arXiv subjects

John P. Perdew

Publications and source records attributed to John P. Perdew.

At least 19 recordsLinked to original sources

arXiv2026

Balanced and broadly-normed meta-generalized gradient approximation

Conventional semi-local density functional approximations for the exchange-correlation energy systematically underestimate fundamental band gaps. Advanced meta-generalized gradient approximations (meta-GGAs) can partially overcome this limitation through their orbital dependence via the kinetic energy density. In principle, the flexible meta-GGA form should allow for balanced performance across diverse properties. However, atomization energies, reaction barrier heights, and lattice constants, as well as band gaps, exhibit different sensitivities to the large parameter space of meta-GGA functionals. In this work, we explore the space of appropriate norms that guide the r$^2$SCAN-type functional construction through the interpolation function connecting two fundamental paradigms: the one-electron limit and the uniform electron gas limit. We introduce local modifications to a strongly-smoothed interpolation function and investigate their impact on functional performance. To further probe the limits of the meta-GGA framework, we include spin-unpolarized bonded systems among the set of norms. The resulting bonded-norm bn-r$^2$SCAN meta-GGA achieves balance: band-gap predictions approaching the accuracy of the ultra-nonlocal LAK meta-GGA and lattice constants of nearly r$^2$SCAN accuracy.

physics.chem-ph↗

arXiv2026

Symmetry breaking transforms strong to normal correlation and false metals to true insulators

Material scientists and condensed matter physicists have long been divided on the issue of choosing the conceptual framework for explaining why open-shell transition-metal oxides tend to be insulators, whereas otherwise successful theories such as DFT often predict them to be (false) metals. Strong correlation becomes the recommended medicine. We point out that strong correlation can be mitigated by allowing DFT to lower the energy by breaking structural, magnetic or dipolar symmetries. Such local motifs are observed experimentally by local probes beyond the 'average structure' determined by X-Ray diffraction. Observed broken symmetries can arise from slow fluctuations that persist over the observation time or longer. The surprising fact is that when symmetry breaking motifs are used as input to electronic structure calculations, false metals are converted into real insulators without the recommended medicine of strong correlation. Consistently, DFT calculations that show energy lowering symmetry breaking correct most cases where DFT, even with advanced exchange-correlation functionals, previously missed the correct metal vs insulator designation. Total energy calculations distinguish systems that support energy-lowering symmetry breaking from those that do not. This approach distinguishes between paramagnetic insulating and metallic phases and shows mass enhancement in Mott metals. The reason is that symmetry breaking removes many of the degeneracies that exist in a symmetry-unbroken system, reducing significantly the need for strong correlation. If one chooses to ignore symmetry breaking, the persistent degeneracies often call for strong correlation treatment. Thus, symmetry breaking transforms strong to normal correlation and false metals to true insulators. This view sheds light on the historic controversy between Mott and Slater that still reverberates today.

cond-mat.mtrl-sci↗

arXiv2026

Identifying strong correlation using only the Kohn-Sham density of one-electron states

Strongly correlated systems have long been a central and highly non-trivial topic in condensed matter physics. At the non-interacting level, strong correlation can be associated with powerful (near) degeneracies between occupied and unoccupied states, which leads to a high density of states near the Fermi level in metallic configurations. Such regimes are commonly treated with beyond-density functional theory (DFT) approaches, such as DFT+U or DFT+DMFT while maintaining symmetric configurations. Here, we explore the hypothesis that symmetry breaking in the Kohn-Sham (KS) non-interacting system can qualitatively account for the energetic effects of strong correlation in the corresponding interacting system within standard DFT. By lifting near-degeneracies around the Fermi level, symmetry breaking diminishes the potential correlation effects, reducing the need for an explicit treatment of electron correlation, transforming an otherwise strongly correlated symmetric configuration into a normally correlated one, thus avoiding the need for interacting methods beyond DFT. This naturally connects nonmagnetic to magnetic states. We apply this idea to both strongly and normally correlated metals and observe that spin symmetry breaking leads to a pronounced reduction of the density of states at the Fermi level and a significant lowering of the total energy in strongly correlated cases. To describe the degree of correlation that the interacting system would have relative to the KS state, we introduce a correlation parameter ($Γ$), defined as the ratio between the Kohn-Sham density of one-electron states at the Fermi level and that of a corresponding uniform electron gas. This parameter distinguishes strongly correlated systems, which would require explicit treatment, from normally correlated ones, which do not.

cond-mat.str-el↗

arXiv2026

Reducing Self-Interaction Error in Transition-Metal Oxides with Different Exact-Exchange Fractions for Energy and Density

Density functional theory (DFT) in chemistry and materials science aims for "chemical accuracy," but this goal is challenged by the need to approximate the exact exchange-correlation (XC) energy functional. The r$^2$SCAN, meta-generalized gradient approximation to the XC functional fulfills 17 exact constraints of the XC energy, and has significantly boosted prediction accuracy for molecules and materials. However, r$^2$SCAN remains inadequate at predicting properties of open \textit{d} and \textit{f} transition-metal strongly correlated compounds, such as band gaps, magnetic moments, and oxidation energies. Prediction inaccuracies of r$^2$SCAN energies arise from functional and density-driven errors, mainly resulting from the DFT self-interaction error. We propose the r$^2$SCANY@r$^2$SCANX method to mitigate the self-interaction error of XC functionals for the accurate simulations of electronic, magnetic, and thermochemical properties of transition metal oxides. r$^2$SCANY@r$^2$SCANX uses different fractions of exact Hartree-Fock exchange: X for the electronic density and Y for the density functional approximation of the total energy, thereby simultaneously addressing functional-driven and density-driven inaccuracies. Building just on 1 (or maximum 2) parameters that apply unchanged to \emph{s-p}-bonded systems, we demonstrate that, r$^2$SCANY@r$^2$SCANX improves upon the r$^2$SCAN predictions for 20 highly correlated oxides and even outperforms the highly parameterized DFT(r$^2$SCAN)+\emph{U} method -- the state-of-the-art approach to predict strongly correlated materials. Prediction uncertainties for oxidation energies and magnetic moments of transition metal oxides are significantly reduced by r$^2$SCAN10@r$^2$SCAN50 and band gaps with r$^2$SCAN10@r$^2$SCAN. r$^2$SCAN10@r$^2$SCAN50 diminishes the density-driven error of the energy in r$^2$SCAN and r$^2$SCAN10.

cond-mat.mtrl-sci↗

arXiv2026

Rotational Phonons Drive Low-Energy Kinks in Cuprate Superconductors

Angle-resolved photoemission spectroscopy (ARPES) reveals ubiquitous quasiparticle ``kinks'' near $\sim$70 meV and $\sim$40 meV across cuprate superconductors, often accompanied by peak--dip--hump (PDH) structures. These features point to strong coupling between electrons and low-energy bosonic excitations, but the microscopic origin has remained elusive due to the limitations of conventional density-functional theory (DFT) and the high cost of beyond-DFT methods. Here, we systematically study the electron--phonon coupling (EPC) in hole-doped infinite-layer CaCuO$_2$ using the Strongly Constrained and Appropriately Normed (SCAN) density functional, explicitly including magnetic effects. We find a substantial EPC strength $λ$ of $\sim$0.5 in the magnetic phase, producing kinks and PDH structures in the 40-80~meV window in excellent agreement with experiments. The dominant contribution arises from rotational oxygen phonons, while breathing modes contribute little. Our results establish strong EPC in cuprates, highlight the key role of rotational phonons, and provide a framework for understanding spectral anomalies in cuprates and beyond.

cond-mat.str-el↗

arXiv2025

Effect of the Gradient of the Spin-Polarization in Density Functional Approximations

The construction of non-empirical density functional approximations is typically guided by the satisfaction of exact constraints. An important constraint is the recovery of the gradient expansion for slowly varying electron densities. In prior constructions of semilocal density functional approximations, the $\nabla ζ$-dependent terms in the gradient expansion of the correlation have been dropped, where $ζ$ is the relative spin polarization. We propose a scheme by which such terms can be reintroduced into already constructed functionals without significantly affecting other constraints and norms. We implement this scheme on the Strongly Constrained and Appropriately Normed (SCAN) functional to construct a $\nabla ζ$-corrected version of SCAN. The resulting functional is shown to provide improvements in transition-metal atoms and molecules without significantly affecting SCAN's accurate description of $sp$-systems. For the binding energy curve of the chromium dimer Cr$_2$, the SCAN underbinding is fully corected at large bond lengths and reduced at short bond lengths.

physics.chem-ph↗

arXiv2025

Electron Localization in Non-Compact Covalent Bonds Captured by the r2SCAN+V Approach

In density functional theory, the SCAN (Strongly Constrained and Appropriately Normed) and r2SCAN functionals significantly improve over generalized gradient approximation functionals such as PBE (Perdew-Burke-Ernzerhof) in predicting electronic, magnetic, and structural properties across various materials, including transition-metal compounds. However, there remain puzzling cases where SCAN and r2SCAN underperform, such as in calculating the band structure of graphene, the magnetic moment of Fe, the potential energy curve of the Cr2 molecule, and the bond length of VO2. This research identifies a common characteristic among these challenging materials: non-compact covalent bonding through s-s, p-p, or d-d electron hybridization. While SCAN and r2SCAN excel at capturing electron localization at local atomic sites, they struggle to accurately describe electron localization in non-compact covalent bonds, resulting in a biased improvement. To address this issue, we propose the r2SCAN+V approach as a practical modification that improves accuracy across all the tested materials. The parameter V is 4 eV for metallic Fe, but substantially lower for the other cases. Our findings provide valuable insights for the future development of advanced functionals.

cond-mat.mtrl-sci↗

arXiv2025

ΔSCF Excitation Energies Up a Ladder of Ground-State Density Functionals

Density functional theory (DFT) is a widespread and effective tool in electronic structure calculations for ground-state electron systems. Its success has prompted exploration into the use of DFT for non-collective excited states. The delta self-consistent field ($Δ$SCF) method allows for the extension of DFT to excited-state energies by restricting the Kohn-Sham orbital occupations, producing an excited-state electron density, and then computing its energy. In this paper, we examine the performance of the LSDA, PBE generalized-gradient approximation (GGA), and SCAN/r2SCAN meta-GGA for the excitation energies of several important systems. We consider the energies of atoms with atomic number 1-18. For the hydrogen atom, where we use the exact electron density and have no multiplet splitting, we find significant improvement up the ladder from LSDA to PBE to SCAN. For the uniform gas, we find an effective mass different from the bare mass only with r2SCAN. We split the case of multi-electron atoms into non-aufbau excitations, where the highest-energy electron is excited to the lowest state in the next nl subshell (where accuracy is least limited by available basis sets), and spin-flip excitations, where the spin of an electorn is flipped, leading to a higher-energy state of the same nl configuration. We find reasonably accurate approximate excitation energies, except for the spin-flip cases where the auxiliary non-interacting wavefunction of a non-Hund's-rule spin state is not well-described by a single determinant.

physics.chem-ph↗

arXiv2025

Atomic Ionization: sd energy imbalance and Perdew-Zunger self-interaction correction energy penalty in 3d atoms

To accurately describe the energetics of transition metal systems, density functional approximations (DFAs) must provide a balanced description of s- and d- electrons. One measure of this is the sd transfer error, which has previously been defined as $E(\mathrm{3d}^{n-1}\mathrm{4s}^{1}) - E(\mathrm{3d}^{n-2}\mathrm{4s}^{2})$. Theoretical concerns have been raised about this definition due to its evaluation of excited-state energies using ground-state DFAs. A more serious concern appears to be strong correlation in the ${4s}^2$ configuration. Here we define a ground-state measure of the sd energy imbalance, based on the errors of s- and d-electron second ionization energies of the 3d atoms, that effectively circumvents the aforementioned problems. We find an improved performance as we move from LSDA to PBE to r$^2$SCAN for first-row transition metal atoms. However, we find large (~ 2 eV) ground-state sd energy imbalances when applying a Perdew-Zunger 1981 self-interaction correction. This is attributed to an "energy penalty" associated with the noded 3d orbitals. A local scaling of the self-interaction correction to LSDA results in a balance of s- and d-errors.

physics.chem-ph↗

arXiv2024

Effect of Strain on the Band Gap of Monolayer MoS$_2$

Monolayer molybdenum disulfide ($\mathrm{MoS_2}$) under strain has many interesting properties and possible applications in technology. A recent experimental study examined the effect of strain on the bandgap of monolayer $\mathrm{MoS_2}$ on a mildly curved graphite surface, reporting that under biaxial strain with a Poisson's ratio of 0.44, the bandgap decreases at a rate of 400 meV/\% strain. In this work, we performed density functional theory (DFT) calculations for a free-standing $\mathrm{MoS_2}$ monolayer, using the generalized gradient approximation (GGA) PBE, the hybrid functional HSE06, and many-body perturbation theory with the GW approximation using PBE wavefunctions (G0W0@PBE). For the unstrained monolayer, we found a standard level of agreement for the bandgap between theory and experiment. For biaxial strain at the experimental Poisson's ratio, we found that the bandgap decreases at rates of 63 meV/\% strain (PBE), 73 meV/\% strain (HSE06), and 43 meV/\% strain (G0W0@PBE), which are significantly smaller than the experimental rate. We also found that PBE predicts a similarly smaller rate (90 meV/\% strain) for a different Poisson's ratio of 0.25. Spin-orbit correction (SOC) has little effect on the gap or its strain dependence. The strong disagreement between theory and experiment may reflect an unexpectedly strong effect of the substrate on the strain dependence of the gap. Additionally, we observed a transition from a direct to an indirect bandgap under strain, and (under an equal biaxial strain of 10\%) a semiconductor-to-metal transition, consistent with previous theoretical work.

cond-mat.mtrl-sci↗

arXiv2024

Comparing Meta-GGAs, +U Corrections, and Hybrid Functionals for Polaronic Point Defects in Layered MnO$_2$, NiO$_2$, and KCoO$_2$

Defects in a material can significantly tune properties and enhance utility. Hybrid functionals like HSE06 are often used to describe solids with such defects. However, geometry optimization (including accounting for effects such as Jahn-Teller distortion) using hybrid functionals is challenging for the large supercells needed for defect study. The proposed r$^2$SCAN+rVV10+U+$\mathrm{U_d}$ method, which is computationally much cheaper and faster than hybrid functionals, can successfully describe defects in materials with the proper choice of U (for the d orbitals of the host atom) and $\mathrm{U_d}$ (for those of the defect atom), as shown here for small polaron defects in layered transition-metal oxides. For a range of U and $\mathrm{U_d}$ around literature values (from solid-state reaction energies) for a given transition-metal ion and its oxidation state, we find that this approach predicts localized polaronic states in band gaps, as hybrid functionals do. The layered materials birnessite ($\mathrm{K_nMnO_2}, n=0.03 $) and $\mathrm{K_nNiO_2},n=0.03$, with one K atom intercalated between layers in a supercell, are found to have one localized occupied $\mathrm{e_g}$ polaronic state on the transition metal ion reduced by the insertion of the K atom, when the geometry is calculated as above using published U values. The expected Jahn-Teller distortion is not observed when U=$\mathrm{U_d}$=0. Layered cobalt oxide with additional potassium ions intercalated ($\mathrm{K_nCoO_2},n=1.03$) is different, due to a dramatic difference in electronic configuration of the defected Co(II) ion: A single extra K atom in the supercell leads to four localized electrons in the band gap, using standard U values, and even for U=$\mathrm{U_d}$=0.

physics.comp-ph↗

arXiv2024

Comparing first-principles density functionals plus corrections for the lattice dynamics of YBa$_2$Cu$_3$O$_6$

The enigmatic mechanism underlying unconventional high-temperature superconductivity, especially the role of lattice dynamics, has remained a subject of debate. Theoretical insights have long been hindered due to the lack of an accurate first-principles description of the lattice dynamics of cuprates. Recently, using the r2SCAN meta-GGA functional, we were able to achieve accurate phonon spectra of an insulating cuprate YBa$_2$Cu$_3$O$_6$, and discover significant magnetoelastic coupling in experimentally interesting Cu-O bond stretching optical modes [Ning et al., Phys. Rev. B 107, 045126 (2023)]. We extend this work by comparing PBE and r2SCAN performances with corrections from the on-site Hubbard U and the D4 van der Waals (vdW) methods, aiming at further understanding on both the materials science side and the density functional side. We demonstrate the importance of vdW and self-interaction corrections for accurate first-principles YBa2 Cu3 O6 lattice dynamics. Since r2SCAN by itself partially accounts for these effects, the good performance of r2SCAN is now more fully explained. In addition, the performances of the Tao-Mo series of meta-GGAs, which are constructed in a different way from SCAN/r2SCAN, are also compared and discussed.

cond-mat.mtrl-sci↗

arXiv2023

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 semi-local approximations for the exchange-correlation energy. The accuracy of a semi-local density functional approximation is strongly boosted for reaction barrier heights by evaluating that approximation non-self-consistently on Hartree-Fock electron densities, as known for about 30 years. The conventional explanation is that Hartree-Fock theory yields the more accurate density. This article presents a benchmark Kohn-Sham inversion of accurate coupled-cluster densities for the reaction H$_2$ + F $\rightarrow$ HHF $\rightarrow$ H + HF, and finds a strong, understandable cancellation between positive (excessively over-corrected) 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 et al., J. Chem. Theory Comput. 19, 532--543 (2023)] based on 76 barrier heights and three less reliable, but less expensive, fully-nonlocal density-functional proxies for the exact density.

physics.chem-ph↗

arXiv2023

Challenges for density functional theory in simulating metal-metal singlet bonding: a case study of dimerized VO2

VO2 is renowned for its electric transition from an insulating monoclinic (M1) phase characterized by V-V dimerized structures, to a metallic rutile (R) phase above 340 Kelvin. This transition is accompanied by a magnetic change: the M1 phase exhibits a non-magnetic spin-singlet state, while the R phase exhibits a state with local magnetic moments. Simultaneous simulation of the structural, electric, and magnetic properties of this compound is of fundamental importance, but the M1 phase alone has posed a significant challenge to density functional theory (DFT). In this study, we show none of the commonly used DFT functionals, including those combined with on-site Hubbard U to better treat 3d electrons, can accurately predict the V-V dimer length. The spin-restricted method tends to overestimate the strength of the V-V bonds, resulting in a small V-V bond length. Conversely, the spin-symmetry-breaking method exhibits the opposite trends. Each bond-calculation method underscores one of the two contentious mechanisms, i.e., Peierls or Mott, involved in the metal-insulator transition in VO2. To elucidate the challenges encountered in DFT, we also employ an effective Hamiltonian that integrates one-dimensional magnetic sites, thereby revealing the inherent difficulties linked with the DFT computations.

cond-mat.mtrl-sci↗

arXiv2022

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 semi-local DFA non-self-consistently on the Hartree-Fock (HF) density is often recommended as a computationally cheap remedy for delocalization errors. For sophisticated meta-GGAs like SCAN, this approach can achieve remarkable accuracy. When this HF-DFT (or DFA@HF) significantly improves over the DFA, it is often presumed that the HF density is more accurate than the self-consistent DFA density. 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 over-correction, 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 transition states confirms this trend. We do not have the exact functional and exact densities that are needed to evaluate the exact density- and functional-driven errors for the large BH76 database of barrier heights. Instead, we have identified and used three non-local proxy functionals (the SCAN 50% global hybrid, the range-separated hybrid LC-$ω$PBE, and SCAN-FLOSIC) and their self-consistent densities. These functionals yield reasonably accurate self-consistent barrier heights, and 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.

physics.chem-ph↗

arXiv2022

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 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, the appearance of net up- and down-spin densities on opposite sides (not ends) of the bond, corrects that under-binding, 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.

physics.chem-ph↗

arXiv2022

Testing the r$^2$SCAN 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 r$^2$SCAN 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 r$^2$SCAN description of solid-state thermodynamics, we evaluate the formation and decomposition enthalpies, equilibrium volumes, and fundamental bandgaps of more than 1,000 solids using r$^2$SCAN, SCAN, and PBE, as well as two dispersion-corrected variants, SCAN+rVV10 and r$^2$SCAN+rVV10. We show that r$^2$SCAN achieves accuracy comparable to SCAN and often improves upon SCAN's already excellent accuracy. Whereas SCAN+rVV10 is often observed to worsen the formation enthalpies of SCAN, and makes no substantial correction to SCAN's cell volume predictions, r$^2$SCAN+rVV10 predicts marginally less-accurate formation enthalpies than r$^2$SCAN, and slightly more-accurate cell volumes than r$^2$SCAN. 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 r$^2$SCAN 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, r$^2$SCAN and r$^2$SCAN+rVV10 can be recommended as stable, general-purpose meta-GGAs for materials discovery.

cond-mat.mtrl-sci↗

arXiv2022

Laplacian-level meta-generalized gradient approximation for solid and liquid metals

We derive and motivate a Laplacian-level, orbital-free meta-generalized-gradient approximation (LL-MGGA) for the exchange-correlation energy, targeting accurate ground-state properties of $sp$ and $sd$ metallic condensed matter, in which the density functional for the exchange-correlation energy is only weakly nonlocal due to perfect long-range screening. Our model for the orbital-free kinetic energy density restores the fourth-order gradient expansion for exchange to the r$^2$SCAN meta-GGA [Furness et al., J. Phys. Chem. Lett. 11, 8208 (2020)], yielding a LL-MGGA we call OFR2. OFR2 matches the accuracy of SCAN for prediction of common lattice constants and improves the equilibrium properties of alkali metals, transition metals, and intermetallics that were degraded relative to the PBE GGA values by both SCAN and r$^2$SCAN. We compare OFR2 to the r$^2$SCAN-L LL-MGGA [D. Mejia-Rodriguez and S.B. Trickey, Phys. Rev. B 102, 121109 (2020)] and show that OFR2 tends to outperform r$^2$SCAN-L for the equilibrium properties of solids, but r$^2$SCAN-L much better describes the atomization energies of molecules than OFR2 does. For best accuracy in molecules and non-metallic condensed matter, we continue to recommend SCAN and r$^2$SCAN. Numerical performance is discussed in detail, and our work provides an outlook to machine learning.

cond-mat.mtrl-sci↗