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The one-particle reduced density-matrix (1-RDM) functional theory is a promising alternative to density-functional theory (DFT) that uses the 1-RDM rather than the electronic density as a basic variable. However, long-standing challenges such as the lack of the Kohn–Sham scheme and the complexity of the pure N-representability conditions are still impeding its wild utilization. Fortunately, ensemble N-representability conditions derived in the natural orbital basis are known and trivial such that almost every functional of the 1-RDM is actually natural orbital functional, which does not perform well for all the correlation regimes. In this work, we propose a variational minimization scheme in the ensemble N-representable domain that is not restricted to the natural orbital representation of the 1-RDM. We show that splitting the minimization into the diagonal and off-diagonal parts of the 1-RDM can open the way toward the development of functionals of the orbital occupations, which remains a challenge for the generalization of site-occupation functional theory in chemistry. Our approach is tested on the uniform Hubbard model using the Müller and the Töws–Pastor functionals, as well as on the dihydrogen molecule using the Müller functional. [ABSTRACT FROM AUTHOR]

We explore the entire search space of 32-layer ZnxCd1−xTe superlattices to find the structures that minimize and maximize the bandgap at each possible zinc concentration. The searching is accomplished through an accurate and efficient combination of valence force field dynamics, the empirical pseudopotential method, and the folded spectrum method. We also describe the use of an alternate preconditioner that improves the robustness and efficiency of the locally optimal preconditioned conjugate gradient's solutions to the folded spectrum method. The physical properties of these superlattices, such as their formation energies, bandgaps, densities of states, effective masses, and optical response functions, are investigated with density functional theory paired with hybrid functionals and compare well to available experimental measurements. It is revealed that the bandgap of ZnxCd1−xTe may change by up to 0.2 eV depending on how the layers in the superlattice are ordered. Stacking order has a large, irregular effect on the effective masses, but optical response functions seem insensitive to it. [ABSTRACT FROM AUTHOR]

Within the framework of natural orbital functional theory, having a convenient representation of the occupation numbers and orbitals becomes critical for the computational performance of the calculations. Recognizing this, we propose an innovative parametrization of the occupation numbers that takes advantage of the electron-pairing approach used in Piris natural orbital functionals through the adoption of the softmax function, a pivotal component in modern deep-learning models. Our approach not only ensures adherence to the N-representability of the first-order reduced density matrix (1RDM) but also significantly enhances the computational efficiency of 1RDM functional theory calculations. The effectiveness of this alternative parameterization approach was assessed using the W4-17-MR molecular set, which demonstrated faster and more robust convergence compared to previous implementations. [ABSTRACT FROM AUTHOR]

We study the accuracy of excited state (ES) geometries using optimally tuned LC-PBE functionals with tuning based on GW quasiparticle energies. We compare the results obtained with the PBE, PBE0, non-tuned, and tuned LC-PBE functionals with available high-level CC reference values as well as experimental data. First, we compare ES geometrical parameters obtained for three different types of systems: molecules composed of a few atoms, 4-(dimethylamino)benzonitrile (DMABN), and conjugated dyes. To this end, we used wave-function results as benchmarks. Next, we evaluate the accuracy of the theoretically simulated spectra as compared to the experimental ones for five large dyes. Our results show that, besides small compact molecules for which tuning LC-PBE does not allow obtaining geometries more accurate than those computed with standard functionals, tuned range-separated functionals are clearly to be favored, not only for ES geometries but also for 0–0 energies, band shapes, and intensities for absorption and emission spectra. In particular, the results indicate that GW-tuned LC-PBE functionals provide improved matching with experimental spectra as compared to conventionally tuned functionals. It is an open question whether TD-DFT with GW-tuned functionals can qualitatively mimic the actual many-body Bethe–Salpeter (BSE/GW) formalism for which analytic ionic gradients remain to be developed. [ABSTRACT FROM AUTHOR]

Incorporating self-interaction corrections (SIC) significantly improves chemical reaction barrier height predictions made using density functional theory methods. We present a detailed orbital-by-orbital analysis of these corrections for three semi-local density functional approximations (DFAs) situated on the three lowest rungs of Jacob's ladder of approximations. The analysis is based on Fermi–Löwdin Orbital Self-Interaction Correction (FLOSIC) calculations performed at several steps along the reaction pathway from the reactants (R) to the transition state (TS) to the products (P) for four representative reactions selected from the BH76 benchmark set. For all three functionals, the major contribution to self-interaction corrections of the barrier heights can be traced to stretched bond orbitals that develop near the TS configuration. The magnitude of the ratio of the self-exchange–correlation energy to the self-Hartree energy (XC/H) for a given orbital is introduced as an indicator of one-electron self-interaction error. XC/H = 1.0 implies that an orbital's self-exchange–correlation energy exactly cancels its self-Hartree energy and that the orbital, therefore, makes no contribution to the SIC in the FLOSIC scheme. For the practical DFAs studied here, XC/H spans a range of values. The largest values are obtained for stretched or strongly lobed orbitals. We show that significant differences in XC/H for corresponding orbitals in the R, TS, and P configurations can be used to identify the major contributors to the SIC of barrier heights and reaction energies. Based on such comparisons, we suggest that barrier height predictions made using the strongly constrained and appropriately normed meta-generalized gradient approximation may have attained the best accuracy possible for a semi-local functional using the Perdew–Zunger SIC approach. [ABSTRACT FROM AUTHOR]

We investigate the native point defects in the long-wavelength infrared (LWIR) detector material H g 0.75 C d 0.25 Te using a dielectric-dependent hybrid density functional combined with spin–orbit coupling. Characterizing these point defects is essential as they are responsible for intrinsic doping and nonradiative recombination centers in the detector material. The dielectric-dependent hybrid functional allows for an accurate description of the bandgap (E g) for Hg 1 − x Cd x Te (MCT) over the entire compositional range, a level of accuracy challenging with standard hybrid functionals. Our comprehensive examination of the native point defects confirms that cation vacancies V Hg (Cd) are the primary sources of p -type conductivity in the LWIR material given their low defect formation energies and the presence of a shallow acceptor level (− /0) near the valence-band maximum. In addition to the shallow acceptor level, the cation vacancies exhibit a deep charge transition level (2 − / −) situated near the midgap, characteristic of nonradiative recombination centers. Our results indicate that Hg interstitial could also be a deep center in the LWIR MCT through a metastable configuration under the Hg-rich growth conditions. While an isolated Te antisite does not show deep levels, the formation of V Hg –Te Hg defect complex introduces a deep acceptor level within the bandgap. [ABSTRACT FROM AUTHOR]

We propose exchanging the energy functionals in ground-state density-functional theory with physically equivalent exact force expressions as a new promising route toward approximations to the exchange–correlation potential and energy. In analogy to the usual energy-based procedure, we split the force difference between the interacting and auxiliary Kohn–Sham system into a Hartree, an exchange, and a correlation force. The corresponding scalar potential is obtained by solving a Poisson equation, while an additional transverse part of the force yields a vector potential. These vector potentials obey an exact constraint between the exchange and correlation contribution and can further be related to the atomic shell structure. Numerically, the force-based local-exchange potential and the corresponding exchange energy compare well with the numerically more involved optimized effective potential method. Overall, the force-based method has several benefits when compared to the usual energy-based approach and opens a route toward numerically inexpensive nonlocal and (in the time-dependent case) nonadiabatic approximations. [ABSTRACT FROM AUTHOR]

EA-EOM-CCSD electron affinities and LUMO energies of various Kohn–Sham density functional theory (DFT) methods are calculated for an a priori IP benchmark set of 64 small, closed-shell molecules. The purpose of these calculations was to investigate whether the QTP KS-DFT functionals can emulate EA-EOM-CC with only a mean-field approximation. We show that the accuracy of DFT—relative to CCSD—improves significantly when elements of correlated orbital theory are introduced into the parameterization to define the QTP family of functionals. In particular, QTP(02), which has only a single range separation parameter, provides results accurate to a MAD of < 0.15 eV for the whole set of 64 molecules compared to EA-EOM-CCSD, far exceeding the results from the non-QTP family of density functionals. [ABSTRACT FROM AUTHOR]

The adiabatic connection interaction strength interpolation (ISI)-like method provides a high-level expression for the correlation energy, being, in principle, exact not only in the weak-interaction limit, where it recovers the second-order Görling–Levy perturbation term, but also in the strong-interaction limit that is described by the strictly correlated electron approach. In this work, we construct a genISI functional made accurate for the uniform electron gas, a solid-state physics paradigm that is a very difficult test for ISI-like correlation functionals. We assess the genISI functional for various jellium spheres with the number of electrons Z ≤ 912 and for the non-relativistic noble atoms with Z ≤ 290. For the jellium clusters, the genISI is remarkably accurate, while for the noble atoms, it shows a good performance, similar to other ISI-like methods. Then, the genISI functional can open the path using the ISI-like method in solid-state calculations. [ABSTRACT FROM AUTHOR]

Parr and Ghosh [Phys. Rev. A. 51 3564 (1995)] demonstrated that when near-exact electron densities and potentials are used, the exchange–correlation energies of first- and second-row atoms are well-described by a combination of the Fermi–Amaldi functional with a functional that is homogeneous of degree one under density scaling. Insight into this observation is provided by considering their work from the perspective of the effective homogeneity of the overall exchange–correlation functional. By considering a general form that combines the Fermi–Amaldi functional with a functional that is homogeneous of degree k, it is shown that for these atoms, the functional of Parr and Ghosh (k = 1) exhibits essentially optimal effective homogeneities on the electron-deficient side of the integer. Percentage errors in effective homogeneities are close to percentage errors in exchange–correlation energies. [ABSTRACT FROM AUTHOR]

The regularized and restored semi-local meta-generalized gradient approximation (meta-GGA) exchange–correlation functional r2SCAN [Furness et al., J. Phys. Chem. Lett. 11, 8208–8215 (2020)] is used to create adiabatic-connection-derived global double-hybrid functionals employing spin-opposite-scaled MP2. The 0-DH, CIDH, QIDH, and 0–2 type double-hybrid functionals are assessed as a starting point for further modification. Variants with 50% and 69% Hartree–Fock exchange (HFX) are empirically optimized (Pr2SCAN50 and Pr2SCAN69), and the effect of MP2-regularization (κPr2SCAN50) and range-separated HFX (ωPr2SCAN50) is evaluated. All optimized functionals are combined with the state-of-the-art London dispersion corrections D4 and NL. The resulting functionals are assessed comprehensively for their performance on main-group and metal-organic thermochemistry on 90 different benchmark sets containing 25 800 data points. These include the extensive GMTKN55 database, additional sets for main-group chemistry, and multiple sets for transition-metal complexes, including the ROST61, the MOR41, and the MOBH35 sets. As the main target of this study is the development of a broadly applicable, robust functional with low empiricism, special focus is put on variants with moderate amounts of HFX (50%), which are compared to the so far successful PWPB95-D4 (50% HFX, 20% MP2 correlation) functional. The overall best variant, ωPr2SCAN50-D4, performs well on main-group and metal-organic thermochemistry, followed by Pr2SCAN69-D4 that offers a slight edge for metal-organic thermochemistry and by the low HFX global double-hybrid Pr2SCAN50-D4 that performs robustly across all tested sets. All four optimized functionals, Pr2SCAN69-D4, Pr2SCAN50-D4, κPr2SCAN50-D4, and ωPr2SCAN50-D4, outperform the PWPB95-D4 functional. [ABSTRACT FROM AUTHOR]

Density functional calculations of Rydberg excited states up to high energy are carried out for several molecules using an approach where the orbitals are variationally optimized by converging on saddle points on the electronic energy surface within a real space grid representation. Remarkably good agreement with experimental estimates of the excitation energy is obtained using the generalized gradient approximation (GGA) functional of Perdew, Burke, and Ernzerhof (PBE) when Perdew–Zunger self-interaction correction is applied in combination with complex-valued orbitals. Even without the correction, the PBE functional gives quite good results despite the fact that corresponding Rydberg virtual orbitals have positive energy in the ground state calculation. Results obtained using the Tao, Perdew, Staroverov, and Scuseria (TPSS) and r2SCAN meta-GGA functionals are also presented, but they do not provide a systematic improvement over the results from the uncorrected PBE functional. The grid representation combined with the projector augmented-wave approach gives a simpler and better representation of diffuse Rydberg orbitals than a linear combination of atomic orbitals with commonly used basis sets, the latter leading to an overestimation of the excitation energy due to confinement of the excited states. [ABSTRACT FROM AUTHOR]

In Paper I [H. Francisco, A. C. Cancio, and S. B. Trickey, J. Chem. Phys. 159, 214102 (2023)], we gave a regularization of the Tao–Mo exchange functional that removes the order-of-limits problem in the original Tao–Mo form and also eliminates the unphysical behavior introduced by an earlier regularization while essentially preserving compliance with the second-order gradient expansion. The resulting simplified, regularized (sregTM) functional delivers performance on standard molecular and solid state test sets equal to that of the earlier revised, regularized Tao–Mo functional. Here, we address de-orbitalization of that new sregTM into a pure density functional. We summarize the failures of the Mejía-Rodríguez and Trickey de-orbitalization strategy [Phys. Rev. A 96, 052512 (2017)] when used with both versions. We discuss how those failures apparently arise in the so-called z′ indicator function and in substitutes for the reduced density Laplacian in the parent functionals. Then, we show that the sregTM functional can be de-orbitalized somewhat well with a rather peculiarly parameterized version of the previously used deorbitalizer. We discuss, briefly, a de-orbitalization that works in the sense of reproducing error patterns but that apparently succeeds by cancelation of major qualitative errors associated with the de-orbitalized indicator functions α and z, hence, is not recommended. We suggest that the same issue underlies the earlier finding of comparatively mediocre performance of the de-orbitalized Tao–Perdew–Staroverov–Scuseri functional. Our work demonstrates that the intricacy of such two-indicator functionals magnifies the errors introduced by the Mejía-Rodríguez and Trickey de-orbitalization approach in ways that are extremely difficult to analyze and correct. [ABSTRACT FROM AUTHOR]

Non-fullerene electron acceptors have emerged as promising alternatives to traditional electron-acceptors in the active layers of organic photovoltaics. This is due to their tunable energy levels, optical response in the visible light spectrum, high electron mobility, and photochemical stability. In this study, the electronic properties of two representative non-fullerene acceptors, ITIC and Y5, have been calculated within the framework of density functional theory using a range of hybrid and non-hybrid density functionals. Screened range-separated hybrid (SRSH) approaches were also tested. The results are analyzed in light of the previously reported experimental outcomes. Specifically, we have calculated the oxidation and reduction potentials, fundamental and optical gaps, the highest occupied molecular orbital and lowest unoccupied molecular orbital energies, and exciton binding energies. Additionally, we have investigated the effects of the medium dielectric constant on these properties employing a universal implicit solvent model. It was found that hybrid functionals generally perform poorly in predicting oxidation potentials, while non-hybrid functionals tend to overestimate reduction potentials. The inclusion of a large Hartree–Fock contribution to the global or long range was identified as the source of inaccuracy for many hybrid functionals in predicting both redox potentials and the fundamental and optical gaps. Corroborating with the available literature, ∼50% of all tested functionals predicted very small exciton binding energies, within the range of ±0.1 eV, that become even smaller by increasing the dielectric constant of the material. Finally, the OHSE2PBE and tHCTHhyb functionals and the optimal tuning SRSH approach emerged as the best-performing methods, with good accuracy in the description of the electronic properties of interest. [ABSTRACT FROM AUTHOR]

The choice of exchange functional is a critical factor in determining the energy bandgap of semiconductors. Ab initio calculations using different exchange functionals, including the conventional generalized-gradient approximation (GGA) functionals, meta-GGA functionals, and hybrid functionals, show significant differences in the calculated energy bandgap for semiconducting half-Heusler compounds. These compounds, which have 18 valence electrons per unit cell, are of great interest due to their thermoelectric properties, making them suitable for energy conversion applications. In addition, accounting for electronic correlations using the G W method also affects the calculated energy bandgaps compared to standard GGA calculations. The variations in calculated energy bandgaps are specific to each material when using different functionals. Hence, a detailed investigation of the electronic properties of each compound is necessary to determine the most appropriate functional for an accurate description of the electronic properties. Our results indicate that no general rules can be established and a comparison with experimental results is required to determine the most appropriate functional. [ABSTRACT FROM AUTHOR]

The H2O and H2O2 molecules resemble each other in a multitude of ways as has been noted in the literature. Here, we present density functional theory (DFT) calculations for the H2O2(s) and H2O2·2H2O(s) crystals and make selected comparisons with ice polymorphs. The performance of a number of dispersion-corrected density functionals—both self-consistent and a posteriori ones—are assessed, and we give special attention to the D3 correction and its effects. The D3 correction to the lattice energies is large: for H2O2(s) the D3 correction constitutes about 25% of the lattice energy using PBE, much more for RPBE, much less for SCAN, and it primarily arises from non-H-bonded interactions out to about 5 Å.The large D3 corrections to the lattice energies are likely a consequence of several effects: correction for missing dispersion interaction, the ability of D3 to capture and correct various other kinds of limitations built into the underlying DFT functionals, and finally some degree of cell-contraction-induced polarization enhancement. We find that the overall best-performing functionals of the twelve examined are optPBEvdW and RPBE-D3. Comparisons with DFT assessments for ices in the literature show that where the same methods have been used, the assessments largely agree. [ABSTRACT FROM AUTHOR]

An efficient implementation of zero-field splitting parameters based on the work of Schmitt et al. [J. Chem. Phys. 134, 194113 (2011)] is presented. Seminumerical integration techniques are used for the two-electron spin–dipole contribution and the response equations of the spin–orbit perturbation. The original formulation is further generalized. First, it is extended to meta-generalized gradient approximations and local hybrid functionals. For these functional classes, the response of the paramagnetic current density is considered in the coupled-perturbed Kohn–Sham equations for the spin–orbit perturbation term. Second, the spin–orbit perturbation is formulated within relativistic exact two-component theory and the screened nuclear spin–orbit (SNSO) approximation. The accuracy of the implementation is demonstrated for transition-metal and diatomic main-group compounds. The efficiency is assessed for Mn and Mo complexes. Here, it is found that coarse integration grids for the seminumerical schemes lead to drastic speedups while introducing clearly negligible errors. In addition, the SNSO approximation substantially reduces the computational demands and leads to very similar results as the spin–orbit mean field Ansatz. [ABSTRACT FROM AUTHOR]

Practical density functional theory (DFT) owes its success to the groundbreaking work of Kohn and Sham that introduced the exact calculation of the non-interacting kinetic energy of the electrons using an auxiliary mean-field system. However, the full power of DFT will not be unleashed until the exact relationship between the electron density and the non-interacting kinetic energy is found. Various attempts have been made to approximate this functional, similar to the exchange–correlation functional, with much less success due to the larger contribution of kinetic energy and its more non-local nature. In this work, we propose a new and efficient regularization method to train density functionals based on deep neural networks, with particular interest in the kinetic-energy functional. The method is tested on (effectively) one-dimensional systems, including the hydrogen chain, non-interacting electrons, and atoms of the first two periods, with excellent results. For atomic systems, the generalizability of the regularization method is demonstrated by training also an exchange–correlation functional, and the contrasting nature of the two functionals is discussed from a machine-learning perspective. [ABSTRACT FROM AUTHOR]

σ-Functionals are promising new developments for the Kohn–Sham correlation energy based upon the direct Random Phase Approximation (dRPA) within the adiabatic connection formalism, providing impressive improvements over dRPA for a broad range of benchmarks. However, σ-functionals exhibit a high amount of self-interaction inherited from the approximations made within dRPA. Inclusion of an exchange kernel in deriving the coupling-strength-dependent density–density response function leads to so-called τ-functionals, which – apart from a fourth-order Taylor series expansion – have only been realized in an approximate fashion so far to the best of our knowledge, most notably in the form of scaled σ-functionals. In this work, we derive, optimize, and benchmark three types of σ- and τ-functionals including approximate exchange effects in the form of an antisymmetrized Hartree kernel. These functionals, based on a second-order screened exchange type contribution in the adiabatic connection formalism, the electron–hole time-dependent Hartree–Fock kernel (eh-TDHF) otherwise known as RPA with exchange (RPAx), and an approximation thereof known as approximate exchange kernel (AXK), are optimized on the ASCDB database using two new parametrizations named A1 and A2. In addition, we report a first full evaluation of σ- and τ-functionals on the GMTKN55 database, revealing our exchange-including functionals to considerably outperform existing σ-functionals while being highly competitive with some of the best double-hybrid functionals of the original GMTKN55 publication. In particular, the σ-functionals based on AXK and τ-functionals based on RPAx with PBE0 reference stand out as highly accurate approaches for a wide variety of chemically relevant problems. [ABSTRACT FROM AUTHOR]

For an electronic system, given a mean field method and a distribution of orbital occupation numbers that are close to the natural occupations of the correlated system, we provide formal evidence and computational support to the hypothesis that the entropy (or more precisely −σS, where σ is a parameter and S is the entropy) of such a distribution is a good approximation to the correlation energy. Underpinning the formal evidence are mild assumptions: the correlation energy is strictly a functional of the occupation numbers, and the occupation numbers derive from an invertible distribution. Computational support centers around employing different mean field methods and occupation number distributions (Fermi–Dirac, Gaussian, and linear), for which our claims are verified for a series of pilot calculations involving bond breaking and chemical reactions. This work establishes a formal footing for those methods employing entropy as a measure of electronic correlation energy (e.g., i-DMFT [Wang and Baerends, Phys. Rev. Lett. 128, 013001 (2022)] and TAO-DFT [J.-D. Chai, J. Chem. Phys. 136, 154104 (2012)]) and sets the stage for the widespread use of entropy functionals for approximating the (static) electronic correlation. [ABSTRACT FROM AUTHOR]

Since the form of the exact functional in density functional theory is unknown, we must rely on density functional approximations (DFAs). In the past, very promising results have been reported by combining semi-local DFAs with exact, i.e. Hartree–Fock, exchange. However, the spin-state energy ordering and the predictions of global minima structures are particularly sensitive to the choice of the hybrid functional and to the amount of exact exchange. This has been already qualitatively described for single conformations, reactions, and a limited number of conformations. Here, we have analyzed the mixing of exact exchange in exchange functionals for a set of several hundred isomers of the transition metal carbide, Mo4C2. The analysis of the calculated energies and charges using PBE0-type functional with varying amounts of exact exchange yields the following insights: (1) The sensitivity of spin-energy splitting is strongly correlated with the amount of exact exchange mixing. (2) Spin contamination is exacerbated when correlation is omitted from the exchange-correlation functional. (3) There is not one ideal value for the exact exchange mixing which can be used to parametrize or choose among the functionals. Calculated energies and electronic structures are influenced by exact exchange at a different magnitude within a given distribution; therefore, to extend the application range of hybrid functionals to the full periodic table the spin-energy splitting energies should be investigated. [ABSTRACT FROM AUTHOR]

Since the form of the exact functional in density functional theory is unknown, we must rely on density functional approximations (DFAs). In the past, very promising results have been reported by combining semi-local DFAs with exact, i.e. Hartree–Fock, exchange. However, the spin-state energy ordering and the predictions of global minima structures are particularly sensitive to the choice of the hybrid functional and to the amount of exact exchange. This has been already qualitatively described for single conformations, reactions, and a limited number of conformations. Here, we have analyzed the mixing of exact exchange in exchange functionals for a set of several hundred isomers of the transition metal carbide, Mo4C2. The analysis of the calculated energies and charges using PBE0-type functional with varying amounts of exact exchange yields the following insights: (1) The sensitivity of spin-energy splitting is strongly correlated with the amount of exact exchange mixing. (2) Spin contamination is exacerbated when correlation is omitted from the exchange-correlation functional. (3) There is not one ideal value for the exact exchange mixing which can be used to parametrize or choose among the functionals. Calculated energies and electronic structures are influenced by exact exchange at a different magnitude within a given distribution; therefore, to extend the application range of hybrid functionals to the full periodic table the spin-energy splitting energies should be investigated. [ABSTRACT FROM AUTHOR]

A recently initiated variant of density functional theory utilizes a set of spherically symmetric densities instead of the density. The exact functionals are unknown in the new theory akin to the standard density functional theory. In order to test approximate functionals exactly solvable models are introduced. A harmonic molecular ion, the analogue to the hydrogen molecule ion and a harmonic two-electron molecule showing analogy to the hydrogen molecule are proposed. It has been found that the wave function and the density can be given analytically. The exact spherical densities and the effective potentials of the Euler equations also have analytical form. It has been shown that the models can be easily extended to several "nuclei." [ABSTRACT FROM AUTHOR]

We update the Quadratic Integrand Double-Hybrid (QIDH) model [J. Chem. Phys. 141, 031101 (2014)] by incorporating the nonempirical restored-regularized Strongly Constrained and Appropriately Normed (r2SCAN) meta-generalized gradient approximation exchange-correlation functional, thus devising a robust density functional approximation free of any empirical parameter and incorporating all the constraints so far known for the exchange-correlation kernel. We assessed the new r2SCAN-QIDH expression on the GMTKN55 database and further extend its application to various types of non-covalent interactions (e.g., S66 × 8, O24 × 5). The assessment done shows that the model becomes very competitive in accuracy with respect to parent exchange-correlation functionals of any type, but without relying on any fitted parameter or numerical training. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Recent measurements of the third harmonic scattering responses of molecules have given a new impetus for computing molecular second hyperpolarizabilities (γ) and for deducing structure–property relationships. This paper has employed a variety of wavefunction and density functional theory methods to evaluate the second hyperpolarizability of the p-nitroaniline prototypical push-pull π-conjugated molecule, addressing also numerical aspects, such as the selection of an integration grid and the impact of the order of differentiation vs the achievable accuracy by using the Romberg quadrature. The reliability of the different methods has been assessed by comparison to reference Coupled-Cluster Singles and Doubles with perturbative treatment of the Triples results. On the one hand, among wavefunction methods, the MP2 scheme offers the best accuracy/cost ratio for computing the static γ. On the other hand, using density functional theory, γ remains a challenging property to compute because all conventional, global hybrid or range-separated hybrid, exchange–correlation functionals underestimate static γ values by at least 15%. Even tuning the range-separating parameter to minimize the delocalization errors does not enable to improve the γ values. Nevertheless, the original double-hybrid B2-PLYP functional, which benefits from 27% of PT2 correlation and 53% Hartree–Fock exchange, provides accurate estimates of static γ values. Unfortunately, the best performing exchange–correlation functionals for γ are not necessarily reliable for the first hyperpolarizability, β, and vice versa. In fact, the β of p-nitroaniline (pNA) could be predicted, with a good accuracy, with several hybrid exchange–correlation functionals (including by tuning the range-separating parameter), but these systematically underestimate γ. As for γ, the MP2 wavefunction method remains the best compromise to evaluate the first hyperpolarizability of pNA at low computational cost. [ABSTRACT FROM AUTHOR]

The swift progression and expansion of machine learning (ML) have not gone unnoticed within the realm of statistical mechanics. In particular, ML techniques have attracted attention by the classical density-functional theory (DFT) community, as they enable automatic discovery of free-energy functionals to determine the equilibrium-density profile of a many-particle system. Within classical DFT, the external potential accounts for the interaction of the many-particle system with an external field, thus, affecting the density distribution. In this context, we introduce a statistical-learning framework to infer the external potential exerted on a classical many-particle system. We combine a Bayesian inference approach with the classical DFT apparatus to reconstruct the external potential, yielding a probabilistic description of the external-potential functional form with inherent uncertainty quantification. Our framework is exemplified with a grand-canonical one-dimensional classical particle ensemble with excluded volume interactions in a confined geometry. The required training dataset is generated using a Monte Carlo (MC) simulation where the external potential is applied to the grand-canonical ensemble. The resulting particle coordinates from the MC simulation are fed into the learning framework to uncover the external potential. This eventually allows us to characterize the equilibrium density profile of the system by using the tools of DFT. Our approach benchmarks the inferred density against the exact one calculated through the DFT formulation with the true external potential. The proposed Bayesian procedure accurately infers the external potential and the density profile. We also highlight the external-potential uncertainty quantification conditioned on the amount of available simulated data. The seemingly simple case study introduced in this work might serve as a prototype for studying a wide variety of applications, including adsorption, wetting, and capillarity, to name a few. [ABSTRACT FROM AUTHOR]

Detailed characterization of deep-level color centers requires understanding their electronic and atomic structure, which is most commonly investigated utilizing the Kohn–Sham density functional theory. Standard semilocal functionals based on the generalized gradient approximation (GGA) are inclined toward an imprecise quantitative description of defects' electronic structure. Hybrid functionals provide an improved prediction of electronic properties, albeit at a much higher computational cost. In this work, we test the newly developed Strongly Constrained and Appropriately Normed (SCAN) family of meta-GGA density functionals for selected color centers in diamond. In particular, we study nitrogen-, silicon-, germanium-, and tin-vacancy centers that have been recently investigated for their use in quantum technological applications. We show that SCAN and its derivatives, the rSCAN and r2SCAN functionals, significantly improve the calculated energies of optical transitions within the delta-self-consistent-field approach, almost reaching the accuracy of the hybrid Heyd–Scuseria–Ernzerhof (HSE) functional. In the case of the NV− center, we also show that the SCAN family of functionals improves the description of the adiabatic potential energy surfaces compared to both GGA and hybrid functionals, improving calculated luminescence lineshapes. As a result of these findings, we recommend using the SCAN family of functionals as a promising alternative for studying color centers in solids. [ABSTRACT FROM AUTHOR]

Kohn–Sham density functional theory (DFT) is nowadays widely used for electronic structure theory simulations, and the accuracy and efficiency of DFT rely on approximations of the exchange-correlation functional. By including the kinetic energy density τ, the meta-generalized-gradient approximation (meta-GGA) family of functionals achieves better accuracy and flexibility while retaining the efficiency of semi-local functionals. For example, the strongly constrained and appropriately normed (SCAN) meta-GGA functional has been proven to yield accurate results for solid and molecular systems. We implement meta-GGA functionals with both numerical atomic orbitals and plane wave bases in the ABACUS package. Apart from the exchange-correlation potential, we also discuss the evaluation of force and stress. To validate our implementation, we perform finite-difference tests and convergence tests with the SCAN, rSCAN, and r2SCAN meta-GGA functionals. We further test water hexamers, weakly interacting molecules from the S22 dataset, as well as 13 semiconductors using the three functionals. The results show satisfactory agreement with previous calculations and available experimental values. [ABSTRACT FROM AUTHOR]

Despite the wide applications, the ab initio modeling of the ceria based catalyst is challenging. The partial occupation in the 4f orbitals creates a fundamental challenge for commonly used density functional theory (DFT) methods, including semilocal functionals with Hubbard U correction to force localization and hybrid functionals. In this work, we benchmark the random phase approximation (RPA) for ceria surface properties, including surface energy and hydrogenation energy, compared to the results utilizing the DFT + U approach or hybrid functionals. We show that, for the latter approaches, different surface properties require opposite directions of parameter tuning. This forms a dilemma for the parameter based DFT methods, as the improvement of a certain property by tuning parameters will inevitably lead to the worsening of other properties. Our results suggest that the parameter-free many-body perturbation theory methods exemplified by RPA are a promising strategy to escape the dilemma and provide highly accurate descriptions, which will allow us to better understand the catalytic reactions in ceria related systems. [ABSTRACT FROM AUTHOR]

We have designed a [Fe(SH)4H]− model with the fifth proton binding either to Fe or S. We show that the energy difference between these two isomers (∆E) is hard to estimate with quantum-mechanical (QM) methods. For example, different density functional theory (DFT) methods give ∆E estimates that vary by almost 140 kJ/mol, mainly depending on the amount of exact Hartree–Fock included (0%–54%). The model is so small that it can be treated by many high-level QM methods, including coupled-cluster (CC) and multiconfigurational perturbation theory approaches. With extrapolated CC series (up to fully connected coupled-cluster calculations with singles, doubles, and triples) and semistochastic heat-bath configuration interaction methods, we obtain results that seem to be converged to full configuration interaction results within 5 kJ/mol. Our best result for ∆E is 101 kJ/mol. With this reference, we show that M06 and B3LYP-D3 give the best results among 35 DFT methods tested for this system. Brueckner doubles coupled cluster with perturbaitve triples seems to be the most accurate coupled-cluster approach with approximate triples. CCSD(T) with Kohn–Sham orbitals gives results within 4–11 kJ/mol of the extrapolated CC results, depending on the DFT method. Single-reference CC calculations seem to be reasonably accurate (giving an error of ∼5 kJ/mol compared to multireference methods), even if the D1 diagnostic is quite high (0.25) for one of the two isomers. [ABSTRACT FROM AUTHOR]

The feature vector mapping used to represent chemical systems is a key factor governing the superior data efficiency of kernel based quantum machine learning (QML) models applicable throughout chemical compound space. Unfortunately, the most accurate representations require a high dimensional feature mapping, thereby imposing a considerable computational burden on model training and use. We introduce compact yet accurate, linear scaling QML representations based on atomic Gaussian many-body distribution functionals (MBDF) and their derivatives. Weighted density functions of MBDF values are used as global representations that are constant in size, i.e., invariant with respect to the number of atoms. We report predictive performance and training data efficiency that is competitive with state-of-the-art for two diverse datasets of organic molecules, QM9 and QMugs. Generalization capability has been investigated for atomization energies, highest occupied molecular orbital–lowest unoccupied molecular orbital eigenvalues and gap, internal energies at 0 K, zero point vibrational energies, dipole moment norm, static isotropic polarizability, and heat capacity as encoded in QM9. MBDF based QM9 performance lowers the optimal Pareto front spanned between sampling and training cost to compute node minutes, effectively sampling chemical compound space with chemical accuracy at a sampling rate of ∼ 48 molecules per core second. [ABSTRACT FROM AUTHOR]

This article focuses on a class of nonstrict feedback systems with input delay, state delays and time‐varying full‐state constraints by proposing an adaptive neural control scheme. To overcome the problems of all state variables effected by time‐varying constraints, the asymmetric time‐varying barrier Lyapunov functions are constructed. The influence of state delays and input delay is eliminated by employing suitable Lyapunov–Krasovskii functionals. Additionally, the process of controller design is based on backstepping method and the unknown functions can be approximated by radial basis function neural networks. Moreover, the problem of repeated differentiations for nonlinear components during controller design is hugely simplified by taking advantage of the dynamic surface control method. The boundness of all the closed‐loop signals can be ensured by the designed controller. Finally, two numerical simulations illustrate that the proposed adaptive neural control scheme is effective. [ABSTRACT FROM AUTHOR]

This work is concerned with the L1$$ {L}_1 $$ bumpless transfer control issue for switched positive delayed systems on finite horizons. A bumpless transfer input scheme and a state‐dependent limited frequency switching scheme, induced by the transient control input discontinuities during controller switching and by the frequent switching for multiple subsystems, respectively, are jointly introduced. The main advantages of the presented schemes are that the amplitude of the control input discontinuities and the frequent switches are all reduced by the dual design of bumpless transfer controllers for subsystems and a state‐dependent limited frequency switching scheme. By utilizing the multiple linear copositive Lyapunov functionals, sufficient conditions are also offered so as to achieve the desirable closed‐loop performances, including positivity, finite‐time boundedness, bumpless transfer property and a prescribed L1$$ {L}_1 $$‐gain property. Finally, a practical example is borrowed which validates the proposed schemes. [ABSTRACT FROM AUTHOR]

In this work, we demonstrate that a functional modeling the self-aggregation of stochastically distributed lipid molecules can be obtained as the Γ -limit of a family of discrete energies driven by a sequence of independent and identically distributed random variables. These random variables are intended to describe the asymptotic behavior of lipid molecules that satisfy an incompressibility condition. The discrete energy keeps into account the interactions between particles. We resort to transportation maps to compare functionals defined on discrete and continuous domains, and we prove that, under suitable conditions on the scaling of these maps as the number of random variables increases, the limit functional features an interfacial term with a Wasserstein-type penalization. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two one-parameter families of quadratic polynomial enrichments designed to enhance the accuracy of the classical Crouzeix–Raviart finite element. These enrichments are realized by using weighted line integrals as enriched linear functionals and quadratic polynomial functions as enrichment functions. To validate the effectiveness of our approach, we conduct numerical experiments that confirm the improvement achieved by the proposed method. [ABSTRACT FROM AUTHOR]

This study investigated an n-species cooperative Lotka-Volterra system with impulsive controls and time delays. In the case of constructing appropriate Lyapunov functionals and using inequality techniques, some conditions for the permanence and global attractiveness of the system are obtained. In addition, sufficient conditions for the existence of positive periodic solutions were established using the coincidence degree theory. Finally, numerical simulation results illustrate the effectiveness of our findings. [ABSTRACT FROM AUTHOR]

This paper deals with a two-species attraction–repulsion chemotaxis system u t = Δ u - ξ 1 ∇ · (u ∇ v) + χ 1 ∇ · (u ∇ z) + f 1 (u , w) , (x , t) ∈ Ω × (0 , ∞) , τ v t = Δ v + w - v , (x , t) ∈ Ω × (0 , ∞) , w t = Δ w - ξ 2 ∇ · (w ∇ z) + χ 2 ∇ · (w ∇ v) + f 2 (u , w) , (x , t) ∈ Ω × (0 , ∞) , τ z t = Δ z + u - z , (x , t) ∈ Ω × (0 , ∞) , under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω ⊆ R n , where τ ∈ { 0 , 1 } , ξ i , χ i > 0 and f i (u , w) (i = 1 , 2) satisfy f 1 (u , w) = u (a 0 - a 1 u - a 2 w + a 3 ∫ Ω u d x + a 4 ∫ Ω w d x) , f 2 (u , w) = w (b 0 - b 1 u - b 2 w + b 3 ∫ Ω u d x + b 4 ∫ Ω w d x) with a i , b i > 0 (i = 0 , 1 , 2) , a j , b j ∈ R (j = 3 , 4) . It is proved that in any space dimension n ≥ 1 , the above system possesses a unique global and uniformly bounded classical solution regardless of τ = 0 or τ = 1 under some suitable assumptions. Moreover, by constructing Lyapunov functionals, we establish the globally asymptotic stabilization of coexistence and semi-coexistence steady states. [ABSTRACT FROM AUTHOR]

We establish laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is different, were obtained by Khoshnevisan [Probab. Theory Related Fields 93 (1992), pp. 377–392]. Our starting point is a version of Strassen's functional law of the iterated logarithm for random walks with drift. For the special case of the area of a planar random walk with drift, we compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the classical Dido problem. For general intrinsic volumes and dimensions, our proof exploits a novel zero–one law for functionals of convex hulls of walks with drift, of some independent interest. As another application of our approach, we obtain iterated-logarithm laws for intrinsic volumes of the convex hull of the centre of mass (running average) process associated to the random walk. [ABSTRACT FROM AUTHOR]

This work investigates the less conservative stability conditions for linear systems with a time-varying delay. At first, augmented Lyapunov–Krasovskii functionals(LKFs) are constructed with state vectors that have not been utilized in the existing works, and an augmented zero equality that can be derived according to the augmented vector is proposed. By utilizing them, a stability condition is proposed in the form of a linear matrix inequality. And, by using novel delay-dependent LKFs and the introduced ones, improved results are obtained than the previous result. The addition of the delay-dependent LKFs increases the number of decision variables in the results. Therefore, any vectors of integral inequalities utilized in the proposed criterion are appropriately adjusted to reduce computational complexity. To check the excellence and validity of the proposed results, several numerical examples are applied. • The improved stability conditions for the linear system with a time-varying delay are proposed. • The proposed Lyapunov-Krasovskii functionals(LKFs) have more extended augmented vectors to obtain less conservative results. And LKFs based on an integral inequality (Lee et al., 2023) is utilized. • The novel LKFs which are the delay-dependent forms are proposed. The superiority of the results is checked according to the presence or absence of the forms. • The computational complexity is reduced by adjusting any vector of the integral inequality utilized in the proposed criteria. [ABSTRACT FROM AUTHOR]

This article investigates the H∞$$ {H}_{\infty } $$ control problem of switched systems with time‐varying delays under asynchronous switching. By reconstructing the switching rule of local controllers and empolying multiple Lyapunov functionals and integral inequalities, sufficient conditions without the constraint between the switching delay and the dwell time of submodes to solve the H∞$$ {H}_{\infty } $$ control problem are derived. The control gain matrices can be conveniently obtained by the proposed linear search algorithm. [ABSTRACT FROM AUTHOR]

In this paper, as a step towards a unified mathematical treatment of the gauge functionals from quantum field theory that have found profound applications in mathematics, we generalize the Seiberg–Witten functional that in particular includes the Kapustin–Witten functional as a special case. We first demonstrate the smoothness of weak solutions to this generalized functional. We then establish the existence of weak solutions under the assumption that the structure group of the bundle is abelian, by verifying the Palais–Smale compactness. [ABSTRACT FROM AUTHOR]

We construct optimal Hardy weights to subcritical energy functionals h associated with quasilinear Schrödinger operators on infinite graphs. Here, optimality means that the weight w is the largest possible with respect to a partial ordering, and that the corresponding shifted energy functional h - w is null-critical. Moreover, we show a decay condition of Hardy weights in terms of their integrability with respect to certain integral weights. As an application of the decay condition, we show that null-criticality implies optimality near infinity. We also briefly discuss an uncertainty-type principle, a Rellich-type inequality and examples. [ABSTRACT FROM AUTHOR]

A directionally‐split volume‐of‐fluid (VOF) methodology for evolving interfaces under curvature‐dependent speed is devised. The interface is reconstructed geometrically and the volume fraction is advected with a technique to incorporate a topological volume conservation constraint. The proposed approach uses the idea that the role of curvature in a speed function V$$ \mathbf{V} $$ is analogous to the role of viscosity in the corresponding hyperbolic conservation law to propagate complex interfaces where singularities may exist. The approach has the advantage of simple implementation and straightforward extension to more complex multiphase systems by formulating the interface evolution problem using energy functionals to derive an expression for the interface‐advecting velocity. The numerical details of the volume‐of‐fluid based formulation are discussed with emphasis on the importance of curvature estimation. Finally, canonical curves and surfaces traditionally investigated by the level set (LS) method are tested with the devised approach and the results are compared with existing work in LS. [ABSTRACT FROM AUTHOR]

The change in local, semi-local and peripheral aromaticity of coronene upon gradual replacement of its peripheral CC bonds with BN bonds, starting from the C3h symmetrical 3-BN derivative, was followed by calculating HOMA, PDI, FLU, EDDB and NICS-xy scans. They were calculated using B3LYP-D3(BJ), M06-2X and ωB97XD functionals, whose results agreed very well. It was found that introduction of BN units instead of CC units can, more or less, isolate aromatic carbocyclic subunits and increase peripheral electron delocalization. These subunits dominate the frontier orbital shapes and energy gap, thus enabling their modulation by applying BN/CC isosterism. [ABSTRACT FROM AUTHOR]

This article investigates the exponential H∞$$ {H}_{\infty } $$ control issue of bidirectional associative memory neural network (BAMNN) with unbounded time‐varying delays. A bounded real lemma (BRL) is first established via a direct method, which is on the basis of the solutions of BAMNN. Second, based on the obtained BRL, the state feedback controller is designed to guarantee the global exponential stability of the resulting closed‐loop BAMNN with an H∞$$ {H}_{\infty } $$ performance index. Since no Lyapunov–Krasovskii functionals is constructed in the proposed method, the computation burden and complexity are reduced. Lastly, the effectiveness of the theoretical results is illustrated through two numerical examples. [ABSTRACT FROM AUTHOR]

We introduce a notion of uniform Ding stability for a projective manifold with big anticanonical class, and prove that the existence of a unique Kähler–Einstein metric on such a manifold implies uniform Ding stability. The main new techniques are to develop a general theory of Deligne functionals – and corresponding slope formulas – for singular metrics, and to prove a slope formula for the Ding functional in the big setting. This extends work of Berman in the Fano situation, when the anticanonical class is actually ample, and proves one direction of the analogue of the Yau–Tian–Donaldson conjecture in this setting. We also speculate about the relevance of uniform Ding stability and K-stability to moduli in the big setting. [ABSTRACT FROM AUTHOR]

In this research work, we examined the decomposition mechanisms of N-substituted diacetamides. We focused on the substituent effect on the nitrogen lone-pair electron delocalization, with electron-withdrawing and electron donor groups. DFT functionals used the following: B1LYP, B3PW91, CAMB3LYP, LC-BLYP, and X3LYP. Dispersion corrections (d3bj) with Becke–Johnson damping were applied when necessary to improve non-covalent interactions in the transition state. Pople basis sets with higher angular moments and def2-TZVP basis sets were also applied and were crucial for obtaining consistent thermodynamic parameters. The proposed mechanism involves a six-membered transition state with the extraction of an α hydrogen. Several conformers of N-diacetamides were used to account for the decrease in entropy in the transition state in the rate-determining state. All calculations, including natural bond orbital (NBO) analyses, were performed using the Gaussian16 computational package and its GaussView 6.0 visualizer, along with VMD and GNUPLOT software. The isosurfaces and IBSIs were calculated using MultiWFN and IGMPlot, respectively. [ABSTRACT FROM AUTHOR]

Under the influence of sensor saturation and packet dropout, this paper is concerned with the fault detection (FD) problem for a class of nonlinear systems with two additive time‐varying delay components. In the addressed measurement model, the sensor is assumed to have two communication channels, which can contribute to improving the probability of successfully delivering the measurements. A FD filter is designed to generate residual signals, and the sufficient criteria are established to ensure the stochastic finite‐time stability (SFTS) of the residual system with H∞$$ {H}_{\infty } $$ performance constraint by constructing new Lyapunov‐Krasovskii functionals. Based on the established criteria, the explicit expression of the desired FD filter is obtained via solving a set of linear matrix inequalities (LMIs). Finally, the usefulness of the proposed FD scheme is verified by a simulation example. [ABSTRACT FROM AUTHOR]

Through the Bayesian lens of four-dimensional variational (4D-Var) data assimilation, uncertainty in model parameters is traditionally quantified through the posterior covariance matrix. However, in modern settings involving high-dimensional and computationally expensive forward models, posterior covariance knowledge must be relaxed to deterministic or stochastic approximations. In the carbon flux inversion literature, proposed a stochastic method capable of approximating posterior variances of linear functionals of the model parameters that is particularly well suited for large-scale Earth-system 4D-Var data assimilation tasks. This note formalizes this algorithm and clarifies its properties. We provide a formal statement of the algorithm, demonstrate why it converges to the desired posterior variance quantity of interest, and provide additional uncertainty quantification allowing incorporation of the Monte Carlo sampling uncertainty into the method's Bayesian credible intervals. The methodology is demonstrated using toy simulations and a realistic carbon flux inversion observing system simulation experiment. [ABSTRACT FROM AUTHOR]