Joachim Paier

Screened hybrid density functionals applied to solids

Hybrid Fock exchange/density functional theory functionals have shown to be very successful in describing a wide range of molecular properties. For periodic systems, however, the long-range nature of the Fock exchange interaction and the resultant large computational requirements present a major drawback. This is especially true for metallic systems, which require a dense Brillouin zone sampling. Recently, a new hybrid functional [HSE03, J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003)] that addresses this problem within the context of methods that evaluate the Fock exchange in real space was introduced. We discuss the advantages the HSE03 functional brings to methods that rely on a reciprocal space description of the Fock exchange interaction, e.g., all methods that use plane wave basis sets. Furthermore, we present a detailed comparison of the performance of the HSE03 and PBE0 functionals for a set of archetypical solid state systems by calculating lattice parameters, bulk moduli, heats of formation, and band gaps. The results indicate that the hybrid functionals indeed often improve the description of these properties, but in several cases the results are not yet on par with standard gradient corrected functionals. This concerns in particular metallic systems for which the bandwidth and exchange splitting are seriously overestimated.

The Perdew–Burke–Ernzerhof exchange-correlation functional applied to the G2-1 test set using a plane-wave basis set

Present local and semilocal functionals show significant errors, for instance, in the energetics of small molecules and in the description of band gaps. One possible solution to these problems is the introduction of exact exchange and hybrid functionals. A plane-wave-based algorithm was implemented in VASP (Vienna ab-initio simulation package) to allow for the calculation of the exact exchange. To systematically assess the precision of the present implementation, calculations for the 55 molecules of the G2-1 quantum chemical test set were performed applying the PBE and PBE0 functionals. Excellent agreement for both atomization energies and geometries compared with the results obtained by GAUSSIAN 03 calculations using large basis sets (augmented correlation consistent polarized valence quadruple zeta for the geometry optimization and augmented correlation-consistent polarized valence quintuple zeta for the energy calculations) was found. The mean absolute error for atomization energies between VASP and the experiment is 8.6 and 3.7 kcalmol, as calculated with the PBE and PBE0 functionals, respectively. The mean deviations between VASP and GAUSSIAN are 0.46 and 0.49 kcalmol for the PBE and PBE0 functionals, respectively.

Why does the B3LYP hybrid functional fail for metals

The B3LYP hybrid functional has shown to successfully predict a wide range of molecular properties. For periodic systems, however, the failure to attain the exact homogeneous electron gas limit as well as the semiempirical construction turns out to be a major drawback of the functional. We rigorously assess the B3LYP functional for solids through calculations of lattice parameters, bulk moduli, and thermochemical properties (atomization energies and reaction energies). The theoretical lattice constants overestimate the experimental ones by approximately 1%, and hence behave similarly to the PBE gradient-corrected exchange-correlation functional. B3LYP atomization energies of solids are drastically worse than those of nonempirical hybrid Hartree-Fock/density functionals (HF/DFT) such as PBE0 and HSE03. These large errors can be traced back to the lack of a proper description of free-electron-like systems with a significant itinerant character (metals and small gap semiconductors). Similar calculations using the popular semiempirical B3PW91 hybrid functional, which fulfills the uniform electron gas limit, show a clear improvement over B3LYP regarding atomization energies. Finally, theoretical values for heats of formation for both the B3LYP as well as the B3PW91 functionals are presented. These document a most likely fortuitously good agreement with experiment for the B3LYP hybrid functional.

Hybrid functionals applied to extended systems

We present an overview of the description of structural, thermochemical, and electronic properties of extended systems using several well known hybrid Hartree-Fock/density-functional-theory functionals (PBE0, HSE03, and B3LYP). In addition we address a few aspects of the evaluation of the Hartree-Fock exchange interactions in reciprocal space, relevant to all methods that employ a plane wave basis set and periodic boundary conditions.

Assessing the performance of recent density functionals for bulk solids

We assess the performance of recent density functionals for the exchange-correlation energy of a nonmolecular solid, by applying accurate calculations with the GAUSSIAN, BAND, and VASP codes to a test set of 24 solid metals and nonmetals. The functionals tested are the modified Perdew-Burke-Ernzerhof generalized gradient approximation PBEsol GGA, the second-order GGA SOGGA, and the Armiento-Mattsson 2005 AM05 GGA. For completeness, we also test more standard functionals: the local density approximation, the original PBE GGA, and the Tao-Perdew-Staroverov-Scuseria meta-GGA. We find that the recent density functionals for solids reach a high accuracy for bulk properties lattice constant and bulk modulus. For the cohesive energy, PBE is better than PBEsol overall, as expected, but PBEsol is actually better for the alkali metals and alkali halides. For fair comparison of calculated and experimental results, we consider the zeropoint phonon and finite-temperature effects ignored by many workers. We show how GAUSSIAN basis sets and inaccurate experimental reference data may affect the rating of the quality of the functionals. The results show that PBEsol and AM05 perform somewhat differently from each other for alkali metal, alkaline-earth metal, and alkali halide crystals where the maximum value of the reduced density gradient is about 2, but perform very similarly for most of the other solids where it is often about 1. Our explanation for this is consistent with the importance of exchange-correlation nonlocality in regions of core-valence overlap.

Second-order Møller–Plesset perturbation theory applied to extended systems. I. Within the projector-augmented-wave formalism using a plane wave basis set

We present an implementation of the canonical formulation of second-order Møller-Plesset (MP2) perturbation theory within the projector-augmented-wave method under periodic boundary conditions using a plane wave basis set. To demonstrate the accuracy of our approach we show that our result for the atomization energy of a LiH molecule at the Hartree-Fock+MP2 level is in excellent agreement with well converged Gaussian-type-orbital calculations. To establish the feasibility of employing MP2 perturbation theory in its canonical form to systems that are periodic in three dimensions we calculated the cohesive energy of bulk LiH.

Photochemistry of ethylene: A multireference configuration interaction investigation of the excited-state energy surfaces

Multireference configuration interaction with singles and doubles (MR-CISD) calculations have been performed for the optimization of conical intersections and stationary points on the ethylene excited-state energy surfaces using recently developed methods for the computation of analytic gradients and nonadiabatic coupling terms. Basis set dependence and the effect of various choices of reference spaces for the MR-CISD calculations have been investigated. The crossing seam between the S0 and S1 states has been explored in detail. This seam connects all conical intersections presently known for ethylene. Major emphasis has been laid on the hydrogen-migration path. Starting in the V state of twisted-orthogonal ethylene, a barrierless path to ethylidene was found. The feasibility of ethylidene formation will be important for the explanation of the relative yield of cis and trans H2 elimination.

The AM05 density functional applied to solids

We show that the AM05 functional [Armiento and Mattsson, Phys. Rev. B 72, 085108 (2005)] has the same excellent performance for solids as the hybrid density functionals tested in Paier et al. [J. Chem. Phys. 124, 154709 (2006); 125, 249901 (2006)]. This confirms the original finding that AM05 performs exceptionally well for solids and surfaces. Hartree-Fock hybrid calculations are typically an order of magnitude slower than local or semilocal density functionals such as AM05, which is of a regular semilocal generalized gradient approximation form. The performance of AM05 is on average found to be superior to selecting the best of local density approximation and PBE for each solid. By comparing data from several different electronic-structure codes, we have determined that the numerical errors in this study are equal to or smaller than the corresponding experimental uncertainties.

CO adsorption on metal surfaces: A hybrid functional study with plane-wave basis set

We present a detailed study of the adsorption of CO on Cu, Rh, and Pt (111) surfaces in top and hollow sites. The study has been performed using the local density approximation, the gradient corrected functional PBE, and the hybrid Hartree-Fock density functionals PBE0 and HSE03 within the framework of generalized Kohn-Sham density functional theory using a plane-wave basis set. As expected, the local density approximation and generalized gradient approximation functionals show a tendency to favor the hollow sites, at variance with experimental findings that give the top site as the most stable adsorption site. The PBE0 and HSE03 functionals reduce this tendency. In fact, they predict the correct adsorption site for Cu and Rh but fail for Pt. However, even in this case, the hybrid functional destabilizes the hollow site by

Assessment of correlation energies based on the random-phase approximation

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