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 GGA͒, the second-order GGA ͑SOGGA͒, and the Armiento-Mattsson 2005 ͑AM05͒ 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 modulus͒. 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 2͒, but perform very similarly for most of the other solids ͑where it is often about 1͒. Our explanation for this is consistent with the importance of exchange-correlation nonlocality in regions of core-valence overlap.

A parametrization scheme for the electronic part of the density-functional based tight-binding (DFTB) method that covers the periodic table is presented. A semiautomatic parametrization scheme has been developed that uses Kohn-Sham energies and band structure curvatures of real and fictitious homoatomic crystal structures as reference data. A confinement potential is used to tighten the Kohn-Sham orbitals, which includes two free parameters that are used to optimize the performance of the method. The method is tested on more than 100 systems and shows excellent overall performance.

We report that an external electric field applied normal to bilayers of transition-metal dichalcogenides T X 2 (T = Mo, W, X = S, Se) creates significant spin-orbit splittings and reduces the electronic band gap linearly with the field strength. Contrary to the T X 2 monolayers, spin-orbit splittings and valley polarization are absent in bilayers due to the presence of inversion symmetry. This symmetry can be broken by an electric field, and the spin-orbit splittings in the valence band quickly reach values similar to those in the monolayers (145 meV for MoS 2 , . . . , 418 meV for WSe 2 ) at saturation fields less than 500 mVÅ −1 . The band gap closure results in a semiconductor-metal transition at field strength between 1.25 (WX 2 ) and 1.50 (MoX 2 ) VÅ −1 . Thus, by using a gate voltage, the spin polarization can be switched on and off in T X 2 bilayers, thus activating them for spintronic and valleytronic applications.

In the first part of this series, we presented a parametrization strategy to obtain high-quality electronic band structures on the basis of density-functional-based tight-binding (DFTB) calculations and published a parameter set called QUASINANO2013.1. Here, we extend our parametrization effort to include the remaining terms that are needed to compute the total energy and its gradient, commonly referred to as repulsive potential. Instead of parametrizing these terms as a two-body potential, we calculate them explicitly from the DFTB analogues of the Kohn-Sham total energy expression. This strategy requires only two further numerical parameters per element. Thus, the atomic configuration and four real numbers per element are sufficient to define the DFTB model at this level of parametrization. The QUASINANO2015 parameter set allows the calculation of energy, structure, and electronic structure of all systems composed of elements ranging from H to Ca. Extensive benchmarks show that the overall accuracy of QUASINANO2015 is comparable to that of well-established methods, including PM7 and hand-tuned DFTB parameter sets, while coverage of a much larger range of chemical systems is available.

In a standard Kohn-Sham density functional calculation, the total energy of a crystal at zero temperature is evaluated for a perfect static lattice of nuclei, and minimized with respect to the lattice constant. Sometimes a zero-point vibrational energy, whose anharmonicity expands the minimizing or equilibrium lattice constant, is included in the calculation or (as here) used to correct the experimental reference value for the lattice constant to that for a static lattice. A simple model for this correction, based on the Debye and Dugdale-MacDonald approximations, requires as input only readily-available parameters of the equation of state, plus the experimental Debye temperature. However, due in particular to the rough Dugdale-MacDonald estimation of Grüneisen parameters for diatomic solids, this simple model is found to overestimate the correction by about a factor of two for some solids in the diamond and zinc-blende structures. Using the quasi-harmonic phonon frequencies calculated from density functional perturbation theory gives a more accurate zero-point anharmonic expansion (ZPAE) correction. The error statistics for the lattice constants of various semilocal density functionals for the exchange-correlation energy are however little changed by improving the ZPAE correction. The PerdewBurke-Ernzerhof generalized gradient approximation (GGA) for solids (PBEsol) and the revised TaoPerdew-Staroverov-Scuseria (revTPSS) meta-GGA, the latter implemented here selfconsistently in BAND, applied to a test set of 58 solids, remain the most accurate of the functionals tested, with mean absolute relative errors below 0.7% for the lattice constants. The most positive and most negative revTPSS relative errors tend to occur for solids where full nonlocality (missing from revTPSS) may be most important.

The influence of an external electric field on single-layer transition-metal dichalcogenides TX 2 with T = Mo, W and X = S, Se (MoWSeS) have been investigated by means of density-functional theory within two-dimensional periodic boundary conditions under consideration of relativistic effects including the spin-orbit interactions. Our results show that the external field modifies the band structure of the monolayers, in particular the conduction band. This modification has, however, very little influence on the band gap and effective masses of holes and electrons at the K point, and also the spin-orbit splitting of these monolayers is almost unaffected. Our results indicate a remarkable stability of the electronic properties of TX 2 monolayers with respect to gate voltages. A reduction of the electronic band gap is observed only starting from field strengths of 2.0 VÅ −1 (3.5 VÅ −1 ) for selenides (sulphides), and the transition to a metallic phase would occur at fields of 4.5Å −1 (6.5Å −1 ).

There are small difference between our calculated lattice constants of the soft solids in Table VI and those in Table III. For the soft solids, the curve of energy versus lattice constant is very flat, and the equilibrium lattice constant is hard to resolve. We believe that the results in Table VI are more precise, because for them the integration for the energy was more accurate (accint = 12 in the BAND code) and because these accurate energies were fitted over a narrower range of lattice constants to the chosen equation of state (which is most reliable near equilibrium). In Table III, the energies of the soft solids displayed some small noise and so their equilibrium lattice constants could only be found by fitting to the equation of state over a larger range.

Pair atomic density fitting (PADF) has been identified as a promising strategy to reduce the scaling with system size of quantum chemical methods for the calculation of the correlation energy like the direct random-phase approximation (RPA) or second-order Møller–Plesset perturbation theory (MP2). PADF can however introduce large errors in correlation energies as the two-electron interaction energy is not guaranteed to be bounded from below. This issue can be partially alleviated by using very large fit sets, but this comes at the price of reduced efficiency and having to deal with near-linear dependencies in the fit set. One posibility is to use global density fitting (DF), but in this work, we introduce an alternative methodology to overcome this problem that preserves the intrinsically favorable scaling of PADF. We first regularize the Fock matrix by projecting out parts of the basis set which gives rise to orbital products that are hard to describe by PADF. After having thus obtained a reliable self-consistent field solution, we then also apply this projector to the orbital coefficient matrix to improve the precision of PADF-MP2 and PADF-RPA. We systematically assess the accuracy of this new approach in a numerical atomic orbital framework using Slater type orbitals (STO) and correlation consistent Gaussian type basis sets up to quintuple-ζ quality for systems with more than 200 atoms. For the small and medium systems in the S66 database we show the maximum deviation of PADF-MP2 and PADF-RPA relative correlation energies to DF-MP2 and DF-RPA reference results to be 0.07 and 0.14 kcal/mol, respectively. When the new projector method is used, the errors only slightly increase for large molecules and also when moderately sized fit sets are used the resulting errors are well under control. Finally, we demonstrate the computational efficiency of our algorithm by calculating the interaction energies of large, non-covalently bound complexes with more than 1000 atoms and 20000 atomic orbitals at the RPA@PBE/CC-pVTZ level of theory.