Derivative learning of tensorial quantities-Predicting finite temperature infrared spectra from first principles.

We develop a strategy that integrates machine learning and first-principles calculations to achieve technically accurate predictions of infrared spectra. In particular, the methodology allows one to predict infrared spectra for complex systems at finite temperatures. The method's effectiveness is demonstrated in challenging scenarios, such as the analysis of water and the organic-inorganic halide perovskite MAPbI3, where our results consistently align with experimental data. A distinctive feature of the methodology is the incorporation of derivative learning, which proves indispensable for obtaining accurate polarization data in bulk materials and facilitates the training of a machine learning surrogate model of the polarization adapted to rotational and translational symmetries. We achieve polarization prediction accuracies of about 1% for the water dimer by training only on the predicted Born effective charges.

Density isobar of water and melting temperature of ice: Assessing common density functionals.

We investigate the density isobar of water and the melting temperature of ice using six different density functionals. Machine-learning potentials are employed to ensure computational affordability. Our findings reveal significant discrepancies between various base functionals. Notably, even the choice of damping can result in substantial differences. Overall, the outcomes obtained through density functional theory are not entirely satisfactory across most utilized functionals. All functionals exhibit significant deviations either in the melting temperature or equilibrium volume, with most of them even predicting an incorrect volume difference between ice and water. Our heuristic analysis indicates that a hybrid functional with 25% exact exchange and van der Waals damping averaged between zero and Becke-Johnson dampings yields the closest agreement with experimental data. This study underscores the necessity for further enhancements in the treatment of van der Waals interactions and, more broadly, density functional theory to enable accurate quantitative predictions for molecular liquids.

Comparing machine learning potentials for water: Kernel-based regression and Behler-Parrinello neural networks.

In this paper, we investigate the performance of different machine learning potentials (MLPs) in predicting key thermodynamic properties of water using RPBE + D3. Specifically, we scrutinize kernel-based regression and high-dimensional neural networks trained on a highly accurate dataset consisting of about 1500 structures, as well as a smaller dataset, about half the size, obtained using only on-the-fly learning. This study reveals that despite minor differences between the MLPs, their agreement on observables such as the diffusion constant and pair-correlation functions is excellent, especially for the large training dataset. Variations in the predicted density isobars, albeit somewhat larger, are also acceptable, particularly given the errors inherent to approximate density functional theory. Overall, this study emphasizes the relevance of the database over the fitting method. Finally, this study underscores the limitations of root mean square errors and the need for comprehensive testing, advocating the use of multiple MLPs for enhanced certainty, particularly when simulating complex thermodynamic properties that may not be fully captured by simpler tests.

Combining Machine Learning and Many-Body Calculations: Coverage-Dependent Adsorption of CO on Rh(111).

Adsorption of carbon monoxide (CO) on transition-metal surfaces is a prototypical process in surface sciences and catalysis. Despite its simplicity, it has posed great challenges to theoretical modeling. Pretty much all existing density functionals fail to accurately describe surface energies and CO adsorption site preference as well as adsorption energies simultaneously. Although the random phase approximation (RPA) cures these density functional theory failures, its large computational cost makes it prohibitive to study the CO adsorption for any but the simplest ordered cases. Here, we address these challenges by developing a machine-learned force field (MLFF) with near RPA accuracy for the prediction of coverage-dependent adsorption of CO on the Rh(111) surface through an efficient on-the-fly active learning procedure and a Δ-machine learning approach. We show that the RPA-derived MLFF is capable to accurately predict the Rh(111) surface energy and CO adsorption site preference as well as adsorption energies at different coverages that are all in good agreement with experiments. Moreover, the coverage-dependent ground-state adsorption patterns and adsorption saturation coverage are identified.

Correction: New insights into the 1D carbon chain through the RPA.

Correction for 'New insights into the 1D carbon chain through the RPA' by Benjamin Ramberger et al., Phys. Chem. Chem. Phys., 2021, 23, 5254-5260, https://doi.org/10.1039/D0CP06607A.

Phaseless auxiliary field quantum Monte Carlo with projector-augmented wave method for solids.

We implement the phaseless auxiliary field quantum Monte Carlo method using the plane-wave based projector augmented wave method and explore the accuracy and the feasibility of applying our implementation to solids. We use a singular value decomposition to compress the two-body Hamiltonian and, thus, reduce the computational cost. Consistent correlation energies from the primitive-cell sampling and the corresponding supercell calculations numerically verify our implementation. We calculate the equation of state for diamond and the correlation energies for a range of prototypical solid materials. A down-sampling technique along with natural orbitals accelerates the convergence with respect to the number of orbitals and crystal momentum points. We illustrate the competitiveness of our implementation in accuracy and computational cost for dense crystal momentum point meshes compared to a well-established quantum-chemistry approach, the coupled-cluster ansatz including singles, doubles, and perturbative triple particle-hole excitation operators.

Approaching the basis-set limit of the dRPA correlation energy with explicitly correlated and projector augmented-wave methods.

The direct random-phase approximation (dRPA) is used to calculate and compare atomization energies for the HEAT set and ten selected molecules of the G2-1 set using both plane waves and Gaussian-type orbitals. We describe detailed procedures to obtain highly accurate and well converged results for the projector augmented-wave method as implemented in the Vienna Ab initio Simulation Package as well as the explicitly correlated dRPA-F12 method as implemented in the TURBOMOLE package. The two approaches agree within chemical accuracy (1 kcal/mol) for the atomization energies of all considered molecules, both for the exact exchange as well as for the RPA. The root mean-square deviation is 0.41 kcal/mol for the exact exchange (evaluated using density functional theory orbitals) and 0.33 kcal/mol for exact exchange plus correlation from the RPA.

New insights into the 1D carbon chain through the RPA.

We investigated the electronic and structural properties of the infinite linear carbon chain (carbyne) using density functional theory (DFT) and the random phase approximation (RPA) to the correlation energy. The studies are performed in vacuo and for carbyne inside a carbon nano tube (CNT). In the vacuum, semi-local DFT and RPA predict bond length alternations of about 0.04 Å and 0.13 Å, respectively. The frequency of the highest optical mode at the Γ point is 1219 cm-1 and about 2000 cm-1 for DFT and the RPA. Agreement of the RPA to previous high level quantum chemistry and diffusion Monte-Carlo results is excellent. For the RPA we calculate the phonon-dispersion in the full Brillouine zone and find marked quantitative differences to DFT calculations not only at the Γ point but also throughout the entire Brillouine zone. To model carbyne inside a carbon nanotube, we considered a (10,0) CNT. Here the DFT calculations are even qualitatively sensitive to the k-points sampling. At the limes of a very dense k-points sampling, semi-local DFT predicts no bond length alternation (BLA), whereas in the RPA a sizeable BLA of 0.09 Å prevails. The reduced BLA leads to a significant red shift of the vibrational frequencies of about 350 cm-1, so that they are in good agreement with experimental estimates. Overall, the good agreement between the RPA and previously reported results from correlated wavefunction methods and experimental Raman data suggests that the RPA provides reliable results at moderate computational costs. It hence presents a useful addition to the repertoire of correlated wavefunction methods and its accuracy clearly prevails for low dimensional systems, where semi-local density functionals struggle to yield even qualitatively correct results.

First-principles hydration free energies of oxygenated species at water-platinum interfaces.

The hydration free energy of atoms and molecules adsorbed at liquid-solid interfaces strongly influences the stability and reactivity of solid surfaces. However, its evaluation is challenging in both experiments and theories. In this work, a machine learning aided molecular dynamics method is proposed and applied to oxygen atoms and hydroxyl groups adsorbed on Pt(111) and Pt(100) surfaces in water. The proposed method adopts thermodynamic integration with respect to a coupling parameter specifying a path from well-defined non-interacting species to the fully interacting ones. The atomistic interactions are described by a machine-learned inter-atomic potential trained on first-principles data. The free energy calculated by the machine-learned potential is further corrected by using thermodynamic perturbation theory to provide the first-principles free energy. The calculated hydration free energies indicate that only the hydroxyl group adsorbed on the Pt(111) surface attains a hydration stabilization. The observed trend is attributed to differences in the adsorption site and surface morphology.

Optimized effective potentials from the random-phase approximation: Accuracy of the quasiparticle approximation.

The optimized effective potential (OEP) method presents an unambiguous way to construct the Kohn-Sham potential corresponding to a given diagrammatic approximation for the exchange-correlation functional. The OEP from the random-phase approximation (RPA) has played an important role ever since the conception of the OEP formalism. However, the solution of the OEP equation is computationally fairly expensive and has to be done in a self-consistent way. So far, large scale solid state applications have, therefore, been performed only using the quasiparticle approximation (QPA), neglecting certain dynamical screening effects. We obtain the exact RPA-OEP for 15 semiconductors and insulators by direct solution of the linearized Sham-Schlüter equation. We investigate the accuracy of the QPA on Kohn-Sham bandgaps and dielectric constants, and comment on the issue of self-consistency.

Local embedding of coupled cluster theory into the random phase approximation using plane waves.

We present an embedding approach to treat local electron correlation effects in periodic environments. In a single consistent framework, our plane wave based scheme embeds a local high-level correlation calculation [here, Coupled Cluster (CC) theory], employing localized orbitals, into a low-level correlation calculation [here, the direct Random Phase Approximation (RPA)]. This choice allows for an accurate and efficient treatment of long-range dispersion effects. Accelerated convergence with respect to the local fragment size can be observed if the low-level and high-level long-range dispersions are quantitatively similar, as is the case for CC in RPA. To demonstrate the capabilities of the introduced embedding approach, we calculate adsorption energies of molecules on a surface and in a chabazite crystal cage, as well as the formation energy of a lattice impurity in a solid at the level of highly accurate many-electron perturbation theories. The absorption energy of a methane molecule in a zeolite chabazite is converged with an error well below 20 meV at the CC level. As our largest periodic benchmark system, we apply our scheme to the adsorption of a water molecule on titania in a supercell containing more than 1000 electrons.

Comparing GIPAW with numerically exact chemical shieldings: The role of two-center contributions to the induced current.

In this study, we benchmark density functional theory gauge-including projector-augmented-wave (GIPAW) chemical shieldings against molecular shieldings for which basis set completeness has been achieved [Jensen et al., Phys. Chem. Chem. Phys. 18, 21145 (2016)]. We demonstrate the importance of two-center corrections for GIPAW hydrogen shieldings. For the other nuclei studied, standard GIPAW is sufficiently accurate. We find that GIPAW can be pushed to closely approach the basis set limit. The only source of small inaccuracies lies in the contribution to the shielding that is caused by surface currents, which we estimate comparing GIPAW susceptibilities to converged molecular magnetizabilities.

Plane wave basis set correction methods for RPA correlation energies.

Electronic correlation energies from the random-phase approximation converge slowly with respect to the plane wave basis set size. We study the conditions under which a short-range local density functional can be used to account for the basis set incompleteness error. Furthermore, we propose a one-shot extrapolation scheme based on the Lindhard response function of the homogeneous electron gas. The different basis set correction methods are used to calculate equilibrium lattice constants for prototypical solids of different bonding types.

Descriptors representing two- and three-body atomic distributions and their effects on the accuracy of machine-learned inter-atomic potentials.

When determining machine-learning models for inter-atomic potentials, the potential energy surface is often described as a non-linear function of descriptors representing two- and three-body atomic distribution functions. It is not obvious how the choice of the descriptors affects the efficiency of the training and the accuracy of the final machine-learned model. In this work, we formulate an efficient method to calculate descriptors that can separately represent two- and three-body atomic distribution functions, and we examine the effects of including only two- or three-body descriptors, as well as including both, in the regression model. Our study indicates that non-linear mixing of two- and three-body descriptors is essential for an efficient training and a high accuracy of the final machine-learned model. The efficiency can be further improved by weighting the two-body descriptors more strongly. We furthermore examine a sparsification of the three-body descriptors. The three-body descriptors usually provide redundant representations of the atomistic structure, and the number of descriptors can be significantly reduced without loss of accuracy by applying an automatic sparsification using a principal component analysis. Visualization of the reduced descriptors using three-body distribution functions in real-space indicates that the sparsification automatically removes the components that are less significant for describing the distribution function.

Phase Transitions of Hybrid Perovskites Simulated by Machine-Learning Force Fields Trained on the Fly with Bayesian Inference.

Realistic finite temperature simulations of matter are a formidable challenge for first principles methods. Long simulation times and large length scales are required, demanding years of computing time. Here we present an on-the-fly machine learning scheme that generates force fields automatically during molecular dynamics simulations. This opens up the required time and length scales, while retaining the distinctive chemical precision of first principles methods and minimizing the need for human intervention. The method is widely applicable to multielement complex systems. We demonstrate its predictive power on the entropy driven phase transitions of hybrid perovskites, which have never been accurately described in simulations. Using machine learned potentials, isothermal-isobaric simulations give direct insight into the underlying microscopic mechanisms. Finally, we relate the phase transition temperatures of different perovskites to the radii of the involved species, and we determine the order of the transitions in Landau theory.

Towards an exact description of electronic wavefunctions in real solids.

The properties of all materials arise largely from the quantum mechanics of their constituent electrons under the influence of the electric field of the nuclei. The solution of the underlying many-electron Schrödinger equation is a 'non-polynomial hard' problem, owing to the complex interplay of kinetic energy, electron-electron repulsion and the Pauli exclusion principle. The dominant computational method for describing such systems has been density functional theory. Quantum-chemical methods--based on an explicit ansatz for the many-electron wavefunctions and, hence, potentially more accurate--have not been fully explored in the solid state owing to their computational complexity, which ranges from strongly exponential to high-order polynomial in system size. Here we report the application of an exact technique, full configuration interaction quantum Monte Carlo to a variety of real solids, providing reference many-electron energies that are used to rigorously benchmark the standard hierarchy of quantum-chemical techniques, up to the 'gold standard' coupled-cluster ansatz, including single, double and perturbative triple particle-hole excitation operators. We show the errors in cohesive energies predicted by this method to be small, indicating the potential of this computationally polynomial scaling technique to tackle current solid-state problems.

RPA natural orbitals and their application to post-Hartree-Fock electronic structure methods.

We present a method to approximate post-Hartree-Fock correlation energies by using approximate natural orbitals obtained by the random phase approximation (RPA). We demonstrate the method by applying it to the helium atom, the hydrogen and fluorine molecule, and to diamond as an example of a periodic system. For these benchmark systems, we show that RPA natural orbitals converge the MP2 correlation energy rapidly. Additionally, we calculated full configuration interaction energies for He and H2, which are in excellent agreement with the literature and experimental values. We conclude that the proposed method may serve as a compromise to reach good approximations to correlation energies at moderate computational cost, and we expect the method to be especially useful for theoretical studies on surface chemistry by providing an efficient basis to correlated wave function based methods.

Copernicium: A Relativistic Noble Liquid.

The chemical nature and aggregate state of superheavy copernicium (Cn) have been subject of speculation for many years. While strong relativistic effects render Cn chemically inert, which led Pitzer to suggest a noble-gas-like behavior in 1975, Eichler and co-workers in 2008 reported substantial interactions with a gold surface in atom-at-a-time experiments, suggesting a metallic character and a solid aggregate state. Herein, we explore the physicochemical properties of Cn by means of first-principles free-energy calculations, which confirm Pitzer's original hypothesis: With predicted melting and boiling points of 283±11 K and 340±10 K, Cn is indeed a volatile liquid and exhibits a density very similar to that of mercury. However, in stark contrast to mercury and the lighter Group 12 metals, we find bulk Cn to be bound by dispersion and to exhibit a large band gap of 6.4 eV, which is consistent with a noble-gas-like character. This non-group-conforming behavior is eventually traced back to strong scalar-relativistic effects, and in the non-relativistic limit, Cn appears as a common Group 12 metal.

Melting Si: Beyond Density Functional Theory.

The melting point of silicon in the cubic diamond phase is calculated using the random phase approximation (RPA). The RPA includes exact exchange as well as an approximate treatment of local as well as nonlocal many body correlation effects of the electrons. We predict a melting temperature of about 1735 and 1640 K without and with core polarization effects, respectively. Both values are within 3% of the experimental melting temperature of 1687 K. In comparison, the commonly used gradient approximation to density functional theory predicts a melting point that is 200 K too low, and hybrid functionals overestimate the melting point by 150 K. We correlate the predicted melting point with the energy difference between cubic diamond and the beta-tin phase of silicon, establishing that this energy difference is an important benchmark for the development of approximate functionals. The current results demonstrate that the RPA can be used to predict accurate finite temperature properties and underlines the excellent predictive properties of the RPA for condensed matter.

Laplace transformed MP2 for three dimensional periodic materials using stochastic orbitals in the plane wave basis and correlated sampling.

We present an implementation and analysis of a stochastic high performance algorithm to calculate the correlation energy of three-dimensional periodic systems in second-order Møller-Plesset perturbation theory (MP2). In particular we measure the scaling behavior of the sample variance and probe whether this stochastic approach is competitive if accuracies well below 1 meV per valence orbital are required, as it is necessary for calculations of adsorption, binding, or surface energies. The algorithm is based on the Laplace transformed MP2 (LTMP2) formulation in the plane wave basis. The time-dependent Hartree-Fock orbitals, appearing in the LTMP2 formulation, are stochastically rotated in the occupied and unoccupied Hilbert space. This avoids a full summation over all combinations of occupied and unoccupied orbitals, as inspired by the work of Neuhauser, Rabani, and Baer [J. Chem. Theory Comput. 9, 24 (2013)]. Additionally, correlated sampling is introduced, accelerating the statistical convergence significantly.