Range Separation and Local Hybridization in Density Functional Theory

Electronic self-interaction poses a fundamental challenge in density-functional theory. It greatly limits, e.g., the physical interpretation of eigenvalues as electron removal energies. We here investigate whether local hybrid functionals that are designed to be free from one-electron self-interaction lead to occupied Kohn-Sham eigenvalues and orbitals that approximate photoemission observables well. We compare the local hybrid results to the ones from global hybrid functionals that only partially counteract the self-interaction, and to the results that are obtained with a Perdew-Zunger-type self-interaction correction. Furthermore, we check whether being nominally free from one-electron self-interaction translates into a reduced many-electron self-interaction error. Our findings show that this is not the case for the local hybrid functionals that we studied: In practice they are similar to global hybrids in many respects, despite being formally superior. This finding indicates that there is a conceptual difference between the Perdew-Zunger way and the local hybrid way of translating the one-electron condition to a many-electron system. We also point out and solve some difficulties that occur when using local hybrid functionals in combination with pseudopotentials.

We present local hybrid functionals that incorporate a position-dependent admixture of short-range (screened) nonlocal exact [Hartree-Fock-type (HF)] exchange. We test two limiting cases: screened local hybrids with no long-range HF exchange and long-range-corrected local hybrids with 100% long-range HF exchange. Long-range-corrected local hybrids provide the exact asymptotic exchange-correlation potential in finite systems, while screened local hybrids avoid the problems inherent to long-range HF exchange in metals and small-bandgap systems. We treat these functionals self-consistently using the nonlocal exchange potential constructed from Kohn-Sham orbital derivatives. Generalized Kohn-Sham calculations with screened and long-range-corrected local hybrids can provide accurate molecular thermochemistry and kinetics, comparable to existing local hybrids of full-range exchange. Generalized Kohn-Sham calculations with existing full-range local hybrids provide results consistent with previous non-self-consistent and "localized local hybrid" calculations. These new functionals appear to provide a promising extension of existing local and range-separated hybrids.

Density functional theory (DFT) uses a density functional approximation (DFA) to add electron correlation to mean-field electronic structure calculations. Standard strategies (generalized gradient approximations GGAs, meta-GGAs, hybrids, etc.) for building DFAs, no matter whether based on exact constraints or empirical parametrization, all face a zero-sum game between overdelocalization (fractional charge error, FC) and underestimation of covalent bonding (fractional spin error, FS). This work presents an alternative strategy. Practical "Rung 3.5" ingredients are used to implement insights from hyper-GGA DFAs that reduce both FS and FC errors. Prototypes of this strategy qualitatively improve FS and FC error over 40 years of standard DFAs while maintaining low cost and practical evaluation of properties. Numerical results ranging from transition metal thermochemistry to absorbance peaks and excited-state geometry optimizations highlight this strategy's promise and indicate areas requiring further development.

Nuclear shielding constants from localized local hybrid exchange-correlation potentials

Local hybrid functionals are a more flexible class of density functional approximations, allowing for a position-dependent admixture of exact exchange. This additional flexibility, however, comes with a more involved mathematical form and a more complicated design. A common denominator for previously constructed local hybrid functionals is the usage of thermochemical benchmark data to construct these functionals. Herein, we design a local hybrid functional without relying on benchmark data. Instead, we construct it in a more ab initio manner, following the principles of modern meta-generalized gradient approximations and considering theoretical constraints. To achieve this, we make use of the density matrix expansion and a local mixing function based on an approximate correlation length. The accuracy of the developed density functional approximation is assessed for thermochemistry, excitation energies, polarizabilities, magnetizabilities, nuclear magnetic resonance (NMR) spin-spin coupling constants, NMR shieldings, and shifts, as well as EPR g-tensors and hyperfine coupling constants. Here, the new exchange functional shows a robust performance and is especially well suited for atomization energies, barrier heights, excitation energies, NMR coupling constants, and EPR properties, whereas it loses some ground for the NMR shifts. Therefore, the designed functional is a major step forward for functionals that have been designed from first principles.

A recently reported coupled-perturbed Kohn-Sham implementation to compute nuclear shielding constants with gauge-including atomic orbitals and local hybrid functionals has been extended to cover higher derivatives of the density in the local mixing function (LMF) of the local hybrid as well as the calibration function (CF) needed to deal with the ambiguity of exchange-energy densities. This allowed the first evaluation of state-of-the-art local hybrids with "calibrated" exchange-energy densities for nuclear shieldings. Compared to previously evaluated simpler local hybrids without a CF, appreciable improvements are found for proton shieldings. Furthermore, the recent LH20t functional is still competitive with the outstanding performance of the uncalibrated LH12ct-SsirSVWN and LH12ct-SsifSVWN LHs for heavier nuclei, suggesting that LH20t is possibly the most robust choice of any rung-four functional for computing the nuclear shieldings of main-group nuclei so far. Interestingly, the presence of a CF in the functional significantly reduces the number of artifacts introduced by the widely used Maximoff-Scuseria framework to treat the local kinetic energy τ. The latter occurs in so-called t-LMFs used in many of the present local hybrids. In any case, the use of Dobson's current-density functional framework is also recommended with more advanced calibrated τ-dependent local hybrid functionals.

Local hybrid functionals are a relatively recent class of exchange-correlation functionals that use a real-space dependent admixture of exact exchange. Here, we present the first implementation of time-dependent density functional theory excited-state gradients for these functionals. Based on the ansatz of a fully variational auxiliary Lagrangian of the excitation energy, the working equations for the case of a local hybrid functional are deduced. For the implementation, we derive the third-order functional derivatives used in the hyper-kernel and kernel-gradients following a seminumerical integration scheme. The first assessment for a test set of small molecules reveals competitive performance for excited-state structural parameters with typical mean absolute errors (MAEs) of 1.2 pm (PBE0: 1.4 pm) for bond lengths as well as for vibrational frequencies with typical MAEs of 81 cm-1 (PBE0: 76 cm-1). Excellent performance was found for adiabatic triplet excitation energies with typical MAEs of 0.08 eV (PBE0: 0.32 eV). In a detailed case analysis of the first singlet and triplet excited states of formaldehyde the conceptional (dis-)advantages of the local hybrid scheme for excited-state gradients are exposed.

Range-separated hybrid functionals with a fitted or tuned global range-separation parameter are frequently used in density functional theory. We here explore the concept of local range separation, i.e., of turning the range-separation parameter into an explicit semilocal density functional. We impose three simple constraints on the local range-separation parameter that are frequently used in density functional construction: uniform density scaling, the homogeneous electron gas limit, and freedom from one-electron self-interaction. We further discuss different ways of how to model the spin dependence in combination with local range separation. We evaluate our local range-separation energy functionals exactly for closed-shell atoms using the previously suggested hypergeneralized gradient approximation for molecules and assess the quality of this approximation. We find a local range-separated hybrid functional that yields accurate binding energies for a set of small molecules.

Lanthanide sesquioxides are strongly correlated materials characterized by highly localized unpaired electrons in the $f$ band. Theoretical descriptions based on standard density functional theory (DFT) formulations are known to be unable to correctly describe their peculiar electronic and magnetic features. In this study, electronic and magnetic properties of the first four lanthanide sesquioxides in the series are characterized through a reliable description of spin localization as ensured by hybrid functionals of the DFT, which include a fraction of nonlocal Fock exchange. Because of the high localization of the $f$ electrons, multiple metastable electronic configurations are possible for their ground state depending on the specific partial occupation of the $f$ orbitals: the most stable configuration is here found and characterized for all systems. Magnetic ordering is explicitly investigated, and the higher stability of an antiferromagnetic configuration with respect to the ferromagnetic one is predicted. The critical role of the fraction of exchange on the description of their electronic properties (notably, on spin localization and on the electronic band gap) is addressed. In particular, a recently proposed theoretical approach based on a self-consistent definition---through the material dielectric response---of the optimal fraction of exchange in hybrid functionals is applied to these strongly correlated materials.

X-ray absorption spectra of carbon, silicon, germanium, and sulfur compounds have been investigated by means of damped four-component density functional response theory. It is demonstrated that a reliable description of relativistic effects is obtained at both K- and L-edges. Notably, an excellent agreement with experimental results is obtained for L2,3-spectra-with spin-orbit effects well accounted for-also in cases when the experimental intensity ratio deviates from the statistical one of 2 : 1. The theoretical results are consistent with calculations using standard response theory as well as recently reported real-time propagation methods in time-dependent density functional theory, and the virtues of different approaches are discussed. As compared to silane and silicon tetrachloride, an anomalous error in the absolute energy is reported for the L2,3-spectrum of silicon tetrafluoride, amounting to an additional spectral shift of ∼1 eV. This anomaly is also observed for other exchange-correlation functionals, but it is seen neither at other silicon edges nor at the carbon K-edge of fluorine derivatives of ethene. Considering the series of molecules SiH4-XFX with X = 1, 2, 3, 4, a gradual divergence from interpolated experimental ionization potentials is observed at the level of Kohn-Sham density functional theory (DFT), and to a smaller extent with the use of Hartree-Fock. This anomalous error is thus attributed partly to difficulties in correctly emulating the electronic structure effects imposed by the very electronegative fluorines, and partly due to inconsistencies in the spurious electron self-repulsion in DFT. Substitution with one, or possibly two, fluorine atoms is estimated to yield small enough errors to allow for reliable interpretations and predictions of L2,3-spectra of more complex and extended silicon-based systems.

A local hybrid divide-and-conquer method (LHDC) which combines the high accuracy of sophisticated wave function theory (WFT) methods and the low cost of density functional theory (DFT) has been proposed for computational studies of medium and large molecules. In the method, a large system is divided into small subsystems for which the coefficients of the exchange functional in a hybrid functional are first optimized according to the energy calculated by an accurate WFT method. The hybrid coefficients are then used to evaluate the energy of the whole system. The method not only can reproduce the total energies of the chosen WFT method in good accuracy but also provides electronic structure information for the entire system.

An old and yet unsolved problem in density-functional theory is the strong dependence of degenerate open-shell atomic energies on the occupancy of the atomic orbitals. This arises from the fact that degenerate atomic orbitals of different ml do not have equivalent densities. Approximate density functionals therefore give energies depending strongly on which orbitals are occupied. This problem is solved in the present work by incorporating current density into the calculations using a current-density dependent functional previously published by the author.

Density-functional theory (DFT) is a rigorous and (in principle) exact framework for the description of the ground state properties of atoms, molecules and solids based on their electron density. While computationally efficient density-functional approximations (DFAs) have become essential tools in computational chemistry, their (semi-)local treatment of electron correlation has a number of well-known pathologies, e.g. related to electron self-interaction. Here, we present a type of machine-learning (ML) based DFA (termed Kernel Density Functional Approximation, KDFA) that is pure, non-local and transferable, and can be efficiently trained with fully quantitative reference methods. The functionals retain the mean-field computational cost of common DFAs and are shown to be applicable to non-covalent, ionic and covalent interactions, as well as across different system sizes. We demonstrate their remarkable possibilities by computing the free energy surface for the protonated water dimer at hitherto unfeasible gold-standard coupled cluster quality on a single commodity workstation.

Local hybrid functionals (LHs) use a position‐dependent admixture of exact exchange in real space, which has previously been shown to offer various advantages over global hybrids and the potential to construct functionals with wide applicability. While main‐group thermochemistry and kinetics and other properties have already been studied extensively, no broad and systematic investigation for the energetics of transition‐metal complexes has been reported so far for LHs. To close this gap, we evaluate eight LHs for three previously reported test sets on barrier heights and reaction energies of real‐world transition metal complexes, in comparison with several other rung‐4 hybrid functionals. The recent LH20t‐D4 method emerges as the top‐performing functional for the overall benchmark, comparable to the simpler TPSS0‐D4 and slightly before the more complex range‐separated ωB97M‐V. But LH20t‐D4 and ωB97M‐V perform much better for main‐group energetics, making them overall more reliable tools to study organometallic reaction mechanisms. Given also its previously observed excellent performance for mixed‐valence systems, LH20t‐D4 is thus a widely applicable functional in both main‐group and transition‐metal chemistry, while also providing convincing accuracy for many spectroscopic properties. The earlier LH14t‐calPBE‐D4 local hybrid also performs well, while some first‐generation LHs fall behind.

A new correlation functional is derived within the Kohn–Sham (KS) Density Functional Theory (DFT) involving the electron kinetic energy density τ and the Laplacian of the electron density as key nonlocal variables. The derivation is based on a direct resolution of the adiabatic connection formula and using an analogy with the local thermodynamic approach in DFT, following the Lap3 theory developed previously. Compared to the latter, the new functional involves higher order τ-dependent energy terms in a form suggesting a possible resummation procedure that could be used for further development. It is combined with the nonlocal exchange functional of Becke, by modifying the latter in an empirical fashion to achieve better synchronization between the two energy components. The resulting exchange-correlation scheme (named “Bmτ1”) is validated on several test systems known as difficult for DFT, at least at the Local Spin Density and Generalized Gradient Approximation levels. The recent nonempirical hybrid scheme PBE1PBE (“PBE0”) is included in the comparative tests as a parameter-free benchmark for the hybrid HF-KS DFT approach. Improved results for relative energies, activation barriers and equilibrium geometries are obtained with the Bmτ1 functional, particularly concerning aromatic compounds, systems with weak hydrogen bonds, proton transfer processes and transition-metal carbonyls.