Toward Pair Atomic Density Fitting for Correlation Energies with Benchmark Accuracy

Abstract

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.

Automatic Generation of Auxiliary Basis Sets in Spherical Representation Using the Cholesky Decomposition

“…We do not explicitly consider Slater-type functions for the radial part, even though the schemes introduced in this work might be easily transferred to that case. We note that recent work making use of Slater-type functions has led to significant progress in this context. − The angular part can be written in terms of the Cartesian coordinates Y l μ m μ ( r ) = x i y j z k where i + j + k = l μ . A different way to represent the angular part is using spherical coordinates resulting in the use of spherical harmonics Y l m ( r ) for the case of real-valued orbitals, resulting in Y l μ m μ ( r ) = r l μ Y l μ m μ ( θ , ϕ ) …”

Two-Component GW Calculations: Cubic Scaling Implementation and Comparison of Vertex-Corrected and Partially Self-Consistent GW Variants

“…To calculate the fitting coefficients, we use the pair-atomic density fitting (PADF) method − in the implementation of ref . The following working equations are however completely general and can be implemented using any type of density fitting (DF).…”

“…This corresponds to a very large auxiliary basis which is typically around 12 times larger than the primary basis and eliminates PADF errors for relative energies of medium molecules almost completely. 120 Imaginary time and imaginary frequency variables are discretized using nonuniform bases…”

“…The auxiliary bases used to expand 4-point correlation functions are automatically generated from products of primary basis functions. For this, we use a variant of an algorithm introduced in ref , which has recently been implemented in ADF and BAND . The size of the auxiliary basis in this approach can be tuned by a single threshold which we set to ϵ aux = 1 × 10 –10 in all partially self-consistent calculations and to ϵ aux = 1 × 10 –8 for G 0 W 0 .…”

Deterministic/Fragmented-Stochastic Exchange for Large-Scale Hybrid DFT Calculations

“…With this intelligent design, one can cap the number of β to be comparable to the number of atomic orbitals needed for the calculation, though this auxiliary basis traditionally still scales with system size. In practice, several advancements such as pair atomic RI, auxiliary density RI, and atomic concentric RI have reduced computational cost for many different kinds of DFT codes. − Most of these advancements depend on using atomic basis sets.…”