The Predictive Power of Exact Constraints and Appropriate Norms in Density Functional Theory.

Ground-state Kohn-Sham density functional theory provides, in principle, the exact ground-state energy and electronic spin densities of real interacting electrons in a static external potential. In practice, the exact density functional for the exchange-correlation (xc) energy must be approximated in a computationally efficient way. About 20 mathematical properties of the exact xc functional are known. In this work, we review and discuss these known constraints on the xc energy and hole. By analyzing a sequence of increasingly sophisticated density functional approximations (DFAs), we argue that (a) the satisfaction of more exact constraints and appropriate norms makes a functional more predictive over the immense space of many-electron systems and (b) fitting to bonded systems yields an interpolative DFA that may not extrapolate well to systems unlike those in the fitting set. We discuss both how the class of well-described systems has grown along with constraint satisfaction and the possibilities for future functional development.

Symmetry breaking and self-interaction correction in the chromium atom and dimer.

Density functional approximations to the exchange-correlation energy can often identify strongly correlated systems and estimate their energetics through energy-minimizing symmetry-breaking. In particular, the binding energy curve of the strongly correlated chromium dimer is described qualitatively by the local spin density approximation (LSDA) and almost quantitatively by the Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA), where the symmetry breaking is antiferromagnetic for both. Here, we show that a full Perdew-Zunger self-interaction-correction (SIC) to LSDA seems to go too far by creating an unphysical symmetry-broken state, with effectively zero magnetic moment but non-zero spin density on each atom, which lies ∼4 eV below the antiferromagnetic solution. A similar symmetry-breaking, observed in the atom, better corresponds to the 3d↑↑4s↑3d↓↓4s↓ configuration than to the standard 3d↑↑↑↑↑4s↑. For this new solution, the total energy of the dimer at its observed bond length is higher than that of the separated atoms. These results can be regarded as qualitative evidence that the SIC needs to be scaled down in many-electron regions.

Interpretations of ground-state symmetry breaking and strong correlation in wavefunction and density functional theories.

Strong correlations within a symmetry-unbroken ground-state wavefunction can show up in approximate density functional theory as symmetry-broken spin densities or total densities, which are sometimes observable. They can arise from soft modes of fluctuations (sometimes collective excitations) such as spin-density or charge-density waves at nonzero wavevector. In this sense, an approximate density functional for exchange and correlation that breaks symmetry can be more revealing (albeit less accurate) than an exact functional that does not. The examples discussed here include the stretched H2 molecule, antiferromagnetic solids, and the static charge-density wave/Wigner crystal phase of a low-density jellium. Time-dependent density functional theory is used to show quantitatively that the static charge-density wave is a soft plasmon. More precisely, the frequency of a related density fluctuation drops to zero, as found from the frequency moments of the spectral function, calculated from a recent constraint-based wavevector- and frequency-dependent jellium exchange-correlation kernel.

Symmetry Breaking with the SCAN Density Functional Describes Strong Correlation in the Singlet Carbon Dimer.

The SCAN (strongly constrained and appropriately normed) meta-generalized gradient approximation (meta-GGA), which satisfies all 17 exact constraints that a meta-GGA can satisfy, accurately describes equilibrium bonds that are normally correlated. With symmetry breaking, it also accurately describes some sd equilibrium bonds that are strongly correlated. While sp equilibrium bonds are nearly always normally correlated, the C2 singlet ground state is known from correlated wave function theory to be a rare case of strong correlation in an sp equilibrium bond. Earlier work that calculated atomization energies of the molecular sequence B2, C2, O2, and F2 in the local spin density approximation (LSDA), the Perdew-Burke-Ernzerhof (PBE) GGA, and the SCAN meta-GGA, without symmetry breaking in the molecule, found that only SCAN was accurate enough to reveal an anomalous under-binding for C2. This work shows that spin symmetry breaking in singlet C2, which involves the appearance of net up- and down-spin densities on opposite sides (not ends) of the bond, corrects that underbinding, with a small SCAN atomization-energy error more like that of the other three molecules, suggesting that symmetry breaking with an advanced density functional might reliably describe strong correlation. This article also discusses some general aspects of symmetry breaking and the insights into strong correlation that symmetry breaking can bring. The normally correlated low-lying triplet excited state has the right vertical excitation energy in SCAN but not in LSDA or PBE, where the triplet is a false ground state. Fractional occupation numbers are found only for the symmetry-unbroken singlet and only in LSDA and PBE GGA.

Revealing quasi-excitations in the low-density homogeneous electron gas with model exchange-correlation kernels.

Time-dependent density functional theory within the linear response regime provides a solid mathematical framework to capture excitations. The accuracy of the theory, however, largely depends on the approximations for the exchange-correlation (xc) kernels. Away from the long-wavelength (or q = 0 short wave-vector) and zero-frequency (ω = 0) limit, the correlation contribution to the kernel becomes more relevant and dominant over exchange. The dielectric function, in principle, can encompass xc effects relevant to describe low-density physics. Furthermore, besides collective plasmon excitations, the dielectric function can reveal collective electron-hole excitations, often dubbed "ghost excitons." Besides collective excitons, the physics of the low-density regime is rich, as exemplified by a static charge-density wave that was recently found for rs > 69, and was shown to be associated with softening of the plasmon mode. These excitations are seen to be present in much higher density 2D homogeneous electron gases of rs ≳ 4. In this work, we perform a thorough analysis with xc model kernels for excitations of various nature. The uniform electron gas, as a useful model of real metallic systems, is used as a platform for our analysis. We highlight the relevance of exact constraints as we display and explain screening and excitations in the low-density region.

Construction of meta-GGA functionals through restoration of exact constraint adherence to regularized SCAN functionals.

The strongly constrained and appropriately normed (SCAN) meta-GGA exchange-correlation functional [Sun et al., Phys. Rev. Lett. 115, 036402 (2015)] is constructed as a chemical environment-determined interpolation between two separate energy densities: one describes single-orbital electron densities accurately and another describes slowly varying densities accurately. To conserve constraints known for the exact exchange-correlation functional, the derivatives of this interpolation vanish in the slowly varying limit. While theoretically convenient, this choice introduces numerical challenges that degrade the functional's efficiency. We have recently reported a modification to the SCAN meta-GGA, termed restored-regularized-SCAN (r2SCAN) [Furness et al., J. Phys. Chem. Lett. 11, 8208 (2020)], that introduces two regularizations into SCAN, which improve its numerical performance at the expense of not recovering the fourth order term of the slowly varying density gradient expansion for exchange. Here, we show the derivation of a progression of density functional approximations [regularized SCAN (rSCAN), r++SCAN, r2SCAN, and r4SCAN] with increasing adherence to exact conditions while maintaining a smooth interpolation. The greater smoothness of r2SCAN seems to lead to better general accuracy than the additional exact constraint of SCAN or r4SCAN does.

r2SCAN-D4: Dispersion corrected meta-generalized gradient approximation for general chemical applications.

We combine a regularized variant of the strongly constrained and appropriately normed semilocal density functional [J. Sun, A. Ruzsinszky, and J. P. Perdew, Phys. Rev. Lett. 115, 036402 (2015)] with the latest generation semi-classical London dispersion correction. The resulting density functional approximation r2SCAN-D4 has the speed of generalized gradient approximations while approaching the accuracy of hybrid functionals for general chemical applications. We demonstrate its numerical robustness in real-life settings and benchmark molecular geometries, general main group and organo-metallic thermochemistry, and non-covalent interactions in supramolecular complexes and molecular crystals. Main group and transition metal bond lengths have errors of just 0.8%, which is competitive with hybrid functionals for main group molecules and outperforms them for transition metal complexes. The weighted mean absolute deviation (WTMAD2) on the large GMTKN55 database of chemical properties is exceptionally small at 7.5 kcal/mol. This also holds for metal organic reactions with an MAD of 3.3 kcal/mol. The versatile applicability to organic and metal-organic systems transfers to condensed systems, where lattice energies of molecular crystals are within the chemical accuracy (errors <1 kcal/mol).

Calcium Signaling Silencing in Atrial Fibrillation: Implications for Atrial Sodium Homeostasis.

Atrial fibrillation (AF) is the most common type of cardiac arrhythmia, affecting more than 33 million people worldwide. Despite important advances in therapy, AF's incidence remains high, and treatment often results in recurrence of the arrhythmia. A better understanding of the cellular and molecular changes that (1) trigger AF and (2) occur after the onset of AF will help to identify novel therapeutic targets. Over the past 20 years, a large body of research has shown that intracellular Ca2+ handling is dramatically altered in AF. While some of these changes are arrhythmogenic, other changes counteract cellular arrhythmogenic mechanisms (Calcium Signaling Silencing). The intracellular Na+ concentration ([Na+])i is a key regulator of intracellular Ca2+ handling in cardiac myocytes. Despite its importance in the regulation of intracellular Ca2+ handling, little is known about [Na+]i, its regulation, and how it might be changed in AF. Previous work suggests that there might be increases in the late component of the atrial Na+ current (INa,L) in AF, suggesting that [Na+]i levels might be high in AF. Indeed, a pharmacological blockade of INa,L has been suggested as a treatment for AF. Here, we review calcium signaling silencing and changes in intracellular Na+ homeostasis during AF. We summarize the proposed arrhythmogenic mechanisms associated with increases in INa,L during AF and discuss the evidence from clinical trials that have tested the pharmacological INa,L blocker ranolazine in the treatment of AF.

Simple hydrogenic estimates for the exchange and correlation energies of atoms and atomic ions, with implications for density functional theory.

Exact density functionals for the exchange and correlation energies are approximated in practical calculations for the ground-state electronic structure of a many-electron system. An important exact constraint for the construction of approximations is to recover the correct non-relativistic large-Z expansions for the corresponding energies of neutral atoms with atomic number Z and electron number N = Z, which are correct to the leading order (-0.221Z5/3 and -0.021Z ln Z, respectively) even in the lowest-rung or local density approximation. We find that hydrogenic densities lead to Ex(N, Z) ≈ -0.354N2/3Z (as known before only for Z ≫ N ≫ 1) and Ec ≈ -0.02N ln N. These asymptotic estimates are most correct for atomic ions with large N and Z ≫ N, but we find that they are qualitatively and semi-quantitatively correct even for small N and N ≈ Z. The large-N asymptotic behavior of the energy is pre-figured in small-N atoms and atomic ions, supporting the argument that widely predictive approximate density functionals should be designed to recover the correct asymptotics. It is shown that the exact Kohn-Sham correlation energy, when calculated from the pure ground-state wavefunction, should have no contribution proportional to Z in the Z → ∞ limit for any fixed N.

Patient-specific induced pluripotent stem-cell-derived models of LEOPARD syndrome.

The generation of reprogrammed induced pluripotent stem cells (iPSCs) from patients with defined genetic disorders holds the promise of increased understanding of the aetiologies of complex diseases and may also facilitate the development of novel therapeutic interventions. We have generated iPSCs from patients with LEOPARD syndrome (an acronym formed from its main features; that is, lentigines, electrocardiographic abnormalities, ocular hypertelorism, pulmonary valve stenosis, abnormal genitalia, retardation of growth and deafness), an autosomal-dominant developmental disorder belonging to a relatively prevalent class of inherited RAS-mitogen-activated protein kinase signalling diseases, which also includes Noonan syndrome, with pleomorphic effects on several tissues and organ systems. The patient-derived cells have a mutation in the PTPN11 gene, which encodes the SHP2 phosphatase. The iPSCs have been extensively characterized and produce multiple differentiated cell lineages. A major disease phenotype in patients with LEOPARD syndrome is hypertrophic cardiomyopathy. We show that in vitro-derived cardiomyocytes from LEOPARD syndrome iPSCs are larger, have a higher degree of sarcomeric organization and preferential localization of NFATC4 in the nucleus when compared with cardiomyocytes derived from human embryonic stem cells or wild-type iPSCs derived from a healthy brother of one of the LEOPARD syndrome patients. These features correlate with a potential hypertrophic state. We also provide molecular insights into signalling pathways that may promote the disease phenotype.

Mechanism of automaticity in cardiomyocytes derived from human induced pluripotent stem cells.

BACKGROUND AND OBJECTIVES: The creation of cardiomyocytes derived from human induced pluripotent stem cells (hiPS-CMs) has spawned broad excitement borne out of the prospects to diagnose and treat cardiovascular diseases based on personalized medicine. A common feature of hiPS-CMs is their spontaneous contractions but the mechanism(s) remain uncertain. METHODS: Intrinsic activity was investigated by the voltage-clamp technique, optical mapping of action potentials (APs) and intracellular Ca(2+) (Cai) transients (CaiT) at subcellular-resolution and pharmacological interventions. RESULTS: The frequency of spontaneous CaiT (sCaiT) in monolayers of hiPS-CMs was not altered by ivabradine, an inhibitor of the pacemaker current, If despite high levels of HCN transcripts (1-4). HiPS-CMs had negligible If and IK1 (inwardly-rectifying K(+)-current) and a minimum diastolic potential of -59.1±3.3mV (n=18). APs upstrokes were preceded by a depolarizing-foot coincident with a rise of Cai. Subcellular Cai wavelets varied in amplitude, propagated and died-off; larger Cai-waves triggered cellular sCaTs and APs. SCaiTs increased in frequency with [Ca(2+)]out (0.05-to-1.8mM), isoproterenol (1µM) or caffeine (100µM) (n≥5, p<0.05). HiPS-CMs became quiescent with ryanodine receptor stabilizers (K201=2µM); tetracaine; Na-Ca exchange (NCX) inhibition (SEA0400=2µM); higher [K(+)]out (5→8mM), and thiol-reducing agents but could still be electrically stimulated to elicit CaiTs. Cell-cell coupling of hiPS-CM in monolayers was evident from connexin-43 expression and CaiT propagation. SCaiTs from an ensemble of dispersed hiPS-CMs were out-of-phase but became synchronous through the outgrowth of inter-connecting microtubules. CONCLUSIONS: Automaticity in hiPS-CMs originates from a Ca(2+)-clock mechanism involving Ca(2+) cycling across the sarcoplasmic reticulum linked to NCX to trigger APs.

Regional variation of the inwardly rectifying potassium current in the canine heart and the contributions to differences in action potential repolarization.

The inward rectifier potassium current, IK1, contributes to the terminal phase of repolarization of the action potential (AP), as well as the value and stability of the resting membrane potential. Regional variation in IK1 has been noted in the canine heart, but the biophysical properties have not been directly compared. We examined the properties and functional contribution of IK1 in isolated myocytes from ventricular, atrial and Purkinje tissue. APs were recorded from canine left ventricular midmyocardium, left atrial and Purkinje tissue. The terminal rate of repolarization of the AP in ventricle, but not in Purkinje, depended on changes in external K(+) ([K(+)]o). Isolated ventricular myocytes had the greatest density of IK1 while atrial myocytes had the lowest. Furthermore, the outward component of IK1 in ventricular cells exhibited a prominent outward component and steep negative slope conductance, which was also enhanced in 10 mM [K(+)]o. In contrast, both Purkinje and atrial cells exhibited little outward IK1, even in the presence of 10 mM [K(+)]o, and both cell types showed more persistent current at positive potentials. Expression of Kir2.1 in the ventricle was 76.9-fold higher than that of atria and 5.8-fold higher than that of Purkinje, whereas the expression of Kir2.2 and Kir2.3 subunits was more evenly distributed in Purkinje and atria. Finally, AP clamp data showed distinct contributions of IK1 for each cell type. IK1 and Kir2 subunit expression varies dramatically in regions of the canine heart and these regional differences in Kir2 expression likely underlie regional distinctions in IK1 characteristics, contributing to variations in repolarization in response to in [K(+)]o changes.

Relaxin suppresses atrial fibrillation by reversing fibrosis and myocyte hypertrophy and increasing conduction velocity and sodium current in spontaneously hypertensive rat hearts.

RATIONALE: Atrial fibrillation (AF) contributes significantly to morbidity and mortality in elderly and hypertensive patients and has been correlated to enhanced atrial fibrosis. Despite a lack of direct evidence that fibrosis causes AF, reversal of fibrosis is considered a plausible therapy. OBJECTIVE: To evaluate the efficacy of the antifibrotic hormone relaxin (RLX) in suppressing AF in spontaneously hypertensive rats (SHR). METHODS AND RESULTS: Normotensive Wistar-Kyoto (WKY) and SHR were treated for 2 weeks with vehicle (WKY+V and SHR+V) or RLX (0.4 mg/kg per day, SHR+RLX) using implantable mini-pumps. Hearts were perfused, mapped optically to analyze action potential durations, intracellular Ca²âº transients, and restitution kinetics, and tested for AF vulnerability. SHR hearts had slower conduction velocity (CV; P<0.01 versus WKY), steeper CV restitution kinetics, greater collagen deposition, higher levels of transcripts for transforming growth factor-ß, metalloproteinase-2, metalloproteinase-9, collagen I/III, and reduced connexin 43 phosphorylation (P<0.05 versus WKY). Programmed stimulation triggered sustained AF in SHR (n=5/5) and SHR+V (n=4/4), but not in WKY (n=0/5) and SHR+RLX (n=1/8; P<0.01). RLX treatment reversed the transcripts for fibrosis, flattened CV restitution kinetics, reduced action potential duration at 90% recovery to baseline, increased CV (P<0.01), and reversed atrial hypertrophy (P<0.05). Independent of antifibrotic actions, RLX (0.1 µmol/L) increased Na⁺ current density, INa (≈2-fold in 48 hours) in human cardiomyocytes derived from inducible pluripotent stem cells (n=18/18; P<0.01). CONCLUSIONS: RLX treatment suppressed AF in SHR hearts by increasing CV from a combination of reversal of fibrosis and hypertrophy and by increasing INa. The study provides compelling evidence that RLX may provide a novel therapy to manage AF in humans by reversing fibrosis and hypertrophy and by modulating cardiac ionic currents.

Unconventional Error Cancellation Explains the Success of Hartree-Fock Density Functional Theory for Barrier Heights.

Energy barriers, which control the rates of chemical reactions, are seriously underestimated by computationally efficient semilocal approximations for the exchange-correlation energy. The accuracy of a semilocal density functional approximation is strongly boosted for reaction barrier heights by evaluating that approximation non-self-consistently on Hartree-Fock electron densities, which has been known for ∼30 years. The conventional explanation is that the Hartree-Fock theory yields the more accurate density. This work presents a benchmark Kohn-Sham inversion of accurate coupled-cluster densities for the reaction H2 + F → HHF → H + HF and finds a strong, understandable cancellation between positive (excessively overcorrected) density-driven and large negative functional-driven errors (expected from stretched radical bonds in the transition state) within this Hartree-Fock density functional theory. This confirms earlier conclusions (Kaplan, A. D., et al. J. Chem. Theory Comput. 2023, 19, 532-543) based on 76 barrier heights and three less reliable, but less expensive, fully nonlocal density functional proxies for the exact density.

Ryanodine Receptor Stabilization Therapy Suppresses Ca 2+ -Based Arrhythmias in a Novel Model of Metabolic HFpEF.

Heart Failure with preserved ejection fraction (HFpEF) is the most prevalent form of heart failure worldwide and its significant mortality is associated with a high rate of sudden cardiac death (SCD; 30% - 40%). Chronic metabolic stress is an important driver of HFpEF, and clinical data show metabolic stress as a significant risk factor for ventricular arrhythmias in HFpEF patients. The mechanisms of SCD and ventricular arrhythmia in HFpEF remain critically understudied and empirical treatment is ineffective. To address this important knowledge gap, we developed a novel preclinical model of metabolic-stress induced HFpEF using Western diet (High fructose and fat) and hypertension induced by nitric oxide synthase inhibition (with L-NAME) in wildtype C57BL6/J mice. After 5 months, mice display all clinical characteristics of HFpEF and present with stress-induced sustained ventricular tachycardia (VT). Mechanistically, we found a novel pattern of arrhythmogenic intracellular Ca 2+ handling that is distinct from the well-characterized changes pathognomonic for heart failure with reduced ejection fraction. In addition, we show that the transverse tubular system remains intact in HFpEF and that arrhythmogenic, intracellular Ca 2+ mobilization becomes hyper-sensitive to ß- adrenergic activation. Finally, in proof-of-concept experiments we show in vivo that the clinically used intracellular calcium stabilizer dantrolene, which acts on the Ca 2+ release channels of the sarcoplasmic reticulum (SR), the ryanodine receptors, acutely prevents stress-induced VT in HFpEF mice. Therapeutic control of SR Ca 2+ leak may present a novel mechanistic treatment approach in metabolic HFpEF.

How Does HF-DFT Achieve Chemical Accuracy for Water Clusters?

Bolstered by recent calculations of exact functional-driven errors (FEs) and density-driven errors (DEs) of semilocal density functionals in the water dimer binding energy [Kanungo, B. J. Phys. Chem. Lett. 2024, 15, 323-328], we investigate approximate FEs and DEs in neutral water clusters containing up to 20 monomers, charged water clusters, and alkali- and halide-water clusters. Our proxy for the exact density is r2SCAN 50, a 50% global hybrid of exact exchange with r2SCAN, which may be less correct than r2SCAN for the compact water monomer but importantly more correct for long-range electron transfers in the noncompact water clusters. We show that SCAN makes substantially larger FEs for neutral water clusters than r2SCAN, while both make essentially the same DEs. Unlike the case for barrier heights, these FEs are small in a relative sense and become large in an absolute sense only due to an increase in cluster size. SCAN@HF, short for SCAN evaluated on the Hartree-Fock (HF) density, produces a cancellation of errors that makes it chemically accurate for predicting the absolute binding energies of water clusters. Likewise, adding a long-range dispersion correction to r2SCAN@HF, as in the composite method HF-r2SCAN-DC4, makes its FE more negative than in r2SCAN@HF, permitting a near-perfect cancellation of FE and DE. r2SCAN by itself (and even more so, r2SCAN evaluated on the r2SCAN 50 density), is almost perfect for the energy differences between water hexamers, and thus probably also for liquid water away from the boiling point. Thus, the accuracy of composite methods like SCAN@HF and HF-r2SCAN-DC4 is not due to the HF density being closer to the exact density, but to a compensation of errors from its greater degree of localization. We also give an argument for the approximate reliability of this unconventional error cancellation for diverse molecular properties. Finally, we confirm this unconventional error cancellation for the SCAN description of the water trimer via Kohn-Sham inversion of the CCSD(T) density.

CoeffNet: predicting activation barriers through a chemically-interpretable, equivariant and physically constrained graph neural network.

Activation barriers of elementary reactions are essential to predict molecular reaction mechanisms and kinetics. However, computing these energy barriers by identifying transition states with electronic structure methods (e.g., density functional theory) can be time-consuming and computationally expensive. In this work, we introduce CoeffNet, an equivariant graph neural network that predicts activation barriers using coefficients of any frontier molecular orbital (such as the highest occupied molecular orbital) of reactant and product complexes as graph node features. We show that using coefficients as features offer several advantages, such as chemical interpretability and physical constraints on the network's behaviour and numerical range. Model outputs are either activation barriers or coefficients of the chosen molecular orbital of the transition state; the latter quantity allows us to interpret the results of the neural network through chemical intuition. We test CoeffNet on a dataset of SN2 reactions as a proof-of-concept and show that the activation barriers are predicted with a mean absolute error of less than 0.025 eV. The highest occupied molecular orbital of the transition state is visualized and the distribution of the orbital densities of the transition states is described for a few prototype SN2 reactions.

Understanding Density-Driven Errors for Reaction Barrier Heights.

Delocalization errors, such as charge-transfer and some self-interaction errors, plague computationally efficient and otherwise accurate density functional approximations (DFAs). Evaluating a semilocal DFA non-self-consistently on the Hartree-Fock (HF) density is often recommended as a computationally inexpensive remedy for delocalization errors. For sophisticated meta-GGAs like SCAN, this approach can achieve remarkable accuracy. This HF-DFT (also known as DFA@HF) is often presumed to work, when it significantly improves over the DFA, because the HF density is more accurate than the self-consistent DFA density in those cases. By applying the metrics of density-corrected density functional theory (DFT), we show that HF-DFT works for barrier heights by making a localizing charge-transfer error or density overcorrection, thereby producing a somewhat reliable cancellation of density- and functional-driven errors for the energy. A quantitative analysis of the charge-transfer errors in a few randomly selected transition states confirms this trend. We do not have the exact functional and electron densities that would be needed to evaluate the exact density- and functional-driven errors for the large BH76 database of barrier heights. Instead, we have identified and employed three fully nonlocal proxy functionals (SCAN 50% global hybrid, range-separated hybrid LC-ωPBE, and SCAN-FLOSIC) and their self-consistent proxy densities. These functionals are chosen because they yield reasonably accurate self-consistent barrier heights and because their self-consistent total energies are nearly piecewise linear in fractional electron number─two important points of similarity to the exact functional. We argue that density-driven errors of the energy in a self-consistent density functional calculation are second order in the density error and that large density-driven errors arise primarily from incorrect electron transfers over length scales larger than the diameter of an atom.

Testing the r2SCAN Density Functional for the Thermodynamic Stability of Solids with and without a van der Waals Correction.

A central aim of materials discovery is an accurate and numerically reliable description of thermodynamic properties, such as the enthalpies of formation and decomposition. The r2SCAN revision of the strongly constrained and appropriately normed (SCAN) meta-generalized gradient approximation (meta-GGA) balances numerical stability with high general accuracy. To assess the r2SCAN description of solid-state thermodynamics, we evaluate the formation and decomposition enthalpies, equilibrium volumes, and fundamental band gaps of more than 1000 solids using r2SCAN, SCAN, and PBE, as well as two dispersion-corrected variants, SCAN+rVV10 and r2SCAN+rVV10. We show that r2SCAN achieves accuracy comparable to SCAN and often improves upon SCAN's already excellent accuracy. Although SCAN+rVV10 is often observed to worsen the formation enthalpies of SCAN and makes no substantial correction to SCAN's cell volume predictions, r2SCAN+rVV10 predicts marginally less accurate formation enthalpies than r2SCAN, and slightly more accurate cell volumes than r2SCAN. The average absolute errors in predicted formation enthalpies are found to decrease by a factor of 1.5 to 2.5 from the GGA level to the meta-GGA level. Smaller decreases in error are observed for decomposition enthalpies. For formation enthalpies r2SCAN improves over SCAN for intermetallic systems. For a few classes of systems-transition metals, intermetallics, weakly bound solids, and enthalpies of decomposition into compounds-GGAs are comparable to meta-GGAs. In total, r2SCAN and r2SCAN+rVV10 can be recommended as stable, general-purpose meta-GGAs for materials discovery.