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.

Insight on Gaussian Basis Set Truncation Errors in Weak to Intermediate Magnetic Fields with an Approximate Hamiltonian.

Strong magnetic fields such as those found on white dwarfs have significant effects on the electronic structures of atoms and molecules. However, the vast majority of molecular studies in the literature in such fields are carried out with Gaussian basis sets designed for zero field, leading to large basis set truncation errors [Lehtola et al., Mol. Phys. 2020, 118, e1597989]. In this work, we aim to identify the failures of the Gaussian basis sets in atomic calculations to guide the design of new basis sets for strong magnetic fields. We achieve this by performing fully numerical electronic structure calculations at the complete basis set (CBS) limit for the ground state and low lying excited states of the atoms 1 ≤ Z ≤ 18 in weak to intermediate magnetic fields. We also carry out finite-field calculations for a variety of Gaussian basis sets, introducing a real-orbital approximation for the magnetic-field Hamiltonian. Our primary focus is on the aug-cc-pVTZ basis set, which has been used in many works in the literature. A study of the differences in total energies of the fully numerical CBS limit calculations and the approximate Gaussian basis calculations is carried out to provide insight into basis set truncation errors. Examining a variety of states over the range of magnetic field strengths from B = 0 to B = 0.6B0, we observe significant differences for the aug-cc-pVTZ basis set, while much smaller errors are afforded by the benchmark-quality AHGBSP3-9 basis set [Lehtola, J. Chem. Phys. 2020, 152, 134108]. This suggests that there is considerable room to improve Gaussian basis sets for calculations at finite magnetic fields.

A call to arms: Making the case for more reusable libraries.

The traditional foundation of science lies on the cornerstones of theory and experiment. Theory is used to explain experiment, which in turn guides the development of theory. Since the advent of computers and the development of computational algorithms, computation has risen as the third cornerstone of science, joining theory and experiment on an equal footing. Computation has become an essential part of modern science, amending experiment by enabling accurate comparison of complicated theories to sophisticated experiments, as well as guiding by triage both the design and targets of experiments and the development of novel theories and computational methods. Like experiment, computation relies on continued investment in infrastructure: it requires both hardware (the physical computer on which the calculation is run) as well as software (the source code of the programs that performs the wanted simulations). In this Perspective, I discuss present-day challenges on the software side in computational chemistry, which arise from the fast-paced development of algorithms, programming models, as well as hardware. I argue that many of these challenges could be solved with reusable open source libraries, which are a public good, enhance the reproducibility of science, and accelerate the development and availability of state-of-the-art methods and improved software.

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.

Reproducibility of density functional approximations: How new functionals should be reported.

Density functional theory is the workhorse of chemistry and materials science, and novel density functional approximations are published every year. To become available in program packages, the novel density functional approximations (DFAs) need to be (re)implemented. However, according to our experience as developers of Libxc [Lehtola et al., SoftwareX 7, 1 (2018)], a constant problem in this task is verification due to the lack of reliable reference data. As we discuss in this work, this lack has led to several non-equivalent implementations of functionals such as Becke-Perdew 1986, Perdew-Wang 1991, Perdew-Burke-Ernzerhof, and Becke's three-parameter hybrid functional with Lee-Yang-Parr correlation across various program packages, yielding different total energies. Through careful verification, we have also found many issues with incorrect functional forms in recent DFAs. The goal of this work is to ensure the reproducibility of DFAs. DFAs must be verifiable in order to prevent the reappearance of the above-mentioned errors and incompatibilities. A common framework for verification and testing is, therefore, needed. We suggest several ways in which reference energies can be produced with free and open source software, either with non-self-consistent calculations with tabulated atomic densities or via self-consistent calculations with various program packages. The employed numerical parameters-especially the quadrature grid-need to be converged to guarantee a ≲0.1 µEh precision in the total energy, which is nowadays routinely achievable in fully numerical calculations. Moreover, as such sub-µEh level agreement can only be achieved when fully equivalent implementations of the DFA are used, the source code of the reference implementation should also be made available in any publication describing a new DFA.

Accuracy of a Recent Regularized Nuclear Potential.

F. Gygi recently suggested an analytic, norm-conserving, regularized nuclear potential to enable all-electron plane-wave calculations [Gygi J. Chem. Theory Comput. 2023, 19, 1300-1309.]. This potential V(r) is determined by inverting the Schrödinger equation for the wave function Ansatz ϕ(r) = exp[-h(r)]/√π with h(r) = r erf(ar) + b exp(-a2r2), where a and b are parameters. Gygi fixes b by demanding ϕ to be normalized, with the value b(a) depending on the strength of the regularization controlled by a. We begin this work by re-examining the determination of b(a) and find that the original 10-decimal tabulations of Gygi are only correct to 5 decimals, leading to normalization errors in the order of 10-10. In contrast, we show that a simple 100-point radial quadrature scheme not only ensures at least 10 correct decimals of b but also leads to machine-precision level satisfaction of the normalization condition. Moreover, we extend Gygi's plane-wave study by examining the accuracy of V(r) with high-precision finite element calculations with Hartree-Fock and LDA, GGA, and meta-GGA functionals on first- to fifth-period atoms. We find that although the convergence of the total energy appears slow in the regularization parameter a, orbital energies and shapes are indeed reproduced accurately by the regularized potential even with relatively small values of a, as compared to results obtained with a point nucleus. The accuracy of the potential is furthermore studied with s-d excitation energies of Sc-Cu as well as ionization potentials of He-Kr, which are found to converge to sub-meV precision with a = 4. The findings of this work are in full support of Gygi's contribution, indicating that all-electron plane-wave calculations can be accurately performed with the regularized nuclear potential.

Atomic Electronic Structure Calculations with Hermite Interpolating Polynomials.

We have recently described the implementation of atomic electronic structure calculations within the finite element method with numerical radial basis functions of the form χµ(r) = r-1Bµ(r), where high-order Lagrange interpolating polynomials (LIPs) were used as the shape functions Bµ(r). In this work, we discuss how χµ(r) can be evaluated in a stable manner at small r and also revisit the choice of the shape functions Bµ(r). Three kinds of shape functions are considered: in addition to the C0 continuous LIPs, we consider the analytical implementation of first-order Hermite interpolating polynomials (HIPs) that are C1 continuous, as well as numerical implementations of n-th order (Cn continuous) HIPs that are expressed in terms of an underlying high-order LIP basis. Furnished with the new implementation, we demonstrate that the first-order HIPs are reliable even with large numbers of nodes and that they also work with nonuniform element grids, affording even better results in atomic electronic structure calculations than LIPs with the same total number of basis functions. We demonstrate that discontinuities can be observed in the spin-σ local kinetic energy τσ in small LIP basis sets, while HIP basis sets do not suffer from such issues; however, either set can be used to reach the complete basis set limit with smooth τσ. Moreover, we discuss the implications of HIPs on calculations with meta-GGA functionals with a number of recent meta-GGA functionals, and we find most Minnesota functionals to be ill-behaved. We also examine the potential usefulness of the explicit control over the derivative in HIPs for forming numerical atomic orbital basis sets, but we find that confining potentials are still likely a better option.

Meta-GGA Density Functional Calculations on Atoms with Spherically Symmetric Densities in the Finite Element Formalism.

Density functional calculations on atoms are often used for determining accurate initial guesses as well as generating various types of pseudopotential approximations and efficient atomic-orbital basis sets for polyatomic calculations. To reach the best accuracy for these purposes, the atomic calculations should employ the same density functional as the polyatomic calculation. Atomic density functional calculations are typically carried out employing spherically symmetric densities, corresponding to the use of fractional orbital occupations. We have described their implementation for density functional approximations (DFAs) belonging to the local density approximation (LDA) and generalized gradient approximation (GGA) levels of theory as well as Hartree-Fock (HF) and range-separated exact exchange [Lehtola, S. Phys. Rev. A 2020, 101, 012516]. In this work, we describe the extension to meta-GGA functionals using the generalized Kohn-Sham scheme, in which the energy is minimized with respect to the orbitals, which in turn are expanded in the finite element formalism with high-order numerical basis functions. Furnished with the new implementation, we continue our recent work on the numerical well-behavedness of recent meta-GGA functionals [Lehtola, S.; Marques, M. A. L. J. Chem. Phys. 2022, 157, 174114]. We pursue complete basis set (CBS) limit energies for recent density functionals and find many to be ill-behaved for the Li and Na atoms. We report basis set truncation errors (BSTEs) of some commonly used Gaussian basis sets for these density functionals and find the BSTEs to be strongly functional dependent. We also discuss the importance of density thresholding in DFAs and find that all of the functionals studied in this work yield total energies converged to 0.1 µEh when densities smaller than 10-11a0-3 are screened out.

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.

How good are recent density functionals for ground and excited states of one-electron systems?

Sun et al. [J. Chem. Phys. 144, 191101 (2016)] suggested that common density-functional approximations (DFAs) should exhibit large energy errors for excited states as a necessary consequence of orbital nodality. Motivated by self-interaction corrected density-functional calculations on many-electron systems, we continue their study with the exactly solvable 1s, 2p, and 3d states of 36 hydrogenic one-electron ions (H-Kr35+) and demonstrate with self-consistent calculations that state-of-the-art DFAs indeed exhibit large errors for the 2p and 3d excited states. We consider 56 functionals at the local density approximation (LDA), generalized gradient approximation (GGA) as well as meta-GGA levels, and several hybrid functionals such as the recently proposed machine-learned DM21 local hybrid functional. The best non-hybrid functional for the 1s ground state is revTPSS. As predicted by Sun et al., the 2p and 3d excited states are more difficult for DFAs, and LDA functionals turn out to yield the most systematic accuracy for these states among non-hybrid functionals. The best performance for the three states overall is observed with the BHandH global hybrid GGA functional, which contains 50% Hartree-Fock exchange and 50% LDA exchange. The performance of DM21 is found to be inconsistent, yielding good accuracy for some states and systems and poor accuracy for others. Based on these results, we recommend including a variety of one-electron cations in future training of machine-learned density functionals.

Many recent density functionals are numerically ill-behaved.

Most computational studies in chemistry and materials science are based on the use of density functional theory. Although the exact density functional is unknown, several density functional approximations (DFAs) offer a good balance of affordable computational cost and semi-quantitative accuracy for applications. The development of DFAs still continues on many fronts, and several new DFAs aiming for improved accuracy are published every year. However, the numerical behavior of these DFAs is an often-overlooked problem. In this work, we look at all 592 DFAs for three-dimensional systems available in Libxc 5.2.2 and examine the convergence of the density functional total energy based on tabulated atomic Hartree-Fock wave functions. We show that several recent DFAs, including the celebrated SCAN family of functionals, show impractically slow convergence with typically used numerical quadrature schemes, making these functionals unsuitable both for routine applications and high-precision studies, as thousands of radial quadrature points may be required to achieve sub-µEh accurate total energies for these functionals, while standard quadrature grids like the SG-3 grid only contain O(100) radial quadrature points. These results are both a warning to users to always check the sufficiency of the quadrature grid when adopting novel functionals, as well as a guideline to the theory community to develop better-behaved density functionals.

Gas-Phase Peroxyl Radical Recombination Reactions: A Computational Study of Formation and Decomposition of Tetroxides.

The recombination ("dimerization") of peroxyl radicals (RO2•) is one of the pathways suggested in the literature for the formation of peroxides (ROOR', often referred to as dimers or accretion products in the literature) in the atmosphere. It is generally accepted that these dimers play a major role in the first steps of the formation of submicron aerosol particles. However, the precise reaction pathways and energetics of RO2• + R'O2• reactions are still unknown. In this work, we have studied the formation of tetroxide intermediates (RO4R'): their formation from two peroxyl radicals and their decomposition to triplet molecular oxygen (3O2) and a triplet pair of alkoxyl radicals (RO•). We demonstrate this mechanism for several atmospherically relevant primary and secondary peroxyl radicals. The potential energy surface corresponds to an overall singlet state. The subsequent reaction channels of the alkoxyl radicals include, but are not limited to, their dimerization into ROOR'. Our work considers the multiconfigurational character of the tetroxides and the intermediate phases of the reaction, leading to reliable mechanistic insights for the formation and decomposition of the tetroxides. Despite substantial uncertainties in the computed energetics, our results demonstrate that the barrier heights along the reaction path are invariably small for these systems. This suggests that the reaction mechanism, previously validated at a multireference level only for methyl peroxyl radicals, is a plausible pathway for the formation of aerosol-relevant larger peroxides in the atmosphere.

DQC: A Python program package for differentiable quantum chemistry.

Automatic differentiation represents a paradigm shift in scientific programming, where evaluating both functions and their derivatives is required for most applications. By removing the need to explicitly derive expressions for gradients, development times can be shortened and calculations can be simplified. For these reasons, automatic differentiation has fueled the rapid growth of a variety of sophisticated machine learning techniques over the past decade, but is now also increasingly showing its value to support ab initio simulations of quantum systems and enhance computational quantum chemistry. Here, we present an open-source differentiable quantum chemistry simulation code and explore applications facilitated by automatic differentiation: (1) calculating molecular perturbation properties, (2) reoptimizing a basis set for hydrocarbons, (3) checking the stability of self-consistent field wave functions, and (4) predicting molecular properties via alchemical perturbations.

Chemical bonding theories as guides for self-interaction corrected solutions: Multiple local minima and symmetry breaking.

Fermi-Löwdin orbitals (FLOs) are a special set of localized orbitals, which have become commonly used in combination with the Perdew-Zunger self-interaction correction (SIC) in the FLO-SIC method. The FLOs are obtained for a set of occupied orbitals by specifying a classical position for each electron. These positions are known as Fermi-orbital descriptors (FODs), and they have a clear relation to chemical bonding. In this study, we show how FLOs and FODs can be used to initialize, interpret, and justify SIC solutions in a common chemical picture, both within FLO-SIC and in traditional variational SIC, and to locate distinct local minima in either of these approaches. We demonstrate that FLOs based on Lewis theory lead to symmetry breaking for benzene-the electron density is found to break symmetry already at the symmetric molecular structure-while ones from Linnett's double-quartet theory reproduce symmetric electron densities and molecular geometries. Introducing a benchmark set of 16 planar cyclic molecules, we show that using Lewis theory as the starting point can lead to artifactual dipole moments of up to 1 D, while Linnett SIC dipole moments are in better agreement with experimental values. We suggest using the dipole moment as a diagnostic of symmetry breaking in SIC and monitoring it in all SIC calculations. We show that Linnett structures can often be seen as superpositions of Lewis structures and propose Linnett structures as a simple way to describe aromatic systems in SIC with reduced symmetry breaking. The role of hovering FODs is also briefly discussed.

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

Density fitting (DF), also known as the resolution of the identity (RI), is a widely used technique in quantum chemical calculations with various types of atomic basis sets─Gaussian-type orbitals, Slater-type orbitals, as well as numerical atomic orbitals─to speed up density functional, Hartree-Fock (HF), and post-HF calculations. Traditionally, custom auxiliary basis sets are hand-optimized for each orbital basis set; however, some automatic schemes have also been suggested. In this work, we propose a simple yet numerically stable automated scheme for forming auxiliary basis sets with the help of a pivoted Cholesky decomposition, which is applicable to any type of atomic basis function. We exemplify the scheme with proof-of-concept calculations with Gaussian basis sets and show that the proposed approach leads to negligible DF/RI errors in HF and second-order Møller-Plesset (MP2) total energies of the non-multireference part of the W4-17 test set when used with orbital basis sets of at least polarized triple-ζ quality. The results are promising for future applications employing Slater-type orbitals or numerical atomic orbitals, as well as schemes based on the use of local fitting approaches. Global fitting approaches can also be used, in which case the high-angular-momentum functions produced by the present scheme can be truncated to minimize the computational cost.

Sulfur Molecules in Space by X-rays: A Computational Study.

X-ray astronomy lacks high resolution spectra of interstellar dust analogues and molecules, severely hampering interstellar medium studies based on upcoming X-ray missions. Various theoretical approaches may be used to address this problem, but they must first be shown to reproduce reliable spectra compared to the experiment. In this work, we calculate the sulfur K edge X-ray absorption spectra of H2S, SO2, and OCS, whose spectra are already known from X-ray experiments and predict the X-ray spectrum of CS, which as far as we are aware has not been measured, thereby hampering its detection by X-ray telescopes. We chose these four molecules as the astrochemistry of sulfur is an unsolved problem and as the four molecules are already known to exist in space. We consider three types of methods for modeling the X-ray spectra: more accurate calculations with the algebraic-diagrammatic construction (ADC) and the CC2, CCSD, and CC3 coupled cluster (CC) approaches as well as more affordable ones with transition potential density functional theory (TP-DFT). A comparison of our computational results to previously reported experimental spectra shows that the core-valence separation (CVS) approaches CVS-ADC(2)-x and CVS-CC3 generally yield a good qualitative level of agreement with the experiment, suggesting that they can be used for interpreting measured spectra, while the TP-DFT method is not reliable for these molecules. However, quantitative agreement with the experiment is still outside the reach of the computational methods studied in this work.

Benchmarking Magnetizabilities with Recent Density Functionals.

We have assessed the accuracy of the magnetic properties of a set of 51 density functional approximations, including both recently published and already established functionals. The accuracy assessment considers a series of 27 small molecules and is based on comparing the predicted magnetizabilities to literature reference values calculated using coupled-cluster theory with full singles and doubles and perturbative triples [CCSD(T)] employing large basis sets. The most accurate magnetizabilities, defined as the smallest mean absolute error, are obtained with the BHandHLYP functional. Three of the six studied Berkeley functionals and the three range-separated Florida functionals also yield accurate magnetizabilities. Also, some older functionals like CAM-B3LYP, KT1, BHLYP (BHandH), B3LYP, and PBE0 perform rather well. In contrast, unsatisfactory performance is generally obtained with Minnesota functionals, which are therefore not recommended for calculations of magnetically induced current density susceptibilities and related magnetic properties such as magnetizabilities and nuclear magnetic shieldings. We also demonstrate that magnetizabilities can be calculated by numerical integration of magnetizability density; we have implemented this approach as a new feature in the gauge-including magnetically induced current (GIMIC) method. Magnetizabilities can be calculated from magnetically induced current density susceptibilities within this approach even when analytical approaches for magnetizabilities as the second derivative of the energy have not been implemented. The magnetizability density can also be visualized, providing additional information that is not otherwise easily accessible on the spatial origin of magnetizabilities.

Spatial Contributions to Nuclear Magnetic Shieldings.

We develop a methodology for calculating, analyzing, and visualizing nuclear magnetic shielding densities which are calculated from the current density via the Biot-Savart relation. Atomic contributions to nuclear magnetic shielding constants can be estimated within our framework with a Becke partitioning scheme. The new features have been implemented in the GIMIC program and are applied in this work to the study of the 1H and 13C nuclear magnetic shieldings in benzene (C6H6) and cyclobutadiene (C4H4). The new methodology allows a visual inspection of the spatial origins of the positive (shielding) and negative (deshielding) contributions to the nuclear magnetic shielding constant of a single nucleus, something which has not been hitherto easily accomplished. Analysis of the shielding densities shows that diatropic and paratropic current-density fluxes yield both shielding and deshielding contributions, as the shielding or deshielding is determined by the direction of the current-density flux with respect to the studied nucleus instead of the tropicity. Becke partitioning of the magnetic shieldings shows that the magnetic shielding contributions mainly originate from the studied atom and its nearest neighbors, confirming the localized character of nuclear magnetic shieldings.

Meta-Local Density Functionals: A New Rung on Jacob's Ladder.

The homogeneous electron gas (HEG) is a key ingredient in the construction of most exchange-correlation functionals of density-functional theory. Often, the energy of the HEG is parameterized as a function of its spin density nσ, leading to the local density approximation (LDA) for inhomogeneous systems. However, the connection between the electron density and kinetic energy density of the HEG can be used to generalize the LDA by evaluating it on a geometric average nσavg(r) = nσ1-x(r)ñσx(r) of the local spin density nσ(r) and the spin density ñσ(r) of a HEG that has the local kinetic energy density τσ(r) of the inhomogeneous system. This leads to a new family of functionals that we term meta-local density approximations (meta-LDAs), which are still exact for the HEG, which are derived only from properties of the HEG and which form a new rung of Jacob's ladder of density functionals [ AIP Conf. Proc. 2001, 577, 1]. The first functional of this ladder, the local τ approximation (LTA) of Ernzerhof and Scuseria [ J. Chem. Phys. 1999, 111, 911] that corresponds to x = 1 is unfortunately not stable enough to be used in self-consistent field calculations because it leads to divergent potentials, as we show in this work. However, a geometric averaging of the LDA and LTA densities with smaller values of x not only leads to numerical stability of the resulting functional but also yields more accurate exchange energies in atomic calculations than the LDA, the LTA, or the tLDA functional (x = 1/4) of Eich and Hellgren [ J. Chem. Phys. 2014, 141, 224107]. We choose x = 0.50, as it gives the best total energy in self-consistent exchange-only calculations for the argon atom. Atomization energy benchmarks confirm that the choice x = 0.50 also yields improved energetics in combination with correlation functionals in molecules, almost eliminating the well-known overbinding of the LDA and reducing its error by two thirds.