PBE0 Hybrid Research Articles

A combined first principles TDDFT and experimental study on the UV-Vis spectra properties of M(p-nitrophenyl azo resorcinol)$_{3}$ complexes (M: Fe, Cr)

UV-Vis absorption data of p-nitrophenyl azo resorcinol (Magneson I) and its 2 Fe(III) and Cr(III) complexes were investigated both experimentally and theoretically. The geometries were optimized at BP86/TZVP level. The most stable spin states were computed as doublet and quartet for Fe(magneson)_3 and Cr(magneson)_3 complexes, respectively. Time-dependent density functional theory (TDDFT) was employed to explore the absorption spectra properties, whereas the solvent effects were taken into account using the polarizable continuum model (PCM). The M06, B3LYP, and PBE0 hybrid functionals together with TZVP/LANL2TZ basis sets were used for comparing the results with experimental data. The theoretical analysis of electronic structure and molecular orbitals demonstrated that the low-lying absorption bands in the UV-Vis spectra are mainly \pi \to d ligand-to-metal charge transfer (LMCT) transition and \pi \to \pi ligand-to-ligand charge transfer (LLCT) transition for Fe(magneson)_3, and, in addition to that of LMCT and LLCT, d \to \pi metal-to-ligand charge transfer (MLCT) transition for Cr(magneson)_3 complexes. The good agreement between the experimental and TDDFT calculation, especially M06 and B3LYP absorption spectra of the metal Magneson I complexes, allowed us to provide a detailed estimation of the main spectral features of ferric and chromic complexes.

Stereochemical and conformational study on fenoterol by ECD spectroscopy and TD-DFT calculations

Fenoterol and its derivatives are selective β2-adrenergic receptor (β2-AR) agonists whose stereoselective biological activities have been extensively investigated in the past decade; a complete stereochemical characterization of fenoterol derivatives is therefore crucial for a better understanding of the effects of stereochemistry on β2-AR binding. In the present project, the relationship between chiroptical properties and absolute stereochemistry of the stereoisomers of fenoterol (1) was investigated by experimental ECD spectroscopy and time-dependent density functional theory (TD-DFT). DFT geometry optimizations were carried out at the RI-B97D/TZVP/IEFPCM(MeOH) level and subsequent TD-DFT calculations were performed using the PBE0 hybrid functional.Despite the large pool of equilibrium conformers found for the investigated compounds and the known limitations of the level of theory employed, the computational protocol was able to reproduce the experimental ECD spectra of the stereoisomers of 1. The main contribution to the overall chiroptical properties was found to arise from the absolute configuration of the chiral center in α-position to the resorcinol moiety. Based on this evidence, a thorough conformational analysis was performed on the optimized DFT conformers, which revealed the occurrence of a different equilibrium between conformational patterns for the diastereomers of fenoterol: the (R,R′)/(S,S′) enantiomeric pair showed a higher population of folded conformations than the (R,S′)/(S,R′) pair.

Energetics of Water Oxidation Catalyzed by Cobalt Oxide Nanoparticles: Assessing the Accuracy of DFT and DFT+U Approaches against Coupled Cluster Methods.

Some of the most promising catalysts for water oxidation rely on crystalline and amorphous cobalt oxide nanoparticles. Density functional theory (DFT) calculations are routinely used to study the electronic and atomic structures of these materials as well as the thermodynamics and mechanisms of the electrochemical oxygen evolution reaction. The accuracy of these theoretical predictions has never been compared to high-level quantum chemistry methods. We perform coupled cluster (CC) quantum chemistry calculations on model cobalt oxide surface sites and use them to benchmark the accuracy of the most popular exchange and correlation functionals. Hybrid B3LYP and PBE0 functionals lead to fair agreement with the CC energies, while standard gradient-corrected functionals show important discrepancies. The inclusion of on-site electronic repulsion (DFT+U) substantially improves the calculated electronic and structural properties, but no value of the U parameter reproduces the CC results. We discuss the implications of these findings for amorphous cobalt phosphate nanoparticles, showing that the reactivity of these catalysts is not altered by surface phosphate groups.

Application of recent double-hybrid density functionals to low-lying singlet-singlet excitation energies of large organic compounds

The present work assesses some recently developed double-hybrid density functionals (B2π-PLYP, PBE0-DH, and PBE0-2) using linear-response Tamm-Dancoff Time-Dependent Density Functional Theory. This assessment is achieved against experimentally derived low-lying excitation energies of large organic dyes of recent interest, including some excitations dominated by charge-transfer transitions. Comparisons are made with some of the best-performing methods established from the literature, such as PBE0 or B3LYP hybrid or the recently proposed B2-PLYP and B2GP-PLYP double-hybrid models, to ascertain their quality and robustness on equal footing. The accuracy of parameter-free or empirical forms of double-hybrid functionals is also briefly discussed. Generally speaking, it turns out that double-hybrid expressions always provide more accurate estimates than corresponding hybrid methods. Double-hybrid functionals actually reach averaged accuracies of 0.2 eV, that can be admittedly considered close to any intended accuracy limit within the present theoretical framework.

Structural, electronic, and magnetic properties of the europium chalcogenides: A hybrid-functional DFT study

We analyze the structural, electronic, and magnetic properties of the europium chalcogenide series EuX (X = O, S, Se, and Te) using density-functional theory (DFT). To describe the localized 4f states of Eu, we utilize the parameter-free PBE0 and HSE hybrid functionals and found a systematic qualitative and quantitative improvement over the conventional local and semilocal functionals. Both hybrid functionals predict the lattice constant and the bulk modulus of all four compounds accurately. The semiconducting behavior, the opening of the band gap, as well as the nature of the band gap across the series is reproduced. By mapping the magnetic interactions to a Heisenberg model up to the next-nearest neighbors, we find the correct trend of the magnetic order of the Eu compounds (in magnetic coupling of the Eu 4f moments), from ferromagnetic in EuO to antiferromagnetic in EuTe, with critical temperatures that are in fair agreement with experiment.

Interpretation of the optical absorption spectrum of Co3O4 with normal spinel structure from first principles calculations

First principles calculations based on density functional theory have been employed to study the electronic, magnetic and optical properties of Co3O4 in a cubic normal spinel structure. Exchange and correlation effects between electrons were treated by a B3PW91 hybrid functional, which produced better results than others scheme, such as GGA+U or PBE0 hybrid functionals or mBJ semilocal potential. The work focuses on clarifying the nature of the optical absorption bands, which have motivated various theoretical and experimental works in the literature. The calculated optical absorption spectrum was compared with available experimental data. On the basis of this calculated electronic and magnetic structure, the optical absorption peaks (theoretical and experimental) could be satisfactorily explained in terms of d3d charge transfer transitions between both CO2+→CO2+ and CO3+→CO3+ ions. The calculations also predicted that the crystal field splittings at both octahedral and tetrahedral sites in the Co3O4 compound are of the same magnitude.

Bandgap bowing in Ta-W-O system for efficient solar energy conversion: Insights from density functional theory and X-ray diffraction

The electronic properties of Ta-W-O system have been studied using density functional theory with PBE0 hybrid functional. The bandgap calculations showed a large and composition-dependent bowing parameter. While the large bowing parameter at low W content (<25%) is related to the charge exchange between the localized W 5d state and the extended states in the conduction band, the volume change is considered the main reason for the large bowing parameter at high W content (>25%). The electron excitation from the Ta2O5 valence band to WO3 conduction band at high W content may contribute to the pronounced decrease in the conduction band energy. The staggered bandgap type between Ta2O5 and WO3, as revealed from the energy band diagram, resulted in efficient charge carriers' separation. The minimum hole effective mass occurs along the y-direction and decreases monotonically with increasing W content. Combining the wide tunability of bandgap with the small effective mass, it is suggested that Ta-W-O alloys can replace the various III-V materials for high efficiency solar cells as well as other optoelectronic devices.

H2 Reactions on Palladium Clusters

Adsorption of an H2 molecule on Pd(N) clusters (N = 2-4, 7, 13, 19, and 55) is investigated using density functional theory with the hybrid PBE0 functional. Low-energy Pd(N) isomers, taken from a large pool of candidate structures for all cluster sizes (except N = 55), are used in systematic searches for the most stable Pd(N)H2 (molecular) and Pd(N)2H (dissociative) adsorption complexes. Molecular adsorption of H2 is found to occur strictly at atop sites, with the strongest binding typically occurring at the site with the smallest coordination. Binding of dissociated H atoms occurs preferentially on 3-fold faces and on certain favorable edge sites, while binding at atop sites is unstable. Dissociative adsorption is energetically preferred to molecular adsorption for all cluster sizes. The dissociative adsorption energy decreases with cluster size, with pronounced variations due to cluster size effects for the smallest clusters. Adsorption reaction pathways are computed for cluster sizes up to N = 13. Molecular adsorption is found to be barrierless in all cases. Dissociative adsorption occurs without a barrier for the pathways studied for N = 7 and 13 and with a small barrier on the smaller clusters. Finally, lowest-energy pathways for the migration of a dissociated hydrogen atom between local minima on a cluster surface are computed for the Pd4, Pd7, and Pd13 clusters. Calculated migration barriers range from 0.05 to 0.25 eV.

Application of Time-Dependent Density Functional Theory and Optical Spectroscopy toward the Rational Design of Novel 3,4,5-Triaryl-1-R-1,2-diphospholes

Twenty 3,4,5-triaryl-1-R-1,2-diphospholes were studied within the framework of density functional theory (DFT) and experimentally by UV/vis spectroscopy to check their suitability for opto-electronic applications. Time-dependent DFT (TD-DFT) calculations employing the PBE0 hybrid density functional combined with moderately sized def-TZVP basis set were shown to excellently reproduce the experimental absorption spectra of various 1,2-diphospholes. Frontier molecular orbital analysis reveals that HOMO and LUMO are mainly localized on the diphosphole ring and, to some extent, on the aryl moieties. The HOMO-LUMO energy gap can be easily tuned by variation of substituents introduced in para-positions of the aryl moieties and, to a lesser extent, by modification of the R group at phosphorus atom. As a result, both position and intensity of the absorption bands with highest wavelength are strongly influenced by the above structural changes. The UV-spectra simulations reveal that the introduction of donor groups like para-OMe, para-NMe2, and para-N(H)Ph, which are in n-π conjugation with the aryl moieties, should result in absorption of visible light by the corresponding 1,2-diphospholes, thus making them promising candidates for new functional materials.

Communication: Singularity-free hybrid functional with a Gaussian-attenuating exact exchange in a plane-wave basis

Integrable singularity in the exact exchange calculations in hybrid functionals is an old and well-known problem in plane-wave basis. Recently, we developed a hybrid functional named Gaussian-attenuating Perdew-Burke-Ernzerhof (Gau-PBE), which uses a Gaussian function as a modified Coulomb potential for the exact exchange. We found that the modified Coulomb potential of Gaussian function enables the exact exchange calculation in plane-wave basis to be singularity-free and, as a result, the Gau-PBE functional shows faster energy convergence on k and q grids for the exact exchange calculations. Also, a tight comparison (same k and q meshes) between Gau-PBE and two other hybrid functionals, i.e., PBE0 and HSE06, indicates Gau-PBE functional as the least computational time consuming. The Gau-PBE functional employed in conjunction with a plane wave basis provides bandgaps with higher accuracy than the PBE0 and HSE06 in agreement with bandgaps previously calculated using Gaussian-type-orbitals.

Decafluorocyclohex-1-ene at 4.2 K – crystal structure and theoretical analysis of weak interactions

The crystal structure of 1,2,3,3,4,4,5,5,6,6-decafluorocyclohex-1-ene (decafluorocyclohex-1-ene, C6F10) was solved in direct space from neutron powder diffraction data previously collected at 4.2 K [Pawley, G. S. (1981). J. Appl. Cryst. 14, 357-361] and refined by energy minimization in the solid state. To optimize the positions of the 64 atoms in the monoclinic computational cell the PBESOL and hybrid PBE0 functionals were used. The crystal structure of the title compound, which is liquid at room temperature, is built of antiparallel pairs of molecules assembled into molecular columns stacked along the a axis. Dominating the crystal-building forces are weak intermolecular dispersion interactions. Bonding conditions in the structure were analysed by theoretical molecular calculations of representative next-neighbor molecular dimers carried out using dispersion-corrected density functional theory (DFT) functionals and the SCS-MP2 wavefunction method. The largest interaction energy is of the order of ∼ 21 kJ mol(-1), above the interaction energy of a benzene dimer (11.3 kJ mol(-1)) and close to that of a water dimer (20.9 kJ mol(-1)). The interaction energy for the second most stable dimer can be compared with either that of a benzene dimer or of a C-H...π hydrogen bond. The remaining five weakly interacting dimers (∼ 4.2-8.4 kJ mol(-1)) can be characterized as having stronger interactions than those of methane dimers (-2.2 kJ mol(-1)), but weaker than those of benzene molecule pairs or weak C-H...C interactions for instance.

The first transition metal iodato peroxido complex: the synthesis, vibrational spectra and crystal structure from powder diffraction data of K3[V2O2(O2)4(IO3)]·H2O

Abstract The first transition metal iodato peroxido complex, K3[V2O2(O2)4(IO3)]·H2O (I), was prepared by crystallization from the KVO3 — KIO3 — H2O2 — H2O — ethanol (HNO3) solution. The dinuclear anion is immediately decomposed in aqueous solution; the 51V NMR spectrum exhibits signals corresponding to [VO(O2)2(H2O)]−, [V2O2(OH)(O2)4]3− and H2VO4− species only. The IR and Raman spectra contain all characteristic bands of the VO(O2)2 group and the coordinated IO3− ligand. Based on the positions of bands assigned to the vibrations of the VO(O2)2 groups a pentagonal pyramidal arrangement around the vanadium atoms can be supposed. The crystal structure was solved from X-ray synchrotron powder data by direct space method and refined by energy minimization in the solid state employing a hybrid PBE0 functional. This crystal and molecular structure, has confirmed the presence of hexacoordinated vanadium atoms and revealed asymmetric dinuclear structure of the [V2O2(O2)4(IO3)]3− ion. The coordination spheres of vanadium atoms are different — the IO3− anion is coordinated only to one vanadium center. A thermal analysis of the complex confirmed the presence of water molecules in the crystal structure and revealed a considerable stability of the dehydrated complex.

The angular dependence of spin-state energy splittings in the core

Spin-state energy splittings are highly relevant for catalysis, molecular magnetism, and materials science, yet continue to pose a challenge for electronic structure methods. For a Fe2O2+ 2 core, we evaluate the bridging angle dependence of energy splittings between ferromagnetically and antiferromagnetically coupled states for different exchange–correlation functionals, and compare with complete active space self-consistent field (CASSCF) values, also including second-order perturbative corrections (CASPT2). CASSCF and CASPT2 yield strong antiferromagnetic coupling, with the smallest coupling at 100°, and a smooth dependence on the angle for Fe−O−Fe angles of 70° to 120°. Interestingly, this is qualitatively the same behaviour as often found for stable dinuclear transition metal complexes. While all functionals show the same angular dependence as CASPT2, they favour the antiferromagnetic state less strongly. Pure functionals such as BP86, BLYP, SSB-D, and TPSS come closer to the CASPT2 results (with energy splittings by about 60 kJ/mol smaller than the CASPT2 ones) than hybrid functionals. The hybrid functionals B3LYP, B3LYP⋆, and PBE0 favour the antiferromagnetic state even less strongly, resulting in ferromagnetic coupling for angles around 100°. The good qualitative agreement between CASPT2 and CASSCF on the one hand and CASPT2 and density functional theory on the other hand for angles between 70° and 110° suggests that the chosen active space of 18 electrons in 14 orbitals may be adequate for spin-state energy splitting of Fe2O2+ 2 in that region (possibly due to error cancellation), while angles of 60° or 120° may require larger active spaces. This study is complemented by an analysis of local spins, local charges, and CASSCF natural orbitals.

Conformational Effects on the Lowest Excited States of Benzoyl-Pyrrolopyridazine: Insights from PCM Time-Dependent DFT

Time-dependent density functional theory (TD-DFT) computations and steady-state electronic spectroscopy measurements are performed on two recently synthesized pyrrolopyridazines to account for the detrimental effect of benzoyl substitution on the blue fluorescence emission. In the case of the highly fluorescent ester derivative, planar in ground state, we show that TD-DFT using the PBE0 and B3LYP hybrid functionals in the state-specific solvation approach provides an accurate description of absorption and emission properties. In benzoyl-pyrrolopyridazine, the (pretwisted) orientations of the benzoyl group and the solvent polarity are both found to modulate the nature of the lowest excited states. The first excited state has nπ* character at ground-state geometry of the main conformer (carbonyl group facing the diazine ring) in nonpolar solvents and become nearly degenerate with a ππ* state in polar solvents. The latter, lower than the nπ* state at the ground state geometry of a minor conformer, relaxes into a twisted intramolecular charge transfer. Experimental absorption and excitation spectra are consistent with the conformational-dependent picture of the lowest excited state (as derived from TD-DFT). A rather qualitative agreement in predicting the fluorescence emission wavelength is achieved in computations employing the CAM-B3LYP and BH&HLYP functionals, whereas global hybrids with low or moderate amounts of exact exchange exhibit the expected TD-DFT failure with up to 1 eV underestimated transition energies.

Temperature-Mediated Magnetism in Fe-Doped ZnO Semiconductors

We have employed first-principles calculations with PBE0 hybrid functional to study the magnetic origin of Fe-doped ZnO semiconductors. Density functional theory predicts antiferromagnetic ordering for Fe2+-substituted ZnO materials. Origins of magnetic ordering are attributed directly to the local ordering of Fe in the ZnO matrix. Fe3+ induced magnetism is studied for models exhibiting a zinc vacancy or an interstitial oxygen atom. In both cases Fe3+ couples antiferromagnetically. Taking into account the temperature-dependent relative Gibbs energy, the magnetic ordering of Fe2+ and Fe3+ with interstitial oxygen atoms is changed from anti- to ferromagnetic with increasing T. This indicates that the Fe-doped ZnO magnetism is highly dependent on temperature.

Density functional theory study of the electronic structure of fluorite Cu2Se

We have investigated the electronic structure of fluorite Cu2Se using density functional theory calculations within the LDA, PBE and AM05 approximations as well as the non-local hybrid PBE0 and HSE approximations. We find that Cu2Se is a zero gap semiconductor when using either a local or semi-local density functional approximation while the PBE0 functional opens up a gap. For the HSE approximation, we find that the presence of a gap depends on the range separation for the non-local exchange. For the occupied part in the density of states we find that LDA, PBE, AM05, PBE0 and HSE agree with regard to the overall electronic structure. However, the hybrid functionals result in peaks shifted towards lower energy compared to LDA, PBE and AM05. The valence bands obtained using the hybrid functionals are in good agreement with experimental valence band spectra. We also find that the PBE, PBE0 and HSE approximations give similar results regarding bulk properties, such as lattice constants and bulk modulus. In addition, we have investigated the localization of the Cu d-states and its effect on the band gap in the material using the LDA + U approach. We find that a sufficiently high U indeed opens up a gap; however, this U leads to valence bands that disagree with experimental observations.

Electronic structure, optical properties and the mechanism of the B3–B8 phase transition of BeSe: insights from hybrid functionals, lattice dynamics and NPH molecular dynamics

We have investigated the electronic structure and the mechanism of the pressure induced phase transition of beryllium selenide (BeSe) by employing a first-principles pseudopotential method within the framework of density functional theory. Our study demonstrates that use of the hybrid PBE0 functional (PBE stands for Perdew, Burke and Ernzerhof) leads to significant improvement in the band gap calculations, compared to those using either of the common density functionals (local density approximation (LDA) and generalized gradient approximation (GGA)), which severely underestimate the band gap of BeSe. The band gap obtained from the hybrid PBE0 functional shows excellent agreement with available experimental data. A constant-pressure (NPH) first-principles molecular dynamics (FPMD) approach has been adopted to characterize the first-order pressure induced phase transition from the zinc blende (ZB) to the nickel arsenide (NiAs) structure. We have shown that the FPMD simulation overestimates the transition pressure PT (compared to static enthalpy and experimental data) due to overpressure in the simulation box. The MD simulation reveals the structural pathway (cubic → orthorhombic → monoclinic → hexagonal), leading from the ZB phase to the NiAs phase. To find an explanation for the phase transition we calculated the vibrational and elastic properties under pressure. Negative Grüneisen parameters were obtained for the transverse acoustic phonon modes at the X and L high symmetry points. However, no mechanical instability or imaginary frequencies were found at pressures near PT. Thus the transition results from a thermodynamic instability rather than an elastic/dynamical one. We have also calculated the optical properties of both the B3 and B8 phases, such as the real and imaginary parts of the dielectric constant, reflectivity, loss function and refractive index, and compared them with the existing experimental and theoretical data. An abrupt decrease is obtained from the reflectivity spectrum of the NiAs phase at PT, which is supported from the peaks in the loss function.

Theoretical study of the benzoquinone–tetrathiafulvalene–benzoquinone triad in neutral and oxidized/reduced states

This work presents a thorough DFT quantumchemical characterization of the structural and electronic properties of the benzoquinone–tetrathiafulvalene–benzoquinone (Q–TTF–Q) triad in its neutral and oxidized/ reduced states. The minimum-energy structure of the triad in its neutral state corresponds to a C2v boat conformation, although the planar D2h structure is also found to be a minimum- energy structure practically degenerate with the boat conformation. Upon oxidation, electrons are mainly extracted from the TTF backbone and the boat-shaped TTF structure evolves toward a planar D2hh and to a twisted D2 conformation in the cation and dication, respectively. The theoretical characterization of the triad in its radical anion and dianion states is performed with several hybrid (B3LYP, PBE0 and BHHLYP), meta-hybrid (M06-2X) and longrange- corrected (CAM-B3LYP, ωB97X and ωB97XD) functionals. In gas phase, BHHLYP, M06-2X, CAMB3LYP, ωB97X and ωB97XD functionals correctly predict the localized Q-–TTF–Q structure as the most stable structure, whereas the hybrid B3LYP and PBE0 functionals lead to a delocalized (Q–cTTF–Q)- structure. The incorporation of solvent effects is crucial to provide a qualitatively correct description of the anion species where the localized solutions are found to be minimum-energy structures for all the functionals evaluated. The (Q–TTF–Q)2- dianion is predicted to be a biradicaloid species in its electronic ground state. Calculations predict that both Q- → Q0 and TTF → Q0 charge-transfer electronic transitions are associated with the low-energy, broad absorption band experimentally observed for the localized Q-–TTF–Q anion.

Collective librations of water molecules in the crystal lattice of rubidium bromide: experiment and simulation

Terahertz spectroscopy of RbBr reveals four prominent absorption lines at room temperature and a further 15 lines at 10 K. Via density-functional-theory (DFT) numerical modelling using the PBE0 hybrid GGA functional, all the absorptions are identified as correlated librations of water molecules in the RbBr lattice. Each libration mode is a combination of rocking, wagging and twisting motions of the water molecules. The number of libration lines and numerical modelling show that the C2v symmetry of water in RbBr is broken. Our modelling shows that the distribution of libration amplitudes and phases for different water molecules in the RbBr unit cell varies greatly between the different modes. All librational lines red-shift with increasing temperature. The rate of change for most lines is in the range 60-90 MHz K(-1) (or (2-3) × 10(-3) cm(-1) K(-1)). Two lines shift more rapidly with temperature, at rates of 240 and 300 MHz K(-1) (or (8 and 10) × 10(-3) cm(-1) K(-1)), respectively. Furthermore, the temperature dependence of the linewidth distinguishes two groups of lines. For one group, with weak linear temperature dependence of linewidth, cubic anharmonic terms in the RbBr crystal field are significant. This group is mainly associated with fully symmetric correlated librations. For the second group, with strong non-linear temperature dependence of the linewidth, quartic anharmonic terms in the RbBr crystal field are significant. However, the distribution of libration amplitudes, as well as the type of libration modes, influence the temperature dependence of the red shift, the linewidth, and the intensity, as well. Our combined experimental and theoretical investigation confirms the necessity of obtaining low-temperature data to observe all the calculated modes; moreover, the richness of detail in the temperature dependence of the data invites further modelling spanning a range of temperatures.

Efficient Computation of Hartree-Fock Exchange Using Recursive Subspace Bisection.

We use a recursive subspace bisection approach introduced in Phys. Rev. Lett.2009, 102, 166406 to accelerate the computation of the Hartree-Fock exchange operator in plane-wave pseudopotential electronic structure calculations. Recursive subspace bisection allows for an unbiased localization of orbitals in domains of varying size and a truncation of orbitals that preserves accuracy in a controlled manner. This representation is used to accelerate the computation of the Hartree-Fock exchange operator, which in turn makes first-principles molecular dynamics simulations based on hybrid density functionals feasible for larger systems than previously possible. We describe a parallel implementation of the method and a load balancing algorithm. The efficiency and accuracy of this approach are demonstrated in electronic structure calculations of a chloride ion solvated in liquid water and calculations of the vacancy formation energy in a 512-atom silicon crystal using the PBE0 hybrid exchange-correlation functional.