Beyond Electrons: Correlation and Self-Energy in Multicomponent Density Functional Theory

Post-Kohn-Sham methods are used to evaluate the ground-state correlation energy and the orbital self-energy of systems consisting of multiple flavors of different fermions. Starting from multicomponent density functional theory, suitable ways to arrive at the corresponding multicomponent random-phase approximation and the multicomponent Green's functionG W ${GW}$ approximation, including relativistic effects, are outlined. Given the importance of both of this methods in the development of modern Kohn-Sham density functional approximations, this work will provide a foundation to design advanced multicomponent density functional approximations. Additionally, theG W ${GW}$ quasiparticle energies are needed to study light-matter interactions with the Bethe-Salpeter equation.

Influence of Electronic Correlations on Electron-Phonon Interactions of Molecular Systems with the GW and Coupled Cluster Methods

Electron-phonon interactions are of great importance to a variety of physical phenomena, and their accurate description is an important goal for first-principles calculations. Isolated examples of materials and molecular systems have emerged where electron-phonon coupling is enhanced over density functional theory (DFT) when using the Green's-function-based ab initio GW method, which provides a more accurate description of electronic correlations. It is, however, unclear how general this enhancement is and how employing high-end quantum chemistry methods, which further improve the description of electronic correlations, might further alter electron-phonon interactions over GW or DFT. Here, we address these questions by computing the renormalization of the highest occupied molecular orbital energies of Thiel's set of organic molecules by harmonic vibrations using DFT, GW, and equation-of-motion coupled-cluster calculations. We find that, depending on the amount of exact exchange included in the DFT starting point, GW can increase the magnitude of the electron-phonon coupling across Thiel's set of molecules by an average factor of 1.1-1.8 compared to the underlying DFT, while equation-of-motion coupled-cluster leads to an increase of 1.4-2. The electron-phonon coupling predicted with the ab initio GW method is generally in much closer agreement to coupled cluster values compared to DFT, establishing GW as a promising route for accurately computing electron-phonon phenomena in molecules and beyond at a much lower computational cost than higher-end quantum chemistry techniques.

Chelate Complexes of 3d Transition Metal Ions─A Challenge for Electronic-Structure Methods?

Different electronic-structure methods were assessed for their ability to predict two important properties of the industrially relevant chelating agent nitrilotriacetic acid (NTA): its selectivity with respect to six different first-row transition metal ions and the spin-state energetics of its complex with Fe(III). The investigated methods encompassed density functional theory (DFT), the random phase approximation (RPA), coupled cluster (CC) theory, and the auxiliary-field quantum Monte Carlo (AFQMC) method, as well as the complete active space self-consistent field (CASSCF) method and the respective on-top methods: second-order N-electron valence state perturbation theory (NEVPT2) and multiconfiguration pair-density functional theory (MC-PDFT). Different strategies for selecting active spaces were explored, and the density matrix renormalization group (DMRG) approach was used to solve the largest active spaces. Despite somewhat ambiguous multi-reference diagnostics, most methods gave relatively good agreement with experimental data for the chemical reactions connected to the selectivity, which only involved transition-metal complexes in their high-spin state. CC methods yielded the highest accuracy followed by range-separated DFT and AFQMC. We discussed in detail that even higher accuracies can be obtained with NEVPT2, under the prerequisite that consistent active spaces along the entire chemical reaction can be selected, which was not the case for reactions involving Fe(III). A bigger challenge for electronic-structure methods was the prediction of the spin-state energetics, which additionally involved lower spin states that exhibited larger multi-reference diagnostics. Conceptually different, typically accurate methods ranging from CC theory via DMRG-NEVPT2 in combination with large active spaces to AFQMC agreed well that the high-spin state is energetically significantly favored over the other spin states. This was in contrast to most DFT functionals and RPA which yielded a smaller stabilization and some common DFT functionals and MC-PDFT even predicting the low-spin state to be energetically most favorable.

Hybrid RPA:DFT Approach for Adsorption on Transition Metal Surfaces: Methane and Ethane on Platinum (111)

The hybrid QM:QM approach is extended to adsorption on transition metal surfaces. The random phase approximation (RPA) as the high-level method is applied to cluster models and, using the subtractive scheme, embedded in periodic models which are treated with density functional theory (DFT) that is the low-level method. The PBE functional, both without dispersion and augmented with the many-body dispersion (MBD), is employed. Adsorption of methane and ethane on the Pt(111) surface is studied. For methane in a 2 × 2 surface cell, the hybrid RPA:PBE and RPA:PBE+MBD results, -14.3 and -16.0 kJ mol-1, respectively, are in close agreement with the periodic RPA value of -13.8 kJ mol-1 at significantly reduced computational cost (factor of ∼50). For methane and ethane, the RPA:PBE results (-14.3 and -17.8 kJ mol-1, respectively) indicate underbinding relative to energies derived from experimental desorption barriers for relevant loadings (-15.6 ± 1.6 and -27.2 ± 2.9 kJ mol-1, respectively), whereas the hybrid RPA:PBE+MBD results (-16.0 and -24.9 kJ mol-1, respectively) agree with the experiment well within experimental uncertainty limits (deviation of -0.4 ± 1.5 and +2.3 ± 2.9 kJ mol-1, respectively). Finding a cluster that adequately and robustly represents the adsorbate at the bulk surface is important for the success of the RPA-based QM:QM scheme for metals.

Rigorous Screened Interactions for Realistic Correlated Electron Systems

We derive a widely applicable first-principles approach for determining two-body, static effective interactions for low-energy Hamiltonians with quantitative accuracy. The algebraic construction rigorously conserves all instantaneous two-point correlation functions in a chosen model space at the level of the random phase approximation, improving upon the traditional uncontrolled static approximations. Applied to screened interactions within a quantum embedding framework, we demonstrate these faithfully describe the relaxation of local subspaces via downfolding high-energy physics in molecular systems, as well as enabling a systematically improvable description of the long-range plasmonic contributions in extended graphene.

Visualizing and characterizing excited states from time-dependent density functional theory

Time-dependent density functional theory (TD-DFT) is the most widely-used electronic structure method for excited states, due to a favorable combination of low cost and semi-quantitative accuracy in many contexts, even if there are well recognized limitations. This Perspective describes various ways in which excited states from TD-DFT calculations can be visualized and analyzed, both qualitatively and quantitatively. This includes not just orbitals and densities but also well-defined statistical measures of electron-hole separation and of Frenkel-type exciton delocalization. Emphasis is placed on mathematical connections between methods that have often been discussed separately. Particular attention is paid to charge-transfer diagnostics, which provide indicators of when TD-DFT may not be trustworthy due to its categorical failure to describe long-range electron transfer. Measures of exciton size and charge separation that are directly connected to the underlying transition density are recommended over more ad hoc metrics for quantifying charge-transfer character.

Generalized perturbative singles corrections to the random phase approximation method: Impact on noncovalent interaction energies of closed- and open-shell dimers

The post-Kohn-Sham (KS) random phase approximation (RPA) method may provide a poor description of interaction energies of weakly bonded molecules due to inherent density errors in approximate KS functionals. To overcome these errors, we develop a generalized formalism to incorporate perturbative singles (pS) corrections to the RPA method using orbital rotations as a perturbation parameter. The pS schemes differ in the choice of orbital-rotation gradient and Hessian. We propose a pS scheme termed RPA singles (RPAS)[Hartree-Fock (HF)] that uses the RPA orbital-rotation gradient and time-dependent HF Hessian. This correction reduces the errors in noncovalent interaction energies of closed- and open-shell dimers. For the open-shell dimers, the RPAS(HF) method leads to a consistent error reduction by 50% or more compared to the RPA method for the cases of hydrogen-bonding, metal-solvent, carbene-solvent, and dispersion interactions. We also find that the pS corrections are more important in error reduction compared to higher-order exchange corrections to the RPA method. Overall, for open shells, the RPAS(HF)-corrected RPA method provides chemical accuracy for noncovalent interactions and is more reliable than other perturbative schemes and dispersion-corrected density functional approximations, highlighting its importance as a reliable beyond-RPA correction.

A DFT perspective on organometallic lanthanide chemistry

Computational studies of the coordination chemistry and bonding of lanthanides have grown in recent decades as the need for understanding the distinct physical, optical, and magnetic properties of these compounds increased. Density functional theory (DFT) methods offer a favorable balance of computational cost and accuracy in lanthanide chemistry and have helped to advance the discovery of novel oxidation states and electronic configurations. This Frontier article examines the scope and limitations of DFT in interpreting structural and spectroscopic data of low-valent lanthanide complexes, elucidating periodic trends, and predicting their properties and reactivity, presented through selected examples.

Avoiding spin contamination and spatial symmetry breaking by exact-exchange-only optimized-effective-potential methods within the symmetrized Kohn-Sham framework

For open-shell atoms and molecules, Kohn-Sham (KS) methods typically resort to spin-polarized approaches that exhibit spin-contamination and often break spatial symmetries. As a result, the KS Hamiltonian operator and the KS orbitals do not exhibit the space and spin symmetry of the physical electron system. The KS formalism can be symmetrized in a rigorous way only in real space, only in spin space, or both in real and spin space. Within such symmetrized KS frameworks, we present exact-exchange-only optimized-effective-potential (OEP) methods that are free of spin contamination and/or spatial symmetry breaking. The effect of symmetrizations on the total energy and its parts and on the exchange potential is analyzed. The presented exact-exchange-only OEP methods may serve as a starting point for high-level symmetrized KS methods based, e.g., on the adiabatic-connection fluctuation-dissipation theorem.

Chemically accurate singlet-triplet gaps of organic chromophores and linear acenes by the random phase approximation and σ-functionals

Predicting the energy differences between different spin-states is challenging for many widely used ab initio electronic structure methods. We here assess the ability of the direct random phase approximation (dRPA), dRPA plus two different screened second-order exchange (SOX) corrections, and σ-functionals to predict adiabatic singlet-triplet gaps. With mean absolute deviations of below 0.1 eV to experimental reference values, independent of the Kohn-Sham starting point, dRPA and σ-functionals accurately predict singlet-triplet gaps of 18 organic chromophores. The addition of SOX corrections to dRPA considerably worsens agreement with experiment, adding to the mounting evidence that dRPA+SOX methods are not generally applicable beyond-RPA methods. Also for a series of linear acene chains with up to ten fused rings, dRPA, and σ-functionals are in excellent agreement with coupled-cluster single double triple reference data. In agreement with advanced multi-reference methods, dRPA@PBE and σ-functional@PBE predict a singlet ground state for all chain lengths, while dRPA@PBE0 and σ-functional@PBE0 predict a triplet ground state for longer acenes. Our work shows dRPA and σ-functionals to be reliable methods for calculating singlet-triplet gaps in aromatic molecules.

Optimizing the regularization in size-consistent second-order Brillouin-Wigner perturbation theory

Despite its simplicity and relatively low computational cost, second-order Møller-Plesset perturbation theory (MP2) is well-known to overbind noncovalent interactions between polarizable monomers and some organometallic bonds. In such situations, the pairwise-additive correlation energy expression in MP2 is inadequate. Although energy-gap dependent amplitude regularization can substantially improve the accuracy of conventional MP2 in these regimes, the same regularization parameter worsens the accuracy for small molecule thermochemistry and density-dependent properties. Recently, we proposed a repartitioning of Brillouin-Wigner perturbation theory that is size-consistent to second order (BW-s2), and a free parameter (α) was set to recover the exact dissociation limit of H2 in a minimal basis set. Alternatively α can be viewed as a regularization parameter, where each value of α represents a valid variant of BW-s2, which we denote as BW-s2(α). In this work, we semi-empirically optimize α for noncovalent interactions, thermochemistry, alkane conformational energies, electronic response properties, and transition metal datasets, leading to improvements in accuracy relative to the ab initio parameterization of BW-s2 and MP2. We demonstrate that the optimal α parameter (α = 4) is more transferable across chemical problems than energy-gap-dependent regularization parameters. This is attributable to the fact that the BW-s2(α) regularization strength depends on all of the information encoded in the t amplitudes rather than just orbital energy differences. While the computational scaling of BW-s2(α) is iterative O(N5), this effective and transferable approach to amplitude regularization is a promising route to incorporate higher-order correlation effects at second-order cost.

Corrected density functional theory and the random phase approximation: Improved accuracy at little extra cost

We recently introduced an efficient methodology to perform density-corrected Hartree-Fock density functional theory [DC(HF)-DFT] calculations and an extension to it we called "corrected" HF DFT [C(HF)-DFT] [Graf and Thom, J. Chem. Theory Comput. 19 5427-5438 (2023)]. In this work, we take a further step and combine C(HF)-DFT, augmented with a straightforward orbital energy correction, with the random phase approximation (RPA). We refer to the resulting methodology as corrected HF RPA [C(HF)-RPA]. We evaluate the proposed methodology across various RPA methods: direct RPA (dRPA), RPA with an approximate exchange kernel, and RPA with second-order screened exchange. C(HF)-dRPA demonstrates very promising performance; for RPA with exchange methods, on the other hand, we often find over-corrections.

TURBOMOLE: Today and Tomorrow

TURBOMOLE is a highly optimized software suite for large-scale quantum-chemical and materials science simulations of molecules, clusters, extended systems, and periodic solids. TURBOMOLE uses Gaussian basis sets and has been designed with robust and fast quantum-chemical applications in mind, ranging from homogeneous and heterogeneous catalysis to inorganic and organic chemistry and various types of spectroscopy, light-matter interactions, and biochemistry. This Perspective briefly surveys TURBOMOLE's functionality and highlights recent developments that have taken place between 2020 and 2023, comprising new electronic structure methods for molecules and solids, previously unavailable molecular properties, embedding, and molecular dynamics approaches. Select features under development are reviewed to illustrate the continuous growth of the program suite, including nuclear electronic orbital methods, Hartree-Fock-based adiabatic connection models, simplified time-dependent density functional theory, relativistic effects and magnetic properties, and multiscale modeling of optical properties.

Equation of state predictions for ScF3 and CaZrF6 with neural network-driven molecular dynamics

In silico property prediction based on density functional theory (DFT) is increasingly performed for crystalline materials. Whether quantitative agreement with experiment can be achieved with current methods is often an unresolved question, and may require detailed examination of physical effects such as electron correlation, reciprocal space sampling, phonon anharmonicity, and nuclear quantum effects (NQE), among others. In this work, we attempt first-principles equation of state prediction for the crystalline materials ScF3 and CaZrF6, which are known to exhibit negative thermal expansion (NTE) over a broad temperature range. We develop neural network (NN) potentials for both ScF3 and CaZrF6 trained to extensive DFT data, and conduct direct molecular dynamics prediction of the equation(s) of state over a broad temperature/pressure range. The NN potentials serve as surrogates of the DFT Hamiltonian with enhanced computational efficiency allowing for simulations with larger supercells and inclusion of NQE utilizing path integral approaches. The conclusion of the study is mixed: while some equation of state behavior is predicted in semiquantitative agreement with experiment, the pressure-induced softening phenomenon observed for ScF3 is not captured in our simulations. We show that NQE have a moderate effect on NTE at low temperature but does not significantly contribute to equation of state predictions at increasing temperature. Overall, while the NN potentials are valuable for property prediction of these NTE (and related) materials, we infer that a higher level of electron correlation, beyond the generalized gradient approximation density functional employed here, is necessary for achieving quantitative agreement with experiment.

Exploring Cuneanes as Potential Benzene Isosteres and Energetic Materials: Scope and Mechanistic Investigations into Regioselective Rearrangements from Cubanes

Cuneane is a strained hydrocarbon that can be accessed via metal-catalyzed isomerization of cubane. The carbon atoms of cuneane define a polyhedron of the C2v point group with six faces─two triangular, two quadrilateral, and two pentagonal. The rigidity, strain, and unique exit vectors of the cuneane skeleton make it a potential scaffold of interest for the synthesis of functional small molecules and materials. However, the limited previous synthetic efforts toward cuneanes have focused on monosubstituted or redundantly substituted systems such as permethylated, perfluorinated, and bis(hydroxymethylated) cuneanes. Such compounds, particularly rotationally symmetric redundantly substituted cuneanes, have limited potential as building blocks for the synthesis of complex molecules. Reliable, predictable, and selective syntheses of polysubstituted cuneanes bearing more complex substitution patterns would facilitate the study of this ring system in myriad applications. Herein, we report the regioselective, AgI-catalyzed isomerization of asymmetrically 1,4-disubstituted cubanes to cuneanes. In-depth DFT calculations provide a charge-controlled regioselectivity model, and direct dynamics simulations indicate that the nonclassical carbocation invoked is short-lived and dynamic effects augment the charge model.

Theoretical and Data-Driven Approaches for Biomolecular Condensates

Biomolecular condensation processes are increasingly recognized as a fundamental mechanism that living cells use to organize biomolecules in time and space. These processes can lead to the formation of membraneless organelles that enable cells to perform distinct biochemical processes in controlled local environments, thereby supplying them with an additional degree of spatial control relative to that achieved by membrane-bound organelles. This fundamental importance of biomolecular condensation has motivated a quest to discover and understand the molecular mechanisms and determinants that drive and control this process. Within this molecular viewpoint, computational methods can provide a unique angle to studying biomolecular condensation processes by contributing the resolution and scale that are challenging to reach with experimental techniques alone. In this Review, we focus on three types of dry-lab approaches: theoretical methods, physics-driven simulations and data-driven machine learning methods. We review recent progress in using these tools for probing biomolecular condensation across all three fields and outline the key advantages and limitations of each of the approaches. We further discuss some of the key outstanding challenges that we foresee the community addressing next in order to develop a more complete picture of the molecular driving forces behind biomolecular condensation processes and their biological roles in health and disease.

Quantifying the Impact of Halogenation on Intermolecular Interactions and Binding Modes of Aromatic Molecules

Halogenation of aromatic molecules is frequently used to modulate intermolecular interactions with ramifications for optoelectronic and mechanical properties. In this work, we accurately quantify and understand the nature of intermolecular interactions in perhalogenated benzene (PHB) clusters. Using benchmark binding energies from the fixed-node diffusion Monte Carlo (FN-DMC) method, we show that generalized Kohn-Sham semicanonical projected random phase approximation (GKS-spRPA) plus approximate exchange kernel (AKX) provides reliable interaction energies with mean absolute error (MAE) of 0.23 kcal/mol. Using the GKS-spRPA+AXK method, we quantify the interaction energies of several binding modes of PHB clusters ((C₆X₆)_n; X = F, Cl, Br, I; n = 2, 3). For a given binding mode, the interaction energies increase 3-4 times from X = F to X = I; the X-X binding modes have energies in the range of 2-4 kcal/mol, while the π-π binding mode has interaction energies in the range of 4-12 kcal/mol. SAPT-DFT-based energy decomposition analysis is then used to show that the equilibrium geometries are dictated primarily by the dispersion and exchange interactions. Finally, we test the accuracy of several dispersion-corrected density functional approximations and show that only the r2SCAN-D4 method has a low MAE and correct long-range behavior, which makes it suitable for large-scale simulations and for developing structure-function relationships of halogenated aromatic systems.

A Similarity Renormalization Group Approach to Green's Function Methods

The family of Green's function methods based on the GW approximation has gained popularity in the electronic structure theory thanks to its accuracy in weakly correlated systems combined with its cost-effectiveness. Despite this, self-consistent versions still pose challenges in terms of convergence. A recent study [Monino and Loos J. Chem. Phys. 2022, 156, 231101.] has linked these convergence issues to the intruder-state problem. In this work, a perturbative analysis of the similarity renormalization group (SRG) approach is performed on Green's function methods. The SRG formalism enables us to derive, from first-principles, the expression of a naturally static and Hermitian form of the self-energy that can be employed in quasiparticle self-consistent GW (qsGW) calculations. The resulting SRG-based regularized self-energy significantly accelerates the convergence of qsGW calculations, slightly improves the overall accuracy, and is straightforward to implement in existing code.

Analytical harmonic vibrational frequencies with VV10-containing density functionals: Theory, efficient implementation, and benchmark assessments

VV10 is a powerful nonlocal density functional for long-range correlation that is used to include dispersion effects in many modern density functionals, such as the meta-generalized gradient approximation (mGGA), B97M-V, the hybrid GGA, ωB97X-V, and the hybrid mGGA, ωB97M-V. While energies and analytical gradients for VV10 are already widely available, this study reports the first derivation and efficient implementation of the analytical second derivatives of the VV10 energy. The additional compute cost of the VV10 contributions to analytical frequencies is shown to be small in all but the smallest basis sets for recommended grid sizes. This study also reports the assessment of VV10-containing functionals for predicting harmonic frequencies using the analytical second derivative code. The contribution of VV10 to simulating harmonic frequencies is shown to be small for small molecules but important for systems where weak interactions are important, such as water clusters. In the latter cases, B97M-V, ωB97M-V, and ωB97X-V perform very well. The convergence of frequencies with respect to the grid size and atomic orbital basis set size is studied, and recommendations are reported. Finally, scaling factors to allow comparison of scaled harmonic frequencies with experimental fundamental frequencies and to predict zero-point vibrational energy are presented for some recently developed functionals (including r2SCAN, B97M-V, ωB97X-V, M06-SX, and ωB97M-V).

Practical Post-Kohn-Sham Methods for Time-Reversal Symmetry Breaking References

The applicability of reduced scaling algorithms based on auxiliary subspace methods for the correlation energy from the random phase approximation (RPA) as well as the correlation part of the self-energy obtained from the GW method is outlined for time-reversal symmetry breaking Kohn-Sham (KS) references. The updated algorithms allow for an efficient evaluation of RPA energies and GW quasiparticle energies for molecular systems with KS references that break time-reversal symmetry. The latter occur, for example, in magnetic fields. Furthermore, KS references for relativistic open-shell molecules also break time-reversal symmetry due to the single determinant ansatz used. Errors of the updated reduced-scaling algorithms are shown to be negligible compared to reference implementations, while the overall computational scaling is reduced by 2 orders of magnitude. Ionization energies obtained from the GW approximation are shown to be robust even for the electronically complicated group of trivalent lanthanoid ions. Starting from GW quasiparticle energies, it is subsequently shown that light-matter interactions of these systems can be calculated using the Bethe-Salpeter equation (BSE). Using the combined GW-BSE method, the absorption and emission spectra of a molecular europium(III) complex can be obtained including spin-orbit coupling.

Toward Pair Atomic Density Fitting for Correlation Energies with Benchmark Accuracy

Pair atomic density fitting (PADF) has been identified as a promising strategy to reduce the scaling with system size of quantum chemical methods for the calculation of the correlation energy like the direct random-phase approximation (RPA) or second-order Møller-Plesset perturbation theory (MP2). PADF can however introduce large errors in correlation energies as the two-electron interaction energy is not guaranteed to be bounded from below. This issue can be partially alleviated by using very large fit sets, but this comes at the price of reduced efficiency and having to deal with near-linear dependencies in the fit set. One posibility is to use global density fitting (DF), but in this work, we introduce an alternative methodology to overcome this problem that preserves the intrinsically favorable scaling of PADF. We first regularize the Fock matrix by projecting out parts of the basis set which gives rise to orbital products that are hard to describe by PADF. After having thus obtained a reliable self-consistent field solution, we then also apply this projector to the orbital coefficient matrix to improve the precision of PADF-MP2 and PADF-RPA. We systematically assess the accuracy of this new approach in a numerical atomic orbital framework using Slater type orbitals (STO) and correlation consistent Gaussian type basis sets up to quintuple-ζ quality for systems with more than 200 atoms. For the small and medium systems in the S66 database we show the maximum deviation of PADF-MP2 and PADF-RPA relative correlation energies to DF-MP2 and DF-RPA reference results to be 0.07 and 0.14 kcal/mol, respectively. When the new projector method is used, the errors only slightly increase for large molecules and also when moderately sized fit sets are used the resulting errors are well under control. Finally, we demonstrate the computational efficiency of our algorithm by calculating the interaction energies of large, non-covalently bound complexes with more than 1000 atoms and 20000 atomic orbitals at the RPA@PBE/CC-pVTZ level of theory.

Geometries and vibrational frequencies with Kohn-Sham methods using σ-functionals for the correlation energy

Recently, Kohn-Sham (KS) methods with new correlation functionals, called σ-functionals, have been introduced. Technically, σ-functionals are closely related to the well-known random phase approximation (RPA); formally, σ-functionals are rooted in perturbation theory along the adiabatic connection. If employed in a post-self-consistent field manner in a Gaussian basis set framework, then, σ-functional methods are computationally very efficient. Moreover, for main group chemistry, σ-functionals are highly accurate and can compete with high-level wave-function methods. For reaction and transition state energies, e.g., chemical accuracy of 1 kcal/mol is reached. Here, we show how to calculate first derivatives of the total energy with respect to nuclear coordinates for methods using σ-functionals and then carry out geometry optimizations for test sets of main group molecules, transition metal compounds, and non-covalently bonded systems. For main group molecules, we additionally calculate vibrational frequencies. σ-Functional methods are found to yield very accurate geometries and vibrational frequencies for main group molecules superior not only to those from conventional KS methods but also to those from RPA methods. For geometries of transition metal compounds, not surprisingly, best geometries are found for RPA methods, while σ-functional methods yield somewhat less good results. This is attributed to the fact that in the optimization of σ-functionals, transition metal compounds could not be represented well due to the lack of reliable reference data. For non-covalently bonded systems, σ-functionals yield geometries of the same quality as the RPA or as conventional KS schemes combined with dispersion corrections.

Connections between many-body perturbation and coupled-cluster theories

Here, we build on the works of Scuseria et al. [J. Chem. Phys. 129, 231101 (2008)] and Berkelbach [J. Chem. Phys. 149, 041103 (2018)] to show connections between the Bethe-Salpeter equation (BSE) formalism combined with the GW approximation from many-body perturbation theory and coupled-cluster (CC) theory at the ground- and excited-state levels. In particular, we show how to recast the GW and Bethe-Salpeter equations as non-linear CC-like equations. Similitudes between BSE@GW and the similarity-transformed equation-of-motion CC method are also put forward. The present work allows us to easily transfer key developments and the general knowledge gathered in CC theory to many-body perturbation theory. In particular, it may provide a path for the computation of ground- and excited-state properties (such as nuclear gradients) within the GW and BSE frameworks.

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.

Assessment of the Second-Order Statically Screened Exchange Correction to the Random Phase Approximation for Correlation Energies

With increasing interelectronic distance, the screening of the electron–electron interaction by the presence of other electrons becomes the dominant source of electron correlation. This effect is described by the random phase approximation (RPA) which is therefore a promising method for the calculation of weak interactions. The success of the RPA relies on the cancellation of errors, which can be traced back to the violation of the crossing symmetry of the 4-point vertex, leading to strongly overestimated total correlation energies. By the addition of second-order screened exchange (SOSEX) to the correlation energy, this issue is substantially reduced. In the adiabatic connection (AC) SOSEX formalism, one of the two electron–electron interaction lines in the second-order exchange term is dynamically screened (SOSEX(W, v_c)). A related SOSEX expression in which both electron–electron interaction lines are statically screened (SOSEX(W(0), W(0))) is obtained from the G3W2 contribution to the electronic self-energy. In contrast to SOSEX(W, v_c), the evaluation of this correlation energy expression does not require an expensive numerical frequency integration and is therefore advantageous from a computational perspective. We compare the accuracy of the statically screened variant to RPA and RPA+SOSEX(W, v_c) for a wide range of chemical reactions. While both methods fail for barrier heights, SOSEX(W(0), W(0)) agrees very well with SOSEX(W, v_c) for charged excitations and noncovalent interactions where they lead to major improvements over RPA.

Scaled σ-functionals for the Kohn-Sham correlation energy with scaling functions from the homogeneous electron gas

The recently introduced σ-functionals constitute a new type of functionals for the Kohn-Sham (KS) correlation energy. σ-Functionals are based on the adiabatic-connection fluctuation-dissipation theorem, are computationally closely related to the well-known direct random phase approximation (dRPA), and are formally rooted in many-body perturbation theory along the adiabatic connection. In σ-functionals, the function of the eigenvalues σ of the Kohn-Sham response matrix that enters the coupling constant and frequency integration in the dRPA is replaced by another function optimized with the help of reference sets of atomization, reaction, transition state, and non-covalent interaction energies. σ-Functionals are highly accurate and yield chemical accuracy of 1 kcal/mol in reaction or transition state energies, in main group chemistry. A shortcoming of σ-functionals is their inability to accurately describe processes involving a change of the electron number, such as ionizations or electron attachments. This problem is attributed to unphysical self-interactions caused by the neglect of the exchange kernel in the dRPA and σ-functionals. Here, we tackle this problem by introducing a frequency- and σ-dependent scaling of the eigenvalues σ of the KS response function that models the effect of the exchange kernel. The scaling factors are determined with the help of the homogeneous electron gas. The resulting scaled σ-functionals retain the accuracy of their unscaled parent functionals but in addition yield very accurate ionization potentials and electron affinities. Moreover, atomization and total energies are found to be exceptionally accurate. Scaled σ-functionals are computationally highly efficient like their unscaled counterparts.

Analytical Solution to the Flory-Huggins Model

A self-consistent analytical solution for binodal concentrations of the two-component Flory-Huggins phase separation model is derived. We show that this form extends the validity of the Ginzburg-Landau expansion away from the critical point to cover the whole phase space. Furthermore, this analytical solution reveals an exponential scaling law of the dilute phase binodal concentration as a function of the interaction strength and chain length. We demonstrate explicitly the power of this approach by fitting experimental protein liquid-liquid phase separation boundaries to determine the effective chain length and solute-solvent interaction energies. Moreover, we demonstrate that this strategy allows us to resolve differences in interaction energy contributions of individual amino acids. This analytical framework can serve as a new way to decode the protein sequence grammar for liquid-liquid phase separation.

An optimally tuned range-separated hybrid starting point for ab initio GW plus Bethe–Salpeter equation calculations of molecules

The ab initio GW plus Bethe–Salpeter equation (GW-BSE, where G is the one particle Green's function and W is the screened Coulomb interaction) approach has emerged as a leading method for predicting excitations in both solids and molecules with a predictive power contingent upon several factors. Among these factors are the (1) generalized Kohn–Sham eigensystem used to construct the GW self-energy and to solve the BSE and (2) the efficacy and suitability of the Tamm–Dancoff approximation. Here, we present a detailed benchmark study of low-lying singlet excitations from a generalized Kohn–Sham (gKS) starting point based on an optimally tuned range-separated hybrid (OTRSH) functional. We show that the use of this gKS starting point with one-shot G0W0 and G0W0-BSE leads to the lowest mean absolute errors (MAEs) and mean signed errors (MSEs), with respect to high-accuracy reference values, demonstrated in the literature thus far for the ionization potentials of the GW100 benchmark set and for low-lying neutral excitations of Thiel’s set molecules in the gas phase, without the need for self-consistency. The MSEs and MAEs of one-shot G0W0-BSE@OTRSH excitation energies are comparable to or lower than those obtained with other functional starting points after self-consistency. Additionally, we compare these results with linear-response time-dependent density functional theory (TDDFT) calculations and find GW-BSE to be superior to TDDFT when calculations are based on the same exchange-correlation functional. This work demonstrates tuned range-separated hybrids used in combination with GW and GW-BSE can greatly suppress starting point dependence for molecules, leading to accuracy similar to that for higher-order wavefunction-based theories for molecules without the need for costlier iterations to self-consistency.

A litmus test for the balanced description of dispersion interactions and coordination chemistry of lanthanoids

The influence of long-range interactions on the structure of complexes of Eu(III) with four 9-hydroxy-phenalen-1-one ligands (HPLN) and one alkaline earth metal dication [Eu(PLN)₄AE]⁺ (AE: Mg, Ca, Sr, and Ba) is analyzed. Through the [Eu(PLN)₄Ca]⁺ complex, which is a charged complex with two metals-one of them a lanthanoid-and with four relatively fluxional π-ligands, the difficulties of describing such systems are identified. The inclusion of the D3(BJ) or D4 corrections to different density functionals introduces significant changes in the structure, which are shown to stem from the interaction between pairs of PLN ligands. This interaction is studied further with a variety of density functionals, wave-function based methods, and by means of the random phase approximation. By comparing the computed results with those from experimental evidence of gas-phase photoluminescence and ion mobility measurements it is concluded that the inclusion of dispersion corrections does not always yield structures that are in agreement with the experimental findings.

Efficient Adiabatic Connection Approach for Strongly Correlated Systems: Application to Singlet-Triplet Gaps of Biradicals

Strong electron correlation can be captured with multireference wave function methods, but an accurate description of the electronic structure requires accounting for the dynamic correlation, which they miss. In this work, a new approach for the correlation energy based on the adiabatic connection (AC) is proposed. The AC_n method accounts for terms up to order n in the coupling constant, and it is size-consistent and free from instabilities. It employs the multireference random phase approximation and the Cholesky decomposition technique, leading to a computational cost growing with the fifth power of the system size. Because of the dependence on only one- and two-electron reduced density matrices, AC_n is more efficient than existing ab initio multireference dynamic correlation methods. AC_n affords excellent results for singlet-triplet gaps of challenging organic biradicals. The development presented in this work opens new perspectives for accurate calculations of systems with dozens of strongly correlated electrons.

Tension between predicting accurate ground state correlation energies and excitation energies from adiabatic approximations in TDDFT

The connection between the adiabatic excitation energy of time-dependent density functional theory and the ground state correlation energy from the adiabatic connection fluctuation–dissipation theorem (ACFDT) is explored in the limiting case of one excited state. An exact expression is derived for any adiabatic Hartree-exchange–correlation kernel that connects the excitation energy and the potential contribution to correlation. The resulting formula is applied to the asymmetric Hubbard dimer, a system where this limit is exact. Results from a hierarchy of approximations to the kernel, including the random phase approximation (RPA) with and without exchange and the adiabatically exact (AE) approximation, are compared to the exact ones. At full coupling, the numerical results indicate a tension between predicting an accurate excitation energy and an accurate potential contribution to correlation. The AE approximation is capable of making accurate predictions of both quantities, but only in parts of the parameter space that classify as weakly correlated, while RPA tends to be unable to accurately predict these properties simultaneously anywhere. For a strongly correlated dimer, the AE approximation greatly overestimates the excitation energy yet continues to yield an accurate ground state correlation energy due to its accurate prediction of the adiabatic connection integrand. If similar trends hold for real systems, the development of correlation kernels will be important for applications of the ACFDT in systems with large potential contributions to correlation.

Staggered Mesh Method for Correlation Energy Calculations of Solids: Random Phase Approximation in Direct Ring Coupled Cluster Doubles and Adiabatic Connection Formalisms

We propose a staggered mesh method for correlation energy calculations of periodic systems under the random phase approximation (RPA), which generalizes the recently developed staggered mesh method for periodic second order Møller-Plesset perturbation theory (MP2) calculations [Xing; Li; Lin J. Chem. Theory Comput. 2021]. Compared to standard RPA calculations, the staggered mesh method introduces negligible additional computational cost. It avoids a significant portion of the finite-size error and can be asymptotically advantageous for quasi-1D systems and certain quasi-2D and 3D systems with high symmetries. We demonstrate the applicability of the method using two different formalisms: the direct ring coupled cluster doubles (drCCD) theory, and the adiabatic-connection (AC) fluctuation-dissipation theory. In the drCCD formalism, the second order screened exchange (SOSEX) correction can also be readily obtained using the staggered mesh method. In the AC formalism, the staggered mesh method naturally avoids the need of performing "head/wing" corrections to the dielectric operator. The effectiveness of the staggered mesh method for insulating systems is theoretically justified by investigating the finite-size error of each individual perturbative term in the RPA correlation energy, expanded as an infinite series of terms associated with ring diagrams. As a side contribution, our analysis provides proof that the finite-size error of each perturbative term of standard RPA and SOSEX calculations scales as O(Nk-1), where N_k is the number of grid points in a Monkhorst-Pack mesh.

NMR Coupling Constants Based on the Bethe-Salpeter Equation in the GW Approximation

We present the first steps to extend the Green's function GW method and the Bethe-Salpeter equation (BSE) to molecular response properties such as nuclear magnetic resonance (NMR) indirect spin-spin coupling constants. We discuss both a nonrelativistic one-component and a quasi-relativistic two-component formalism. The latter describes scalar-relativistic and spin-orbit effects and allows us to study heavy-element systems with reasonable accuracy. Efficiency is maintained by the application of the resolution of the identity approximation throughout. The performance is demonstrated using conventional central processing units (CPUs) and modern graphics processing units (GPUs) for molecules involving several thousand basis functions. Our results show that a large amount of Hartree-Fock exchange is vital to provide a sufficient Kohn-Sham starting point to compute the GW quasi-particle energies. As the GW-BSE approach is generally less accurate for triplet excitations or related properties such as the Fermi-contact interaction, the admixture of the Kohn-Sham correlation kernel through the contracted BSE (cBSE) method improves the results for NMR coupling constants. This leads to remarkable results when combined with the eigenvalue-only self-consistent variant (evGW) and Becke's half and half functional (BH&HLYP) or the CAM-QTP family. The developed methodology is used to calculate the Karplus curve of tin molecules, illustrating its applicability to extended chemically relevant molecules. Here, the GW-cBSE method improves upon the chosen BH&HLYP Kohn-Sham starting points.

Resolving Chemical Dynamics in Biological Energy Conversion: Long-Range Proton-Coupled Electron Transfer in Respiratory Complex I

Biological energy conversion is catalyzed by membrane-bound proteins that transduce chemical or light energy into energy forms that power endergonic processes in the cell. At a molecular level, these catalytic processes involve elementary electron-, proton-, charge-, and energy-transfer reactions that take place in the intricate molecular machineries of cell respiration and photosynthesis. Recent developments in structural biology, particularly cryo-electron microscopy (cryoEM), have resolved the molecular architecture of several energy transducing proteins, but detailed mechanistic principles of their charge transfer reactions still remain poorly understood and a major challenge for modern biochemical research. To this end, multiscale molecular simulations provide a powerful approach to probe mechanistic principles on a broad range of time scales (femtoseconds to milliseconds) and spatial resolutions (10¹–10⁶ atoms), although technical challenges also require balancing between the computational accuracy, cost, and approximations introduced within the model. Here we discuss how the combination of atomistic (aMD) and hybrid quantum/classical molecular dynamics (QM/MM MD) simulations with free energy (FE) sampling methods can be used to probe mechanistic principles of enzymes responsible for biological energy conversion. We present mechanistic explorations of long-range proton-coupled electron transfer (PCET) dynamics in the highly intricate respiratory chain enzyme Complex I, which functions as a redox-driven proton pump in bacterial and mitochondrial respiratory chains by catalyzing a 300 Å fully reversible PCET process. This process is initiated by a hydride (H^–) transfer between NADH and FMN, followed by long-range (>100 Å) electron transfer along a wire of 8 FeS centers leading to a quinone biding site. The reduction of the quinone to quinol initiates dissociation of the latter to a second membrane-bound binding site, and triggers proton pumping across the membrane domain of complex I, in subunits up to 200 Å away from the active site. Our simulations across different size and time scales suggest that transient charge transfer reactions lead to changes in the internal hydration state of key regions, local electric fields, and the conformation of conserved ion pairs, which in turn modulate the dynamics of functional steps along the reaction cycle. Similar functional principles, which operate on much shorter length scales, are also found in some unrelated proteins, suggesting that enzymes may employ conserved principles in the catalysis of biological energy transduction processes.

Adsorption of CH4 on the Pt(111) surface: Random phase approximation compared to density functional theory

We investigate the adsorption of CH₄ on the Pt(111) surface for two adsorption modes, hcp (hexagonal closed packed) hollow tripod and top monopod in a (√3 × √3)R30° surface cell that corresponds to experimental surface coverage. Surface structures are optimized with density functional theory using the Perdew-Burke-Ernzerhof (PBE) functional augmented with the many-body dispersion scheme of Tkatchenko (PBE+MBD). Whereas the Random Phase Approximation (RPA) predicts a clear preference of about 5 kJ mol^-1 for the hcp tripod compared to the top monopod structure, in agreement with vibrational spectra, PBE+MBD predicts about equal stability for the two adsorption structures. For the hcp tripod, RPA yields an adsorption energy of -14.5 kJ mol^-1, which is converged to within 1.0 ± 0.5 kJ mol^-1 with respect to the plane wave energy cutoff (500 eV), the k-point mesh (4 × 4 × 1), the vacuum layer (about 10.3 Å, with extrapolation to infinite distance), and the number of Pt layers (3). Increments for increasing the number of Pt layers to 4 (+1.6 kJ mol^-1) and the k-point mesh to 6 × 6 × 1 (-0.6 kJ mol^-1) yield a final estimate of -13.5 ± 2.1 kJ mol^-1, which agrees to within 2.2 ± 2.1 kJ mol^-1 with experiment (-15.7 ± 1.6), well within the chemical accuracy range.

High-Resolution X-ray Photoelectron Spectroscopy of Organometallic (C5H4SiMe3)3LnIII and [(C5H4SiMe3)3LnII]1- Complexes (Ln = Sm, Eu, Gd, Tb)

The capacity of X-ray photoelectron spectroscopy (XPS) to provide information on the electronic structure of molecular organometallic complexes of Ln(II) ions (Ln = lanthanide) has been examined for the first time. XPS spectra were obtained on the air-sensitive molecular trivalent 4fⁿ Cp'₃Ln^III complexes (Ln = Sm, Eu, Gd, Tb; Cp' = C₅H₄SiMe₃) and compared to those of the highly reactive divalent complexes, [K(crypt)][Cp'₃Ln^II] (crypt = 2.2.2-cryptand), which have either 4fⁿ⁺¹ (Sm, Eu) or 4fⁿ5d¹ electron configurations (Gd, Tb). The Ln 4d, Si 2p, and C 1s regions of the Ln(III) and Ln(II) complexes were identified and compared. The metal 4d peaks of these molecular lanthanide complexes were used diagnostically to compare oxidation states. The valence region of the Gd(III) and Gd(II) complexes was also examined with XPS and density function theory/random phase approximation (DFT/RPA) calculations, and this led to the tentative assignment of a signal from the 5d¹ electron consistent with a 4f⁷5d¹ electron configuration for Gd(II).

Chemical accuracy with σ-functionals for the Kohn-Sham correlation energy optimized for different input orbitals and eigenvalues

Recently, a new type of orbital-dependent functional for the Kohn-Sham (KS) correlation energy, σ-functionals, was introduced. Technically, σ-functionals are closely related to the well-known direct random phase approximation (dRPA). Within the dRPA, a function of the eigenvalues σ of the frequency-dependent KS response function is integrated over purely imaginary frequencies. In σ-functionals, this function is replaced by one that is optimized with respect to reference sets of atomization, reaction, transition state, and non-covalent interaction energies. The previously introduced σ-functional uses input orbitals and eigenvalues from KS calculations with the generalized gradient approximation (GGA) exchange-correlation functional of Perdew, Burke, and Ernzerhof (PBE). Here, σ-functionals using input orbitals and eigenvalues from the meta-GGA TPSS and the hybrid-functionals PBE0 and B3LYP are presented and tested. The number of reference sets taken into account in the optimization of the σ-functionals is larger than in the first PBE based σ-functional and includes sets with 3d-transition metal compounds. Therefore, also a reparameterized PBE based σ-functional is introduced. The σ-functionals based on PBE0 and B3LYP orbitals and eigenvalues reach chemical accuracy for main group chemistry. For the 10 966 reactions from the highly accurate W4-11RE reference set, the B3LYP based σ-functional exhibits a mean average deviation of 1.03 kcal/mol compared to 1.08 kcal/mol for the coupled cluster singles doubles perturbative triples method if the same valence quadruple zeta basis set is used. For 3d-transition metal chemistry, accuracies of about 2 kcal/mol are reached. The computational effort for the post-self-consistent evaluation of the σ-functional is lower than that of a preceding PBE0 or B3LYP calculation for typical systems.

Exploring the Limits of Second- and Third-Order Møller-Plesset Perturbation Theories for Noncovalent Interactions: Revisiting MP2.5 and Assessing the Importance of Regularization and Reference Orbitals

This work systematically assesses the influence of reference orbitals, regularization, and scaling on the performance of second- and third-order Møller-Plesset perturbation theory wave function methods for noncovalent interactions (NCIs). Testing on 19 data sets (A24, DS14, HB15, HSG, S22, X40, HW30, NC15, S66, AlkBind12, CO2Nitrogen16, HB49, Ionic43, TA13, XB18, Bauza30, CT20, XB51, and Orel26rad) covers a wide range of different NCIs including hydrogen bonding, dispersion, and halogen bonding. Inclusion of potential energy surfaces from different hydrogen bonds and dispersion-bound complexes gauges accuracy for nonequilibrium geometries. Fifteen methods are tested. In notation where nonstandard choices of orbitals are denoted as methods:orbitals, these are MP2, κ-MP2, SCS-MP2, OOMP2, κ-OOMP2, MP3, MP2.5, MP3:OOMP2, MP2.5:OOMP2, MP3:κ-OOMP2, MP2.5:κ-OOMP2, κ-MP3:κ-OOMP2, κ-MP2.5:κ-OOMP2, MP3:ωB97X-V, and MP2.5:ωB97X-V. Furthermore, we compare these methods to the ωB97M-V and B3LYP-D3 density functionals, as well as CCSD. We find that the κ-regularization (κ = 1.45 au was used throughout) improves the energetics in almost all data sets for both MP2 (in 17 out of 19 data sets) and OOMP2 (16 out of 19). The improvement is significant (e.g., the root-mean-square deviation (RMSD) for the S66 data set is 0.29 kcal/mol for κ-OOMP2 versus 0.67 kcal/mol for MP2) and for interactions between stable closed-shell molecules, not strongly dependent on the reference orbitals. Scaled MP3 (with a factor of 0.5) using κ-OOMP2 reference orbitals (MP2.5:κ-OOMP2) provides significantly more accurate results for NCIs across all data sets with noniterative O(N⁶) scaling (S66 data set RMSD: 0.10 kcal/mol). Across the entire data set of 356 points, the improvement over standard MP2.5 is approximately a factor of 2: RMSD for MP3:κ-OOMP2 is 0.25 vs 0.50 kcal/mol for MP2.5. The use of high-quality density functional reference orbitals (ωB97X-V) also significantly improves the results of MP2.5 for NCI over a Hartree-Fock orbital reference. All our assessments and conclusions are based on the use of the medium-sized aug-cc-pVTZ basis to yield results that are directly compared against complete basis set limit reference values.

Energies, structures, and harmonic frequencies of small water clusters from the direct random phase approximation

The binding energies, structures, and vibrational frequencies of water clusters up to 20 molecules are computed at the direct random phase approximation (RPA) level of theory and compared to theoretical benchmarks. Binding energies of the WATER27 set, which includes neutral and positively and negatively charged clusters, are predicted to be too low in the complete basis set limit by an average of 7 kcal/mol (9%) and are worse than the results from the best density functional theory methods or from the Møller-Plesset theory. The RPA shows significant basis set size dependence for binding energies. The order of the relative energies of the water hexamer and dodecamer isomers is predicted correctly by the RPA. The mean absolute deviation for angles and distances for neutral clusters up to the water hexamer are 0.2° and 0.6 pm, respectively, using quintuple-ζ basis sets. The relative energetic order of the hexamer isomers is preserved upon optimization. Vibrational frequencies for these systems are underestimated by several tens of wavenumbers for large basis sets, and deviations increase with the basis set size. Overall, the direct RPA method yields accurate structural parameters but systematically underestimates binding energies and shows strong basis set size dependence.

Quantum-electrodynamical time-dependent density functional theory within Gaussian atomic basis

Inspired by the formulation of quantum-electrodynamical time-dependent density functional theory (QED-TDDFT) by Rubio and co-workers [Flick et al., ACS Photonics 6, 2757-2778 (2019)], we propose an implementation that uses dimensionless amplitudes for describing the photonic contributions to QED-TDDFT electron-photon eigenstates. This leads to a Hermitian QED-TDDFT coupling matrix that is expected to facilitate the future development of analytic derivatives. Through a Gaussian atomic basis implementation of the QED-TDDFT method, we examined the effect of dipole self-energy, rotating-wave approximation, and the Tamm-Dancoff approximation on the QED-TDDFT eigenstates of model compounds (ethene, formaldehyde, and benzaldehyde) in an optical cavity. We highlight, in the strong coupling regime, the role of higher-energy and off-resonance excited states with large transition dipole moments in the direction of the photonic field, which are automatically accounted for in our QED-TDDFT calculations and might substantially affect the energies and compositions of polaritons associated with lower-energy electronic states.

Numerically stable optimized effective potential method with standard Gaussian basis sets

We present a numerically stable optimized effective potential (OEP) method based on Gaussian basis sets. The key point of the approach is a sequence of preprocessing steps of the auxiliary basis set used to represent exchange or correlation potentials, the Kohn-Sham (KS) response function, and the right-hand side of the OEP equation in conjunction with a representation of exchange or correlation potentials via exchange or correlation charge densities whose electrostatic potentials generate the potentials. Due to the preprocessing, standard Gaussian basis sets from basis set libraries can be used in OEP calculations. As examples, we present numerical stable computational setups based on aux-cc-pwCVXZ basis sets with X = T, Q, 5 for the orbitals and aux-cc-pVDZ/mp2fit and aux-cc-pVTZ/mp2fit auxiliary basis sets and use them to calculate KS exchange potentials with the exact exchange-only KS method for various atoms and molecules. The resulting exchange potentials not only are numerically stable and physically reasonable but also show convergence with increasing quality of the orbital basis sets. The effect of incorporating exact conditions that the KS exchange potential has to obey is discussed. Moreover, it is briefly demonstrated that the presented approach not only works for KS exchange potentials but equally well for correlation potentials within the direct random phase approximation. Besides for OEP methods, the introduced preprocessing of auxiliary basis sets should also be beneficial in procedures to calculate back effective KS potentials from given electron densities.

Selfconsistent random phase approximation methods

This Perspective reviews recent efforts toward selfconsistent calculations of ground-state energies within the random phase approximation (RPA) in the (generalized) Kohn-Sham (KS) density functional theory context. Since the RPA correlation energy explicitly depends on the non-interacting KS potential, an additional condition to determine the energy as a functional of the density is necessary. This observation leads to the concept of functional selfconsistency (FSC), which requires that the KS density equals the interacting density defined as the functional derivative of the ground-state energy with respect to the external potential. While all existing selfconsistent RPA schemes violate FSC, the recent generalized KS semicanonical projected RPA (GKS-spRPA) method takes a step toward satisfying it. This leads to systematic improvements in densities, binding energy curves, reference state stability, and molecular properties compared to non-selfconsistent RPA as well as optimized effective potential RPA. GKS-spRPA orbital energies accurately approximate valence and core ionization potentials, and even electron affinities of non-valence bound anions. The computational cost and performance of GKS-spRPA are compared to those of related selfconsistent schemes, including GW and orbital optimization methods, and limitations are discussed. Large differences between KS and interacting densities observed in the absence of FSC and the well-rounded performance of GKS-spRPA suggest that the KS potential as a density functional should be defined via the FSC condition for explicitly potential-dependent density functionals.

Revealing the nature of electron correlation in transition metal complexes with symmetry breaking and chemical intuition

In this work, we provide a nuanced view of electron correlation in the context of transition metal complexes, reconciling computational characterization via spin and spatial symmetry breaking in single-reference methods with qualitative concepts from ligand-field and molecular orbital theories. These insights provide the tools to reliably diagnose the multi-reference character, and our analysis reveals that while strong (i.e., static) correlation can be found in linear molecules (e.g., diatomics) and weakly bound and antiferromagnetically coupled (monometal-noninnocent ligand or multi-metal) complexes, it is rarely found in the ground-states of mono-transition-metal complexes. This leads to a picture of static correlation that is no more complex for transition metals than it is, e.g., for organic biradicaloids. In contrast, the ability of organometallic species to form more complex interactions, involving both ligand-to-metal σ-donation and metal-to-ligand π-backdonation, places a larger burden on a theory's treatment of dynamic correlation. We hypothesize that chemical bonds in which inter-electron pair correlation is non-negligible cannot be adequately described by theories using MP2 correlation energies and indeed find large errors vs experiment for carbonyl-dissociation energies from double-hybrid density functionals. A theory's description of dynamic correlation (and to a less important extent, delocalization error), which affects relative spin-state energetics and thus spin symmetry breaking, is found to govern the efficacy of its use to diagnose static correlation.

Assessment of Performance of Density Functionals for Predicting Potential Energy Curves in Hydrogen Storage Applications

The availability of accurate computational tools for modeling and simulation is vital to accelerate the discovery of materials capable of storing hydrogen (H₂) under given parameters of pressure swing and temperature. Previously, we compiled the H2Bind275 data set consisting of equilibrium geometries and assessed the performance of 55 density functionals over this data set (Veccham, S. P.; Head-Gordon, M. J. Chem. Theory Comput. 2020, 16, 4963-4982). As it is crucial for computational tools to accurately model the entire potential energy curve (PEC), in addition to the equilibrium geometry, we extended this data set with 389 new data points to include two compressed and three elongated geometries along 78 PECs for H₂ binding, forming the H2Bind78 × 7 data set. By assessing the performance of 55 density functionals on this significantly larger and more comprehensive H2Bind78 × 7 data set, we identified the best performing density functionals for H₂ binding applications: PBE0-DH, ωB97X-V, ωB97M-V, and DSD-PBEPBE-D3(BJ). The addition of Hartree-Fock exchange improves the performance of density functionals, albeit not uniformly throughout the PEC. We recommend the usage of ωB97X-V and ωB97M-V density functionals as they offer good performance for both geometries and energies. In addition, we also identified B97M-V and B97M-rV as the best semilocal density functionals for predicting H₂ binding energy at its equilibrium geometry.

Excited states in the adiabatic connection fluctuation-dissipation theory: Recovering missing correlation energy from the negative part of the density response spectrum

The adiabatic connection (AC) theory offers an alternative to the perturbation theory methods for computing correlation energy in the multireference wavefunction framework. We show that the AC correlation energy formula can be expressed in terms of the density linear response function as a sum of components related to positive and negative parts of the transition energy spectrum. Consequently, generalization of the adiabatic connection fluctuation-dissipation theory to electronically excited states is obtained. The component of the linear response function related to the negative-transition energy enters the correlation energy expression with an opposite sign to that of the positive-transition part and is non-negligible in the description of excited states. To illustrate this, we analyze the approximate AC model in which the linear response function is obtained in the extended random phase approximation (ERPA). We demonstrate that AC can be successfully combined with the ERPA for excited states, provided that the negative-excitation component of the response function is rigorously accounted for. The resulting AC0D model, an extension of the AC0 scheme introduced in our earlier works, is applied to a benchmark set of singlet excitation energies of organic molecules. AC0D constitutes a significant improvement over AC0 by bringing the excitation energies of the lowest excited states to a satisfactory agreement with theoretical best estimates, which parallels or even exceeds the accuracy of the n-electron valence state perturbation theory method. For higher excitations, AC0D is less reliable due to the gradual deterioration of the underlying ERPA linear response.

Computational approaches to dissociative chemisorption on metals: towards chemical accuracy

We review the state-of-the-art in the theory of dissociative chemisorption (DC) of small gas phase molecules on metal surfaces, which is important to modeling heterogeneous catalysis for practical reasons, and for achieving an understanding of the wealth of experimental information that exists for this topic, for fundamental reasons. We first give a quick overview of the experimental state of the field. Turning to the theory, we address the challenge that barrier heights (E_b, which are not observables) for DC on metals cannot yet be calculated with chemical accuracy, although embedded correlated wave function theory and diffusion Monte-Carlo are moving in this direction. For benchmarking, at present chemically accurate E_b can only be derived from dynamics calculations based on a semi-empirically derived density functional (DF), by computing a sticking curve and demonstrating that it is shifted from the curve measured in a supersonic beam experiment by no more than 1 kcal mol^-1. The approach capable of delivering this accuracy is called the specific reaction parameter (SRP) approach to density functional theory (DFT). SRP-DFT relies on DFT and on dynamics calculations, which are most efficiently performed if a potential energy surface (PES) is available. We therefore present a brief review of the DFs that now exist, also considering their performance on databases for E_b for gas phase reactions and DC on metals, and for adsorption to metals. We also consider expressions for SRP-DFs and briefly discuss other electronic structure methods that have addressed the interaction of molecules with metal surfaces. An overview is presented of dynamical models, which make a distinction as to whether or not, and which dissipative channels are modeled, the dissipative channels being surface phonons and electronically non-adiabatic channels such as electron-hole pair excitation. We also discuss the dynamical methods that have been used, such as the quasi-classical trajectory method and quantum dynamical methods like the time-dependent wave packet method and the reaction path Hamiltonian method. Limits on the accuracy of these methods are discussed for DC of diatomic and polyatomic molecules on metal surfaces, paying particular attention to reduced dimensionality approximations that still have to be invoked in wave packet calculations on polyatomic molecules like CH₄. We also address the accuracy of fitting methods, such as recent machine learning methods (like neural network methods) and the corrugation reducing procedure. In discussing the calculation of observables we emphasize the importance of modeling the properties of the supersonic beams in simulating the sticking probability curves measured in the associated experiments. We show that chemically accurate barrier heights have now been extracted for DC in 11 molecule-metal surface systems, some of which form the most accurate core of the only existing database of E_b for DC reactions on metal surfaces (SBH10). The SRP-DFs (or candidate SRP-DFs) that have been derived show transferability in many cases, i.e., they have been shown also to yield chemically accurate E_b for chemically related systems. This can in principle be exploited in simulating rates of catalyzed reactions on nano-particles containing facets and edges, as SRP-DFs may be transferable among systems in which a molecule dissociates on low index and stepped surfaces of the same metal. In many instances SRP-DFs have allowed important conclusions regarding the mechanisms underlying observed experimental trends. An important recent observation is that SRP-DFT based on semi-local exchange DFs has so far only been successful for systems for which the difference of the metal work function and the molecule's electron affinity exceeds 7 eV. A main challenge to SRP-DFT is to extend its applicability to the other systems, which involve a range of important DC reactions of e.g. O₂, H₂O, NH₃, CO₂, and CH₃OH. Recent calculations employing a PES based on a screened hybrid exchange functional suggest that the road to success may be based on using exchange functionals of this category.

Advanced First-Principle Modeling of Relativistic Ruddlesden—Popper Strontium Iridates

In this review, we provide a survey of the application of advanced first-principle methods on the theoretical modeling and understanding of novel electronic, optical, and magnetic properties of the spin-orbit coupled Ruddlesden–Popper series of iridates Srn+1IrnO3n+1 (n = 1, 2, and ∞). After a brief description of the basic aspects of the adopted methods (noncollinear local spin density approximation plus an on-site Coulomb interaction (LSDA+U), constrained random phase approximation (cRPA), GW, and Bethe–Salpeter equation (BSE)), we present and discuss select results. We show that a detailed phase diagrams of the metal–insulator transition and magnetic phase transition can be constructed by inspecting the evolution of electronic and magnetic properties as a function of Hubbard U, spin–orbit coupling (SOC) strength, and dimensionality n, which provide clear evidence for the crucial role played by SOC and U in establishing a relativistic (Dirac) Mott–Hubbard insulating state in Sr2IrO4 and Sr3Ir2O7. To characterize the ground-state phases, we quantify the most relevant energy scales fully ab initio—crystal field energy, Hubbard U, and SOC constant of three compounds—and discuss the quasiparticle band structures in detail by comparing GW and LSDA+U data. We examine the different magnetic ground states of structurally similar n = 1 and n = 2 compounds and clarify that the origin of the in-plane canted antiferromagnetic (AFM) state of Sr2IrO4 arises from competition between isotropic exchange and Dzyaloshinskii–Moriya (DM) interactions whereas the collinear AFM state of Sr3Ir2O7 is due to strong interlayer magnetic coupling. Finally, we report the dimensionality controlled metal–insulator transition across the series by computing their optical transitions and conductivity spectra at the GW+BSE level from the the quasi two-dimensional insulating n = 1 and 2 phases to the three-dimensional metallic n=∞ phase.

Toward chemical accuracy at low computational cost: Density-functional theory with σ-functionals for the correlation energy

We introduce new functionals for the Kohn-Sham correlation energy that are based on the adiabatic-connection fluctuation-dissipation (ACFD) theorem and are named σ-functionals. Like in the well-established direct random phase approximation (dRPA), σ-functionals require as input exclusively eigenvalues σ of the frequency-dependent KS response function. In the new functionals, functions of σ replace the σ-dependent dRPA expression in the coupling-constant and frequency integrations contained in the ACFD theorem. We optimize σ-functionals with the help of reference sets for atomization, reaction, transition state, and non-covalent interaction energies. The optimized functionals are to be used in a post-self-consistent way using orbitals and eigenvalues from conventional Kohn-Sham calculations employing the exchange-correlation functional of Perdew, Burke, and Ernzerhof. The accuracy of the presented approach is much higher than that of dRPA methods and is comparable to that of high-level wave function methods. Reaction and transition state energies from σ-functionals exhibit accuracies close to 1 kcal/mol and thus approach chemical accuracy. For the 10 966 reactions of the W4-11RE reference set, the mean absolute deviation is 1.25 kcal/mol compared to 3.21 kcal/mol in the dRPA case. Non-covalent binding energies are accurate to a few tenths of a kcal/mol. The presented approach is highly efficient, and the post-self-consistent calculation of the total energy requires less computational time than a density-functional calculation with a hybrid functional and thus can be easily carried out routinely. σ-Functionals can be implemented in any existing dRPA code with negligible programming effort.

A range-separated generalized Kohn-Sham method including a long-range nonlocal random phase approximation correlation potential

Based on our recently published range-separated random phase approximation (RPA) functional [Kreppel et al., "Range-separated density-functional theory in combination with the random phase approximation: An accuracy benchmark," J. Chem. Theory Comput. 16, 2985-2994 (2020)], we introduce self-consistent minimization with respect to the one-particle density matrix. In contrast to the range-separated RPA methods presented so far, the new method includes a long-range nonlocal RPA correlation potential in the orbital optimization process, making it a full-featured variational generalized Kohn-Sham (GKS) method. The new method not only improves upon all other tested RPA schemes including the standard post-GKS range-separated RPA for the investigated test cases covering general main group thermochemistry, kinetics, and noncovalent interactions but also significantly outperforms the popular G₀W₀ method in estimating the ionization potentials and fundamental gaps considered in this work using the eigenvalue spectra obtained from the GKS Hamiltonian.