Analytic Interatomic Forces in the Random Phase Approximation

Massively parallel implementation of gradients within the random phase approximation: Application to the polymorphs of benzene

The Random-Phase approximation (RPA) provides an appealing framework for semi-local density functional theory. In its Resolution-of-the-Identity (RI) approach, it is a very accurate and more cost-effective method than most other wavefunction-based correlation methods. For widespread applications, efficient implementations of nuclear gradients for structure optimizations and data sampling of machine learning approaches are required. We report a well scaling implementation of RI-RPA nuclear gradients on massively parallel computers. The approach is applied to two polymorphs of the benzene crystal obtaining very good cohesive and relative energies. Different correction and extrapolation schemes are investigated for further improvement of the results and estimations of error bars.

Analytical Second-Order Properties for the Random Phase Approximation: Nuclear Magnetic Resonance Shieldings

A method for the analytical computation of nuclear magnetic resonance (NMR) shieldings within the direct random phase approximation (RPA) is presented. As a starting point, we use the RPA ground-state energy expression within the resolution-of-the-identity approximation in the atomic-orbital formalism. As has been shown in a recent benchmark study using numerical second derivatives [Glasbrenner, M. J. Chem. Theory Comput. 2022, 18, 192], RPA based on a Hartree-Fock reference shows accuracies comparable to coupled cluster singles and doubles (CCSD) for NMR chemical shieldings. Together with the much lower computational cost of RPA, it has emerged as an accurate method for the computation of NMR shieldings. Therefore, we aim to extend the applicability of RPA NMR to larger systems by introducing analytical second-order derivatives, making it a viable method for the accurate and efficient computation of NMR chemical shieldings.

Machine Learning Density Functionals from the Random-Phase Approximation

Kohn-Sham density functional theory (DFT) is the standard method for first-principles calculations in computational chemistry and materials science. More accurate theories such as the random-phase approximation (RPA) are limited in application due to their large computational cost. Here, we use machine learning to map the RPA to a pure Kohn-Sham density functional. The machine learned RPA model (ML-RPA) is a nonlocal extension of the standard gradient approximation. The density descriptors used as ingredients for the enhancement factor are nonlocal counterparts of the local density and its gradient. Rather than fitting only RPA exchange-correlation energies, we also include derivative information in the form of RPA optimized effective potentials. We train a single ML-RPA functional for diamond, its surfaces, and liquid water. The accuracy of ML-RPA for the formation energies of 28 diamond surfaces reaches that of state-of-the-art van der Waals functionals. For liquid water, however, ML-RPA cannot yet improve upon the standard gradient approximation. Overall, our work demonstrates how machine learning can extend the applicability of the RPA to larger system sizes, time scales, and chemical spaces.

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.

Efficient Method for the Computation of Frozen-Core Nuclear Gradients within the Random Phase Approximation

A method for the evaluation of analytical frozen-core gradients within the random phase approximation is presented. We outline an efficient way to evaluate the response of the density of active electrons arising only when introducing the frozen-core approximation and constituting the main difficulty, together with the response of the standard Kohn-Sham density. The general framework allows to extend the outlined procedure to related electron correlation methods in the atomic orbital basis that require the evaluation of density responses, such as second-order Møller-Plesset perturbation theory or coupled cluster variants. By using Cholesky decomposed densities─which reintroduce the occupied index in the time-determining steps─we are able to achieve speedups of 20-30% (depending on the size of the basis set) by using the frozen-core approximation, which is of similar magnitude as for molecular orbital formulations. We further show that the errors introduced by the frozen-core approximation are practically insignificant for molecular geometries.

Localized Resolution of Identity Approach to the Analytical Gradients of Random-Phase Approximation Ground-State Energy: Algorithm and Benchmarks

We develop and implement a formalism which enables calculating the analytical gradients of particle-hole random-phase approximation (RPA) ground-state energy with respect to the atomic positions within the atomic orbital basis set framework. Our approach is based on a localized resolution of identity (LRI) approximation for evaluating the two-electron Coulomb integrals and their derivatives, and the density functional perturbation theory for computing the first-order derivatives of the Kohn-Sham (KS) orbitals and orbital energies. Our implementation allows one to relax molecular structures at the RPA level using both Gaussian-type orbitals (GTOs) and numerical atomic orbitals (NAOs). Benchmark calculations against previous implementations show that our approach delivers adequate numerical precision, highlighting the usefulness of LRI in the context of RPA gradient evaluations. A careful assessment of the quality of RPA geometries for small molecules reveals that post-KS RPA systematically overestimates the bond lengths. We furthermore optimized the geometries of the four low-lying water hexamers-cage, prism, cyclic, and book isomers, and determined the energy hierarchy of these four isomers using RPA. The obtained RPA energy ordering is in good agreement with that yielded by the coupled cluster method with single, double and perturbative triple excitations, despite that the dissociation energies themselves are appreciably underestimated. The underestimation of the dissociation energies by RPA is well corrected by the renormalized single excitation correction.

Assessing the Accuracy of Machine Learning Thermodynamic Perturbation Theory: Density Functional Theory and Beyond

Machine learning thermodynamic perturbation theory (MLPT) is a promising approach to compute finite temperature properties when the goal is to compare several different levels of ab initio theory and/or to apply highly expensive computational methods. Indeed, starting from a production molecular dynamics trajectory, this method can estimate properties at one or more target levels of theory from only a small number of additional fixed-geometry calculations, which are used to train a machine learning model. However, as MLPT is based on thermodynamic perturbation theory (TPT), inaccuracies might arise when the starting point trajectory samples a configurational space which has a small overlap with that of the target approximations of interest. By considering case studies of molecules adsorbed in zeolites and several different density functional theory approximations, in this work we assess the accuracy of MLPT for ensemble total energies and enthalpies of adsorption. It is shown that problematic cases can be detected even without knowing reference results and that even in these situations it is possible to recover target level results within chemical accuracy by applying a machine-learning-based Monte Carlo (MLMC) resampling. Finally, on the basis of the ideas developed in this work, we assess and confirm the accuracy of recently published MLPT-based enthalpies of adsorption at the random phase approximation level, whose high computational cost would completely hinder a direct molecular dynamics simulation.

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.

Ultrafast Spectroscopy of Photoactive Molecular Systems from First Principles: Where We Stand Today and Where We Are Going

Computational spectroscopy is becoming a mandatory tool for the interpretation of the complex, and often congested, spectral maps delivered by modern non-linear multi-pulse techniques. The fields of Electronic Structure Methods, Non-Adiabatic Molecular Dynamics, and Theoretical Spectroscopy represent the three pillars of the virtual ultrafast optical spectrometer, able to deliver transient spectra in silico from first principles. A successful simulation strategy requires a synergistic approach that balances between the three fields, each one having its very own challenges and bottlenecks. The aim of this Perspective is to demonstrate that, despite these challenges, an impressive agreement between theory and experiment is achievable now regarding the modeling of ultrafast photoinduced processes in complex molecular architectures. Beyond that, some key recent developments in the three fields are presented that we believe will have major impacts on spectroscopic simulations in the very near future. Potential directions of development, pending challenges, and rising opportunities are illustrated.

Assessing model-dielectric-dependent hybrid functionals on the antiferromagnetic transition-metal monoxides MnO, FeO, CoO, and NiO

Recently, two nonempirical hybrid functionals, dielectric-dependent range-separated hybrid functional based on the Coulomb-attenuating method (DD-RSH-CAM) and doubly screened hybrid functional (DSH), have been suggested by Chen et al (2018 Phys. Rev. Mater. 2 073803) and Cui et al (2018 J. Phys. Chem. Lett. 9 2338), respectively. These two hybrid functionals are both based on a common model dielectric function approach, but differ in the way how to non-empirically obtain the range-separation parameter. By retaining the full short-range Fock exchange and a fraction of the long-range Fock exchange that equals the inverse of the dielectric constant, both DD-RSH-CAM and DSH turn out to perform very well in predicting the band gaps for a large variety of semiconductors and insulators. Here, we assess how these two hybrid functionals perform on challenging antiferromagnetic transition-metal monoxides MnO, FeO, CoO, and NiO by comparing them to other conventional hybrid functionals and the GW method. We find that single-shot DD0-RSH-CAM and DSH0 improve the band gaps towards experiments as compared to conventional hybrid functionals. The magnetic moments are slightly increased, but the predicted dielectric constants are decreased. The valence band density of states (DOS) predicted by DD0-RSH-CAM and DSH0 are as satisfactory as HSE03 in comparison to experimental spectra, however, the conduction band DOS are shifted to higher energies by about 2 eV compared to HSE03. Self-consistent DD-RSH-CAM and DSH deteriorate the results with a significant overestimation of band gaps.

Efficient and accurate description of adsorption in zeolites

Accurate theoretical methods are needed to correctly describe adsorption on solid surfaces or in porous materials. The random phase approximation (RPA) with singles corrections scheme and the second order Møller-Plesset perturbation theory (MP2) are two schemes, which offer high accuracy at affordable computational cost. However, there is little knowledge about their applicability and reliability for different adsorbates and surfaces. Here, we calculate adsorption energies of seven different molecules in zeolite chabazite to show that RPA with singles corrections is superior to MP2, not only in terms of accuracy but also in terms of computer time. Therefore, RPA with singles is a suitable scheme for obtaining highly accurate adsorption energies in porous materials and similar systems.

Tuning the balance between dispersion and entropy to design temperature-responsive flexible metal-organic frameworks

Temperature-responsive flexibility in metal-organic frameworks (MOFs) appeals to the imagination. The ability to transform upon thermal stimuli while retaining a given crystalline topology is desired for specialized sensors and actuators. However, rational design of such shape-memory nanopores is hampered by a lack of knowledge on the nanoscopic interactions governing the observed behavior. Using the prototypical MIL-53(Al) as a starting point, we show that the phase transformation between a narrow-pore and large-pore phase is determined by a delicate balance between dispersion stabilization at low temperatures and entropic effects at higher ones. We present an accurate theoretical framework that allows designing breathing thermo-responsive MOFs, based on many-electron data for the dispersion interactions and density-functional theory entropy contributions. Within an isoreticular series of materials, MIL-53(Al), MIL-53(Al)-FA, DUT-4, DUT-5 and MIL-53(Ga), only MIL-53(Al) and MIL-53(Ga) are proven to switch phases within a realistic temperature range.

Rational design of metal organic frameworks (MOFs) with shape-memory nanopores is a formidable challenge. Here the authors use an accurate theoretical approach to design thermo-responsive MOFs based on a balance of van der Waals and entropy contributions.

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.

Hydrogen adsorption on Pt(111) revisited from random phase approximation

Hydrogen adsorption on Pt(111) has been actively studied using semilocal approximations within the density functional theory featuring simultaneous adsorption of hydrogen on multiple sites, i.e., fcc, atop, and hcp. Considering the accuracy needed to detail the feature, we revisit this problem with the help of higher level of theory, the adiabatic connection fluctuation dissipation theorem within the random phase approximation. Our simulation emphasizes important roles played by the equilibrium lattice parameter of the surface, mass of the hydrogen isotope, and hydrogen coverage. The insight acquired in this study provides a way to consistently interpret electrochemical and spectroscopic data.

Toward a predictive theory of correlated materials

Correlated electron materials display a rich variety of notable properties ranging from unconventional superconductivity to metal-insulator transitions. These properties are of interest from the point of view of applications but are hard to treat theoretically, as they result from multiple competing energy scales. Although possible in more weakly correlated materials, theoretical design and spectroscopy of strongly correlated electron materials have been a difficult challenge for many years. By treating all the relevant energy scales with sufficient accuracy, complementary advances in Green's functions and quantum Monte Carlo methods open a path to first-principles computational property predictions in this class of materials.

Density anomaly of water at negative pressures from first principles

Using molecular dynamics simulations based on ab initio trained high-dimensional neural network potentials, we study the equation of state of liquid water at negative pressures. From density isobars computed for various pressures down to p  =  -230 MPa we determine the line of density maxima for two potentials based on the BLYP and the RPBE functionals, respectively. In both cases, dispersion corrections are included to account for non-local long-range correlations that give rise to van der Waals forces. We have followed the density maximum down to negative pressures close to the spinodal instability. For both functionals, the temperature of maximum density increases with decreasing pressure under moderate stretching, but changes slope at [Formula: see text] MPa and [Formula: see text] MPa for BLYP and RPBE, respectively. Our calculations confirm qualitatively the retracing shape of the line of density maxima found for empirical water models, indicating that the spinodal line maintains a positive slope even at strongly negative pressures.

Simplified DFT methods for consistent structures and energies of large systems

Kohn-Sham density functional theory (DFT) is routinely used for the fast electronic structure computation of large systems and will most likely continue to be the method of choice for the generation of reliable geometries in the foreseeable future. Here, we present a hierarchy of simplified DFT methods designed for consistent structures and non-covalent interactions of large systems with particular focus on molecular crystals. The covered methods are a minimal basis set Hartree-Fock (HF-3c), a small basis set screened exchange hybrid functional (HSE-3c), and a generalized gradient approximated functional evaluated in a medium-sized basis set (B97-3c), all augmented with semi-classical correction potentials. We give an overview on the methods design, a comprehensive evaluation on established benchmark sets for geometries and lattice energies of molecular crystals, and highlight some realistic applications on large organic crystals with several hundreds of atoms in the primitive unit cell.

The Bethe–Salpeter equation in chemistry: relations with TD-DFT, applications and challenges

We review the Bethe–Salpeter formalism and analyze its performances for the calculation of the excited state properties of molecular systems.

Assessing Density Functionals Using Many Body Theory for Hybrid Perovskites

Which density functional is the "best" for structure simulations of a particular material? A concise, first principles, approach to answer this question is presented. The random phase approximation (RPA)-an accurate many body theory-is used to evaluate various density functionals. To demonstrate and verify the method, we apply it to the hybrid perovskite MAPbI_{3}, a promising new solar cell material. The evaluation is done by first creating finite temperature ensembles for small supercells using RPA molecular dynamics, and then evaluating the variance between the RPA and various approximate density functionals for these ensembles. We find that, contrary to recent suggestions, van der Waals functionals do not improve the description of the material, whereas hybrid functionals and the strongly constrained appropriately normed (SCAN) density functional yield very good agreement with the RPA. Finally, our study shows that in the room temperature tetragonal phase of MAPbI_{3}, the molecules are preferentially parallel to the shorter lattice vectors but reorientation on ps time scales is still possible.