Gradient Approximation For Exchange Research Articles

Hardness of molecules and bandgap of solids from a generalized gradient approximation exchange energy functional.

The deviations from linearity of the energy as a function of the number of electrons that arise with current approximations to the exchange-correlation (XC) energy functional have important consequences for the frontier eigenvalues of molecules and the corresponding valence-band maxima for solids. In this work, we present an analysis of the exact theory that allows one to infer the effects of such approximations on the highest occupied and lowest unoccupied molecular orbital eigenvalues. Then, we show the importance of the asymptotic behavior of the XC potential in the generalized gradient approximation (GGA) in the case of the NCAPR functional (nearly correct asymptotic potential revised) for determining the shift of the frontier orbital eigenvalues toward the exact values. Thereby we establish a procedure at the GGA level of refinement that allows one to make a single calculation to determine the ionization potential, the electron affinity, and the hardness of molecules (and its solid counterpart, the bandgap) with an accuracy equivalent to that obtained for those properties through energy differences, a procedure that requires three calculations. For solids, the accuracy achieved for the bandgap lies rather close to that which is obtained through hybrid XC energy functionals, but those also demand much greater computational effort than what is required with the simple NCAPR GGA calculation.

Investigation of structural, morphological, and optoelectronic properties of Ga-doped TiO2 nanoparticles for electron transport layer in solar cell applications: An experimental and theoretical study

In the current work, the sol-gel synthesis method was used to synthesize both undoped and gallium (1%-Ga) doped TiO2 nanoparticles. The as-deposited materials were calcined at 400–500 °C with a temperature variation rate of 5 °C per minute. The crystallographic properties of the semiconductor materials were determined from X-ray diffraction (XRD) which showed the formation of anatase TiO2. The average crystallite sizes, determined from the widths of the XRD reflections, were 11 and 4 nm for undoped and Ga-doped TiO2, respectively. The materials were further studied by density functional theory with meta-generalized gradient approximation exchange–correlation functional. Supercell systems consisting of 24 and 48 atoms were created to form 12.5% and 6.25% Ga dopant substitution, respectively. These structures were optimized to obtain the ground state properties. Transmission electron microscopy showed polycrystalline, spherical powders, with grain sizes comparable to those found using XRD. The surface morphology of the doped TiO2 obtained from scanning electron microscopy showed an improved surface structure compared to pristine TiO2. Moreover, the electronic states of the materials were investigated using X-ray photoelectron spectroscopy. Energy dispersive X-ray spectroscopy verified the elemental composition of titanium, oxygen, and Ga present in the nanoparticles. The surface chemical bonds were analyzed by Fourier-transform infrared spectroscopy. Ultraviolet diffused reflectance spectra analysis was used to examine their optical properties. The experimentally obtained bandgaps of pristine and 1% Ga-doped TiO2 nanoparticles were 2.9 and 3.09 eV, respectively. The optical properties were further analyzed by computing the band structures and density of states using density functional theory. Similar to the experimental results, these electronic structures showed an increase in bandgap after 6.25% Ga incorporation. However, with 12.5% doping, there was a decrement in the bandgap. The analysis showed that, with an appropriate amount of substitution, Ga-doped TiO2 semiconductor material can be used in the electron transport layer and many other fields of photoelectric cells.

Negative Electron Affinities and Derivative Discontinuity Contribution from a Generalized Gradient Approximation Exchange Functional.

Two methods to calculate negative electron affinities systematically from ground-state density functional methods are presented. One makes use of the lowest unoccupied molecular orbital energy shift provided by approximate inclusion of derivative discontinuity in the nearly correct asymptotic potential (NCAP) nonempirical, constraint-based generalized gradient approximation exchange functional. The other uses a second-order perturbation calculation of the derivative discontinuity based on the NCAP exchange-correlation potential. On a set of thirty-eight molecules, NCAP leads to a rather accurate description that is improved further through the perturbation correction. The results presented show the importance of the asymptotic behavior of the exchange-correlation potential in the calculation of negative electron affinities as well as demonstrating the versatility of the NCAP functional.

Optical properties of lead-free perovskite Cs2InAg(BrxCl1−x)6 for solar cell applications: A DFT study

Organic-inorganic metal halide perovskites have emerged as promising materials for low-cost, high-efficiency solar cells. Chemical compositional tuning along with different device structures for fabrication and interfacial engineering made the record PCE of perovskite solar-cells reaching 25.2%, which is close to that of single crystal silicon solar cells. We investigated the structural, electronic and optical properties of bromine (Br) doped Cs2InAgCl6 in the cubic phase for various doping percentages. The lattice constants were optimized using Birch Murnaghan fit and band gap were found to be increasing as the doping percentage of Br is increased. The theoretical calculations were performed using generalized gradient approximation exchange correlation for all calculations. The optical properties are studied by comparing the absorption coefficients. Contour plots of charge density shows that the nature of chemical bonding is ionic. Band structure plot along high symmetry points revealed a direct band.

Generalized Gradient Approximation Exchange Energy Functional with Near-Best Semilocal Performance.

We develop and validate a nonempirical generalized gradient approximation (GGA) exchange (X) density functional that performs as well as the SCAN (strongly constrained and appropriately normed) meta-GGA on standard thermochemistry tests. Additionally, the new functional (NCAP, nearly correct asymptotic potential) yields Kohn-Sham eigenvalues that are useful approximations of the density functional theory (DFT) ionization potential theorem values by inclusion of a systematic derivative discontinuity shift of the X potential. NCAP also enables time-dependent DFT (TD-DFT) calculations of good-quality polarizabilities, hyper-polarizabilities, and one-Fermion excited states without modification (calculated or ad hoc) of the long-range behavior of the exchange potential or other patches. NCAP is constructed by reconsidering the imposition of the asymptotic correctness of the X potential (-1/ r) as a constraint. Inclusion of derivative discontinuity and approximate integer self-interaction correction treatments along with first-principles determination of the effective second-order gradient expansion coefficient yields a major advance over our earlier correct asymptotic potential functional [CAP; J. Chem. Phys. 2015 , 142 , 054105 ]. The new functional reduces a spurious bump in the CAP atomic exchange potential and moves it to distances irrelevantly far from the nucleus (outside the tail of essentially all practical basis functions). It therefore has nearly correct atomic exchange-potential behavior out to rather large finite distances r from the nucleus but eventually goes as - c/ r with an estimated value for the constant c of around 0.3, so as to achieve other important properties of exact DFT exchange within the restrictions of the GGA form. We illustrate the results with the Ne atom optimized effective potentials and with standard molecular benchmark test data sets for thermochemical, structural, and response properties.

Does the gradient‐regulated connection improve the description of correlated metal bond properties?

AbstractGradient‐regulated connection (GRAC) is a generalized gradient approximation exchange density functional designed by combining the revPBE and PW91 exchange functionals to impose their behaviors in the slowly‐ and fast‐varying density regions, respectively. Such a construction allows one single density functional to accurately estimate both covalent and weak interactions occurring in main‐group‐based molecular systems. For the first time, the assessment of the performance of the GRAC exchange functional is extended to the modeling of various metal bond energy and structure properties. This assessment shows that when GRAC is coupled with the Perdew, Burke, Ernzerhof (PBE) correlation, the resulting exchange‐correlation density functional is an excellent alternative to global hybrids to model bond dissociation energy, atomic electronic excitation energy, and bond length structure properties of single‐reference metal bonds. It also shows that coupling with the Tognetti, Cortona, Adamo (TCA) correlation constitutes a robust approach to tackle energy bond properties of organometallic complexes with multi‐reference character.

Molecular excited states from the SCAN functional

ABSTRACTThe performance of the strongly constrained and appropriately normed (SCAN) meta-generalised gradient approximation exchange–correlation functional is investigated for the calculation of time-dependent density-functional theory molecular excitation energies of local, charge-transfer and Rydberg character, together with the excited potential energy curve in H2. The SCAN results frequently resemble those obtained using a global hybrid functional, with either a standard or increased fraction of exact orbital exchange. For local excitations, SCAN can exhibit significant triplet instability problems, resulting in imaginary triplet excitation energies for a number of cases. The Tamm–Dancoff approximation offers a simple approach to improve the situation, but the excitation energies are still significantly underestimated. Understanding the origin of these (near)-triplet instabilities may provide useful insight into future functional development.

Long-range exchange limit and dispersion in pure silica zeolites

The roles of the large exchange dimensionless gradient limit, s→∞, with $$s = \left| {\nabla n\left( \varvec{r} \right)} \right|/\left( {2k_{\text{f}} \left( \varvec{r} \right)n\left( \varvec{r} \right)} \right)$$ , $$k_{\text{f}} = \left( {3\pi^{2} n\left( \varvec{r} \right)} \right)^{1/3}$$ , and of dispersion interactions computed by Grimme’s scheme in the context of solids are considered. Two families of recently developed generalized gradient approximation exchange functionals in combination with a suitably calibrated dispersion contribution are studied. Furthermore, the effects of changing the correlation functional or including exact exchange in the calculations also are explored. The results indicate that the large exchange dimensionless gradient limit has a small influence and that the most important contribution for a better description of the structure and energetics of porous materials is dispersion. The functional that provides best overall agreement with the experimental stability trend of a large set of pure silica zeolites is an exchange functional (denoted lsRPBE) based on the modified version of PBE, the exchange functional RPBE, corrected to satisfy the large exchange dimensionless limit, combined with PBE correlation and including a calibrated Grimme dispersion contribution. It outperforms any of the functionals that include exact exchange which were tested. Remarkably, the simple local density approximation does almost as well.

Impact of S doping on the structural, electronic and magnetic properties of Cr n (n = 2 − 6) clusters

The impact of sulphur doping on the geometrical and electronic properties of small chromium clusters with up to six atoms is investigated within the density functional theory in the generalized gradient approximation for exchange and correlation. Neutral and charged states are considered in order to analyze the behavior of the vertical ionization potential and adiabatic electron affinity as a function of cluster size. We find good agreement with experimental data for the above electronic indicators in the case of pure chromium clusters, and show that S doping enhances their absolute stability, without destroying for the smaller ones, or not completely for the larger ones, the dimerization pattern typical of small particles made of elements with exact half-band filling. Thus, the chromium skeleton is largely preserved in general. Moreover, S doping does not destroy the odd-even behavior of the electronic quantities either. We find lower relative stability in clusters with odd number of Cr atoms than in those with even number, while only odd-Cr clusters retain a total magnetic moment which results slightly quenched upon S doping. The moment of the anionic clusters remains unchanged upon doping. We show that doping with an S impurity is a way to increase the stability of small Cr nanoparticles, without substantially modifying their magnetic properties and other electronic indicators.

Generalized gradient approximation exchange energy functional with correct asymptotic behavior of the corresponding potential.

A new non-empirical exchange energy functional of the generalized gradient approximation (GGA) type, which gives an exchange potential with the correct asymptotic behavior, is developed and explored. In combination with the Perdew-Burke-Ernzerhof (PBE) correlation energy functional, the new CAP-PBE (CAP stands for correct asymptotic potential) exchange-correlation functional gives heats of formation, ionization potentials, electron affinities, proton affinities, binding energies of weakly interacting systems, barrier heights for hydrogen and non-hydrogen transfer reactions, bond distances, and harmonic frequencies on standard test sets that are fully competitive with those obtained from other GGA-type functionals that do not have the correct asymptotic exchange potential behavior. Distinct from them, the new functional provides important improvements in quantities dependent upon response functions, e.g., static and dynamic polarizabilities and hyperpolarizabilities. CAP combined with the Lee-Yang-Parr correlation functional gives roughly equivalent results. Consideration of the computed dynamical polarizabilities in the context of the broad spectrum of other properties considered tips the balance to the non-empirical CAP-PBE combination. Intriguingly, these improvements arise primarily from improvements in the highest occupied and lowest unoccupied molecular orbitals, and not from shifts in the associated eigenvalues. Those eigenvalues do not change dramatically with respect to eigenvalues from other GGA-type functionals that do not provide the correct asymptotic behavior of the potential. Unexpected behavior of the potential at intermediate distances from the nucleus explains this unexpected result and indicates a clear route for improvement.

Role of Oxygen Atoms in Bonding Properties of Semiconducting Tungsten Trioxide

We have computed the Mulliken’s population (MP) to deduce charge transfer from WO in semiconducting WO3 using density functional theory (DFT) within pseudopotential scheme. In the DFT scheme, second order generalized gradient approximation for exchange and correlation has been implemented for the first time. The MP data show significant difference in charge transfer between W and six non–equivalent O atoms. In addition, the full potential linearized augmented plane wave method has been applied to compute the partial and total density of states. The MP data have also been explained in terms of partial DOS.

Improved CO Adsorption Energies, Site Preferences, and Surface Formation Energies from a Meta-Generalized Gradient Approximation Exchange-Correlation Functional, M06-L.

A notorious failing of approximate exchange-correlation functionals when applied to problems involving catalysis has been the inability of most local functionals to predict the correct adsorption site for CO on metal surfaces or to simultaneously predict accurate surface formation energies and adsorption energies for transition metals. By adding the kinetic energy density τ to the density functional, the revTPSS density functional was shown recently to achieve a balanced description of surface energies and adsorption energies. Here, we show that the older M06-L density functional, also containing τ, provides improved surface formation energies and CO adsorption energies over revTPSS for five transition metals and correctly predicted the on-top/hollow site adsorption preferences for four of the five metals, which was not achieved by most other local functionals. Because M06-L was entirely designed on the basis of atomic and molecular energies, its very good performance is a confirmation of the reasonableness of its functional form. Two GGA functionals with an expansion in the reduced gradient that is correct through second order, namely, SOGGA and SOGGA11, were also tested and found to produce the best surface formation energies of all tested GGA functionals, although they significantly overestimate the adsorption energies.

Electronic structure and magnetic properties of X2YZ (X = Co, Y = Mn, Z = Ge, Sn) type Heusler compounds: a first principle study

We performed the structure optimization followed by the calculation of electronic structure and magnetic properties on Co2MnGe and Co2MnSn. The structure optimization was based on generalized gradient approximation exchange correlation and full potential linearized augmented plane wave (FP-LAPW) method. The calculation of electronic structure was based on FP-LAPW method using local spin density approximation. We have studied the electronic structure and magnetic properties. The calculated density of states and band structures shows the half-metallic ferromagnets character of Co2MnGe and Co2MnSn.

Improved constraint satisfaction in a simple generalized gradient approximation exchange functional

Though there is fevered effort on orbital-dependent approximate exchange-correlation functionals, generalized gradient approximations, especially the Perdew-Burke-Ernzerhof (PBE) form, remain the overwhelming choice in calculations. A simple generalized gradient approximation (GGA) exchange functional [A. Vela, V. Medel, and S. B. Trickey, J. Chem. Phys. 130, 244103 (2009)] was developed that improves substantially over PBE in energetics (on a typical test set) while being almost as simple in form. The improvement came from constraining the exchange enhancement factor to be below the Lieb-Oxford bound for all but one value of the exchange dimensionless gradient, s, and to go to the uniform electron gas limit at both s = 0 and s → ∞. Here we discuss the issue of asymptotic constraints for GGAs and show that imposition of the large s constraint, lim(s→∞)s(1/2)F(xc)(n,s)<∞, where F(xc)(n, s) is the enhancement factor and n is the electron density, upon the Vela-Medel-Trickey (VMT) exchange functional yields modest further improvement. The resulting exchange functional, denoted VT{8,4}, is only slightly more complicated than VMT and easy to program. Additional improvement is obtained by combining VT{8,4} or VMT exchange with the Lee-Yang-Parr correlation functional. Extensive computational results on several datasets are provided as verification of the overall performance gains of both versions.

Generalized gradient exchange functionals based on the gradient-regulated connection: a new member of the TCA family

A new type of generalized gradient approximation exchange functional coupled with the TCA correlation model has been proposed. Based on gradient-regulated connection, this exchange functional is able to mix performances of a modified PBE for the bulk region, to those of the PW91 for the asymptotic one, leading to a significant improvement on the modelization of weak interacting systems, while keeping a good accuracy for the atomization energies.

On the Importance of Gradient-Corrected Correlation for van der Waals Density Functionals

The van der Waals density functional (vdW-DF) family of exchange–correlation functionals is a promising step towards accounting for van der Waals interactions in density functional theory. This approach consists of a nonlocal correlation term in addition to semilocal generalized gradient approximation exchange and local density approximation correlation. It has proven useful for describing vdW bonded complexes but unfortunately deteriorates the prediction of solid-state properties such as bulk lattice parameters and cohesive energies, as compared to the underlying GGA functional. By considering a broad range of different condensed matter systems including weakly interacting complexes as well as strongly bonded molecules and bulk solids, we show that inclusion of gradient-corrected correlations in vdW-DF-type calculations may not only improve the accuracy for vdW bonded systems, but also amend vdW-DF deficiencies in predicting structural properties of solids. Based on this insight we construct a prototype vdW-DF which demonstrates high accuracy in describing the dispersive interactions responsible for benzene adsorption on the noble Au(111) surface.

Range-separated local hybrids

We present two range-separated (screened and long-range-corrected) extensions to our recent work on local hybrids of generalized gradient approximation exchange. Our screened local hybrid improves over HSE06 for thermochemistry and barrier heights. The long-range-corrected (LC) local hybrid improves over LC-omegaPBE for heats of formation and nonhydrogen transfer reaction barriers but does not reach the accuracy of LC-omegaPBE for hydrogen transfer barrier heights.

Relation between single-particle kinetic energy and exchange energy in DFT for the inhomogeneous electron liquid in the Be atom

In a recent study, the authors have used the semi empirical fine-tuned Hartree–Fock ground-state electron density n(r) of Cordero et al. [Phys. Rev. A 75, 052502 (2007)] for the Be atom to calculate the phase θ(r) from a non-linear pendulum-like equation. Since the density amplitude n(r)1/2 plus θ(r) determine, in turn, the idempotent Dirac density matrix γ(r, r′), we use n(r) and θ(r) first of all to calculate the exchange energy density e X (r) of the density functional theory (DFT). This enables us to obtain the Slater (Sl) approximation to the exchange-only potential. A comparison can then be made, by integrating the earlier predicted exchange-correlation force −∂V XC (r)/∂r, of V XC (r) with . Relationship to the Becke semiempirical density gradient approximation for exchange is also established. Some brief discussion of the Perdew–Burke–Ernzerhof density functional is added.

Local hybrids as a perturbation to global hybrid functionals

We present new local hybrids of generalized gradient approximation exchange, designed to be small perturbations to the corresponding global hybrid. In general, local hybrids include a position-dependent admixture of nonlocal Hartree-Fock exchange. These new local hybrids incorporate a constant fraction of nonlocal exchange, plus additional nonlocal exchange contributions near nuclei. These functionals predict molecular thermochemistry and reaction barriers on average more accurately than their "parent" global hybrid.

Variable Lieb–Oxford bound satisfaction in a generalized gradient exchange-correlation functional

We propose a different way to satisfy both gradient expansion limiting behavior and the Lieb-Oxford bound in a generalized gradient approximation exchange functional by extension of the Perdew-Burke-Ernzerhof (PBE) form. Motivation includes early and recent exploration of modified values for the gradient expansion coefficient in the PBE exchange-correlation functional (cf. the PBEsol functional) and earlier experience with a numerical cutoff for large-s (s proportional to absolute value(vector differential n)/n(4/3)) in a version of the deMon molecular code. For either the original PBE or the PBEsol choice of the gradient coefficient, we find improved performance from using an s-dependent (spatially varying) satisfaction of the Lieb-Oxford bound which quenches to uniform electron gas behavior at large s. The mean absolute deviations (MADs) in atomization energies for a widely used test set of 20 small molecules are reduced by about 22% relative to PBE and PBEsol. For these small molecules, the bond length MADs are essentially unchanged.