Exact-exchange Methods Research Articles

Assessment of quality and reliability of band structures from exact-exchange-only Kohn–Sham, hybrid, and GW methods

Band gaps for a large number of semiconductors obtained with the exact-exchange(EXX)-only Kohn–Sham method are compared to those from various generalized Kohn–Sham methods based on hybrid functionals and with experimental values. EXX band gaps are shown to be of superior quality than the band gaps from most of the considered hybrid methods, band gaps obtained with the hybrid functional HSE06 have a similar quality as EXX ones. The EXX method is found to yield d -electron binding energies that are clearly superior to those determined within the local density approximation and to be comparable to d -electron binding energies from GW approaches. EXX plane-wave pseudopotential calculations and EXX all-electron calculations are shown to yield very similar KS transition energies demonstrating that the plane-wave pseudopotential approach is well-suited for EXX methods. Previous discrepancies between pseudopotential and all-electron calculations, most likely, have to be attributed to technical problems in early all-electron EXX calculations.

Improved electronic structure and magnetic exchange interactions in transition metal oxides

We discuss the application of the Agapito Curtarolo and Buongiorno Nardelli (ACBN0) pseudo-hybrid Hubbard density functional to several transition metal oxides. For simple binary metal oxides, ACBN0 is found to be a fast, reasonably accurate and parameter-free alternative to traditional DFT + U and hybrid exact exchange methods. In ACBN0, the Hubbard energy of DFT + U is calculated via the direct evaluation of the local Coulomb and exchange integrals in which the screening of the bare Coulomb potential is accounted for by a renormalization of the density matrix. We demonstrate the success of the ACBN0 approach for the electronic properties of a series technologically relevant mono-oxides (MnO, CoO, NiO, FeO, both at equilibrium and under pressure). We also present results on two mixed valence compounds, Co3O4 and Mn3O4. Our results for these binary oxides and all the materials we have investigated, obtained at the computational cost of a standard LDA/PBE calculation, are in excellent agreement with hybrid functionals, the GW approximation and experimental measurements.

Stability conditions for exact-exchange Kohn-Sham methods and their relation to correlation energies from the adiabatic-connection fluctuation-dissipation theorem.

The occurrence of instabilities, in particular singlet-triplet and singlet-singlet instabilities, in the exact-exchange (EXX) Kohn-Sham method is investigated. Hessian matrices of the EXX electronic energy with respect to the expansion coefficients of the EXX effective Kohn-Sham potential in an auxiliary basis set are derived. The eigenvalues of these Hessian matrices determine whether or not instabilities are present. Similar as in the corresponding Hartree-Fock case instabilities in the EXX method are related to symmetry breaking of the Hamiltonian operator for the EXX orbitals. In the EXX methods symmetry breaking can easily be visualized by displaying the local multiplicative exchange potential. Examples (N2, O2, and the polyyne C10H2) for instabilities and symmetry breaking are discussed. The relation of the stability conditions for EXX methods to approaches calculating the Kohn-Sham correlation energy via the adiabatic-connection fluctuation-dissipation (ACFD) theorem is discussed. The existence or nonexistence of singlet-singlet instabilities in an EXX calculation is shown to indicate whether or not the frequency-integration in the evaluation of the correlation energy is singular in the EXX-ACFD method. This method calculates the Kohn-Sham correlation energy through the ACFD theorem theorem employing besides the Coulomb kernel also the full frequency-dependent exchange kernel and yields highly accurate electronic energies. For the case of singular frequency-integrands in the EXX-ACFD method a regularization is suggested. Finally, we present examples of molecular systems for which the self-consistent field procedure of the EXX as well as the Hartree-Fock method can converge to more than one local minimum depending on the initial conditions.

Hyperfine coupling constants of the nitrogen and phosphorus atoms: A challenge for exact-exchange density-functional and post-Hartree–Fock methods

The isotropic hyperfine coupling constants of the free N(S4) and P(S4) atoms have been evaluated with high-level post-Hartree–Fock and density-functional methods. The phosphorus hyperfine coupling presents a significant challenge to both types of methods. With large basis sets, MP2 and coupled-cluster singles and doubles calculations give much too small values for the phosphorus atom. Triple excitations are needed in coupled-cluster calculations to achieve reasonable agreement with experiment. None of the standard density functionals reproduce even the correct sign of this hyperfine coupling. Similarly, the computed hyperfine couplings depend crucially on the self-consistent treatment in exact-exchange density-functional theory within the optimized effective potential (OEP) method. Well-balanced auxiliary and orbital basis sets are needed for basis-expansion exact-exchange-only OEP approaches to come close to Hartree–Fock or numerical OEP data. Results from the localized Hartree–Fock and Krieger–Li–Iafrate approximations deviate notably from exact OEP data in spite of very similar total energies. Of the functionals tested, only full exact-exchange methods augmented by a correlation functional gave at least the correct sign of the P(S4) hyperfine coupling but with too low absolute values. The subtle interplay between the spin-polarization contributions of the different core shells has been analyzed, and the influence of even very small changes in the exchange-correlation potential could be identified.

Density-Functional Theory with Orbital-Dependent Functionals: Exact-exchange Kohn-Sham and Density-Functional Response Methods

Abstract Kohn-Sham methods with orbital-dependent exchange-correlation functionals, in particular exact-exchange methods, are discussed. The numerical stability of optimized-effective-potential methods to construct the exchange-correlation potential in the case of orbital-dependent energy functionals is considered. A numerically stable exact-exchange Kohn-Sham method is briefly presented. A new ansatz for time-dependent density-functional methods in the response regime is considered and an exact-exchange time-dependent density-functional method based on it is discussed and it is demonstrated that this methods is able to treat qualitatively correctly charge-transfer excitations.

Understanding and correcting the self-interaction error in the electrical response of hydrogen chains

Semilocal density functionals such as the local-spin-density and generalized-gradient approximations are known to overestimate the polarizabilities and especially the hyperpolarizabilities of long-chain molecules, the latter by as much as a factor of 10 or more in model hydrogen chains. These quantities are much better predicted by exact-exchange methods such as Hartree-Fock or optimized effective potential. We show here that the semilocal functionals, after full or scaled-down Perdew-Zunger self-interaction correction (SIC), are about as good as the exact-exchange methods for these quantities. As is the case for the exact-exchange methods, SIC is fully nonlocal and exact for all one-electron densities, and (more relevantly to the electrical response) tends to maintain an integer number of electrons on each ${\mathrm{H}}_{2}$ chain unit to a greater extent than the semilocal functionals do. In this study, the SIC energy is minimized directly, without an optimized effective potential.

Electrical response of molecular chains in density functional theory: Ultranonlocal response from a semilocal functional

An exchange potential functional is constructed from semi-local quantities and is shown to reproduce hydrogen chain polarizabilities with the same accuracy as exact exchange methods. We discuss the exchange potential features that are essential for accurate polarizability calculations, i.e., derivative discontinuities and the potential step structure. The possibility of a future generalization of the methods into a complete semi-local exchange-correlation functional is discussed.

General treatment of the singularities in Hartree-Fock and exact-exchange Kohn-Sham methods for solids

We present a general scheme for treating the integrable singular terms within exact exchange (EXX) Kohn-Sham or Hartree-Fock (HF) methods for periodic solids. We show that the singularity corrections for treating these divergencies depend only on the total number and the positions of $\mathbf{k}$ points and on the lattice vectors, in particular, the unit cell volume, but not on the particular positions of atoms within the unit cell. The method proposed here to treat the singularities constitutes a stable, simple to implement, and general scheme that can be applied to systems with arbitrary lattice parameters within either the EXX Kohn-Sham or the HF formalism. We apply the singularity correction to a typical symmetric structure, diamond, and to a more general structure, trans-polyacetylene. We consider the effect of the singularity corrections on volume optimizations and $\mathbf{k}$-point convergence. While the singularity correction clearly depends on the total number of $\mathbf{k}$ points, it exhibits a remarkably small dependence upon the choice of the specific arrangement of the $\mathbf{k}$ points.

Theoretical modelling of intermediate band solar cell materials based on metal-doped chalcopyrite compounds

Electronic structure calculations are carried out for CuGaS 2 partially substituted with Ti, V, Cr or Mn to ascertain if some of these systems could provide an intermediate band material able to give a high efficiency photovoltaic cell. Trends in electronic level positions are analyzed and more accurate advanced theory levels (exact exchange or Hubbard-type methods) are used in some cases. The Ti-substituted system seems more likely to yield an intermediate band material with the desired properties, and furthermore seems realizable from the thermodynamic point of view, while those with Cr and Mn might give half-metal structures with applications in spintronics.

Competing crystal structures inLa0.5Ca0.5MnO3: Conventional charge order versus Zener polarons

Equilibrium crystal structures for La0.5Ca0.5MnO3 have been calculated using hybrid exact exchange and density functional methods. Two distinct ground states with either conventional checkerboard charge ordering or Zener polaron formation are found depending on the proportion of exact exchange used. The checkerboard state has mixed x^2-y^2, 3x^2-r^2 and 3y^2-r^2 orbital ordering and CE-type magnetic order while the Zener polaron state has 3x^2-r^2 and 3y^2-r^2 ordering and A-type magnetic order.

Asymptotic behavior of the Kohn-Sham exchange potential.

The Kohn-Sham exchange potential of finite systems is shown to approach different asymptotic limits on nodal surfaces of the energetically highest-occupied orbital than in other regions. This leads to barrier-well structures in the near asymptotic region, which have a strong influence on virtual orbitals and thus on excitation energies. Common approximations for the exchange potential do not exhibit these features. These asymptotic structures, however, can be correctly described by effective exact-exchange methods. Conditions for the presence of an asymptotic barrier well in the full exchange-correlation potential are discussed.

Calculation of ionization energies, electron affinities, electronegativities, and hardnesses using density functional methods

The performance of two exact exchange methods is tested in the calculation of ionization energies, electron affinities, electronegativities, and hardnesses using Dunning’s correlation consistent basis sets. Comparison is made to experiment and other density functional methods, including the local density approximation and two gradient corrected functionals. The obtained electronegativities and hardnesses are also compared with high level coupled cluster results. Both the exact exchange methods show an excellent performance in the calculation of all four properties, yielding mean absolute deviations from experiment below 0.20 eV for all basis sets.