Towards Pair Atomic Density Fitting for Correlation Energies with Benchmark Accuracy

Pair atomic density fitting (PADF) is 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 M{\o}ller-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. In this work, we introduce an alternative methodology to overcome this problem that preserves the intrinsically favourable 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. 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-$\zeta$ 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 non-covalently bound complexes with more than 1000 atoms and 20000 atomic orbitals at the RPA@PBE/CC-pVTZ level of theory

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 non-metals. 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 (TPSS) 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 zero-point 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.Comment: 32 pages, single pdf fil

Toward Pair Atomic Density Fitting for Correlation Energies with Benchmark Accuracy

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

DFTB Parameters for the Periodic Table, Part 2: Energies and Energy Gradients from Hydrogen to Calcium

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

DFTB Parameters for the Periodic Table: Part 1, Electronic Structure

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