Density Functional Approximations Research Articles

Adapting hybrid density functionals with machine learning.

Exact exchange contributions significantly affect electronic states, influencing covalent bond formation and breaking. Hybrid density functional approximations, which average exact exchange admixtures empirically, have achieved success but fall short of high-level quantum chemistry accuracy due to delocalization errors. We propose adaptive hybrid functionals, generating optimal exact exchange admixture ratios on the fly using data-efficient quantum machine learning models with negligible overhead. The adaptive Perdew-Burke-Ernzerhof hybrid density functional (aPBE0) improves energetics, electron densities, and HOMO-LUMO gaps in QM9, QM7b, and GMTKN55 benchmark datasets. A model uncertainty-based constraint reduces the method smoothly to PBE0 in extrapolative regimes, ensuring general applicability with limited training. By tuning exact exchange fractions for different spin states, aPBE0 effectively addresses the spin gap problem in open-shell systems such as carbenes. We also present a revised QM9 (revQM9) dataset with more accurate quantum properties, including stronger covalent binding, larger bandgaps, more localized electron densities, and larger dipole moments.

Systematic Study of Hard-Wall Confinement-Induced Effects on Atomic Electronic Structure.

We point out that although a litany of studies have been published on atoms in hard-wall confinement, they have either not been systematic, having only looked at select atoms and/or select electron configurations, or they have not used robust numerical methods. To remedy the situation, we perform in this work a methodical study of atoms in hard-wall confinement with the HelFEM program, which employs the finite element method that trivially implements the hard-wall potential, guarantees variational results, and allows for easily finding the numerically exact solution. Our fully numerical calculations are based on nonrelativistic density functional theory and spherically averaged densities. We consider three levels of density functional approximations: the local density approximation employing the Perdew-Wang (PW92) functional, the generalized-gradient approximation (GGA) employing the Perdew-Burke-Ernzerhof (PBE) functional, and the meta-GGA approximation employing the r2SCAN functional. Importantly, the completely dissimilar density functional approximations are in excellent agreement, suggesting that the observed results are not artifacts of the employed level of theory. We systematically examine low-lying configurations of the H-Xe atoms and their monocations and investigate how the configurations─especially the ground-state configuration─behave as a function of the position of the hard-wall boundary. We perform calculations with both spin-polarized as well as spin-restricted densities and demonstrate that spin-polarization effects are significant in open-shell configurations, even though some previous studies have only considered the spin-restricted model. We demonstrate the importance of considering ground-state changes for confined atoms by computing the ionization radii for the H-Xe atoms and observe significant differences to earlier studies. Confirming previous observations, we identify electron shifts on the outermost shells for a majority of the elements: valence s electrons are highly unfavored under strong confinement, and the high-lying 3d and 4f orbitals become occupied in atoms of periods 2-3 and 3-4, respectively. We also comment on deficiencies of a commonly used density-based estimate for the van der Waals (vdW) radius of atoms and propose a better behaved variant in terms of the number of electrons outside the vdW radius that we expect will prove useful in future studies.

Effect of Exact Exchange on the Energy Landscape in Self-Consistent Field Theory.

Density functional approximations can reduce the spin symmetry breaking observed for self-consistent field (SCF) solutions compared to Hartree-Fock theory, but the amount of exact Hartree-Fock (HF) exchange appears to be a key determinant in broken Ŝ2 symmetry. To elucidate the precise role of exact exchange, we investigate the energy landscape of unrestricted Hartree-Fock and Kohn-Sham density functional theory for benzene and square cyclobutadiene, which provide paradigmatic examples of closed-shell and open-shell electronic structures, respectively. We find that increasing the amount of exact exchange leads to more local SCF minima, which can be characterized as combinatorial arrangements of unpaired electrons in the carbon π system. Furthermore, we studied the pathways connecting local minima to understand the relationships between different solutions. Our analysis reveals a subtle balance between one- and two-body interactions in determining SCF symmetry breaking, shedding new light on the physical driving forces for spin-symmetry-broken solutions in SCF approaches.

Many-body perturbation-theory calculation of electronic and optical properties of β-phase Be3N2

Abstract The hexagonal β-phase of bulk Be3N2 is a wide band gap semiconductor with potential ultraviolet (UV) optoelectronics applications. Here, we have performed the first theoretical study of the electronic and optical properties of the β phase of Be3N2 considering many-body effects. In particular, we have obtained the electronic band structure and spectra of the dielectric function, complex refractive index and electron energy loss function. For the calculation of the band structure, both Density Functional Theory and G0W0 approximation plus Wannier interpolation of the bands have been applied. For the calculation of the optical spectra, we have employed both the independent particle approach as well as the formalism that incorporates many-body and excitonic effects, by applying G0W0 approximation and solving the Bethe-Salpeter equation (BSE). The results show, for instance, that the spectrum of the imaginary part of the dielectric function of β-Be3N2 has an anisotropic behavior. Even more, the material shows its highest absorption peak in the UV range, which makes it interesting due to its possible high-temperature photodetection applications. Also, in the visible energy range of the spectrum, the index of refraction along the z-axis (see figure 1) is predicted to be around 1.6, which is a high value for a solid.

Global Aromatic Ring Currents in Neutral Porphyrin Nanobelts.

The ability of a ring-shaped molecule to sustain a global aromatic or antiaromatic ring current when placed in a magnetic field indicates that its electronic wave function is coherently delocalized around its whole circumference. Large molecules that display this behavior are attractive components for molecular electronic devices, but this phenomenon is rare in neutral molecules with circuits of more than 40 π-electrons. Here, we use theoretical methods to investigate how the global ring currents evolve with increasing ring size in cyclic molecular nanobelts built from edge-fused porphyrins. Our results indicate that a global ring current persists in neutral nanobelts with Hückel circuits of 220 π-electrons (22 porphyrin units, circumference 18.6 nm). Our predictions are validated by using coupled clusters to construct a density functional approximation (denoted as OX-B3LYP) that accurately describes these nanobelts and by checking compliance with Koopmans' theorem.

A General and Transferable Local Hybrid Functional for Electronic Structure Theory and Many-Fermion Approaches.

Density functional theory has become the workhorse of quantum physics, chemistry, and materials science. Within these fields, a broad range of applications needs to be covered. These applications range from solids to molecular systems, from organic to inorganic chemistry, or even from electrons to other Fermions, such as protons or muons. This is emphasized by the plethora of density functional approximations that have been developed for various cases. In this work, two new local hybrid exchange-correlation density functionals are constructed from first-principles, promoting generality and transferability. We show that constraint satisfaction can be achieved even for admixtures with full exact exchange, without sacrificing accuracy. The performance of the new functionals CHYF-PBE and CHYF-B95 is assessed for thermochemical properties, excitation energies, Mössbauer isomer shifts, NMR spin-spin coupling constants, NMR shieldings and shifts, magnetizabilities, and EPR hyperfine coupling constants. Here, the new density functional shows excellent performance throughout all tests and is numerically robust only requiring small grids for converged results. Additionally, both functionals can easily be generalized to arbitrary Fermions as shown for electron-proton correlation energies. Therefore, we outline that density functionals generated in this way are general purpose tools for quantum mechanical studies.

Perturbative ensemble density functional theory applied to charge transfer excitations.

Charge transfer excitation energies are known to be challenging for standard time-dependent (TD) density functional theory (DFT) calculations. Perturbative ensemble DFT (pEDFT) was suggested as an easy-to-implelemt, low-cost alternative to TDDFT, because it is an in principle exact theory for calculating excitation energies that produces useful valence excitation energies. Here, we examine analytically and numerically (based on the benzene-tetracyanoethylene complex) how well pEDFT performs in the charge transfer limit. We find that pEDFT is qualitatively correct in that it follows the Mulliken limit while being only weakly dependent on the underlying density functional approximation. We observe, however, that quantitatively pEDFT is not as accurate as TDDFT. We attribute this to the emergence of a new type of self-interaction-like term that adversely affects the computation.

On the increase of the melting temperature of water confined in one-dimensional nano-cavities.

Water confined in nanoscale cavities plays a crucial role in everyday phenomena in geology and biology, as well as technological applications at the water-energy nexus. However, even understanding the basic properties of nano-confined water is extremely challenging for theory, simulations, and experiments. In particular, determining the melting temperature of quasi-one-dimensional ice polymorphs confined in carbon nanotubes has proven to be an exceptionally difficult task, with previous experimental and classical simulation approaches reporting values ranging from ∼180K up to ∼450K at ambient pressure. In this work, we use a machine learning potential that delivers first principles accuracy (trained to the density functional theory approximation revPBE0-D3) to study the phase diagram of water for confinement diameters 9.5 < d < 12.5 Å. We find that several distinct ice polymorphs melt in a surprisingly narrow range between ∼280 and ∼310K, with a melting mechanism that depends on the nanotube diameter. These results shed new light on the melting of ice in one-dimension and have implications for the operating conditions of carbon-based filtration and desalination devices.

Cavity-enhanced superconductivity in MgB2 from first-principles quantum electrodynamics (QEDFT)

Strong laser pulses can control superconductivity, inducing nonequilibrium transient pairing by leveraging strong-light matter interaction. Here, we demonstrate theoretically that equilibrium ground-state phonon-mediated superconductive pairing can be affected through the vacuum fluctuating electromagnetic field in a cavity. Using the recently developed ab initio quantum electrodynamical density-functional theory approximation, we specifically investigate the phonon-mediated superconductive behavior of MgB[Formula: see text] under different cavity setups and find that in the strong light-matter coupling regime its superconducting transition temperature T[Formula: see text] can be enhanced at most by [Formula: see text]10% in an in-plane (or out-of-plane) polarized and realistic cavity via photon vacuum fluctuations. The results highlight that strong light-matter coupling in extended systems can profoundly alter material properties in a nonperturbative way by modifying their electronic structure and phononic dispersion at the same time. Our findings indicate a pathway to the experimental realization of light-controlled superconductivity in solid-state materials at equilibrium via cavity materials engineering.

Comparison of intermediate-range order in GeO2 glass: Molecular dynamics using machine-learning interatomic potential vs reverse Monte Carlo fitting to experimental data.

The short-range order and intermediate-range order in GeO2 glass are investigated by molecular dynamics using machine-learning interatomic potential trained on abinitio calculation data and compared with the reverse Monte Carlo fitting of neutron diffraction data. To characterize the structural differences in each model, the total/partial structure factors, coordination number, ring size and shape distributions, and persistent homology analysis were performed. These results show that although the two approaches yield similar two-body correlations, they can lead to three-dimensional models with different short- and intermediate-range ordering. A clear difference was observed especially in the ring distributions; RMC models exhibit a broad distribution in the ring size distribution, while neural network potential molecular dynamics yield much narrower ring distributions. This confirms that the density functional approximation in the abinitio calculations determines the preferred network assembly more strictly than RMC with simple coordination constraints even when using multiple diffraction data.

Slope of the Delocalization Function Is Proportional to Analytical Hardness.

Conceptual Density Functional Theory (CDFT) has been extended beyond its traditional role in elucidating chemical reactivity to the development of density functional theory methods, e.g., the investigation of the delocalization error. This delocalization error causes the dependence of the energy on the number of electrons (N) to deviate from its exact piecewise linear behavior, an error which is the basis of many well-known limitations of commonly used density-functional approximations (DFAs). Following our previous work on the analytical hardness η± for pure functionals, we extend its application to hybrid and range-separated functionals. A comparison is made between the analytical hardness and the slope of the delocalization function introduced by Hait and Head-Gordon. Our results show that there is a linear relationship between its slope and the analytical hardness. An approximate scheme is presented to construct the energy vs N curve without fractional occupation number calculations. The extension to densities is discussed.

Effect of Fluorine Substitution on Magnetizabilities: Insights from Density Functional Theory and Benchmark Coupled-Cluster Calculations.

The quantitative calculation of the magnetizability tensor of fluorine-containing molecules has been a longstanding challenge for quantum chemistry. We report a benchmark study on the effect of fluorine substitution on the magnetizability (both isotropic and anisotropic) of 24 small closed-shell molecules using coupled-cluster (CCSD(T)) theory, large basis sets, and London atomic orbitals. By extrapolation to a complete basis set (CBS) result, we establish the CCSD(T)/CBS limit and take the opportunity to assess the performance of various density functional theory approximations. Correcting for zero-point vibration further allows us to directly compare theory with (gas-phase) experimental data. We revisit Flygare's hypothesis on molecular magnetizabilities, which relates the successive replacement of hydrogen by fluorine atoms to changes in the magnetizability anisotropy. Some exceptions to this hypothesis are documented, and we rationalize these observations on the basis of hybridization on carbon.

Structural and energetic properties of cluster models of anatase-supported single late transition metal atoms: a density functional theory benchmark study.

Single-atom catalytic systems constitute an intriguing research topic due to their inherently different chemical behavior as compared to classic heterogeneous catalysts. In this study, cluster systems representing single late transition metal atoms adsorbed on anatase were constructed starting from previously generated periodic models and subjected to a density functional theory (DFT) benchmark study. The ability of different density functional approximations representing all rungs of the Jacob's Ladder classification to accurately describe bond lengths and adsorption energies was assessed for these clusters with the aim of revealing the functional that allows to retain the structural characteristics of the initial periodic system, while also delivering reliable energetics. In this regard, our results indicate that optimisation of the clusters with the meta-GGA functionals TPSS or RevTPSS provides the lowest mean unsigned error and root-mean-square deviations with respect to the periodic models. Moreover, these functionals and, to a slightly lesser degree, PW91 were also found to provide adsorption energies that are statistically the least deviating from the CCSD(T) reference data. More complex hybrid functionals appear to be performing less well. Cluster geometries were determined at the Kohn-Sham DFT level using the LANL2DZ basis set for the transition metals and the Pople 6-31G(d) basis set for O and H. The density functional approximations considered were SVWN, PBE, BP86, BLYP, PW91, TPSS, RevTPSS, M06L, M11L, B3LYP, PBE0, M06, M06-2X, MN15, ωB97X-D, CAM-B3LYP, M11, and MN12-SX. Reference adsorption energies of the metals on the support cluster were obtained at the CCSD(T)/LANL2TZ (transition metals)/6-311 + + G(d,p)//RevTPSS/LANLD2DZ (transition metals)/6-31G*. Besides the above-mentioned functionals, energy calculations using the double-hybrid functionals, DSDPBEP86, PBE0-DH, and B2PLYP, were also performed. All adsorption energy calculations were carried out on the RevTPSS geometries.

Ground and Excited Electronic Structure Analysis of FeH with Correlated Wave Function Theory and Density Functional Approximations.

FeH is one of the most challenging diatomic molecules to study under electronic structure theory. Here, we have successfully studied 22 electronic states of FeH using ab initio multireference configuration interaction (MRCI), Davidson-corrected MRCI (MRCI+Q), and coupled cluster singles, doubles, and perturbative triples [CCSD(T)] levels of theory. We report their potential energy curves (PECs), excitation energies, dissociation energies, equilibrium electronic configurations, and a series of spectroscopic constants with the use of augmented triple-ζ, quadruple-ζ, and quintuple-ζ quality correlation consistent basis sets. The scalar relativistic effects and active space and core electron correlation contribution on the properties of FeH are also explored. The use of a large CASSCF active space that includes 4s, 4p, 3d, and 4d orbitals of Fe and the 1s of H is critical for producing accurate full PECs with proper dissociations and predicting the exact order of the electronic states. Our findings are in harmony with the experimental results available in the literature and will serve as reference values for future studies of FeH. Furthermore, with the use of PECs, the total internal partition function sum (TIPS) of FeH was calculated across a range of temperatures. Finally, we exploited the single-reference nature of the a6Δ of FeH and its ionized product FeH+ (X5Δ) to evaluate the associated density functional theory (DFT) errors on their dissociation energies and spectroscopic parameters.

Bridging electronic and classical density-functional theory using universal machine-learned functional approximations.

The accuracy of density-functional theory (DFT) calculations is ultimately determined by the quality of the underlying approximate functionals, namely the exchange-correlation functional in electronic DFT and the excess functional in the classical DFT formalism of fluids. For both electrons and fluids, the exact functional is highly nonlocal, yet most calculations employ approximate functionals that are semi-local or nonlocal in a limited weighted-density form. Machine-learned (ML) nonlocal density-functional approximations show promise in advancing applications of both electronic and classical DFTs, but so far these two distinct research areas have implemented disparate approaches with limited generality. Here, we formulate a universal ML framework and training protocol to learn nonlocal functionals that combine features of equivariant convolutional neural networks and the weighted-density approximation. We prototype this new approach for several 1D and quasi-1D problems and demonstrate that functionals with exactly the same hyperparameters achieve excellent accuracy for a diverse set of systems, including the hard-rod fluid, the inhomogeneous Ising model, the exact exchange energy of electrons, the electron kinetic energy for orbital-free DFT, as well as for liquid water with 1D inhomogeneities. These results lay the foundation for a universal ML approach to approximate exact 3D functionals spanning electronic and classical DFTs.

Phase transition in the shape memory alloy NiTi described by the SCAN meta-GGA functional.

Climbing the ladder of density functional approximations has long been proposed to systematically improve the accuracy of first-principles calculations employing the density functional theory (DFT); however, up until now, the Perdew-Burke-Ernzerhof (PBE) functional at the second rung of the ladder, has dominated. Here, we present a study of the martensitic phase transition in NiTi based on abinitio molecular dynamics simulations and thermodynamic integration using the third-rung approximation of the strongly constrained and appropriately normalized (SCAN) meta-generalized gradient approximation (GGA). Although the predicted equilibrium lattice constants and formation enthalpy agree well with experimental data, the martensitic transition temperature (MTT) is overestimated by 94% (or 324K too high), compared with only 22% (77K) overestimation by PBE. The latent heat (q) is severely overestimated by SCAN as well. This deteriorated performance originates from the enlarged energy difference (ΔE) between the austenite and martensite phases, compared with the PBE result. Furthermore, a large variation (over 50meV/atom) in ΔE using different meta-GGAs indicates large variations in computed MTTs (∼400-500 K) and q, i.e., the predicted thermodynamic properties depend sensitively on the choice of meta-GGA. This would pose a serious problem when upgrading DFT calculations to the third rung. One possible solution is to add NiTi as a normsystem so that the revised SCAN meta-GGA could reproduce the PBE results of the relevant energy difference.

A Critical Evaluation of the Hybrid KS DFT Functionals Based on the KS Exchange-Correlation Potentials.

We have developed a critical methodology for the evaluation of the quality of hybrid exchange-correlation (XC) density functional approximations (DFAs) based on very fundamental quantities, i.e., Kohn-Sham (KS) XC potentials, self-consistent electron densities, first ionization potentials (IPs), and total energies. Since the XC potentials, the primary objects in the current study, are not directly accessible for the hybrids, we calculate them by inverting the KS electron densities. Utilizing this methodology, we tested 155 hybrid DFAs available in the LIBXC library using FCI and CCSD(T) methods as a reference. We have found that a group of functionals produces very decent XC potentials, mainly those with a large mixture of Hartree-Fock exchange. Moreover, the value of IP strongly depends on the XC potential quality. On the other hand, we show that the XC energy is dominated by functional-driven error, which in some cases leads to substantial errors in electronic densities. The study shows new directions for constructing more accurate XC functionals within the KS-DFT framework.

Electron Localization Function for Noncollinear Spins.

Understanding of bonding is key to modeling materials and predicting properties thereof. A widely adopted indicator of bonds and atomic shells is the electron localization function (ELF). The building blocks of the ELF are also used in the construction of modern density functional approximations. Here, we demonstrate that the ELF breaks down when applied beyond regular nonrelativistic quantum states. We show that for tackling general noncollinear open-shell solutions, it is essential to address both the U(1) gauge invariance, i.e., invariance under a multiplication by a position dependent phase factor, and SU(2) gauge invariance, i.e., invariance under local spin rotations, conjointly. Remarkably, we find that the extended ELF also improves the description of paradigmatic collinear states.

Ab Initio Calculation of Coupling-Constant Averaged Exchange-Correlation Holes for Spherically Symmetric Atoms.

Accurate approximation of the exchange-correlation (XC) energy in density functional theory (DFT) calculations is essential for reliably modeling electronic systems. Many such approximations are developed from models of the XC hole; accurate reference XC holes for real electronic systems are crucial for evaluating the accuracy of these models however the availability of reliable reference data is limited to a few systems. In this study, we employ the Lieb optimization with a coupled cluster singles and doubles (CCSD) reference to construct accurate coupling-constant averaged XC holes, resolved into individual exchange and correlation components, for five spherically symmetric atoms: He, Li, Be, N, and Ne. Alongside providing a new set of reference data for the construction and evaluation of model XC holes, we compare our data against the exchange and correlation hole models of the established local density approximation (LDA) and Perdew-Burke-Ernzerhof (PBE) density functional approximations. Our analysis confirms the established rationalization for the limitations of LDA and the improvement observed with PBE in terms of the hole depth and its long-range decay, which is demonstrated in real-space for this series of spherically symmetric atoms.

Key Role of Density Functional Approximation in Predicting M–N–C Catalyst Activities for Oxygen Reduction

Key Role of Density Functional Approximation in Predicting M–N–C Catalyst Activities for Oxygen Reduction