XC Functionals Research Articles

Correct Structural Phase Stability of FeS2, TiO2, and MnO2 from a Semilocal Density Functional

The structural phase stabilities of iron disulfide (FeS2), titanium dioxide (TiO2), and polymorphs of manganese oxides (MnO2) are known to be challenging problems for the semilocal exchange-correlation (xc) approximations of the Density Functional Theory. Popular and nonempirical semilocal xc functionals fail badly for at least one of these solids. For example, the strongly constrained and appropriately normed (SCAN) meta-GGA xc functional fails to predict the correct ground state for FeS2 (pyrite and marcasite polymorphs) and TiO2 (rutile and anatase polymorphs) structures but yields accurate formation energies and the correct ground state of all MnO2 polymorphs. In this work, we find that the recently proposed meta-GGA constructed from the cuspless hydrogen exchange hole density (MGGAC) [Phys. Rev. B 2019, 100, 155140] can predict the correct ground state of polymorphs of FeS2, TiO2, and MnO2. Moreover, when compared to the experimental results, we also obtain the quantitatively good agreement from the MGGAC functional for the structural properties and bandgaps of those solids, and this agreement indicates the feasibility of this method.

Dependency of XC functionals and role of 3s(2p) orbitals of Co(Si) as core/valence states on the vibrational and thermodynamic properties of CoSi

Here, we investigate the effect of exchange correlation (XC) functionals and role of 3s(2p) orbitals of Co(Si) as core/valence states on the vibrational and thermodynamic properties of CoSi. Phonon calculations are carried out by finite displacement method with supercell approach using equilibrium crystal structures obtained from DFT calculations. Three XC functionals, viz. LDA, PBEsol and SCAN are employed. SCAN gives the highest phonon frequency of ~56 meV in good agreement with experiment, while LDA (PBEsol) gives ~54 (~55) meV. The zero-point energy is calculated as ~10 kJ/mol for all the functionals. The Debye temperatures are computed as ~626 K, ~638 K and ~650 K for LDA, PBEsol and SCAN, respectively. The phonon dispersion and phonon DOS are found to be dependent whether 3s(2p) orbitals of Co(Si) are considered as core or valence states. But, this orbital dependency is seemed to be insignificant on the thermal properties of this compound.

Structural and Electronic Calculations of CdTe Using DFT: Exchange–Correlation Functionals and DFT-1/2 Corrections

Ab initio calculations were performed to investigate the structural and electronic properties of bulk CdTe using various exchange–correlation (XC) functionals available. Among the selected XC functionals include the local density approximation (LDA), generalized gradient approximation (GGA), meta-generalized gradient approximation (MGGA) (using the linear combination of atomic orbitals basis scheme) and Heyd–Scuseria–Ernzerhof-06 (HSE06) (using the plane-wave basis scheme). Further computational studies were performed based on the local density approximation-1/2 (LDA-1/2) and generalized gradient approximation-1/2 (GGA-1/2) self-energy correction schemes to verify their effect on the CdTe band gap in comparison to the other traditional XC functionals. The lattice parameter values obtained using different XC functionals (LDA, GGA and MGGA) were well in agreement with experimental value, with LDA predicting 6.548 A. This is 1.02% greater than the experimental value of 6.482 A. The electronic structure of CdTe was calculated for the fixed 6.482 A lattice parameter of bulk CdTe and resulted in a band gap ranging between 0.68 and 1.56 eV for LDA, GGA, MGGA, and HSE06. The band gap values predicted by the LDA-1/2 and GGA-1/2 corrections were 1.47 eV and 1.50 eV, respectively, and are found to be in good agreement with experimental values. The influence of XC functionals and semi-empirical correction schemes are expected to have important implications on the prediction and understanding of bulk CdTe thin-films found in photovoltaic applications.

Equation of state measurements of dense krypton up to the insulator-metal transition regime: Evaluating the exchange-correlation functionals

Motivated by the poor understanding of the applicability of new exchange-correlation (XC) functionals to warm dense matter (WDM), we designed and performed multiple-shock reverberation compression experiments on dense krypton to evaluate explicitly the implications of recently derived XC functionals. The equation of states of krypton up to 155 GPa and 45 000 K, which ranges from an initial dense gaseous state up to the insulator-metal transition regime, were determined accurately. It is found that the experimental data are better reproduced by the strongly constrained and appropriately normed (SCAN) XC functional compared to the conventional Perdew-Burke-Ernzerhof and Van der Waals (vdW) DF1 functionals, elucidating that the introduction of the kinetic energy density and the intermediate-range vdW interaction is decisive. However, the incorporation of long-range interactions into the SCAN ($\mathrm{SCAN}+\mathrm{rVV}10$ XC functional) results in a noticeably stiffer prediction due to an overestimation of the density and internal energy of the system at low densities and temperatures. Our evaluation of the Karasiev-Sjostrom-Dufty-Trickey free-energy functional experimentally validates the XC thermal effect in the WDM regime, verifies the previous predictions, and sheds light on a direction for future theoretical efforts. Finally, a phase diagram of krypton is given, which provides a clear picture for understanding the thermophysical behavior of krypton in a wider temperature-pressure range.

Adiabatic connection in spin-current density functional theory

The spin-current density functional theory (SCDFT), when formulated in a basis of Pauli spinors, provides a proper theoretical framework for the study of materials in an arbitrarily oriented external magnetic field and/or upon inclusion of spin-dependent relativistic effects, such as spin-orbit coupling. The SCDFT is formulated in terms of the particle-number density $n$, the Cartesian components of the magnetization ${m}_{x},$ ${m}_{y},$ and ${m}_{z}$, the orbital-current density $\mathbit{j}$, and the three spin-current densities ${\mathbit{J}}^{x},$ ${\mathbit{J}}^{y},$ and ${\mathbit{J}}^{z}$, where each of these density variables depends on specific blocks of the density matrix. Exchange-correlation (xc) functionals within the SCDFT should therefore depend on all of these eight fundamental density variables: ${F}_{xc}[n,{m}_{x},{m}_{y},{m}_{z},\mathbit{j},{\mathbit{J}}^{x},{\mathbit{J}}^{y},{\mathbit{J}}^{z}]$, which makes their parametrization a formidable task. Here, we formulate the adiabatic connection of the SCDFT for a treatment of exact Fock exchange in the theory. We show how the inclusion of a fraction of Fock exchange in standard functionals of the (spin) DFT (either in their collinear or noncollinear versions: ${F}_{xc}[n],$ ${F}_{xc}[n,{m}_{z}]$ and ${F}_{xc}[n,{m}_{x},{m}_{y},{m}_{z}]$) allows for the two-electron potential to depend on all those blocks of the density matrix that correspond to the eight density variables of the SCDFT, in a sensible and yet practical way. In particular, in the local-density and generalized-gradient approximations of the SCDFT, the treatment of the current densities solely from the Fock exchange term is formally justified by the short-range behavior of the exchange hole. We discuss that the adiabatic coupling strength parameter modulates the two-electron coupling of the orbital- and spin-current densities with the particle-number density and magnetization. Formal considerations are complemented by numerical tests on a periodic model system in the presence of spin-orbit coupling and in the absence of an external magnetic field.

Large Impact of Approximate Exchange-Correlation Functionals on Modeling the Water Gas Shift Reaction on Copper

We investigate the water gas shift reaction (WGSR) on copper using three levels of exchange-correlation (XC) functionals of increasing complexity: the Perdew–Burke–Ernzerhof (PBE) functional, the Heyd–Scuseria–Ernzerhof (HSE) hybrid functional, and the exact exchange plus the random phase approximation (RPA) correlation functional. We show that DFT predictions for the kinetics of the WGSR strongly depend on the choice of XC functionals. It is important to have accurate CO adsorption energy. Due to PBE’s overestimation of CO’s adsorption energy, it predicts small free surface and a negative (incorrect) reaction order for CO. HSE and RPA largely avoid such overestimation and predict large free surface and positive reaction orders for CO. The key finding in this work is that the prediction for the WGSR mechanism also depends on the choice of XC functionals. PBE and HSE predict the carboxyl mechanism, while RPA predicts that both redox and carboxyl mechanisms are important. These results suggest that caution should be paid when using approximate XC functionals to model heterogeneous catalysis (such as the WGSR investigated here) in which several mechanisms compete. In addition, we also observed one problem for RPA: It underestimates the overall WGSR energy and predicts a negative reaction order for CO2. In addition, we examine formate’s role in the WGSR over copper. Both DFT and previous experiments suggest that formate does not participate much in the reaction and cannot cause a negative reaction for CO2 for the WGSR over copper.

Generalized gradient approximation adjusted to transition metals properties: Key roles of exchange and local spin density.

Perdew-Burke-Ernzerhof (PBE) and PBE adapted for solids (PBEsol) are exchange-correlation (xc) functionals widely used in density functional theory simulations. Their differences are the exchange, μ, and correlation, β, coefficients, causing PBEsol to lose the Local Spin Density (LSD) response. Here, the μ/β two-dimensional (2D) accuracy landscape is analyzed between PBE and PBEsol xc functional limits for 27 transition metal (TM) bulks, as well as for 81 TM surfaces. Several properties are analyzed, including the shortest interatomic distances, cohesive energies, and bulk moduli for TM bulks, and surface relaxation degree, surface energies, and work functions for TM surfaces. The exploration, comparing the accuracy degree with respect experimental values, reveals that the found xc minimum, called VV, being a PBE variant, represents an improvement of 5% in mean absolute percentage error terms, whereas this improvement reaches ~11% for VVsol, a xc resulting from the restoration of LSD response in PBEsol, and so regarded as its variant.

Reproducibility of potential energy surfaces of organic/metal interfaces on the example of PTCDA on Ag(111).

Molecular adsorption at organic/metal interfaces depends on a range of mechanisms: covalent bonds, charge transfer, Pauli repulsion, and van der Waals (vdW) interactions shape the potential energy surface (PES), making it key to understanding organic/metal interfaces. Describing such interfaces with density functional theory requires carefully selecting the exchange correlation (XC) functional and vdW correction scheme. To explore the reproducibility of the PES with respect to the choice of method, we present a benchmark of common local, semi-local, and non-local XC functionals in combination with various vdW corrections. We benchmark these methods using perylene-tetracarboxylic dianhydride on Ag(111), one of the most frequently studied organic/metal interfaces. For each method, we determine the PES using a Gaussian process regression algorithm, which requires only about 50 density functional theory calculations as input. This allows a detailed analysis of the PESs' features, such as the positions and energies of minima and saddle points. Comparing the results from different combinations of XC functionals and vdW corrections enables us to identify trends and differences between the approaches. PESs for different computation methods are in qualitative agreement but also display significant quantitative differences. In particular, the lateral positions of adsorption geometries agree well with experiment, while adsorption heights, energies, and barriers show larger discrepancies.

Uncertainty quantification in first-principles predictions of phonon properties and lattice thermal conductivity

We present a framework for quantifying the uncertainty that results from the choice of exchange-correlation (XC) functional in predictions of phonon properties and thermal conductivity that use density functional theory (DFT) to calculate the atomic force constants. The energy ensemble capabilities of the BEEF-vdW XC functional are first applied to determine an ensemble of interatomic force constants, which are then used as inputs to lattice dynamics calculations and a solution of the Boltzmann transport equation. The framework is applied to isotopically-pure silicon. We find that the uncertainty estimates bound property predictions (e.g., phonon dispersions, specific heat, thermal conductivity) from other XC functionals and experiments. We distinguish between properties that are correlated with the predicted thermal conductivity [e.g., the transverse acoustic branch sound speed ($R^2=0.89$) and average Gr\"uneisen parameter ($R^2=0.85$)] and those that are not [e.g., longitudinal acoustic branch sound speed ($R^2=0.23$) and specific heat ($R^2=0.00$)]. We find that differences in ensemble predictions of thermal conductivity are correlated with the behavior of phonons with mean free paths between $100$ and $300$ nm. The framework systematically accounts for XC uncertainty in phonon calculations and should be used whenever it is suspected that the choice of XC functional is influencing physical interpretations.

Exchange-correlation functional challenges in modeling quaternary chalcogenides

The development of next-generation quaternary chalcogenides, such as $\mathrm{C}{\mathrm{u}}_{2}\mathrm{ZnSn}{\mathrm{S}}_{4}$ (CZTS) and $\mathrm{C}{\mathrm{u}}_{2}\mathrm{ZnGe}{\mathrm{S}}_{4}$ (CZGS), for solar energy and thermoelectric applications hinges upon both careful experimentation and accurate quantum mechanical modeling. To address the latter, many have turned to density functional theory (DFT), which offers several choices for the approximate treatment of electron exchange and correlation (XC). Popular XC functionals include the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) and the recently developed strongly constrained and appropriately normed (SCAN) meta-GGA. Extensions of DFT functionals, such as adding a Hubbard $U$ correction and introducing a fraction of the Fock exchange (hybrid functionals), have been used widely to model systems containing $3d$ metal ions. However, no studies yet have compared comprehensively PBE($+U$) and SCAN($+U$) in the quality of their predictions of the bulk and defect thermodynamics of quaternary chalcogenides, which play a critical role in device fabrication and performance. Hence, here we calculate the (i) 0 K formation energies of bulk Ge compounds and (ii) neutral defect formation energies including charge-balanced (e.g., $\mathrm{C}{\mathrm{u}}_{\mathrm{Zn}}+\mathrm{Z}{\mathrm{n}}_{\mathrm{Cu}}$) and charge-imbalanced (e.g., $\mathrm{C}{\mathrm{u}}_{\mathrm{Sn}}$) combinations of antisites and vacancies in CZTS and CZGS using the PBE, $\mathrm{PBE}+U$, SCAN, $\mathrm{SCAN}+U$, and the hybrid Heyd-Scuseria-Ernzerhof XC frameworks. We find that the formation energies of charge-imbalanced defects are more sensitive to the choice of the XC functional than those of charge-balanced defects, which can be explained by the differences in the extent of penalization of defect-generated delocalized electrons/holes by PBE, $\mathrm{PBE}+U$, SCAN, and $\mathrm{SCAN}+U$. Additionally, our results show that SCAN systematically underbinds Ge-containing compounds, thus highlighting the need for even further improvement of XC functionals. Based on our findings, we recommend the use of SCAN for modeling quaternary chalcogenides because its errors are systematic, and it has the firmest theoretical underpinning. Our work provides guidance for future modeling of quaternary chalcogenides.

Electronic structure benchmark calculations of CO2 fixing elementary chemical steps in RuBisCO using the projector-based embedding approach.

Ribulose 1,5-bisphosphate carboxylase-oxygenase (RuBisCO) is the main enzyme involved in atmospheric carbon dioxide (CO2 ) fixation in the biosphere. This enzyme catalyzes a set of five chemical steps that take place in the same active-site within magnesium (II) coordination sphere. Here, a set of electronic structure benchmark calculations have been carried out on a reaction path proposed by Gready et al. by means of the projector-based embedding approach. Activation and reaction energies for all main steps catalyzed by RuBisCO have been calculated at the MP2, SCS-MP2, CCSD, and CCSD(T)/aug-cc-pVDZ and cc-pVDZ levels of theory. The treatment of the magnesium cation with post-HF methods is explored to determine the nature of its involvement in the mechanism. With the high-level ab initio values as a reference, we tested the performance of a set of density functional theory (DFT) exchange-correlation (xc) functionals in reproducing the reaction energetics of RuBisCO carboxylase activity on a set of model fragments. Different DFT xc-functionals show large variation in activation and reaction energies. Activation and reaction energies computed at the B3LYP level are close to the reference SCS-MP2 results for carboxylation, hydration and protonation reactions. However, for the carbon-carbon bond dissociation reaction, B3LYP and other functionals give results that differ significantly from the ab initio reference values. The results show the applicability of the projector-based embedding approach to metalloenzymes. This technique removes the uncertainty associated with the selection of different DFT xc-functionals and so can overcome some of inherent limitations of DFT calculations, complementing, and potentially adding to modeling of enzyme reaction mechanisms with DFT methods.

Thermal hybrid exchange-correlation density functional for improving the description of warm dense matter

Finite-temperature density functional theory has become a standard tool for first-principles calculations of the properties of warm dense matter (WDM) relevant to high-energy-density physics (HEDP) applications. Here we present theoretical grounds of thermal hybrid exchange-correlation (XC) functionals within the generalized Mermin-Kohn-Sham scheme for an improved description of WDM. Building on the previously developed KSDT (Karasiev-Sjostrom-Dufty-Trickey) [Karasiev et al., Phys. Rev. Lett. 112, 076403 (2014)] local density approximation (LDA) and the KDT16 (Karasiev-Dufty-Trickey 2016) [Karasiev et al., Phys. Rev. Lett. 120, 076401 (2018)] generalized-gradient approximation (GGA) XC free-energy density functionals, we construct a novel thermal hybrid XC functional, referred to here as KDT0. The KDT0 model at low temperature reduces to the popular ground-state PBE0 hybrid due to properties of the used KDT16 density functional approximation. Application to static calculations of electronic band gap and band structure at a wide range of temperatures for various systems of interest to HEDP show that KDT0 provides a significant improvement to the lower LDA and GGA rung XC functionals and to the ground-state PBE0 hybrid.

Influence of finite temperature exchange-correlation effects in hydrogen

We use density functional molecular dynamics (DFT-MD) to study the effect of finite temperature exchange-correlation (xc) in Hydrogen. Using the Kohn-Sham approach, the xc energy of the system, $E_{xc}(r_{s})$ is replaced by the xc free energy $f_{xc}(r_{s},\Theta)$ within the local density approximation (LDA) based on parametrized path integral Monte Carlo (PIMC) data for the uniform electron gas (UEG) at warm dense matter (WDM) conditions. We observe insignificant changes in the equation of state (EOS) at the region of metal-insulator transition compared to the regular LDA form, whereas significant changes are observed for T$>$10000 K, i.e., in the important WDM regime. Thus, our results further corroborate the need for temperature-dependent xc functionals for DFT simulations of WDM systems. Moreover, we present the first finite-temperature DFT results for the EOS of Hydrogen in the electron liquid regime up to $r_{s}=14$ and find a drastic impact (about $20\%$) of thermal xc effects, which manifests at lower temperatures compared to WDM. We expect our results to be important for many applications beyond DFT, like quantum hydrodynamics and astrophysical models.

Neural network interpolation of exchange-correlation functional

Density functional theory (DFT) is one of the most widely used tools to solve the many-body Schrodinger equation. The core uncertainty inside DFT theory is the exchange-correlation (XC) functional, the exact form of which is still unknown. Therefore, the essential part of DFT success is based on the progress in the development of XC approximations. Traditionally, they are built upon analytic solutions in low- and high-density limits and result from quantum Monte Carlo numerical calculations. However, there is no consistent and general scheme of XC interpolation and functional representation. Many different developed parametrizations mainly utilize a number of phenomenological rules to construct a specific XC functional. In contrast, the neural network (NN) approach can provide a general way to parametrize an XC functional without any a priori knowledge of its functional form. In this work, we develop NN XC functionals and prove their applicability to 3-dimensional physical systems. We show that both the local density approximation (LDA) and generalized gradient approximation (GGA) are well reproduced by the NN approach. It is demonstrated that the local environment can be easily considered by changing only the number of neurons in the first layer of the NN. The developed NN XC functionals show good results when applied to systems that are not presented in the training/test data. The generalizability of the formulated NN XC framework leads us to believe that it could give superior results in comparison with traditional XC schemes provided training data from high-level theories such as the quantum Monte Carlo and post-Hartree-Fock methods.

Electronic structures of 24-valence-electron full Heusler compounds investigated by density functional and GW calculations

The electronic structures of Fe-based and Ru-based full Heusler compounds have been investigated systematically by density functional theory (DFT) with PBE, PBE + U, and HSE06 exchange-correlation (XC) functionals. In order to have a better systematic and quantitative comparison between the results of different approximations, the average deviation of eigenvalues (ADE) between any two electronic band structures were calculated. From quantitative analysis of the ADEs, we have shown that different XC functionals used in the DFT calculations will result in very different and inconsistent electronic band structures. However, the discrepancies are dramatically reduced and get more consistent band structures after the GW calculations. Furthermore, comparing the experimental and calculated Seebeck coefficients and band-gap values of Fe2VAl, it implies that the GW methods including dynamically screened Coulomb interactions are more reliable than DFT with PBE or HSE06 functionals. Conclusively, contrast to the fact that DFT methods give inconsistent band structures when using different XC functionals, the GW methods have better predictive power for the band structures of Fe-based and Ru-based full Heusler compounds.

Density functional theory study on two D-π-A-type organic dyes containing different anchoring groups for dye-sensitized solar cells

The electronic structure and absorption spectra of two D-π-A-type organic dyes with different anchoring groups have been investigated using density functional theory (DFT) and time-dependent density functional theory (TD-DFT). The effect of anchoring groups on the electronic absorption of the free dyes on (TiO2)9 has been studied for the two carbazole dyes (MK1 and MK2). Results from DFT calculations indicate that hydroxamic acid anchoring group in MK2 lead to much stronger intermolecular charge transfer and adsorption energies on (TiO2)9 cluster. The effect of four different XC functionals (B3LYP, ωB97xD, M06-2X, and CAM-B3LYP) on the transition energies has been tested in order to explore the valid functional for the studied system. The wavelength values from the ωB97xD/6-31+G** level of theory are in excellent agreement with experimental data so this functional was considered to calculate the electronic absorption of the two studied dyes. The highest occupied molecular orbital (HOMO), the lowest unoccupied molecular orbital (LUMO), and the gap energy (H–L) of the studied dyes are slightly influenced by change of anchoring group. Results reveal that the LUMO energy levels of all studied dyes are higher than the conduction band (CB) of TiO2 (− 4.00 eV). Deprotonation process enhances the efficiency of dye-sensitized solar cells during decreasing adsorption energy of dyes with (TiO2)9 cluster.

Orbital localization error of density functional theory in shear properties of vanadium and niobium.

It is believed that the density functional theory (DFT) describes most elements with s, p, and d orbitals very well, except some materials that have strongly localized and correlated valence electrons. In this work, we find that the widely employed exchange-correlation (XC) functionals, including local-density approximation (LDA), generalized gradient approximation (GGA), and meta-GGA, underestimate the shear modulus and phase stability of V and Nb greatly. The advanced hybrid functional that is usually better for correlated systems, on the other hand, completely fails in these two simple metals. This striking failure is revealed due to the orbital localization error in GGA, which is further deteriorated by hybrid functionals. This observation is corroborated by a similar failure of DFT+U and van der Waals functionals when applied to V and Nb. To remedy this problem, a semiempirical approach of DFT+J is proposed, which can delocalize electrons by facilitating the on-site exchange. Furthermore, it is observed that including density derivatives slightly improves the performance of the semilocal functionals, with meta-GGA outperforms GGA, and the latter is better than LDA. This discovery indicates the possibility and necessity to include higher-order density derivatives beyond the Laplacian level for the purpose of removing the orbital localization error (mainly from d orbitals) and delocalization error (mainly from s and p orbitals) completely in V and Nb so that a better description of their electronic structures is achieved. The same strategy can be applied to the other d electron system and f electron system.

Finding the optimal exchange-correlation functional to describe the excited state properties of push-pull organic dyes designed for thermally activated delayed fluorescence.

To gauge the suitability of an organic dye for thermally activated delayed fluorescence (TADF), its excited state properties are often calculated using density functional theory. For this purpose, the choice of the exchange-correlation (XC) functional is crucial as it heavily influences the quality of the obtained results. In this work, 19 different XC functionals with various amounts of Hartree-Fock (HF) exchange and/or long-range correction parameters are benchmarked versus resolution-of-the-identity second-order coupled cluster (riCC2) calculations for a set of 10 prototype intramolecular donor-acceptor compounds. For the time-dependent density functional theory (TD-DFT) calculations, LC-BLYP(ω = 0.20) and M06-2X are the better performing XC functionals when looking at singlet and triplet excitation energies, respectively. For the singlet-triplet energy gap, LC-BLYP(ω = 0.17), LC-ωPBE(ω = 0.17) and a hybrid LC-BLYP(ω = 0.20)/M06-2X method give the smallest mean average errors (MAEs). Using the Tamm-Dancoff approximation (TD-DFT/TDA), the MAEs are further reduced for the triplet vertical excitation energies and the singlet-triplet energy gaps.

Density functional theory calculations of the radial breathing mode in graphene quantum dots

The luminescence emission of graphene quantum dots (GQDs) is around the visible range, so it could be used as light-emitting diodes or biological markers. Also, the diameter of the GQDs determines the luminescence emission by the variation of the energy gap. For this reason, it is important to establish models that help us to estimate the size of the GQDs. In particular, the radial breathing mode (RBM) allows the correlation of the diameter in some nanostructures with a specific vibrational frequency that could be obtained by Raman spectroscopy. For this reason, we performed the calculation of the RBM in GQDs using the density functional theory. We performed the calculations with different exchange and correlation (XC) functionals to compare the values obtained and select which XC functional could be more accurate in the calculation of the RBM. Also, we found that in the case of the energy gap the values obtained from the various functionals are very similar for standard XC functionals. By contrast, the RBM within the local density approximation is very similar to the other XC functionals, but with small differences to that obtained within the generalized gradient approximation (GGA) to the XC functional. Finally, we found that the dependency of the RBM behavior on the diameter could be described by the inverse of its radius as in the case of nanowires.

Pseudodiagonalization-based wavefunction optimization with contracted planewave basis functions.

Electronic structure calculations representing the molecular orbitals (MOs) with contracted planewave basis functions (CPWBFs) have been reported recently. CPWBFs are Fourier-series representations of atom-centered basis functions. The mathematical features of CPWBFs permit the construction of matrix-vector products, FC o , involving the application of the Fock matrix, F, to the set of occupied MOs, C o , without the explicit evaluation of F. This approach offers a theoretical speed-up of M/n over F-based methods, where M and n are the number of basis functions and occupied MOs, respectively. The present study reports methodological advances that permit FC o -based optimization of wavefunction formed from CPWBFs. In particular, a technique is reported for optimizing wavefunctions by combining pseudodiagonalization techniques based on an exact representation of FC o , approximate information regarding the virtual orbital energies, and direct inversion of the iterative subspace optimization schemes to guide the wavefunction to a converged solution. This method is found to speed-up wavefunction optimizations by factors of up to ~6 - 8 over F-based optimization methods while providing identical results. Further, the computational cost of this technique is relatively insensitive to basis set size, thus providing further benefits in calculations using large CPWBF basis sets. The results of density functional theory calculations show that this method permits the use of hybrid exchange-correlation (XC) functionals with a small increase in effort over analogous calculations using generalized gradient approximation XC functionals. © 2019 Wiley Periodicals, Inc.