Local Density Approximation Errors Research Articles

On the self-consistency of DFT-1/2.

Density functional theory (DFT)-1/2 is an efficient bandgap rectification method for DFT under local density approximation (LDA) or generalized gradient approximation. It was suggested that non-self-consistent DFT-1/2 should be used for highly ionic insulators like LiF, whereas self-consistent DFT-1/2 should still be used for other compounds. Nevertheless, there is no quantitative criterion prescribed for which implementation should work for an arbitrary insulator, which leads to severe ambiguity in this method. In this work, we analyze the impact of self-consistency in DFT-1/2 and shell DFT-1/2 calculations in insulators or semiconductors with ionic bonds, covalent bonds, and intermediate cases and show that self-consistency is required even for highly ionic insulators for globally better electronic structure details. The self-energy correction renders electrons more localized around the anions in self-consistent LDA-1/2. The well-known delocalization error of LDA is rectified, but with strong overcorrection, due to the presence of additional self-energy potential. However, in non-self-consistent LDA-1/2 calculations, the electron wave functions indicate that such localization is much more severe and beyond a reasonable range because the strong Coulomb repulsion is not counted in the Hamiltonian. Another common drawback of non-self-consistent LDA-1/2 is that the ionicity of the bonding gets substantially enhanced, and the bandgap can be enormously high in mixed ionic-covalent compounds like TiO2.

Testing density functional theory in a quantum Ising chain

By using the quantum Ising chain as a test bed and treating the spin polarization along the external transverse field as the "generalized density", we examine the performance of different levels of density functional approximations parallel to those widely used for interacting electrons, such as local density approximation (LDA) and generalized gradient approximation (GGA). We show that by adding the lowest-order and nearest-neighbor density variation correction to the simple LDA, a semi-local energy functional in the spirit of GGA is almost exact over a wide range of inhomogeneous density distribution. In addition, the LDA and GGA error structures bear a high level of resemblance to the quantum phase diagram of the system. These results provide insights into the triumph and failure of these approximations in a general context.

Calculations of quasi-particle spectra of semiconductors under pressure

Different approximations in calculations of electronic quasi-particle states in semiconductors are compared and evaluated with respect to their validity in predictions of optical properties. The quasi-particle self-consistent GW (QSGW) approach yields values of the band gaps which are close to experiments and represents a significant improvement over single-shot GW calculations using local density approximation (LDA) start wavefunctions. The QSGW approximation is compared to LDA bands for a wide-gap material (CuAlO 2 ) and materials with very small gaps, PbX (X=S, Se, and Te). For wide-gap materials QSGW overestimates the gaps by 0.3―0.8 eV, an error which is ascribed to the omission of vertex corrections. This is confirmed by calculations of excitonic effects, by solving the Bethe-Salpeter equation. The LDA error in predicting the binding energy of the Cu-3d states is examined and the QSGW and LDA + U approximations are compared. For PbX the spinorbit coupling is included, and it is shown that although LDA gives a reasonable magnitude of the gap at L, only QSGW predicts the correct order of the L + 6 and L ― 6 states and thus the correct sign (negative) of the gap pressure coefficient. The pressure-induced gap closure leads to linear (Dirac-type) band dispersions around the L point.

Which electronic structure method for the study of defects: A commentary

AbstractA historic perspective is provided to the choice of methodologies for point defects in semiconductors. A summary and commentary on the highlights of the CECAM workshop: “Which electronic structure method for the study of defects?” is given, attempting to provide a link between the different contributions. To this purpose different themes running through the compilation are identified and rather than discussing individual contributions one by one, the discussion is focused around these themes. The first theme is the problem of correcting for finite size effects. The second theme is the problem of the underestimate of the band gaps and how to correct for it in defect calculations. The third theme is the self‐interaction error of local density approximation (LDA) and its repercussions for polaronic defects. The fourth theme is progress in methods beyond LDA that are becoming applicable to point defects, either for ground states or excited state properties.

Van der Waals Interactions for Isolated Systems Calculated Using Density Functional Theory within Plasmon-Pole Approximation

We have developed a new method of calculating the van der Waals interaction in isolated systems on the bases of the van der Waals density functional theory. This method is designed to be absolutely nonempirical, applicable to any three-dimensional isolated system and implementable into various first-principles electronic state calculation codes utilizing local density approximation (LDA). The validity of this method is tested by comparing the adiabatic potentials of an Ar dimer and a Kr dimer obtained by this method to those obtained in experiments and other theoretical studies. We find that the LDA error is largely mitigated in both equilibrium bond length and cohesive energy.

“Quantum Coaxial Cables” for Solar Energy Harvesting

Type II core-shell nanowires based on III-V and II-VI semiconductors are designed to provide the highly desirable but not readily available feature--efficient charge separation--and concurrently address the different material challenges specific for a few key renewable energy applications: including hydrogen generation via photoelectrochemical water splitting, dye-sensitized solar cells, and conventional solar cells. They also open up new avenues for studying novel physics and material sciences in reduced dimensionality of very unusual quasi-one-dimensional systems. A first-principles density function theory within the local density approximation (LDA) is used for the electronic structure calculation and a valence-force-field method for the structural relaxation, and empirical corrections to the LDA errors are applied.

Exchange-correlation energy in molecules: A variational quantum Monte Carlo study

We have used the combination of the coupling-constant integration procedure and the variational quantum Monte Carlo method to study the exchange-correlation (XC) interaction in small molecules: ${\mathrm{Si}}_{2}$, ${\mathrm{C}}_{2}{\mathrm{H}}_{2}$, ${\mathrm{C}}_{2}{\mathrm{H}}_{4}$, and ${\mathrm{C}}_{2}{\mathrm{H}}_{6}$. In this paper we report the calculated XC energy density, a central quantity in density functional theory, as deduced from the interaction between the electron and its XC hole integrated over the interaction strength. Comparing these ``exact'' XC energy densities with results using the local-density approximation (LDA), one can analyze the errors in this widely used approximation. Since the XC energy is an integrated quantity, error cancellation among the XC energy density in different regions is possible. Indeed we find a general error cancellation between the high-density and low-density regions. Moreover, the error distribution of the exchange contribution is out of phase with the error distribution of the correlation contribution. Similar to what is found for bulk silicon and an isolated silicon atom, the spatial variation of the errors of the LDA XC energy density in these molecules largely follows the sign and shape of the Laplacian of the electron density. Some noticeable deviations are found in ${\mathrm{Si}}_{2}$ in which the Laplacian peaks between the atoms, while the LDA error peaks in the regions ``behind'' atoms where a good portion of the charge density originates from an occupied $1{\ensuremath{\sigma}}_{u}$ antibonding orbital. Our results indicate that, although the functional form could be quite complex, an XC energy functional containing the Laplacian of the energy is a promising possibility for improving LDA.

Testing the local density approximation with energy-versus-separation curves of jellium slab pairs

We explore the successes and failures of the local-density approximation (LDA) in predicting the attractive force and energetics of attraction between two jellium metals. We model the metals by two very thin slabs, and follow the energetics at all separations $D$ from the van der Waals (vdW) regime at large $D$ through to metallic bonding as $D\ensuremath{\rightarrow}0$. We compare LDA results with calculations including exact exchange and microscopic nonlocal random-phase-approximation-type correlation energy: only these latter calculations include dispersion (van der Waals) forces correctly. We show how this error of LDA at large separations leads also to errors even at smaller separations outside the vdW regime.

Electronic structure of calcium hexaboride within the weighted density approximation

We report calculations of the electronic structure of CaB$_6$ using the weighted density approximation (WDA) to density functional theory. We find a semiconducting band structure with a sizable gap, in contrast to local density approximation (LDA) results, but in accord with recent experimental data. In particular, we find an $X$-point band gap of 0.8 eV. The WDA correction of the LDA error in describing the electronic structure of CaB$_6$ is discussed in terms of the orbital character of the bands and the better cancelation of self-interactions within the WDA.

Exchange-correlation energy and the phase diagram of Si

Previous first-principles calculations of the melting properties of Si, based on the local-density approximation (LDA) for electronic exchange-correlation energy, underpredict the melting temperature by $\ensuremath{\sim}20$ %. We present new first-principles results indicating that a large part of this problem is due to noncancellation of exchange-correlation errors between the semiconducting solid and the metallic liquid. It is shown that other sources of error, particularly those due to system size and Brillouin-zone sampling, can be made negligible. The same LDA errors cause an underprediction of the pressure of the diamond-Si $\ensuremath{\rightarrow}$ beta-tin-Si transition. The generalized-gradient approximation considerably improves both features of the Si phase diagram.

S−dcoupling in zinc-blende semiconductors

Most zinc blende semiconductors have a single anion-like s state near the bottom of the valence band, found in density-of-states (DOS) calculations, and seen in photoemission. Here, we discuss the case where two s-like peaks appear, due to strong $s\ensuremath{-}d$ coupling. Indeed, away from the $\mathbf{k}=0$ Brillouin zone center, cation d states and anion s states can couple in zinc blende symmetry. Depending on the energy difference $\ensuremath{\Delta}{E}_{\mathrm{sd}}{=E}_{s}^{\mathrm{anion}}\ensuremath{-}{E}_{d}^{\mathrm{cation}},$ this interaction can lead to either a single or two s-like peaks in the DOS and photoemission. We find four types of behaviors. (i) In GaP, GaAs, InP, and InAs, $\ensuremath{\Delta}{E}_{\mathrm{sd}}$ is large, giving rise to a single cation d peak well below the single anion s peak. (ii) Similarly, in CdS, CdSe, ZnS, ZnSe, and ZnTe, we see also a single s peak, but now the cation d is above the anion s. In both (i) and (ii) the $s\ensuremath{-}d$ coupling is very weak. (iii) In GaN and InN, the local density approximation (LDA) predicts two s-like peaks bracketing below and above the cation d-like state. Correcting the too low binding energies of LDA by LDA+SIC (self-interaction correction) still leaves the two s-like peaks. The occurrence of two s-like peaks represents the fingerprint of strong $s\ensuremath{-}d$ coupling. (iv) In CdTe, LDA predicts a single s-like peak just as in case (ii) above. However, LDA+SIC correction shifts down the cation d state closer to the anion s band, enhancing the $s\ensuremath{-}d$ coupling, and leading to the appearance of two s-like peaks. Case (iv) is a remarkable situation where LDA errors cause not only quantitative energetic errors, but actually leads to a qualitative effect of a DOS peak that exists in LDA+SIC but is missing in LDA. We predict that the double-$s$ peak should be observed in photoemission for GaN, InN, and CdTe.

Exchange-correlation errors at harmonic and anharmonic orders: the case of bulk Cu

As an aid towards improving the treatment of exchange and correlation effects in electronic structure calculations, it is desirable to have a clear picture of the errors introduced by currently popular approximate exchange-correlation functionals. We have performed ab initio density functional theory and density functional perturbation theory calculations to investigate the thermal properties of bulk Cu, using both the local density approximation (LDA) and the generalized gradient approximation (GGA). Thermal effects are treated within the quasiharmonic approximation. We find that the LDA and GGA errors for anharmonic quantities are an order of magnitude smaller than for harmonic quantities; we argue that this might be a general feature. We also obtain much closer agreement with experiment than earlier, more approximate calculations.

Density-functional-theory calculations of the total energies, ionization potentials, and optical response properties with the van Leeuwen–Baerends potential

van Leeuwen and Baerends proposed a Becke-like nonlocal correction to the local-density-approximation (LDA) exchange-correlation potential so that its asymptotic structure becomes exact i.e., $\ensuremath{-}1/r$ [Phys. Rev. A 49, 2421 (1994)]. They showed that it significantly improves the value of the highest occupied orbital eigenvalue of atoms and molecules. However, the correction is exchangelike in nature. With this in mind, in this paper we investigate how this correction affects the total energies and highest eigenvalues within the exchange-only approximation. We show that the potential also corrects the LDA errors substantially within this approximation, and leads to total energies and high eigenvalues which compare well with their Hartree-Fock counterparts. Improvement in the asymptotic behavior of the potential should also result in better values of the response properties of these systems. We show that with this correction one obtains better estimates, both within the exchange-only approximation and with correlation included, of the linear and nonlinear polarizabilities of inert gas atoms. This is quite significant, since the LDA is known to overestimate the nonlinear polarizabilities of these atoms by roughly 100%. On the other hand, for alkaline-earth atoms the values of polarizabilities obtained with this correction are not satisfactory. Nonetheless, hyperpolarizabilities show a marked improvement over the LDA results.

Thermodynamical properties of thallium-based III-V materials

Thermodynamical factors that affect growth of the thallium-bearing zinc-blende alloys InTIP and InTlAs are examined within the local density approximation (LDA), using the linear muffin-tin orbital method. The LDA predicts TIP and TIAs to be unstable with respect to decomposition into the elemental constituent solids, or marginally stable if conservative estimates of the LDA errors are made. Several thallium-rich and anion-rich compounds have also been examined; some are found to have excess energies per atom comparable to the zinc-blende phase. The equilibrium partial pressures over InTIP and InTlAs have also been calculated as a function of composition. Even with conservative error estimates, we predict that only low concentrations of thallium can be achieved in InTIP (<5% at 350°C) using gas source molecular beam epitaxy (GSMBE), far less than the 67% needed for long-wave infrared (LWIR) applications. Although much less than 1% thallium is predicted to be soluble in InTlAs for GSMBE growth at 350 °C, the addition of error estimates into the calculation indicates that obtaining the 15% thallium needed for LWIR applications may be possible. Native defect populations have been calculated for alloy compositions corresponding to band gaps in the LWIR, and the anion antisite densities are predicted to be quite high, especially in InTIP, reflecting the comparable stability of the TIP and T1P3 phases.

Cohesive energy of 3d transition metals: Density functional theory atomic and bulk calculations.

We report generalized gradient approximation (GGA) cohesive energies for 3d metals. The problem of obtaining atomic reference energies in density-functional theory is considered. The effect of going to nonspherical atomic charge distributions is much larger at the GGA than at the local-density approximation (LDA) level, but allowing fractional occupations of 3d and 4s shells has negligible effect. When nonsphericity effects are taken into account in the atomic reference energies, the average absolute error of 0.3 eV in the GGA cohesive energies is much smaller than the LDA error of 1.3 eV. The working of the GGA is analyzed in terms of the cohesive energy density and the charge inhomogeneity division of the exchange energy. The low-gradient limit in the GGA functionals is not important as regions with charge inhomogeneity s0.2 have negligible contributions. \textcopyright{} 1996 The American Physical Society.

D-band excitations in II-VI semiconductors: A broken-symmetry approach to the core hole.

Local-density approximation (LDA) band-structure calculations place the 3d band of zinc-blende ZnO, ZnS, ZnSe, and ZnTe at 5.4, 6.4, 6.8, and 7.5 eV below the valence-band maximum (VBM), while photoemission measurements place them at 7.8, 9.0, 9.4, and 9.8 eV below the VBM, respectively. We show that this \ensuremath{\sim}3-eV LDA error can be accounted for using a ``broken symmetry'' band-structure approach. In this approach, a d core hole is placed in an impuritylike splitoff d subband resulting from the creation of the hole on a particular Zn sublattice. Self-consistent solutions to such a constrained LDA problem reveal that the final hole state is sufficiently localized to trigger a self-interaction correction of 3--4 eV, needed to explain the discrepancy with experiment. This 3--4 eV shift is reduced, by screening effects, from the 9.7-eV value in a free Zn atom. Finally, we calculated the binding energy ${\mathit{E}}_{\mathrm{Mn}}$ for Mn 3d states in ZnTe:Mn and the effective Coulomb interaction parameter ${\mathit{U}}_{\mathrm{eff}}$. Significant improvements over the results of local-spin-density calculations were found. The calculated ${\mathit{E}}_{\mathrm{Mn}}$=${\mathit{E}}_{\mathrm{VBM}}$-3.93 eV and ${\mathit{U}}_{\mathrm{eff}}$=6.85 eV are in good agreement with experiments.

Pseudopotentials for non-local-density functionals.

Ab initio pseudopotentials for the IIA and IIB elements have been generated in the framework of density-functional theory, including gradient corrections to the local-density approximation (LDA). Their accuracy and transferability have been tested by extensive atomic computations. We applied these pseudopotentials to the evaluation of bonding properties of homonuclear dimers. Our molecular computations, not restricted to light elements, allow a wide assessment of the importance of gradient corrections in finite systems. For all the dimers considered here, the LDA error for bond energies is reduced by roughly 50%. The relative improvement on equilibrium distances and vibrational frequencies is less impressive, but still important and systematic. We discuss the computational cost for the evaluation of gradient corrections in computer programs based on the plane-wave (or plane-wave-like) basis-set expansion.

Bonding properties of IIA and IIB dimers in density functional theory with gradient corrections to the local density approximation

We have computed bonding properties of the IIA and IIB homonuclear dimers within the Pseudopotential — Density Functional scheme, including Gradient Corrections to the Local Density Approximation (LDA). Our computations confirm that Gradient Corrections offer a significant improvement over LDA for what concerns ground state properties of finite systems: The LDA error on bond energies is reduced by more than 50%. Although less impressive, the relative improvement on equilibrium distances and vibrational frequencies is also systematic and important. We briefly discuss the relevance of our results to the description of the non-metal to metal transition expected in clusters of these elements as a function of size.