Local Density Approximation Exchange Research Articles

Investigations of the exchange energy of neutral atoms in the large-Z limit.

The non-relativistic large-Z expansion of the exchange energy of neutral atoms provides an important input to modern non-empirical density functional approximations. Recent works report results of fitting the terms beyond the dominant term, given by the local density approximation (LDA), leading to an anomalous Z ln Z term that cannot be predicted from naïve scaling arguments. Here, we provide much more detailed data analysis of the mostly smooth asymptotic trend describing the difference between exact and LDA exchange energy, the nature of oscillations across rows of the Periodic Table, and the behavior of the LDA contribution itself. Special emphasis is given to the successes and difficulties in reproducing the exchange energy and its asymptotics with existing density functional approximations.

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

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

Leading Correction to the Local Density Approximation for Exchange in Large-Z Atoms.

The large-Z asymptotic expansion of atomic energies has been useful in determining exact conditions for corrections to the local density approximation in density functional theory. The correction for exchange is fit well with a leading ZlnZ term, and we find its coefficient numerically. The gradient expansion approximation also has such a term, but with a smaller coefficient. Analytic results in the limit of vanishing interaction with hydrogenic orbitals (a Bohr atom) lead to the conjecture that the coefficients are precisely 2.7 times larger than their gradient expansion counterparts, yielding an analytic expression for the exchange-energy correction which is accurate to ∼5% for all Z.

Migration processes of the As interstitial in GaAs

Thermal migration processes of the As interstitial in GaAs were investigated using density-functional theory and the local-density approximation for exchange and correlation. The lowest-energy processes were found to involve the −1, 0, and +1 charge states, and to produce migration along ⟨110⟩-type directions. In the −1 and 0 charge states, migration proceeds via hops between split-interstitial stable configurations at bulk As sites through bridging saddle-point configurations in which the interstitial atom is equidistant from two adjacent bulk As sites. In the +1 charge state, the roles of these two configurations are approximately reversed and migration proceeds via hops between bridging stable configurations through higher-energy split-interstitial stable configurations bounded by a pair of distorted split-interstitial saddle-point configurations. The predicted activation energies for migration in the 0 and +1 charge states agree well with measurements in semi-insulating and p-type material, respectively. Also consistent with experiments, the approximate reversal of the stable and saddle-point configurations between the 0 and +1 charge states is predicted to enable carrier-induced migration with a residual activation energy of 0.05 eV.

On the errors of local density (LDA) and generalized gradient (GGA) approximations to the Kohn-Sham potential and orbital energies.

In spite of the high quality of exchange-correlation energies Exc obtained with the generalized gradient approximations (GGAs) of density functional theory, their xc potentials vxc are strongly deficient, yielding upshifts of ca. 5 eV in the orbital energy spectrum (in the order of 50% of high-lying valence orbital energies). The GGAs share this deficiency with the local density approximation (LDA). We argue that this error is not caused by the incorrect long-range asymptotics of vxc or by self-interaction error. It arises from incorrect density dependencies of LDA and GGA exchange functionals leading to incorrect (too repulsive) functional derivatives (i.e., response parts of the potentials). The vxc potential is partitioned into the potential of the xc hole vxchole (twice the xc energy density ϵxc), which determines Exc, and the response potential vresp, which does not contribute to Exc explicitly. The substantial upshift of LDA/GGA orbital energies is due to a too repulsive LDA exchange response potential vxresp (LDA) in the bulk region. Retaining the LDA exchange hole potential plus the B88 gradient correction to it but replacing the response parts of these potentials by the model orbital-dependent response potential vxresp (GLLB) of Gritsenko et al. [Phys. Rev. A 51, 1944 (1995)], which has the proper step-wise form, improves the orbital energies by more than an order of magnitude. Examples are given for the prototype molecules: dihydrogen, dinitrogen, carbon monoxide, ethylene, formaldehyde, and formic acid.

Connection between Hybrid Functionals and Importance of the Local Density Approximation.

The exchange-correlation (XC) local density approximation (LDA) is the original density functional used to investigate the electronic structure of molecules and solids within the formulation of Kohn and Sham. The LDA is fundamental for the development of density-functional approximations. In this work we consider the generalized Kohn-Sham (GKS) theory of hybrid functionals. The GKS formalism is an extension of the Kohn-Sham theory for electronic ground states and leads to a vast set of alternative density functionals, which can be estimated by the LDA and related methods. Herein we study auxiliary electronic systems with parametrized interactions and derive (i) a set of exact equations relating the GKS XC energies in the parameter space and (ii) a formal relation between the parameters and the standard XC derivative discontinuity. In view of the new results and previously reported findings, we discuss why the inclusion of Fock exchange, and its long-range-corrected form (in the ground-state calculations and in linear-response Kohn-Sham equations), dominate over the generalized gradient corrections to enhance the quality of the fundamental gap and to enhance excitation-energy estimations. As an example, we show that the adiabatic CAM-LDA0 (a functional with 1/4 global and 1/2 long-range Hartree-Fock interaction, respectively, a range separation factor of 1/3, and pure LDA exchange and correlation) works for electronic excitations as well as the adiabatic CAM-B3LYP functional.

Existence of the β-tin structure in Sr: First evidence from computational approach

Molecular Dynamics (MD) calculation is one of the most powerful theoretical methods widely used to predict and to confirm structural phase transitions. In this work, the MD method has been used to verify phase transition from body-centered cubic (bcc) to β-tin structure, then, to the Cmcm and hexagonal close-packed (hcp) structure, respectively. The transition sequence from previous theoretical works has been confirmed. In this study, Density Functional Theory (DFT), has been used to calculate phonon dispersion to confirm the stability of β-tin and hcp phases. The long time discrepancies in transition sequence between the calculation and the experimental works has been explained by conventional DFT calculation using screened exchange local density approximation (sX-LDA). More importantly, the existence of β-tin structure is finally predicted and the transition nature of Sr has also been revealed.

Distortion and electric-field control of the band structure of silicene

Density functional theory with local density approximation for exchange and correlation functional is used to tune the electronic band structure of silicene monolayer. The cohesive energy of the free-standing monolayer increases (decreases) with the external electric field (distortion). Electrons in silicene behave like Dirac fermions, when the bond angle between the Si atoms is larger than . Large distortions destroy the electronic structure of silicene so that it is no longer a semi-metallic material, and the distorted silicene acts like an n-doped system. The electric field opens a band gap around the K-point in the Brillouin zone, which increases with electric field. The bond angle between the Si atoms is a key player to determine the presence or absence of Dirac cones in silicene.

Accurate screened exchange band structures for the transition metal monoxides MnO, FeO, CoO and NiO

We report calculations of the band structures and density of states of the four transition metal monoxides MnO, FeO, CoO and NiO using the hybrid density functional sX-LDA (‘screened exchange local density approximation’). Late transition metal oxides are prototypical examples of strongly correlated materials, which pose challenges for electronic structure methods. We compare our results with available experimental data and show that our calculations generally yield accurate predictions for the fundamental band gaps and valence bands, in favourable agreement with previously reported theoretical studies. For MnO, the band gaps are still underestimated, suggesting additional many-body effects that are not captured by our screened hybrid functional approach.

Screened-exchange electronic structures and thermodynamic properties of the cubic perovskite BiAlO3

The screened exchange local density approximation (sX-LDA) hybrid functional within the pseudopotential plane wave method is performed to improve the electronic structures of cubic BiAlO3 and its thermodynamic properties are investigated using the density functional perturbation theory (DFPT). The equilibrium lattice constant of cubic BiAlO3 is 3.67Å which is applied to calculate its electronic structures. The calculated sX-LDA electronic structures are compared to standard generalized gradient approximation (GGA) approach. It is found that BiAlO3 is an indirect band gap material at M(1/2,1/2,0)-X(0,0,1/2) point with the gap value of 1.84, and 3.43eV for GGA, and sX-LDA calculations, respectively. The linear response method is employed in order to calculate the phonon dispersion relations and phonon density of states. The predicted thermodynamic quantities, specific heat and Debye temperature, of cubic perovskite BiAlO3 are also reported.

First-Principles Study of Electronic, Elastic, and Lattice Vibrational Properties of Pbnm Orthorhombic SrHfO3

Through first-principles pseudopotential calculations based on density functional theory, the electronic structure and lattice vibrational properties of Pbnm orthorhombic SrHfO3 were investigated in the framework of standard functional approximation and density functional perturbation theory, respectively. The calculated equilibrium lattice constants of Pbnm orthorhombic SrHfO3 are in good agreement with available experimental and theoretical results. The results show that Pbnm orthorhombic SrHfO3 is an insulator with a direct band gap of 3.9 eV and 4.0 eV within the calculations using local density approximation (LDA) and generalized gradient approximation (GGA), respectively. Use of the screened exchange local density approximation (sX-LDA) as a functional in a successive band calculation has also been performed. The band gap is predicted to be 6.7 eV within sX-LDA, somewhat higher than the gap values of 6.1 ± 0.1 eV and 6.5 eV obtained from recent x-ray photoelectron spectroscopy. The phonon dispersion curves of Pbnm orthorhombic SrHfO3 were also calculated. All-positive phonon frequencies were observed in the whole Brillouin zone, indicating stability of the Pbnm orthorhombic SrHfO3 structure. In addition, the infrared-active and Raman-active vibrational modes of SrHfO3 were calculated and compared with available theoretical and experimental investigations.

Electronic and lattice vibrational properties of cubic BaHfO3 from first principles calculations

The electronic structure and lattice vibrational properties of cubic SrHfO3 were investigated by first-principles calculations based on density functional theory in the framework of the local density approximation (LDA), generalized gradient approximation (GGA), and density functional perturbation theory (DFPT), respectively. The calculated equilibrium lattice constant of cubic SrHfO3 is in good agreement with available experimental and theoretical results. The results show that cubic SrHfO3 is an insulator with an indirect LDA (GGA) band gap of 3.6 (3.7) eV. Use of the screened exchange local density approximation (sX-LDA) as a functional in successive band calculation has also been performed. The band gap is predicted to be 6.27 eV within sX-LDA, in excellent agreement with the gap value of 6.1 ± 0.1 eV obtained from x-ray photoelectron spectroscopy. The phonon dispersion curves and LO–TO splitting of cubic SrHfO3 were also calculated. Negative phonon frequencies were observed along the M–C–R–M line in the Brillouin zone, indicating instability of the SrHfO3 structure, consistent with previous theoretical investigation.

Consistent LDA’ + DMFT—an unambiguous way to avoid double counting problem: NiO test

We present a consistent way of treating a double counting problem unavoidably arising within the LDA + DMFT combined approach to realistic calculations of electronic structure of strongly correlated systems. The main obstacle here is the absence of systematic (e.g., diagrammatic) way to express LDA (local density approximation) contribution to exchange correlation energy appearing in the density functional theory. It is not clear then, which part of interaction entering DMFT (dynamical mean-field theory) is already taken into account through LDA calculations. Because of that, up to now there is no accepted unique expression for the double counting correction in LDA + DMFT. To avoid this problem we propose here the consistent LDA’ + DMFT approach, where LDA exchange correlation contribution is explicitly excluded for correlated states (bands) during self-consistent band structure calculations. What is left out of Coulomb interaction for those strongly correlated states (bands) is its non-local part, which is not included in DMFT, and the local Hartreelike contribution. Then the double counting correction is uniquely reduced to the local Hartree contribution. Correlations for strongly correlated states are then directly accounted for via the standard DMFT. We further test the consistent LDA’ + DMFT scheme and compare it with conventional LDA + DMFT calculating the electronic structure of NiO. Opposite to the conventional LDA + DMFT our consistent LDA’ + DMFT approach unambiguously produces the insulating band structure in agreement with experiments.

Density functional theory calculations for two-dimensional silicene with halogen functionalization

The electronic structures and band gaps of silicene (the Si analogue of graphene) adsorbed with halogen elements are studied using the density functional theory based screened exchange local density approximation method. It is found that the band gaps of silicene adsorbed with F, Cl, Br and I have a nonmonotonic change as the periodic number of the halogen elements increases. This is attributed to the transfer of contributions to band gaps from Si-Si bonding to Si-halogen bonding.

Electronic and Lattice Vibrational Properties of Cubic SrHfO3 from First-Principles Calculations

The electronic structure and lattice vibrational properties of cubic SrHfO3 were investigated by first-principles calculations based on density functional theory in the framework of the local density approximation (LDA), generalized gradient approximation (GGA), and density functional perturbation theory (DFPT), respectively. The calculated equilibrium lattice constant of cubic SrHfO3 is in good agreement with available experimental and theoretical results. The results show that cubic SrHfO3 is an insulator with an indirect LDA (GGA) band gap of 3.6 (3.7) eV. Use of the screened exchange local density approximation (sX-LDA) as a functional in successive band calculation has also been performed. The band gap is predicted to be 6.27 eV within sX-LDA, in excellent agreement with the gap value of 6.1 ± 0.1 eV obtained from x-ray photoelectron spectroscopy. The phonon dispersion curves and LO–TO splitting of cubic SrHfO3 were also calculated. Negative phonon frequencies were observed along the M–Γ–R–M line in the Brillouin zone, indicating instability of the SrHfO3 structure, consistent with previous theoretical investigation.

ELECTROMAGNETIC AND ISOVECTOR TERMS IN STANDARD RELATIVISTIC MEAN FIELD MODEL

The effects of auxiliary contribution in forms of electromagnetic tensors and relativistic electromagnetic exchange in local density approximation as well as δ meson and isovector density-dependent nonlinear terms in standard relativistic mean field model constrained by nuclear matter stability criteria in some selected finite nuclei and nuclear matter properties are studied. It is found that in the case of finite nuclei, the electromagnetic tensors play the most dominant part compared to other auxiliary terms. Due to the presence of electromagnetic tensors, the binding energies prediction of the model can be improved quite significantly. However, these terms do not yield demanded effects for rms radii prediction. In the case of nuclear matter properties, the isovector density-dependent nonlinear term plays the most crucial role in providing predictions which are quite compatible with experimental constraints. We have also shown these auxiliary contributions are indeed unable to improve the single particle spectrum results of the model.

Structural, elastic, and electronic properties of cubic perovskite obtained from first principles

We investigated the structural, elastic, and electronic properties of the cubic perovskite-type BaHfO3 using a first-principles method based on the plane-wave basis set. Analysis of the band structure shows that perovskite-type BaHfO3 is a wide gap indirect semiconductor. The band-gap is predicted to be 3.94 eV within the screened exchange local density approximation (sX-LDA). The calculated equilibrium lattice constant of this compound is in good agreement with the available experimental and theoretical data reported in the literatures. The independent elastic constants (C-11, C-12, and C-44), bulk modules B and its pressure derivatives B', compressibility beta, shear modulus G, Young's modulus Y, Poisson's ratio nu, and Lame constants (mu, lambda) are obtained and analyzed in comparison with the available theoretical and experimental data for both the singlecrystalline and polycrystalline BaHfO3. The bonding-charge density calculation make it clear that the covalent bonds exist between the Hf and 0 atoms and the ionic bonds exist between the Ba atoms and HfO3 ionic groups in BaHfO3. (C) 2009 Elsevier B.V. All rights reserved.

Structural and energetic properties of α-amino acids: a first principles density functional study

The structures, energetics and ionisation potentials of the 20 standard α-amino acids, in both neutral and positively charged states, are investigated using density functional theory by using the Becke–Lee–Yang–Parr (BLYP) generalised gradient corrections to the local density approximation exchange and correlation energy, norm-conserving pseudopotentials, and a plane-wave expansion of Kohn–Sham orbitals. There is a good agreement between our theory and the available experiments, increasing the confidence in density functional theoretical descriptions of this important class of bio-molecules.

Theoretical study of the structural, elastic and electronic properties of the GeX2O4 (X = Mg, Zn, Cd) compounds under pressure

The effect of high pressures, up to 30 GPa, on the structural, elastic and electronic properties of GeX2O4, with X = Mg, Zn and Cd, were studied by means of the pseudo-potential plane waves method within the local density approximation for exchange and correlation. The results of bulk properties, including lattice constants, internal parameters, bulk modulus and derivatives are obtained. The band structures show a direct (Γ–Γ) band gap for the three compounds. All the calculated band gaps increase with increasing pressure and fit well to a quadratic function. The analysis of the density of states revealed that the lowering of the direct gap Γ–Γ from GeMg2O4 to GeZn2O4 to GeCd2O4 can be attributed to the p–d mixing in the upper valence band of GeZn2O4 and GeCd2O4. The elastic constants and their pressure dependence are calculated using the static finite strain technique. We have observed a linear dependence of the elastic moduli on the pressure. We derived the bulk modulus, shear modulus, Young's modulus and Poisson's ratio for ideal polycrystalline GeX2O3 aggregates. We estimated the Debye temperature of GeX2O4 from the average sound velocity.

First-principles theory of tunneling currents in metal-oxide-semiconductor structures

Ultrathin gate-oxide films and alternate dielectrics call for accurate modeling of tunneling currents. Available models, based on the effective-mass approximation, yield good fits to finite-bias data, but fail for infinitesimal biases. Here we report a first-principles theory of tunneling currents. We show that the conductance at infinitesimal bias is a ground-state property and can be calculated accurately using density-functional theory and the local-density approximation for exchange and correlation. At finite biases, a discontinuity in the exchange-correlation potential must be properly included. Challenges for both theory and experiments are identified.