Reproducibility of Hybrid Density Functional Calculations for Equation-of-State Properties and Band Gaps.

Hybrid density functional (HDF) approximations usually deliver higher accuracy than local and semilocal approximations to the exchange-correlation functional, but this comes with drastically increased computational cost. Practical implementations of HDFs inevitably involve numerical approximations─even more so than their local and semilocal counterparts due to the additional numerical complexity arising from treating the exact-exchange component. This raises the question regarding the reproducibility of the HDF results yielded by different implementations. In this work, we benchmark the numerical precision of four independent implementations of the popular Heyd-Scuseria-Ernzerhof (HSE) range-separated HDF on describing key materials' properties, including both properties derived from equations of state (EOS) and band gaps of 20 crystalline solids. We find that the energy band gaps obtained by the four codes agree with each other rather satisfactorily. However, for lattice constants and bulk moduli, the deviations between the results computed by different codes are of the same order of magnitude as the deviations between the computational and experimental results. On the one hand, this means that the HSE functional is rather accurate for describing the cohesive properties of simple insulating solids. On the other hand, this also suggests that the numerical precision achieved with current major HSE implementations is not sufficiently high to unambiguously assess the physical accuracy of HDFs. It is found that the pseudopotential treatment of the core electrons is a major factor that contributes to this uncertainty.

Efficient Hybrid Density Functional Calculations for Large Periodic Systems Using Numerical Atomic Orbitals.

We present an efficient, linear-scaling implementation for building the (screened) Hartree-Fock exchange (HFX) matrix for periodic systems within the framework of numerical atomic orbital (NAO) basis functions. Our implementation is based on the localized resolution of the identity approximation by which two-electron Coulomb repulsion integrals can be obtained by only computing two-center quantities-a feature that is highly beneficial to NAOs. By exploiting the locality of basis functions and efficient prescreening of the intermediate three- and two-index tensors, one can achieve a linear scaling of the computational cost for building the HFX matrix with respect to the system size. Our implementation is massively parallel, thanks to a MPI/OpenMP hybrid parallelization strategy for distributing the computational load and memory storage. All these factors add together to enable highly efficient hybrid functional calculations for large-scale periodic systems. In this work, we describe the key algorithms and implementation details for the HFX build as implemented in the ABACUS code package. The performance and scalability of our implementation with respect to the system size and the number of CPU cores are demonstrated for selected benchmark systems up to 4096 atoms.

Accuracy of Localized Resolution of the Identity in Periodic Hybrid Functional Calculations with Numerical Atomic Orbitals.

We present an implementation of hybrid density functional approximations for periodic systems within a pseudopotential-based, numerical atomic orbital (NAO) framework. The two-electron Coulomb repulsion integrals (ERIs) are evaluated using the localized resolution-of-the-identity (LRI) approximation. The accuracy of the LRI approximation is benchmarked unambiguously against independent reference results obtained via a computational scheme whereby the ERIs are accurately evaluated by expanding the products of NAOs in terms of plane waves. An alternative strategy for constructing auxiliary basis sets is proposed, and its accuracy is assessed and compared to the previously used procedure. Finally, the reliability of our algorithm and implementation is benchmarked against other established implementations within different numerical frameworks in terms of the calculated band gap values of a set of semiconductors and insulators.