Martijn Marsman

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.

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.

Accurate surface and adsorption energies from many-body perturbation theory

Kohn-Sham density functional theory is the workhorse computational method in materials and surface science. Unfortunately, most semilocal density functionals predict surfaces to be more stable than they are experimentally. Naively, we would expect that consequently adsorption energies on surfaces are too small as well, but the contrary is often found: chemisorption energies are usually overestimated. Modifying the functional improves either the adsorption energy or the surface energy but always worsens the other aspect. This suggests that semilocal density functionals possess a fundamental flaw that is difficult to cure, and alternative methods are urgently needed. Here we show that a computationally fairly efficient many-electron approach, the random phase approximation to the correlation energy, resolves this dilemma and yields at the same time excellent lattice constants, surface energies and adsorption energies for carbon monoxide and benzene on transition-metal surfaces.

Reproducibility in density functional theory calculations of solids

A comparison of DFT methods Density functional theory (DFT) is now routinely used for simulating material properties. Many software packages are available, which makes it challenging to know which are the best to use for a specific calculation. Lejaeghere et al. compared the calculated values for the equation of states for 71 elemental crystals from 15 different widely used DFT codes employing 40 different potentials (see the Perspective by Skylaris). Although there were variations in the calculated values, most recent codes and methods converged toward a single value, with errors comparable to those of experiment. Science, this issue p. 10.1126/science.aad3000; see also p. 1394 A survey of recent density functional theory methods shows a convergence to more accurate property calculations. [Also see Perspective by Skylaris] INTRODUCTION The reproducibility of results is one of the underlying principles of science. An observation can only be accepted by the scientific community when it can be confirmed by independent studies. However, reproducibility does not come easily. Recent works have painfully exposed cases where previous conclusions were not upheld. The scrutiny of the scientific community has also turned to research involving computer programs, finding that reproducibility depends more strongly on implementation than commonly thought. These problems are especially relevant for property predictions of crystals and molecules, which hinge on precise computer implementations of the governing equation of quantum physics. RATIONALE This work focuses on density functional theory (DFT), a particularly popular quantum method for both academic and industrial applications. More than 15,000 DFT papers are published each year, and DFT is now increasingly used in an automated fashion to build large databases or apply multiscale techniques with limited human supervision. Therefore, the reproducibility of DFT results underlies the scientific credibility of a substantial fraction of current work in the natural and engineering sciences. A plethora of DFT computer codes are available, many of them differing considerably in their details of implementation, and each yielding a certain “precision” relative to other codes. How is one to decide for more than a few simple cases which code predicts the correct result, and which does not? We devised a procedure to assess the precision of DFT methods and used this to demonstrate reproducibility among many of the most widely used DFT codes. The essential part of this assessment is a pairwise comparison of a wide range of methods with respect to their predictions of the equations of state of the elemental crystals. This effort required the combined expertise of a large group of code developers and expert users. RESULTS We calculated equation-of-state data for four classes of DFT implementations, totaling 40 methods. Most codes agree very well, with pairwise differences that are comparable to those between different high-precision experiments. Even in the case of pseudization approaches, which largely depend on the atomic potentials used, a similar precision can be obtained as when using the full potential. The remaining deviations are due to subtle effects, such as specific numerical implementations or the treatment of relativistic terms. CONCLUSION Our work demonstrates that the precision of DFT implementations can be determined, even in the absence of one absolute reference code. Although this was not the case 5 to 10 years ago, most of the commonly used codes and methods are now found to predict essentially identical results. The established precision of DFT codes not only ensures the reproducibility of DFT predictions but also puts several past and future developments on a firmer footing. Any newly developed methodology can now be tested against the benchmark to verify whether it reaches the same level of precision. New DFT applications can be shown to have used a sufficiently precise method. Moreover, high-precision DFT calculations are essential for developing improvements to DFT methodology, such as new density functionals, which may further increase the predictive power of the simulations. Recent DFT methods yield reproducible results. Whereas older DFT implementations predict different values (red darts), codes have now evolved to mutual agreement (green darts). The scoreboard illustrates the good pairwise agreement of four classes of DFT implementations (horizontal direction) with all-electron results (vertical direction). Each number reflects the average difference between the equations of state for a given pair of methods, with the green-to-red color scheme showing the range from the best to the poorest agreement. The widespread popularity of density functional theory has given rise to an extensive range of dedicated codes for predicting molecular and crystalline properties. However, each code implements the formalism in a different way, raising questions about the reproducibility of such predictions. We report the results of a community-wide effort that compared 15 solid-state codes, using 40 different potentials or basis set types, to assess the quality of the Perdew-Burke-Ernzerhof equations of state for 71 elemental crystals. We conclude that predictions from recent codes and pseudopotentials agree very well, with pairwise differences that are comparable to those between different high-precision experiments. Older methods, however, have less precise agreement. Our benchmark provides a framework for users and developers to document the precision of new applications and methodological improvements.

Making the random phase approximation to electronic correlation accurate

We show that the inclusion of second-order screened exchange to the random phase approximation allows for an accurate description of electronic correlation in atoms and solids clearly surpassing the random phase approximation, but not yet approaching chemical accuracy. From a fundamental point of view, the method is self-correlation free for one-electron systems. From a practical point of view, the approach yields correlation energies for atoms, as well as for the jellium electron gas within a few kcal/mol of exact values, atomization energies within typically 2-3 kcal/mol of experiment, and excellent lattice constants for ionic and covalently bonded solids (0.2% error). The computational complexity is only O(N(5)), comparable to canonical second-order Møller-Plesset perturbation theory, which should allow for routine calculations on many systems.

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.

Structural and magnetic isomers of small Pd and Rh clusters: an ab initio density functional study

We present a comprehensive investigation of the structural, electronic, and magnetic properties of PdN and RhN clusters with up to N = 13 atoms, based on ab initio density functional calculations. The novel aspects of our investigation are the following. (i) The structural optimization of the cluster by a symmetry-unconstrained static total-energy minimization has been supplemented for larger clusters (N≥7) with a search for the ground state structure by dynamical simulated annealing. The dynamical structural optimization has led to the discovery of highly unexpected ground state configurations. (ii) The spin-polarized calculations were performed in a fixed-moment mode. This allowed us to study coexisting magnetic isomers and led to a deeper insight into the importance of magnetostructural effects. For both Pd and Rh the larger clusters adopt ground state structures that can be considered as fragments of the face-centred cubic crystal structure of the bulk phase. For Pd clusters, the magnetic ground state is a spin triplet (S = 1) for N≤9, a spin quintuplet (S = 2) for N = 10, and a spin septet (S = 3) for the largest clusters. Large magnetic moments with up to S = 8 have been found for Rh clusters. Magnetic energy differences have been found to be small, such that there is an appreciable probability of finding excited magnetic isomers even at ambient temperatures. In several cases, the structural energy differences are also sufficiently small to permit the coexistence of polytypes.

Second-order Møller–Plesset perturbation theory applied to extended systems. II. Structural and energetic properties

Results for the lattice constants, atomization energies, and band gaps of typical semiconductors and insulators are presented for Hartree-Fock and second-order Moller-Plesset perturbation theory (MP2). We find that MP2 tends to undercorrelate weakly polarizable systems and overcorrelates strongly polarizable systems. As a result, lattice constants are overestimated for large gap systems and underestimated for small gap systems. The volume dependence of the MP2 correlation energy and the dependence of the MP2 band gaps on the static dielectric screening properties are discussed in detail. Moreover, the relationship between MP2 and the G(0)W(0) quasiparticle energies is elucidated and discussed. Finally, we demonstrate explicitly that the correlation energy diverges with decreasing k-point spacing for metals.

The multiferroic phase of DyFeO3: an ab initio study

By performing accurate ab initio density functional theory (DFT) calculations, we study the role of 4f electrons in stabilizing the magnetic-field-induced ferroelectric state of DyFeO3. We confirm that the ferroelectric polarization is driven by an exchange-strictive mechanism, working between adjacent spin-polarized Fe and Dy layers, as suggested by Y Tokunaga (2008 Phys. Rev. Lett. 101 097205). A careful electronic structure analysis suggests that coupling between Dy and Fe spin sublattices is mediated by Dy–d and O–2p hybridization. Our results are robust with respect to the different computational schemes used for d and f localized states, such as the DFT+U method, the Heyd–Scuseria–Ernzerhof (HSE) hybrid functional and the GW approach. Our findings indicate that the interaction between the f and d sublattices might be used to tailor the ferroelectric and magnetic properties of multiferroic compounds.