Common Density Functional Approximations Research Articles

How good are recent density functionals for ground and excited states of one-electron systems?

Sun et al. [J. Chem. Phys. 144, 191101 (2016)] suggested that common density-functional approximations (DFAs) should exhibit large energy errors for excited states as a necessary consequence of orbital nodality. Motivated by self-interaction corrected density-functional calculations on many-electron systems, we continue their study with the exactly solvable 1s, 2p, and 3d states of 36 hydrogenic one-electron ions (H-Kr35+) and demonstrate with self-consistent calculations that state-of-the-art DFAs indeed exhibit large errors for the 2p and 3d excited states. We consider 56 functionals at the local density approximation (LDA), generalized gradient approximation (GGA) as well as meta-GGA levels, and several hybrid functionals such as the recently proposed machine-learned DM21 local hybrid functional. The best non-hybrid functional for the 1s ground state is revTPSS. As predicted by Sun et al., the 2p and 3d excited states are more difficult for DFAs, and LDA functionals turn out to yield the most systematic accuracy for these states among non-hybrid functionals. The best performance for the three states overall is observed with the BHandH global hybrid GGA functional, which contains 50% Hartree-Fock exchange and 50% LDA exchange. The performance of DM21 is found to be inconsistent, yielding good accuracy for some states and systems and poor accuracy for others. Based on these results, we recommend including a variety of one-electron cations in future training of machine-learned density functionals.

Describing Chemical Reactivity with Frontier Molecular Orbitalets.

Locality in physical space is critical in understanding chemical reactivity in the analysis of various phenomena and processes in chemistry, biology, and materials science, as exemplified in the concepts of reactive functional groups and active sites. Frontier molecular orbitals (FMOs) pinpoint the locality of chemical bonds that are chemically reactive because of the associated orbital energies and thus have achieved great success in describing chemical reactivity, mainly for small systems. For large systems, however, the delocalization nature of canonical molecular orbitals makes it difficult for FMOs to highlight the locality of the chemical reactivity. To obtain localized molecular orbitals that also reflect the frontier nature of the chemical processes, we develop the concept of frontier molecular orbitalets (FMOLs) for describing the reactivity of large systems. The concept of orbitalets was developed recently in the localized orbital scaling correction method, which aims for eliminating the delocalization error in common density functional approximations. Orbitalets are localized in both physical and energy spaces and thus contain both orbital locality and energy information. The FMOLs are thus the orbitalets with energies highest among occupied orbitalets and lowest among unoccupied ones. The applications of FMOLs to hexadeca-1,3,5,7,9,11,13,15-octaene in its equilibrium geometry, inter- and intra-molecular charge-transfer systems, and two transition states of a bifurcating reaction demonstrate that FMOLs can connect quantum mechanical treatments of chemical systems and chemical reactivities by locating the reactive region of large chemical systems. Therefore, FMOLs extend the role of FMOs for small systems and describe the chemical reactivity of large systems with energy and locality insight, with potentially broad applications.

Benchmark Study on the Calculation of 119Sn NMR Chemical Shifts.

A new benchmark set termed SnS51 for assessing quantum chemical methods for the computation of 119Sn NMR chemical shifts is presented. It covers 51 unique 119Sn NMR chemical shifts for a selection of 50 tin compounds with diverse bonding motifs and ligands. The experimental reference data are in the spectral range of ±2500 ppm measured in seven different solvents. Fifteen common density functional approximations, two scalar- and one spin-orbit relativistic approach are assessed based on conformer ensembles generated using the CREST/CENSO scheme and state-of-the-art semiempirical (GFN2-xTB), force field (GFN-FF), and composite DFT methods (r2SCAN-3c). Based on the results of this study, the spin-orbit relativistic method combinations of SO-ZORA with PBE0 or revPBE functionals are generally recommended. Both yield mean absolute deviations from experimental data below 100 ppm and excellent linear regression determination coefficients of ≤0.99. If spin-orbit calculations are not affordable, the use of SR-ZORA with B3LYP or X2C with ωB97X or M06 may be considered to obtain qualitative predictions if no severe spin-orbit effects, for example, due to heavy nuclei containing ligands, are expected. An empirical linear scaling correction is demonstrated to be applicable for further improvement, and respective empirical parameters are given. Conformational effects on chemical shifts are studied in detail but are mostly found to be small. However, in specific cases when the ligand sphere differs substantially between conformers, chemical shifts can change by up to several hundred ppm.

Gold(i) sulfide: unusual bonding and an unexpected computational challenge in a simple solid.

We report the experimental high-pressure crystal structure and equation of state of gold(i) sulfide (Au2S) determined using diamond-anvil cell synchrotron X-ray diffraction. Our data shows that Au2S has a simple cubic structure with six atoms in the unit cell (four Au in linear, and two S in tetrahedral, coordination), no internal degrees of freedom, and relatively low bulk modulus. Despite its structural simplicity, Au2S displays very unusual chemical bonding. The very similar and relatively high electronegativities of Au and S rule out any significant metallic or ionic character. Using a simple valence bond (Lewis) model, we argue that the Au2S crystal possesses two different types of covalent bonds: dative and shared. These bonds are distributed in such a way that each Au atom engages in one bond of each kind. The multiple arrangements in space of dative and shared bonds are degenerate, and the multiplicity of configurations imparts the system with multireference character, which is highly unusual for an extended solid. The other striking feature of this system is that common computational (DFT) methods fail quite spectacularly to describe it, with 20% and 400% errors in the equilibrium volume and bulk modulus, respectively. We explain this by the poor treatment of static correlation in common density-functional approximations. The fact that the solid is structurally very simple, yet presents unique chemical bonding and is unmodelable using current DFT methods, makes it an interesting case study and a computational challenge.

Many-electron expansion: A density functional hierarchy for strongly correlated systems

Density functional theory (DFT) is the de facto method for the electronic structure of weakly correlated systems. But for strongly correlated materials, common density functional approximations break down. Here, we derive a many-electron expansion (MEE) in DFT that accounts for successive one-, two-, three-, ... particle interactions within the system. To compute the correction terms, the density is first decomposed into a sum of localized, nodeless one-electron densities (${\ensuremath{\rho}}_{i}$). These one-electron densities are used to construct relevant two- (${\ensuremath{\rho}}_{i}+{\ensuremath{\rho}}_{j}$), three- (${\ensuremath{\rho}}_{i}+{\ensuremath{\rho}}_{j}+{\ensuremath{\rho}}_{k}$), ... electron densities. Numerically exact results for these few-particle densities can then be used to correct an approximate density functional via any of several many-body expansions. We show that the resulting hierarchy gives accurate results for several important model systems: the Hubbard and Peierls-Hubbard models in 1D and the pure Hubbard model in 2D. We further show that the method is numerically convergent for strongly correlated systems: applying successively higher order corrections leads to systematic improvement of the results. MEE thus provides a hierarchy of density functional approximations that applies to both weakly and strongly correlated systems.

Halogen Bonding from Dispersion-Corrected Density-Functional Theory: The Role of Delocalization Error.

Halogen bonds are formed when a Lewis base interacts with a halogen atom in a different molecule, which acts as an electron acceptor. Due to its charge transfer component, halogen bonding is difficult to model using many common density-functional approximations because they spuriously overstabilize halogen-bonded dimers. It has been suggested that dispersion-corrected density functionals are inadequate to describe halogen bonding. In this work, we show that the exchange-hole dipole moment (XDM) dispersion correction coupled with functionals that minimize delocalization error (for instance, BH&HLYP, but also other half-and-half functionals) accurately model halogen-bonded interactions, with average errors similar to other noncovalent dimers with less charge-transfer effects. The performance of XDM is evaluated for three previously proposed benchmarks (XB18 and XB51 by Kozuch and Martin, and the set proposed by Bauzá et al.) spanning a range of binding energies up to ∼50 kcal/mol. The good performance of BH&HLYP-XDM is comparable to M06-2X, and extends to the "extreme" cases in the Bauzá set. This set contains anionic electron donors where charge transfer occurs even at infinite separation, as well as other charge transfer dimers belonging to the pnictogen and chalcogen bonding classes. We also show that functional delocalization error results in an overly delocalized electron density and exact-exchange hole. We propose intermolecular Bader delocalization indices as an indicator of both the donor-acceptor character of an intermolecular interaction and the delocalization error coming from the underlying functional.

Excitation energies and Stokes shifts from a restricted open-shell Kohn-Sham approach

Restricted open-shell Kohn-Sham (ROKS) theory provides a powerful computational tool for calculating singlet excited state energies and dynamics. However, the possibility of multiple solutions to the ROKS equations - with the associated difficulty of automatically selecting the physically meaningful solution - limits its usefulness for intensive applications such as long-time Born-Oppenheimer molecular dynamics. We present an implementation of ROKS for excited states which prescribes the physically correct solution from an overlap criterion and guarantees that this solution is stationary, allowing for straightforward evaluation of nuclear gradients. The method is used to benchmark ROKS for vertical excitation energies of small and large organic dyes and for the calculation of Stokes shifts. With common density functional approximations, ROKS vertical excitation energies, and Stokes shifts show similar accuracy to those from time-dependent density functional theory and Δ-self-consistent-field approaches. Advantages of the ROKS approach for excited state structure and molecular dynamics are discussed.

Structural manifestation of the delocalization error of density functional approximations: C4N+2 rings and C20 bowl, cage, and ring isomers

The ground state structure of C(4N+2) rings is believed to exhibit a geometric transition from angle alternation (N < or = 2) to bond alternation (N > 2). All previous density functional theory (DFT) studies on these molecules have failed to reproduce this behavior by predicting either that the transition occurs at too large a ring size, or that the transition leads to a higher symmetry cumulene. Employing the recently proposed perspective of delocalization error within DFT we rationalize this failure of common density functional approximations (DFAs) and present calculations with the rCAM-B3LYP exchange-correlation functional that show an angle-to-bond-alternation transition between C(10) and C(14). The behavior exemplified here manifests itself more generally as the well known tendency of DFAs to bias toward delocalized electron distributions as favored by Huckel aromaticity, of which the C(4N+2) rings provide a quintessential example. Additional examples are the relative energies of the C(20) bowl, cage, and ring isomers; we show that the results from functionals with minimal delocalization error are in good agreement with CCSD(T) results, in contrast to other commonly used DFAs. An unbiased DFT treatment of electron delocalization is a key for reliable prediction of relative stability and hence the structures of complex molecules where many structure stabilization mechanisms exist.

Assessing common density functional approximations for theab initiodescription of monovacancies in metals

Using the exact muffin-tin orbitals method, we investigate the accuracy of five common density functional approximations for the theoretical description of the formation energy of monovacancies in ...

Excited state surfaces in density functional theory: A new twist on an old problem

Excited state surfaces in density functional theory and the problem of charge transfer are considered from an orbital overlap perspective. For common density functional approximations, the accuracy of the surface will not be uniform if the spatial overlap between the occupied and virtual orbitals involved in the excitation has a strong conformational dependence; the excited state surface will collapse toward the ground state in regions where the overlap is very low. This characteristic is used to predict and to provide insight into the breakdown of excited state surfaces in the classic push-pull 4-(dimethylamino)benzonitrile molecule, as a function of twist angle. The breakdown is eliminated using a Coulomb-attenuated functional. Analogous situations will arise in many molecules.

Self-consistent implementation of a nonlocal van der Waals density functional with a Gaussian basis set

Nearly all common density functional approximations fail to properly describe dispersion interactions responsible for binding in van der Waals complexes. Empirical corrections can fix some of the failures but cannot fully grasp the complex physics and may not be reliable for systems dissimilar to the fitting set. In contrast, the recently proposed nonlocal van der Waals density functional (vdW-DF) was derived from first principles, describes dispersion interactions in a seamless fashion, and yields the correct asymptotics. Implementation of this functional is somewhat cumbersome: Nonlocal dependence on the electron density requires numerical double integration over the space variables and functional derivatives are nontrivial. This paper shows how vdW-DF can be implemented self-consistently with Gaussian basis functions. The gradients of the energy with respect to nuclear displacements have also been derived and coded, enabling efficient geometry optimizations. We test the vdW-DF correlation functional in combination with several exchange approximations. We also study the sensitivity of the method to the basis set size and to the quality of the numerical quadrature grid. For weakly interacting systems, acceptable accuracy in semilocal exchange is achieved only with fine grids, whereas for nonlocal vdW-DF correlation even rather coarse grids are sufficient. The current version of vdW-DF is not well suited for pairing with Hartree-Fock exchange, leading to considerable overbinding.