Standard Generalized Gradient Approximation Research Articles

Excess electron states in reduced bulk anatase TiO2: Comparison of standard GGA, GGA+U, and hybrid DFT calculations

The removal of lattice O atoms, as well as the addition of interstitial H atoms, in TiO(2) is known to cause the reduction in the material and the formation of "Ti(3+)" ions. By means of electronic structure calculations we have studied the nature of such oxygen vacancy and hydrogen impurity states in the bulk of the anatase polymorph of TiO(2). The spin polarized nature of these centers, the localized or delocalized character of the extra electrons, the presence of defect-induced states in the gap, and the polaronic distortion around the defect have been investigated with different theoretical methods: standard density functional theory (DFT) in the generalized-gradient approximation (GGA), GGA+U methods as a function of the U parameter, and two hybrid functionals with different admixtures of Hartree-Fock exchange. The results are found to be strongly dependent on the method used. Only GGA+U or hybrid functionals are able to reproduce the presence of states at about 1 eV below the conduction band, which are experimentally observed in reduced titania. The corresponding electronic states are localized on Ti 3d levels, but partly delocalized solutions are very close in energy. These findings show the limited predictive power of these theoretical methods to describe the electronic structure of reduced titania in the absence of accurate experimental data.

Jahn–Teller distortion and electronic structure of LiMn 2O 4

LiMn 2O 4 is one of the important cathode material for rechargeable lithium batteries. Standard ab initio studies within local density approximation (LDA) and generalized gradient approximation (GGA) are not able to reproduce the intrinsic insulating electronic structure and Jahn–Teller-type local atomic distortion of LiMn 2O 4. In the present work, GGA+U method is shown to be able to localize electrons to Mn 3 d z 2 orbits in the majority spin channel, and open a gap of about 0.5 eV between the Mn 3 d z 2 and Mn 3 d x 2 − y 2 . Results from GGA+U calculation showed that Mn ions are in the mixed valence state of Mn 3+ and Mn 4+. This is in good agreements with experimental observations and different to Mn 3.5+ obtained from standard GGA calculations. While Jahn–Teller distortion is not observed in Mn 4+O 6 octahedra, it is clearly seen in Mn 3+O 6 octahedra, in which Mn O bond lengths along the z-axis direction are obviously elongated.

Electronic and magnetic properties of YBa 2Fe 3O 8 from a first-principles study

The electronic and magnetic properties of Y Ba 2Fe 3O 8 have been systematically investigated within the framework of density-functional theory using the standard generalized gradient approximation (GGA) as well as the GGA plus Hubbard U ( GGA + U ) method. The GGA results show that the G-type antiferromagnetic (AFM) state is preferred among the considered magnetic configurations. The striking ionic character is shown for Y and Ba atoms while very strong hybridization is found between Fe 3d and O 2p orbitals. Furthermore, the Fe–O–Fe superexchange interaction should be responsible for the stability of the AFM magnetic structure in this case. In addition, our theoretical calculations reveal that the ground state of Y Ba 2Fe 3O 8 is a strongly correlated charge-transfer insulator with a finite band gap above the Fermi level obtained by the GGA + U scheme, which is in agreement with the experimental observations.

First principles calculations of structural, electronic and optical properties of various phases of CaS

First principles calculations have been performed within the framework of density functional theory to investigate the structural, electronic and optical properties of all four possible B1, B2, B3 and B4 phases of CaS. Apart from the standard local density approximation (LDA) and GGA (PBE), a more accurate nonempirical density functional generalized gradient approximation (GGA), as proposed by Wu and Cohen [Phys. Rev. B 73, 235116 (2006)] for the exchange-correlation energy, EXC, has been attempted in these calculations. Calculated electronic structure and the density of states are analyzed in terms of the contribution of Ca d states and S s and p states in determining the nature of the fundamental band gap in various phases. Reflectivity, R (ω), the real and imaginary part of the dielectric functions, e(ω), have been calculated for all the phases and the results have been discussed and compared with the existing experimental data.

Origin of the light green color and electronic ground state ofLaCrO3

The electronic structure and optical properties of LaCrO3 have been calculated and the origin of the light green color of LaCrO3 has been identified. An optical gap of 1=3.4 eV was reproduced in agreement with experiment, which is a charge transfer gap. Another optical gap, 0=2.15 eV, which is not present in experiments has also been found and explains the light green color of this compound. The energy band gap between the top of the valence band and the bottom of the conduction band is =1.40 eV and has Mott character. Our calculated x-ray photoemission spectra XPS and x-ray absorption spectra at the oxygen K edge O-K XAS are in very good agreement with experiments. In contrast to previous reports, these results have been obtained using standard generalized gradient approximation calculations, while additional correlation effects for Cr 3d states due to an effective Hubbard U would result in a colorless LaCrO3 and a disagreement with XPS and XAS experiments. Thus, LaCrO3 should not be considered as highly correlated transition metal oxide, a conclusion also valid for many Cr3+ compounds.

Excitation energies in density functional theory: An evaluation and a diagnostic test

Electronic excitation energies are determined using the CAM-B3LYP Coulomb-attenuated functional [T. Yanai et al. Chem. Phys. Lett. 393, 51 (2004)], together with a standard generalized gradient approximation (GGA) and hybrid functional. The degree of spatial overlap between the occupied and virtual orbitals involved in an excitation is measured using a quantity Lambda, and the extent to which excitation energy errors correlate with Lambda is quantified. For a set of 59 excitations of local, Rydberg, and intramolecular charge-transfer character in 18 theoretically challenging main-group molecules, CAM-B3LYP provides by far the best overall performance; no correlation is observed between excitation energy errors and Lambda, reflecting the good quality, balanced description of all three categories of excitation. By contrast, a clear correlation is observed for the GGA and, to a lesser extent, the hybrid functional, allowing a simple diagnostic test to be proposed for judging the reliability of a general excitation from these functionals--when Lambda falls below a prescribed threshold, excitations are likely to be in very significant error. The study highlights the ambiguous nature of the term "charge transfer," providing insight into the observation that while many charge-transfer excitations are poorly described by GGA and hybrid functionals, others are accurately reproduced.

Double-hybrid density functional theory for excited electronic states of molecules

Double-hybrid density functionals are based on a mixing of standard generalized gradient approximations (GGAs) for exchange and correlation with Hartree-Fock (HF) exchange and a perturbative second-order correlation part (PT2) that is obtained from the Kohn-Sham (GGA) orbitals and eigenvalues. This virtual orbital-dependent functional (dubbed B2PLYP) contains only two empirical parameters that describe the mixture of HF and GGA exchange (ax) and of the PT2 and GGA correlation (ac), respectively. Extensive testing has recently demonstrated the outstanding accuracy of this approach for various ground state problems in general chemistry applications. The method is extended here without any further empirical adjustments to electronically excited states in the framework of time-dependent density functional theory (TD-DFT) or the closely related Tamm-Dancoff approximation (TDA-DFT). In complete analogy to the ground state treatment, a scaled second-order perturbation correction to configuration interaction with singles (CIS(D)) wave functions developed some years ago by Head-Gordon et al. [Chem. Phys. Lett. 219, 21 (1994)] is computed on the basis of density functional data and added to the TD(A)-DFTGGA excitation energy. The method is implemented by applying the resolution of the identity approximation and the efficiency of the code is discussed. Extensive tests for a wide variety of molecules and excited states (of singlet, triplet, and doublet multiplicities) including electronic spectra are presented. In general, rather accurate excitation energies (deviations from reference data typically <0.2 eV) are obtained that are mostly better than those from standard functionals. Still, systematic errors are obtained for Rydberg (too low on average by about 0.3 eV) and charge-transfer transitions but due to the relatively large ax parameter (0.53), B2PLYP outperforms most other functionals in this respect. Compared to conventional HF-based CIS(D), the method is more robust in electronically complex situations due to the implicit account of static correlation effects by the GGA parts. The (D) correction often works in the right direction and compensates for the overestimation of the transition energy at the TD level due to the elevated fraction of HF exchange in the hybrid GGA part. Finally, the limitations of the method are discussed for challenging systems such as transition metal complexes, cyanine dyes, and multireference cases.

Van der Waals density functional: Self-consistent potential and the nature of the van der Waals bond

We derive the exchange-correlation potential corresponding to the nonlocal van der Waals density functional [M. Dion, H. Rydberg, E. Schroder, D. C. Langreth, and B. I. Lundqvist, Phys. Rev. Lett. 92, 246401 (2004)]. We use this potential for a self-consistent calculation of the ground state properties of a number of van der Waals complexes as well as crystalline silicon. For the latter, where little or no van der Waals interaction is expected, we find that the results are mostly determined by semilocal exchange and correlation as in standard generalized gradient approximations (GGA), with the fully nonlocal term giving little effect. On the other hand, our results for the van der Waals complexes show that the self-consistency has little effect at equilibrium separations. This finding validates previous calculations with the same functional that treated the fully nonlocal term as a post GGA perturbation. A comparison of our results with wave-function calculations demonstrates the usefulness of our approach. The exchange-correlation potential also allows us to calculate Hellmann-Feynman forces, hence providing the means for efficient geometry relaxations as well as unleashing the potential use of other standard techniques that depend on the self-consistent charge distribution. The nature of the van der Waals bond is discussed in terms of the self-consistent bonding charge.

Semiempirical hybrid density functional with perturbative second-order correlation

A new hybrid density functional for general chemistry applications is proposed. It is based on a mixing of standard generalized gradient approximations (GGAs) for exchange by Becke (B) and for correlation by Lee, Yang, and Parr (LYP) with Hartree-Fock (HF) exchange and a perturbative second-order correlation part (PT2) that is obtained from the Kohn-Sham (GGA) orbitals and eigenvalues. This virtual orbital-dependent functional contains only two global parameters that describe the mixture of HF and GGA exchange (a(x)) and of the PT2 and GGA correlation (c), respectively. The parameters are obtained in a least-squares-fit procedure to the G297 set of heat of formations. Opposed to conventional hybrid functionals, the optimum a(x) is found to be quite large (53% with c=27%) which at least in part explains the success for many problematic molecular systems compared to conventional approaches. The performance of the new functional termed B2-PLYP is assessed by the G297 standard benchmark set, a second test suite of atoms, molecules, and reactions that are considered as electronically very difficult (including transition-metal compounds, weakly bonded complexes, and reaction barriers) and comparisons with other hybrid functionals of GGA and meta-GGA types. According to many realistic tests, B2-PLYP can be regarded as the best general purpose density functional for molecules (e.g., a mean absolute deviation for the two test sets of only 1.8 and 3.2 kcal/mol compared to about 3 and 5 kcal/mol, respectively, for the best other density functionals). Very importantly, also the maximum and minimum errors (outliers) are strongly reduced (by about 10-20 kcal/mol). Furthermore, very good results are obtained for transition state barriers but unlike previous attempts at such a good description, this definitely comes not at the expense of equilibrium properties. Preliminary calculations of the equilibrium bond lengths and harmonic vibrational frequencies for diatomic molecules and transition-metal complexes also show very promising results. The uniformity with which B2-PLYP improves for a wide range of chemical systems emphasizes the need of (virtual) orbital-dependent terms that describe nonlocal electron correlation in accurate exchange-correlation functionals. From a practical point of view, the new functional seems to be very robust and it is thus suggested as an efficient quantum chemical method of general purpose.

Density Functional Theory Study of Tetrathiafulvalene and Thianthrene in Acetonitrile: Structure, Dynamics, and Redox Properties

The redox potentials of the organic compounds tetrathiafulvalene (TTF) and thianthrene (TH) in an explicit aprotic polar solvent, acetonitrile, have been computed using ab initio molecular dynamics simulation based on a Gaussian basis set methodology. The density functional description of the pure solvent yields a diffuse and mobile liquid, with structural and dynamical properties that are in good agreement with earlier classical models and experiment. Molecular dynamics simulation of both solute species in their neutral and radical cation states combined with free energy difference calculations result in estimates for the redox potentials of the reactions TH*+ + TTF --> TH + TTF*+ and TH2+ + TTF*+ --> TH*+ + TTF2+. The obtained values are 0.95 +/- 0.06 and 1.09 +/- 0.06 V, respectively, in excellent agreement with experimental data of 0.93 and 1.08 V. Our computational approach is based on Marcus theory, assuming quadratic free energy surfaces. We show that this approximation can still be accurate in systems, such as TH, that undergo a significant change in geometry upon oxidation. Furthermore, despite the different localization of the spin density in the radical cations, results based on self-interaction-corrected functionals and on standard generalized gradient approximations are identical to within 10 meV.

Ground- and excited-state diatomic bond lengths, vibrational levels, and potential-energy curves from conventional and localized Hartree–Fock-based density-functional theory

Ground- and excited-state diatomic bond lengths, vibrational levels, and potential-energy curves are determined using conventional and localized Hartree-Fock (LHF)-based density-functional theory. Exchange only and hybrid functionals (with various fractions of exchange) are considered, together with a standard generalized gradient approximation (GGA). Ground-state bond lengths and vibrational wave numbers are relatively insensitive to whether orbital exchange is treated using the conventional or LHF approach. Excited-state calculations are much more sensitive. For a standard fraction of orbital exchange, N2 and CO vertical excitation energies at experimental bond lengths are accurately described by both conventional and LHF-based approaches, providing an asymptotic correction is present. Excited-state bond lengths and vibrational levels are more accurate with the conventional approach. The best quality, however, is obtained with an asymptotically corrected GGA functional. For the ground and lowest four singlet excited states, the GGA mean absolute errors in bond lengths are 0.006 A (0.5%) and 0.011 A (0.8%) for N2 and CO, respectively. Mean absolute errors in fundamental vibrational wavenumbers are 49 cm(-1) (2.7%) and 68 cm(-1) (5.0%), respectively. The GGA potential-energy curves are compared with near-exact Rydberg-Klein-Rees curves. Agreement is very good for the ground and first excited state, but deteriorates for the higher states.

The electronic structure and band gap of LiFePO 4 and LiMnPO 4

Materials with the olivine Li x MPO 4 structure form an important new class of materials for rechargeable Li batteries. There is significant interest in their electronic properties because of the importance of electronic conductivity in batteries for high-rate applications. The density of states of Li x MPO 4 ( x=0, 1 and M=Fe, Mn) has been determined with the ab initio generalized gradient approximation (GGA)+ U method, appropriate for these correlated electron systems. Computed results are compared with the optical gap of LiFePO 4, as measured using UV–Vis-NIR diffuse reflectance spectroscopy. The results obtained from experiment (3.8–4.0 eV) and GGA+ U computations (3.7 eV) are in very good agreement. However, standard GGA, without the same level of treatment of electron correlation, is shown to make large errors in predicting the electronic structure. It is argued that olivines are likely to be polaronic conductors with extrinsically determined carrier levels and that their electronic conductivity is therefore not simply related to the band gap.

Improved Description of Chemical Barriers with Generalized Gradient Approximations (GGAs) and Meta-GGAs

The performance of exchange and correlation (xc) functionals of the generalized gradient approximation (GGA) type and of the meta-GGA type in the calculation of chemical reactions is related to topological features of the electron density which, in turn, are connected to the orbital structure of chemical bonds within the Kohn−Sham (KS) theory. Seventeen GGA and meta-GGA xc functionals are assessed for 15 hydrogen abstraction reactions and 3 symmetrical SN2 reactions. Systems that are problematic for standard GGAs characteristically have enhanced values of the dimensionless gradient argument sσ2 with local maxima in the bonding region. The origin of this topological feature is the occupation of valence KS orbitals with an antibonding or essentially nonbonding character. The local enhancement of sσ2 yields too negative exchange−correlation energies with standard GGAs for the transition state of the SN2 reaction, which leads to the reduced calculated reaction barriers. The unwarranted localization of the effective xc hole of the standard GGAs, i.e., the nondynamical correlation that is built into them but is spurious in this case, wields its effect by their sσ2 dependence. Barriers are improved for xc functionals with the exchange functional OPTX as x component, which has a modified dependence on sσ2. Standard GGAs also underestimate the barriers for the hydrogen abstraction reactions. In this case the barriers are improved by correlation functionals, such as the Laplacian-dependent (LAP3) functional, which has a modified dependence on the Coulomb correlation of the opposite- and like-spin electrons. The best overall performance is established for the combination OLAP3 of OPTX and LAP3.

Functionals of the square kinetic energy density

We construct exchange energy functionals that depend on the non-interacting square kinetic energy density τ2(r)=14∑k|Δφk(r)|2 rather than the electron density. Since τ2 is a non-local functional of the electron density, it may describe non-local effects beyond standard generalized gradient approximation models. These effects may be essential in dealing with such a non-local quantity as the exchange energy. The problem of v-representability of τ2 for a slightly perturbed electron gas and Coulomb systems is discussed. A gradient expansion technique that permits construction of gradient-corrected functionals of τ2 is developed, and the performance of the new exchange functionals is analyzed.

Exchange potential from the common energy denominator approximation for the Kohn–Sham Green’s function: Application to (hyper)polarizabilities of molecular chains

An approximate Kohn–Sham (KS) exchange potential vxσCEDA is developed, based on the common energy denominator approximation (CEDA) for the static orbital Green’s function, which preserves the essential structure of the density response function. vxσCEDA is an explicit functional of the occupied KS orbitals, which has the Slater vSσ and response vrespσCEDA potentials as its components. The latter exhibits the characteristic step structure with “diagonal” contributions from the orbital densities |ψiσ|2, as well as “off-diagonal” ones from the occupied–occupied orbital products ψiσψj(≠1)σ*. Comparison of the results of atomic and molecular ground-state CEDA calculations with those of the Krieger–Li–Iafrate (KLI), exact exchange (EXX), and Hartree–Fock (HF) methods show, that both KLI and CEDA potentials can be considered as very good analytical “closure approximations” to the exact KS exchange potential. The total CEDA and KLI energies nearly coincide with the EXX ones and the corresponding orbital energies εiσ are rather close to each other for the light atoms and small molecules considered. The CEDA, KLI, EXX–εiσ values provide the qualitatively correct order of ionizations and they give an estimate of VIPs comparable to that of the HF Koopmans’ theorem. However, the additional off-diagonal orbital structure of vxσCEDA appears to be essential for the calculated response properties of molecular chains. KLI already considerably improves the calculated (hyper)polarizabilities of the prototype hydrogen chains Hn over local density approximation (LDA) and standard generalized gradient approximations (GGAs), while the CEDA results are definitely an improvement over the KLI ones. The reasons of this success are the specific orbital structures of the CEDA and KLI response potentials, which produce in an external field an ultranonlocal field-counteracting exchange potential.