molecular dynamics (MD) simulations employing classical force fields constitute the cornerstone of contemporary atomistic modeling in chemistry, biology, and materials science. However, the predictive power of these simulations is only as good as the underlying interatomic potential. Classical potentials often fail to faithfully capture key quantum effects in molecules and materials. Here we enable the direct construction of flexible molecular force fields from high-level ab initio calculations by incorporating spatial and temporal physical symmetries into a gradient-domain machine learning (sGDML) model in an automatic data-driven way. The developed sGDML approach faithfully reproduces global force fields at quantum-chemical CCSD(T) level of accuracy and allows converged molecular dynamics simulations with fully quantized electrons and nuclei. We present MD simulations, for flexible molecules with up to a few dozen atoms and provide insights into the dynamical behavior of these molecules. Our approach provides the key missing ingredient for achieving spectroscopic accuracy in molecular simulations.

Machine learning of accurate energy-conserving molecular force fields.

Using conservation of energy-a fundamental property of closed classical and quantum mechanical systems-we develop an efficient gradient-domain machine learning (GDML) approach to construct accurate molecular force fields using a restricted number of samples from ab initio molecular dynamics (AIMD) trajectories. The GDML implementation is able to reproduce global potential energy surfaces of intermediate-sized molecules with an accuracy of 0.3 kcal mol-1 for energies and 1 kcal mol-1 Å̊-1 for atomic forces using only 1000 conformational geometries for training. We demonstrate this accuracy for AIMD trajectories of molecules, including benzene, toluene, naphthalene, ethanol, uracil, and aspirin. The challenge of constructing conservative force fields is accomplished in our work by learning in a Hilbert space of vector-valued functions that obey the law of energy conservation. The GDML approach enables quantitative molecular dynamics simulations for molecules at a fraction of cost of explicit AIMD calculations, thereby allowing the construction of efficient force fields with the accuracy and transferability of high-level ab initio methods.