Exchange Enhancement Factor Research Articles

Customizing PBE exchange-correlation functionals: a comprehensive approach for band gap prediction in diverse semiconductors.

Accurate band gap prediction in semiconductors is crucial for materials science and semiconductor technology advancements. This paper extends the Perdew-Burke-Ernzerhof (PBE) functional for a wide range of semiconductors, tackling the exchange and correlation enhancement factor complexities within density functional theory (DFT). Our customized functionals offer a clearer and more realistic alternative to DFT+U methods, which demand large negative U values for elements like sulfur (S), selenium (Se), and phosphorus (P). Moreover, these functionals are more cost-effective than GW or Heyd-Scuseria-Ernzerhof (HSE) hybrid functional methods, therefore, significantly facilitating the way for unified workflows in analyzing electronic structure, dielectric constants, effective masses, and further transport and elastic properties, allowing for seamless calculations across various properties. We point out that such development could be helpful in the creation of comprehensive databases of band gap and dielectric properties of the materials without expensive calculations. Furthermore, for the semiconductors studied, we show that these customized functionals and the strongly constrained and appropriately normed semilocal density functional (SCAN) perform similarly in terms of the band gap.

Towards a universal exchange enhancement factor in density functional theory

A broadly used strategy to go beyond the well-known local-density approximation of density functional theory relies on the choice of a so-called exchange-correlation (xc) enhancement factor ${F}_{\mathrm{xc}}$, defined as the enhancement of a realistic xc energy density over its local exchange-only counterpart. To date, this density functional, ${F}_{\mathrm{xc}}$, has been constructed by following either semiempirical strategies or nonempirical schemes that impose the fulfillment of exact constraints. Here, we follow a totally different route, which is based on an attempt to construct a universal exchange enhancement factor ${F}_{x}$ from the exact exchange energy density of a given family of electron density profiles and which we implement on the basis of jellium-slab exact-exchange self-consistent calculations. We find that such an enhancement factor can, indeed, be built which obeys, within our sample of electron-density profiles, most exact constraints and thus represents a benchmark towards the construction of a universal exchange enhancement factor suitable for all electron densities. We provide, in particular, an analytical parametrization of our ab initio calculations at the level of the generalized gradient approximation.

Shell DFT-1/2 method towards engineering accuracy for semiconductors: GGA versus LDA

The Kohn-Sham gaps of density functional theory (DFT) obtained in terms of local density approximation (LDA) or generalized gradient approximation (GGA) cannot be directly linked to the fundamental gaps of semiconductors, but in engineering there is a strong demand to match them through certain rectification methods. Shell DFT-1/2 (shDFT-1/2), as a variant of DFT-1/2, is a potential candidate to yield much improved band gaps for covalent semiconductors, but its accuracy depends on the LDA/GGA ground state, including optimized lattice parameters, basic Kohn-Sham gap before self-energy correction and the amount of self-energy correction that is specific to the exchange-correlation (XC) functional. In this work, the LDA/GGA as well as shDFT-1/2 results of six technically important covalent semiconductors Si, Ge, GaN, GaP, GaAs and GaSb were tested, with an additional ionic insulator LiF for comparison. The impact of XC flavor (LDA, PBEsol, PBE and RPBE), either directly on the gap value, or indirectly through the optimized lattice constant, is examined comprehensively. Moreover, the impact of XC flavor on LDA/GGA and shDFT-1/2 gaps was investigated under the condition of fixed experimental lattice constants. In-depth analysis reveals the rule of reaching the best accuracy in calculating the electronic band structures of typical covalent semiconductors. Relevant parameters like lattice constant, self-consistency in shDFT-1/2 runs, as well as the exchange enhancement factor of GGA, are discussed in details.

Reduced-gradient analysis of van der Waals complexes

Different methods to describe dispersion interactions within density functional theory have been developed, which is essential to describe binding in van der Waals complexes. However, several key aspects of such complexes—including binding energies, lattice constants, and binding distances—also depend on the exchange description that is paired with the description of dispersion interactions. This is particularly true for the vdW-DF family of van der Waals density functionals, which has a clear division between truly non-local correlations and semi-local generalized-gradient exchange. Here, we present a systematic analysis of the reduced-gradient values that determine the semi-local exchange for different classes of van der Waals complexes. In particular, we analyze molecular dimers, layered structures, surface adsorption, and molecular crystals. We find that reduced-gradient values of less than ∼1 to ∼1.5—depending on the system—contribute attractively to the exchange binding, while reduced gradients above those values are repulsive. We find that the attractive contributions can be attributed to low-density regions between the constituents with disk-like iso-surfaces. We further identify a mechanism wherein the surface area of these disks decreases through merging with other iso-surfaces and switches the gradient-correction to exchange from attractive to repulsive. This analysis allows us to explain some of the differences in performance of vdW-DF variants and initiates a discussion of desirable features of the exchange enhancement factor. While our analysis is focused on vdW-DF, it also casts light on van der Waals binding in a broader context and can be used to understand why methods perform differently for different classes of van der Waals systems.

Quantitative and qualitative performance of density functional theory rationalized by reduced density gradient distributions

We evaluate the qualitative and quantitative accuracy of various flavors of density functionals with and without accounting for dispersion corrections. Our test system is nickel in the form of bulk, surfaces, and nanoparticles for which we compute structural properties, bulk cohesive energies, surface energies, and work functions and compare to experimental data. We find that the inclusion of any dispersion, either by an a posteriori correction or by a self-consistent treatment by explicitly computing the nonlocal correlation contribution to the total energy, has a significant effect on the calculated properties and improves the quantitative comparison to experiments. Besides the quantitative agreement, we also investigate qualitative features by comparing Wulff shapes of metal nanoparticles as obtained using the different density functionals. We find that all tested functionals predict similar Wulff shapes for nickel nanoparticles but still have some small differences. These results show that the relative energies calculated using the semilocal GGA and meta-GGA functionals, with and without dispersion, are quite similar. Our findings can also be generalized to other systems when rationalized in terms of the computed reduced density gradients. We find that the distribution of reduced density gradients in a material is correlated to the steepness of the exchange enhancement factor and propose that this information can be used as a quantitative guide when it comes to picking the most appropriate density functional for specific target systems as well as when it comes to extrapolating DFT data to predict experiments.

Application of XDM to ionic solids: The importance of dispersion for bulk moduli and crystal geometries.

Dispersion corrections are essential in the description of intermolecular interactions; however, dispersion-corrected functionals must also be transferrable to hard solids. The exchange-hole dipole moment (XDM) model has demonstrated excellent performance for non-covalent interactions. In this article, we examine its ability to describe the relative stability, geometry, and compressibility of simple ionic solids. For the specific cases of the cesium halides, XDM-corrected functionals correctly predict the energy ranking of the B1 and B2 forms, and a dispersion contribution is required to obtain this result. Furthermore, for the lattice constants of the 20 alkali halides, the performance of XDM-corrected functionals is excellent, provided that the base functional's exchange enhancement factor properly captures non-bonded repulsion. The mean absolute errors in lattice constants obtained with B86bPBE-XDM and B86bPBE-25X-XDM are 0.060 Å and 0.039 Å, respectively, suggesting that delocalization error also plays a minor role in these systems. Finally, we considered the calculation of bulk moduli for alkali halides and alkaline-earth oxides. Previous claims in the literature that simple generalized gradient approximations, such as PBE, can reliably predict experimental bulk moduli have benefited from large error cancellations between neglecting both dispersion and vibrational effects. If vibrational effects are taken into account, dispersion-corrected functionals are quite accurate (4 GPa-5 GPa average error), again, if non-bonded repulsion is correctly represented. Careful comparisons of the calculated bulk moduli with experimental data are needed to avoid systematic biases and misleading conclusions.

Analysis of Density-Functional Errors for Noncovalent Interactions between Charged Molecules.

The study of the structure and chemistry of biological systems with density-functional theory requires an accurate description of intermolecular interactions involving charged moieties. While dispersion-corrected functionals accurately model noncovalent interactions in neutral systems, a systematic study of the performance and errors associated with intermolecular interactions between charged fragments is missing. We undertake this study by examining the performance of a series of dispersion-corrected functionals with varying degrees of exact exchange for the side-chain protein interactions from the BioFragment Database (BFDb) of Burns et al. (the SSI set). In general, hybrid functionals with 20-30% exact exchange are accurate across the board, with the lowest mean absolute errors of 0.11 kcal/mol obtained from the 20% exact-exchange BLYP and PW86PBE hybrids coupled with the exchange-hole dipole moment (XDM) dispersion model. In addition, our analysis shows that functionals with higher exact-exchange fractions overestimate the electrostatic contributions to the binding energies, and that GGA functionals overestimate zwitterion binding energies due to delocalization error and overestimated charge transfer. In addition, the (quite large) repulsion in the dications is systematically overestimated by all functionals, and the trends for the monoanionic and dianionic dimers can be successfully explained by appealing to the ability of the underlying GGA to describe Pauli repulsion, as given by its exchange enhancement factor. Going beyond studies of biomolecules, this latter result has important implications for selecting appropriate GGA functionals for applications to ionic solids and layered materials containing anion-anion interactions.

Quantum Chemical Methods for Modeling Covalent Modification of Biological Thiols.

Targeted covalent inhibitor drugs require computational methods that go beyond simple molecular-mechanical force fields in order to model the chemical reactions that occur when they bind to their targets. Here, several semiempirical and density-functional theory (DFT) methods are assessed for their ability to describe the potential energy surface and reaction energies of the covalent modification of a thiol by an electrophile. Functionals such as PBE and B3LYP fail to predict a stable enolate intermediate. This is largely due to delocalization error, which spuriously stabilizes the prereaction complex, in which excess electron density is transferred from the thiolate to the electrophile. Functionals with a high-exact exchange component, range-separated DFT functionals, and variationally optimized exact exchange (i.e., the LC-B05minV functional) correct this issue to various degrees. The large gradient behavior of the exchange enhancement factor is also found to significantly affect the results, leading to the improved performance of PBE0. While ωB97X-D and M06-2X were reasonably accurate, no method provided quantitative accuracy for all three electrophiles, making this a very strenuous test of functional performance. Additionally, one drawback of M06-2X was that molecular dynamics (MD) simulations using this functional were only stable if a fine integration grid was used. The low-cost semiempirical methods, PM3, AM1, and PM7, provide a qualitatively correct description of the reaction mechanism, although the energetics is not quantitatively reliable. As a proof of concept, the potential of mean force for the addition of methylthiolate to methylvinyl ketone was calculated using quantum mechanical/molecular mechanical MD in an explicit polarizable aqueous solvent. © 2019 Wiley Periodicals, Inc.

Simple Modifications of the SCAN Meta-Generalized Gradient Approximation Functional.

We analyzed various possibilities to improve upon the SCAN meta-generalized gradient approximation density functional obeying all known properties of the exact functional that can be satisfied at this level of approximation. We examined the necessity of locally satisfying a strongly tightened lower bound for the exchange energy density in single-orbital regions, the nature of the error cancellation between the exchange and correlation parts in two-electron regions, and the effect of the fourth-order term in the gradient expansion of the correlation energy density. We have concluded that the functional can be modified to separately reproduce the exchange and correlation energies of the helium atom by locally releasing the strongly tightened lower bound for the exchange energy density in single-orbital regions, but this leads to an unbalanced improvement in the single-orbital electron densities. Therefore, we decided to keep the FX ≤ 1.174 exact condition for any single-orbital density, where FX is the exchange enhancement factor. However, we observed a general improvement in the single-orbital electron densities by revising the correlation functional form to follow the second-order gradient expansion in a wider range. Our new revSCAN functional provides more-accurate atomization energies for the systems with multireference character, compared to the SCAN functional. The nonlocal VV10 dispersion-corrected revSCAN functional yields more-accurate noncovalent interaction energies than the VV10-corrected SCAN functional. Furthermore, its global hybrid version with 25% of exact exchange, called revSCAN0, generally performs better than the similar SCAN0 for reaction barrier heights. Here, we also analyzed the possibility of the construction of a local hybrid from the SCAN exchange and a specific locally bounded nonconventional exact exchange energy density. We predict compatibility problems since this nonconventional exact exchange energy density does not really obey the strongly tightened lower bound for the exchange energy density in single-orbital regions.

Orbital nodal surfaces: Topological challenges for density functionals

Nodal surfaces of orbitals, in particular of the highest occupied one, play a special role in Kohn-Sham density-functional theory. The exact Kohn-Sham exchange potential, for example, shows a protruding ridge along such nodal surfaces, leading to the counterintuitive feature of a potential that goes to different asymptotic limits in different directions. We show here that nodal surfaces can heavily affect the potential of semilocal density-functional approximations. For the functional derivatives of the Armiento-K\"ummel (AK13) [Phys. Rev. Lett. 111, 036402 (2013)] and Becke88 [Phys. Rev. A 38, 3098 (1988)] energy functionals, i.e., the corresponding semilocal exchange potentials, as well as the Becke-Johnson [J. Chem. Phys. 124, 221101 (2006)] and van Leeuwen--Baerends (LB94) [Phys. Rev. A 49, 2421 (1994)] model potentials, we explicitly demonstrate exponential divergences in the vicinity of nodal surfaces. We further point out that many other semilocal potentials have similar features. Such divergences pose a challenge for the convergence of numerical solutions of the Kohn-Sham equations. We prove that for exchange functionals of the generalized gradient approximation (GGA) form, enforcing correct asymptotic behavior of the potential or energy density necessarily leads to irregular behavior on or near orbital nodal surfaces. We formulate constraints on the GGA exchange enhancement factor for avoiding such divergences.

Towards an accurate description of perovskite ferroelectrics: exchange and correlation effects

Using the van der Waals density functional with C09 exchange (vdW-DF-C09), which has been applied to describing a wide range of dispersion-bound systems, we explore the physical properties of prototypical ABO3 bulk ferroelectric oxides. Surprisingly, vdW-DF-C09 provides a superior description of experimental values for lattice constants, polarization and bulk moduli, exhibiting similar accuracy to the modified Perdew-Burke-Erzenhoff functional which was designed specifically for bulk solids (PBEsol). The relative performance of vdW-DF-C09 is strongly linked to the form of the exchange enhancement factor which, like PBEsol, tends to behave like the gradient expansion approximation for small reduced gradients. These results suggest the general-purpose nature of the class of vdW-DF functionals, with particular consequences for predicting material functionality across dense and sparse matter regimes.

Hartree potential dependent exchange functional.

We introduce a novel non-local ingredient for the construction of exchange density functionals: the reduced Hartree parameter, which is invariant under the uniform scaling of the density and represents the exact exchange enhancement factor for one- and two-electron systems. The reduced Hartree parameter is used together with the conventional meta-generalized gradient approximation (meta-GGA) semilocal ingredients (i.e., the electron density, its gradient, and the kinetic energy density) to construct a new generation exchange functional, termed u-meta-GGA. This u-meta-GGA functional is exact for the exchange of any one- and two-electron systems, is size-consistent and non-empirical, satisfies the uniform density scaling relation, and recovers the modified gradient expansion derived from the semiclassical atom theory. For atoms, ions, jellium spheres, and molecules, it shows a good accuracy, being often better than meta-GGA exchange functionals. Our construction validates the use of the reduced Hartree ingredient in exchange-correlation functional development, opening the way to an additional rung in the Jacob's ladder classification of non-empirical density functionals.

Development of an exchange–correlation functional with uncertainty quantification capabilities for density functional theory

This paper presents the development of a new exchange–correlation functional from the point of view of machine learning. Using atomization energies of solids and small molecules, we train a linear model for the exchange enhancement factor using a Bayesian approach which allows for the quantification of uncertainties in the predictions. A relevance vector machine is used to automatically select the most relevant terms of the model. We then test this model on atomization energies and also on bulk properties. The average model provides a mean absolute error of only 0.116 eV for the test points of the G2/97 set but a larger 0.314 eV for the test solids. In terms of bulk properties, the prediction for transition metals and monovalent semiconductors has a very low test error. However, as expected, predictions for types of materials not represented in the training set such as ionic solids show much larger errors.

Kohn-Sham kinetic energy density in the nuclear and asymptotic regions: Deviations from the von Weizsäcker behavior and applications to density functionals

We show that the Kohn-Sham positive-definite kinetic energy (KE) density significantly differs from the von Weizs\"acker (VW) one at the nuclear cusp as well as in the asymptotic region. At the nuclear cusp, the VW functional is shown to be linear and the contribution of p-type orbitals to the KE density is theoretically derived and numerically demonstrated in the limit of infinite nuclear charge, as well in the semiclassical limit of neutral large atoms. In the latter case, it reaches 12 of the KE density. In the asymptotic region we find new exact constraints for meta Generalized Gradient Approximation (meta-GGA) exchange functionals: with an exchange enhancement factor proportional to $\sqrt{\alpha}$, where $\alpha$ is the common meta-GGA ingredient, both the exchange energy density and the potential are proportional to the exact ones. In addition, this describes exactly the large-gradient limit of quasi-two dimensional systems.

Semilocal density functional obeying a strongly tightened bound for exchange

Because of its useful accuracy and efficiency, density functional theory (DFT) is one of the most widely used electronic structure theories in physics, materials science, and chemistry. Only the exchange-correlation energy is unknown, and needs to be approximated in practice. Exact constraints provide useful information about this functional. The local spin-density approximation (LSDA) was the first constraint-based density functional. The Lieb-Oxford lower bound on the exchange-correlation energy for any density is another constraint that plays an important role in the development of generalized gradient approximations (GGAs) and meta-GGAs. Recently, a strongly and optimally tightened lower bound on the exchange energy was proved for one- and two-electron densities, and conjectured for all densities. In this article, we present a realistic "meta-GGA made very simple" (MGGA-MVS) for exchange that respects this optimal bound, which no previous beyond-LSDA approximation satisfies. This constraint might have been expected to worsen predicted thermochemical properties, but in fact they are improved over those of the Perdew-Burke-Ernzerhof GGA, which has nearly the same correlation part. MVS exchange is however radically different from that of other GGAs and meta-GGAs. Its exchange enhancement factor has a very strong dependence upon the orbital kinetic energy density, which permits accurate energies even with the drastically tightened bound. When this nonempirical MVS meta-GGA is hybridized with 25% of exact exchange, the resulting global hybrid gives excellent predictions for atomization energies, reaction barriers, and weak interactions of molecules.

Towards an Optimal Gradient-dependent Energy Functional of the PZ-SIC Form

Results of Perdew–Zunger self-interaction corrected (PZ-SIC) density functional theory calculations of the atomization energy of 35 molecules are compared to those of high-level quantum chemistry calculations. While the PBE functional, which is commonly used in calculations of condensed matter, is known to predict on average too high atomization energy (overbinding of the molecules), the application of PZ-SIC gives a large overcorrection and leads to significant underestimation of the atomization energy. The exchange enhancement factor that is optimal for the generalized gradient approximation within the Kohn-Sham (KS) approach may not be optimal for the self-interaction corrected functional. The PBEsol functional, where the exchange enhancement factor was optimized for solids, gives poor results for molecules in KS but turns out to work better than PBE in PZ-SIC calculations. The exchange enhancement is weaker in PBEsol and the functional is closer to the local density approximation. Furthermore, the drop in the exchange enhancement factor for increasing reduced gradient in the PW91 functional gives more accurate results than the plateaued enhancement in the PBE functional. A step towards an optimal exchange enhancement factor for a gradient dependent functional of the PZ-SIC form is taken by constructing an exchange enhancement factor that mimics PBEsol formore » small values of the reduced gradient, and PW91 for large values. The average atomization energy is then in closer agreement with the high-level quantum chemistry calculations, but the variance is still large, the F2 molecule being a notable outlier.« less

Testing the Jacob's ladder of density functionals for electronic structure and magnetism of rutileVO2

We employ semilocal density functionals [local spin-density approximation (LSDA), Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA), and meta-GGAs)], LSDA plus Hubbard $U$ (LSDA+$U$) theory, a nonlocal range-separated Heyd-Scuseria-Ernzerhof hybrid functional (HSE06), and the random-phase approximation (RPA) to assess their performances for the ground-state magnetism and electronic structure of a strongly correlated metal, rutile ${\mathrm{VO}}_{2}$. Using recent quantum Monte Carlo results as the benchmark, all tested semilocal and hybrid functionals as well as the RPA (with PBE inputs) predict the correct magnetic ground states for rutile ${\mathrm{VO}}_{2}$. The observed paramagnetism could arise from temperature-disordered local spin moments or from the thermal destruction of these moments. All semilocal functionals also give the correct ground-state metallicity for rutile ${\mathrm{VO}}_{2}$. However, in the ferromagnetic (FM) and antiferromagnetic (AFM) phases, LSDA+$U$ and HSE06 incorrectly predict rutile ${\mathrm{VO}}_{2}$ to be a Mott-Hubbard insulator. For the computed electronic structures of FM and AFM phases, we find that the Tao-Perdew-Staroverov-Scuseria (TPSS) and revised TPSS (revTPSS) meta-GGAs give strong $2p\text{\ensuremath{-}}3d$ hybridizations, resulting in a depopulation of the $2p$ bands of O atoms, in comparison with other tested meta-GGAs. The regularized TPSS (regTPSS) and meta-GGAs made simple, i.e., MGGA_MS0 and MGGA_MS2, which are free of the spurious order-of-limits problem of TPSS and revTPSS, give electronic states close to those of the PBE GGA and LSDA. In comparison to experiment, semilocal functionals predict better equilibrium cell volumes for rutile ${\mathrm{VO}}_{2}$ in FM and AFM states than in the spin-unpolarized state. For meta-GGAs, a monotonic decrease of the exchange enhancement factor ${F}_{x}(s,\ensuremath{\alpha})$ with \ensuremath{\alpha} for small $s$, as in the MGGA_MS functionals, leads to large (probably too large) local magnetic moments in spin-polarized states.

Gedanken densities and exact constraints in density functional theory.

Approximations to the exact density functional for the exchange-correlation energy of a many-electron ground state can be constructed by satisfying constraints that are universal, i.e., valid for all electron densities. Gedanken densities are designed for the purpose of this construction, but need not be realistic. The uniform electron gas is an old gedanken density. Here, we propose a spherical two-electron gedanken density in which the dimensionless density gradient can be an arbitrary positive constant wherever the density is non-zero. The Lieb-Oxford lower bound on the exchange energy can be satisfied within a generalized gradient approximation (GGA) by bounding its enhancement factor or simplest GGA exchange-energy density. This enhancement-factor bound is well known to be sufficient, but our gedanken density shows that it is also necessary. The conventional exact exchange-energy density satisfies no such local bound, but energy densities are not unique, and the simplest GGA exchange-energy density is not an approximation to it. We further derive a strongly and optimally tightened bound on the exchange enhancement factor of a two-electron density, which is satisfied by the local density approximation but is violated by all published GGA's or meta-GGA's. Finally, some consequences of the non-uniform density-scaling behavior for the asymptotics of the exchange enhancement factor of a GGA or meta-GGA are given.

De Haas-van Alphen effect in the band antiferromagnet FeGe2: Development of spin splitting

The de Haas-van Alphen (dHvA) effect in the antiferromagnetic compound FeGe2 is studied experimentally. A sharp suppression of the amplitude of the first harmonic of the dHvA oscillations is observed for a number of orbits on the Fermi surface of FeGe2. The change in the conditions for suppression of the amplitude (spin zeroes) when FeGe2 is doped with cobalt is determined. A method for analyzing the experimental data is proposed in which the g-factors of the conduction electrons are computed on the basis of calculations of the spin-polarized electron structure of the band antiferromagnet taking spin-orbital interactions into account. The exchange enhancement factor for spin paramagnetism in FeGe2 is also calculated.

Testing density functionals for structural phase transitions of solids under pressure: Si, SiO2, and Zr

We have investigated the structural phase transitions of crystalline Si (insulator-metal), SiO2 (insulator-insulator), and Zr (metal-metal) under pressure, as a test of several density functionals for the exchange-correlation energy. While meta-generalized gradient approximations (meta-GGAs) such as revTPSS (revised Tao-Perdew-Staroverov-Scuseria) are more sophisticated than GGAs such as PBE (Perdew-Burke-Ernzerhof), and are more accurate without empiricism for atomization energies of molecules, lattice constants of solids, and surface energies, we confirm that these meta-GGAs tend to give smaller and less realistic transition pressures than the PBE GGA does. But we also show that the recent functionals of the meta-GGA made simple family (MGGA_MS) behave differently, predicting larger and often more realistic transition pressures. We suggest that further refinement of the meta-GGA can lead to a functional that is more accurate for properties of molecules and solids at equilibrium or under compression. We also show that, contrary to recent suggestions but in line with older ones, an accurate fundamental gap in the noninteracting band structure is not necessary for an accurate prediction of the transition pressure. Unlike the semilocal GGAs and meta-GGAs, and unlike the local density approximation also tested here, the screened hybrid functional HSE06 (Heyd-Scuseria-Ernzerhof) is fully nonlocal and predicts more realistic fundamental gaps. HSE06 is better than the semilocal functionals for the transition pressures of Si and SiO2, but seriously overestimates the transition pressure in Zr. Besides the transition pressures, we report the transition energies and volumes, binding energy curves, and structural parameters at zero and transition pressure. Finally, we discuss how the performance of a functional can reflect its plottable exchange enhancement factor, and why the structural phase transitions are especially challenging for approximate density functionals.