Straightforward and Accurate Automatic Auxiliary Basis Set Generation for Molecular Calculations with Atomic Orbital Basis Sets

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.

Going for Gold(-Standard): Attaining Coupled Cluster Accuracy in Oxide-Supported Nanoclusters

The structure of oxide-supported metal nanoclusters plays an essential role in their sharply enhanced catalytic activity over that of bulk metals. Simulations provide the atomic-scale resolution needed to understand these systems. However, the sensitive mix of metal-metal and metal-support interactions, which govern their structure, puts stringent requirements on the method used, requiring calculations beyond standard density functional theory (DFT). The method of choice is coupled cluster theory [specifically CCSD(T)], but its computational cost has so far prevented its application to these systems. In this work, we showcase two approaches to make CCSD(T) accuracy readily achievable in oxide-supported nanoclusters. First, we leverage the SKZCAM protocol to provide the first benchmarks of oxide-supported nanoclusters, revealing that it is specifically metal-metal interactions that are challenging to capture with DFT. Second, we propose a CCSD(T) correction (ΔCC) to the metal-metal interaction errors in DFT, reaching accuracy comparable to that of the SKZCAM protocol at significantly lower cost. This approach forges a path toward studying larger systems at reliable accuracy, which we highlight by identifying a ground-state structure in agreement with experiments for Au20 on MgO, a challenging system where DFT models have yielded conflicting predictions.

Revisiting Gauge-Independent Kinetic Energy Densities in Meta-GGAs and Local Hybrid Calculations of Magnetizabilities

In a recent study [J. Chem. Theory Comput. 2021, 17, 1457-1468], some of us examined the accuracy of magnetizabilities calculated with density functionals representing the local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA (mGGA), as well as global hybrid (GH) and range-separated (RS) hybrid functionals by assessment against accurate reference values obtained with coupled-cluster theory with singles, doubles, and perturbative triples [CCSD(T)]. Our study was later extended to local hybrid (LH) functionals by Holzer et al. [J. Chem. Theory Comput. 2021, 17, 2928-2947]; in this work, we examine a larger selection of LH functionals, also including range-separated LH (RSLH) functionals and strong-correlation LH (scLH) functionals. Holzer et al. also studied the importance of the physically correct handling of the magnetic gauge dependence of the kinetic energy density (τ) in mGGA calculations by comparing the Maximoff-Scuseria formulation of τ used in our aforementioned study to the more physical current-density extension derived by Dobson. In this work, we also revisit this comparison with a larger selection of mGGA functionals. We find that the newly tested LH, RSLH, and scLH functionals outperform all of the functionals considered in the previous studies. The various LH functionals afford the seven lowest mean absolute errors while also showing remarkably small standard deviations and mean errors. Most strikingly, the best two functionals are scLHs that also perform remarkably well in cases with significant multiconfigurational character, such as the ozone molecule, which is traditionally excluded from statistical error evaluations due to its large errors with common density functionals.

Basis-Set-Error-Free Random-Phase Approximation Correlation Energies for Atoms Based on the Sternheimer Equation

The finite basis set errors for all-electron random-phase approximation (RPA) correlation energy calculations are analyzed for isolated atomic systems. We show that, within the resolution-of-identity (RI) RPA framework, the major source of the basis set errors is the incompleteness of the single-particle atomic orbitals used to expand the Kohn-Sham eigenstates, instead of the auxiliary basis set (ABS) to represent the density response function χ0 and the bare Coulomb operator v. By solving the Sternheimer equation for the first-order wave function on a dense radial grid, we are able to eliminate the major error─the incompleteness error of the single-particle atomic basis set─for atomic RPA calculations. The error stemming from a finite ABS can be readily rendered vanishingly small by increasing the size of the ABS, or by iteratively determining the eigenmodes of the χ0v operator. The variational property of the RI-RPA correlation energy can be further exploited to optimize the ABS in order to achieve fast convergence of the RI-RPA correlation energy. These numerical techniques enable us to obtain basis-set-error-free RPA correlation energies for atoms, and in this work, such energies for atoms from H to Kr are presented. The implications of the numerical techniques developed in the present work for addressing the basis set issue for molecules and solids are discussed.

Automatic Generation of Auxiliary Basis Sets in Spherical Representation Using the Cholesky Decomposition

Density fitting techniques that use automatically generated auxiliary basis sets generally rely on the formation of basis function products. Recently, Lehtola [ J. Chem. Theory Comput. 2021, 17, 6886-6900] presented a procedure making use of a purely spherical representation by adding auxiliary basis functions coupled to the required angular momentum quantum numbers for the product of spherical harmonics and then removing linear dependencies by means of a Cholesky decomposition. In this work, we extend this idea by making use of the explicit equations for the product of two spherical harmonics in the angular part of the basis function product. Some of the resulting terms are not directly accessible when popular standard integral libraries are used, which could prevent the widespread use of the exact product form. For these terms, we introduce four approximations of increasing sophistication that require integrals involving only standard Gaussian-type orbitals and thus can be computed with standard libraries. We assess the accuracy of the different schemes in the context of the aCD for the reconstruction of the electron repulsion integral matrix and absolute and relative single point energies and in the framework of optimally tuned range-separated hybrid functionals.

Automatic Generation of Accurate and Cost-Efficient Auxiliary Basis Sets

We have recently discussed an algorithm to automatically generate auxiliary basis sets (ABSs) of the standard form for density fitting (DF) or resolution-of-the-identity (RI) calculations in a given atomic orbital basis set (OBS) of any form, such as Gaussian-type orbitals, Slater-type orbitals, or numerical atomic orbitals [J. Chem. Theory Comput. 2021, 17, 6886]. In this work, we study two ways to reduce the cost of such automatically generated ABSs without sacrificing their accuracy. We contract the ABS with a singular value decomposition proposed by Kállay [J. Chem. Phys. 2014, 141, 244113], used here in a somewhat different setting. We also drop the high-angular momentum functions from the ABS, as they are unnecessary for global fitting methods. Studying the effect of these two types of truncations on Hartree-Fock (HF) and second-order Møller-Plesset perturbation theory (MP2) calculations on a chemically diverse set of first- and second-row molecules within the RI/DF approach, we show that accurate total and atomization energies can be achieved by a combination of the two approaches with significant reductions in the size of the ABS. While the original approach yields ABSs whose number of functions NbfABS scales with the number of functions in the OBS, NOBSbf, as NABSbf = γNOBSbf with the prefactor γ ≈ O ( 10 ) , the reduction schemes of this work afford results of essentially the same quality as the original unpruned and uncontracted ABS with γ ≈ 5-6, while an accuracy that may suffice for routine applications is achievable with a further reduced ABS with γ ≈ 3-4. The observed errors are similar at HF and MP2 levels of theory, suggesting that the generated ABSs are highly transferable and can also be applied to model challenging properties with high-level methods.

Two local minima for structures of [4Fe-4S] clusters obtained with density functional theory methods

[4Fe-4S] clusters are essential cofactors in many proteins involved in biological redox-active processes. Density functional theory (DFT) methods are widely used to study these clusters. Previous investigations have indicated that there exist two local minima for these clusters in proteins. We perform a detailed study of these minima in five proteins and two oxidation states, using combined quantum mechanical and molecular mechanical (QM/MM) methods. We show that one local minimum (L state) has longer Fe-Fe distances than the other (S state), and that the L state is more stable for all cases studied. We also show that some DFT methods may only obtain the L state, while others may obtain both states. Our work provides new insights into the structural diversity and stability of [4Fe-4S] clusters in proteins, and highlights the importance of reliable DFT methods and geometry optimization. We recommend r2SCAN for optimizing [4Fe-4S] clusters in proteins, which gives the most accurate structures for the five proteins studied.

Correlation consistent auxiliary basis sets in density fitting Hartree-Fock: The atoms sodium through argon revisited

We present a series of auxiliary basis sets, for the elements Na to Ar, for use in density-fitted Hartree-Fock calculations with the correlation consistent cc-pV(n + d)Z orbital basis sets. Benchmarking on total molecular energies, reaction energies and the spectroscopic constants of the SO molecule demonstrate that the new sets address the deficiencies of using existing auxiliary sets in combination with these orbital basis sets. We also report auxiliary basis sets for Na and Mg matched to cc-pVnZ, along with recommendations for pairing auxiliary basis sets to the cc-pVnZ-F12 basis sets for Hartree-Fock calculations.

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.

Accurate Hellmann-Feynman forces from density functional calculations with augmented Gaussian basis sets

The Hellmann-Feynman (HF) theorem provides a way to compute forces directly from the electron density, enabling efficient force calculations for large systems through machine learning (ML) models for the electron density. The main issue holding back the general acceptance of the HF approach for atom-centered basis sets is the well-known Pulay force which, if naively discarded, typically constitutes an error upward of 10 eV/Å in forces. In this work, we demonstrate that if a suitably augmented Gaussian basis set is used for density functional calculations, the Pulay force can be suppressed, and HF forces can be computed as accurately as analytical forces with state-of-the-art basis sets, allowing geometry optimization and molecular dynamics to be reliably performed with HF forces. Our results pave a clear path forward for the accurate and efficient simulation of large systems using ML densities and the HF theorem.

Automatic generation of complementary auxiliary basis sets for explicitly correlated methods

Explicitly correlated calculations, aside from the orbital basis set, typically require three auxiliary basis sets: Coulomb‐exchange fitting (JK), resolution of the identity MP2 (RI‐MP2), and complementary auxiliary basis set (CABS). If unavailable for the orbital basis set and chemical elements of interest, the first two can be auto‐generated on the fly using existing algorithms, but not the third. In this paper, we present a quite simple algorithm named autoCABS; a Python implementation under a free software license is offered at Github. For the cc‐pVnZ‐F12 (n = D,T,Q,5), the W4‐08 thermochemical benchmark, and the HFREQ2014 set of harmonic frequencies, we demonstrate that autoCABS‐generated CABS basis sets are comparable in quality to purpose‐optimized OptRI basis sets from the literature, and that the quality difference becomes entirely negligible as n increases.

Automated Generation of Optimized Auxiliary Basis Sets for Long-Range-Corrected TDDFT Using the Cholesky Decomposition

Range-separated hybrid functionals making use of a smooth separation of the Coulomb operator in terms of the error function and its complement have proven to be a valuable tool for improving Kohn-Sham density functional theory (DFT) calculations. This holds in particular for obtaining accurate excitation energies from linear-response time-dependent DFT. Evaluating the long-range exchange contributions represents one of the most time-consuming tasks in such calculations. Prefitted auxiliary basis sets can be employed to speed up this step. Here, we present a way to generate auxiliary basis sets optimized to fit the long-range exchange contributions only, contrary to the common optimization strategies on the basis of the full Coulomb operator. For this purpose, we use the atomic Cholesky decomposition technique. The basis sets are generated on-the-fly using the specific range-separation parameter defined in the exchange-correlation functional. We obtain excitation energies and oscillator strengths which are of similar or better accuracy than those obtained with conventional resolution-of-the-identity auxiliary basis sets while drastically reducing the number of auxiliary functions required. This is demonstrated for the QUESTDB#5 benchmark set. In addition, we outline the benefits of this approach in sequences of calculations employing varying range-separation parameters, as is the case in the optimally tuned range-separation strategy. Finally, we illustrate the efficiency of this approach for real-world examples, namely, a chlorophyll tetramer from photosystem II and a carotenoid-porphyrin-C₆₀ triad.

General embedded cluster protocol for accurate modeling of oxygen vacancies in metal-oxides

The O vacancy (Ov) formation energy, EOv, is an important property of a metal-oxide, governing its performance in applications such as fuel cells or heterogeneous catalysis. These defects are routinely studied with density functional theory (DFT). However, it is well-recognized that standard DFT formulations (e.g., the generalized gradient approximation) are insufficient for modeling the Ov, requiring higher levels of theory. The embedded cluster method offers a promising approach to compute EOv accurately, giving access to all electronic structure methods. Central to this approach is the construction of quantum(-mechanically treated) clusters placed within suitable embedding environments. Unfortunately, current approaches to constructing the quantum clusters either require large system sizes, preventing application of high-level methods, or require significant manual input, preventing investigations of multiple systems simultaneously. In this work, we present a systematic and general quantum cluster design protocol that can determine small converged quantum clusters for studying the Ov in metal-oxides with accurate methods, such as local coupled cluster with single, double, and perturbative triple excitations. We apply this protocol to study the Ov in the bulk and surface planes of rutile TiO2 and rock salt MgO, producing the first accurate and well-converged determinations of EOv with this method. These reference values are used to benchmark exchange–correlation functionals in DFT, and we find that all the studied functionals underestimate EOv, with the average error decreasing along the rungs of Jacob’s ladder. This protocol is automatable for high-throughput calculations and can be generalized to study other point defects or adsorbates.