Plane wave basis set correction methods for RPA correlation energies

Contributions beyond direct random-phase approximation in the binding energy of solid ethane, ethylene, and acetylene

The random-phase approximation (RPA) includes a subset of higher than second-order correlation-energy contributions, but stays in the same complexity class as the second-order Møller-Plesset perturbation theory (MP2) in both Gaussian-orbital and plane-wave codes. This makes RPA a promising ab initio electronic structure approach for the binding energies of molecular crystals. Still, some issues stand out in practical applications of RPA. Notably, compact clusters of nonpolar molecules are poorly described, and the interaction energies strongly depend on the reference single-determinant state. Using the many-body expansion of the binding energy of a crystal, we investigate those issues and the effect of beyond-RPA corrections. We find the beneficial effect of quartic-scaling exchange and non-ring coupled-cluster doubles corrections. The nonadditive interactions in compact trimers of molecules are improved by using the self-consistent Hartree-Fock orbitals instead of the usual Kohn-Sham states, but this kind of orbital input also leads to underestimated dimer energies. Overall, a substantial improvement over the RPA with a renormalized singles approach is possible at a modest quartic-scaling cost, which encourages further research into additional RPA corrections.

Machine Learning Density Functionals from the Random-Phase Approximation

Kohn-Sham density functional theory (DFT) is the standard method for first-principles calculations in computational chemistry and materials science. More accurate theories such as the random-phase approximation (RPA) are limited in application due to their large computational cost. Here, we use machine learning to map the RPA to a pure Kohn-Sham density functional. The machine learned RPA model (ML-RPA) is a nonlocal extension of the standard gradient approximation. The density descriptors used as ingredients for the enhancement factor are nonlocal counterparts of the local density and its gradient. Rather than fitting only RPA exchange-correlation energies, we also include derivative information in the form of RPA optimized effective potentials. We train a single ML-RPA functional for diamond, its surfaces, and liquid water. The accuracy of ML-RPA for the formation energies of 28 diamond surfaces reaches that of state-of-the-art van der Waals functionals. For liquid water, however, ML-RPA cannot yet improve upon the standard gradient approximation. Overall, our work demonstrates how machine learning can extend the applicability of the RPA to larger system sizes, time scales, and chemical spaces.

Quasiparticle Self-Consistent GW-Bethe-Salpeter Equation Calculations for Large Chromophoric Systems

The GW-Bethe–Salpeter equation (BSE) method is promising for calculating the low-lying excitonic states of molecular systems. However, so far it has only been applied to rather small molecules and in the commonly implemented diagonal approximations to the electronic self-energy, it depends on a mean-field starting point. We describe here an implementation of the self-consistent and starting-point-independent quasiparticle self-consistent (qsGW)-BSE approach, which is suitable for calculations on large molecules. We herein show that eigenvalue-only self-consistency can lead to an unfaithful description of some excitonic states for chlorophyll dimers while the qsGW-BSE vertical excitation energies (VEEs) are in excellent agreement with spectroscopic experiments for chlorophyll monomers and dimers measured in the gas phase. Furthermore, VEEs from time-dependent density functional theory calculations tend to disagree with experimental values and using different range-separated hybrid (RSH) kernels does change the VEEs by up to 0.5 eV. We use the new qsGW-BSE implementation to calculate the lowest excitation energies of the six chromophores of the photosystem II (PSII) reaction center (RC) with nearly 2000 correlated electrons. Using more than 11,000 (6000) basis functions, the calculation could be completed in less than 5 (2) days on a single modern compute node. In agreement with previous TD-DFT calculations using RSH kernels on models that also do not include environmental effects, our qsGW-BSE calculations only yield states with local characters in the low-energy spectrum of the hexameric complex. Earlier works with RSH kernels have demonstrated that the protein environment facilitates the experimentally observed interchromophoric charge transfer. Therefore, future research will need to combine correlation effects beyond TD-DFT with an explicit treatment of environmental electrostatics.

GW100: A Slater-Type Orbital Perspective

We calculate complete basis set (CBS) limit-extrapolated ionization potentials (IPs) and electron affinities (EA) with Slater-type basis sets for the molecules in the GW100 database. To this end, we present two new Slater-type orbital (STO) basis sets of triple-(TZ) and quadruple-ζ (QZ) quality, whose polarization is adequate for correlated-electron methods and which contain extra diffuse functions to be able to correctly calculate EAs of molecules with a positive lowest unoccupied molecular orbital (LUMO). We demonstrate that going from TZ to QZ quality consistently reduces the basis set error of our computed IPs and EAs, and we conclude that a good estimate of these quantities at the CBS limit can be obtained by extrapolation. With mean absolute deviations (MAD) from 70 to 85 meV, our CBS limit-extrapolated IP are in good agreement with results from FHI-AIMS, TURBOMOLE, VASP, and WEST, while they differ by more than 130 meV on average from nanoGW. With a MAD of 160 meV, our EA are also in good agreement with the WEST code. Especially for systems with positive LUMOs, the agreement is excellent. With respect to other codes, the STO-type basis sets generally underestimate EAs of small molecules with strongly bound LUMOs. With 62 meV for IPs and 93 meV for EAs, we find much better agreement with CBS limit-extrapolated results from FHI-AIMS for a set of 250 medium to large organic molecules.

Low-Order Scaling G0W0 by Pair Atomic Density Fitting

We derive a low-scaling G0W0 algorithm for molecules using pair atomic density fitting (PADF) and an imaginary time representation of the Green's function and describe its implementation in the Slater type orbital (STO)-based Amsterdam density functional (ADF) electronic structure code. We demonstrate the scalability of our algorithm on a series of water clusters with up to 432 atoms and 7776 basis functions and observe asymptotic quadratic scaling with realistic threshold qualities controlling distance effects and basis sets of triple-ζ (TZ) plus double polarization quality. Also owing to a very small prefactor, a G0W0 calculation for the largest of these clusters takes only 240 CPU hours with these settings. We assess the accuracy of our algorithm for HOMO and LUMO energies in the GW100 database. With errors of 0.24 eV for HOMO energies on the quadruple-ζ level, our implementation is less accurate than canonical all-electron implementations using the larger def2-QZVP GTO-type basis set. Apart from basis set errors, this is related to the well-known shortcomings of the GW space-time method using analytical continuation techniques as well as to numerical issues of the PADF approach of accurately representing diffuse atomic orbital (AO) products. We speculate that these difficulties might be overcome by using optimized auxiliary fit sets with more diffuse functions of higher angular momenta. Despite these shortcomings, for subsets of medium and large molecules from the GW5000 database, the error of our approach using basis sets of TZ and augmented double-ζ (DZ) quality is decreasing with system size. On the augmented DZ level, we reproduce canonical, complete basis set limit extrapolated reference values with an accuracy of 80 meV on average for a set of 20 large organic molecules. We anticipate our algorithm, in its current form, to be very useful in the study of single-particle properties of large organic systems such as chromophores and acceptor molecules.