Even Faster Exact Exchange for Solids via Tensor Hypercontraction

Abstract

Hybrid density functional theory (DFT) remains intractable for large periodic systems due to the demanding computational cost of exact exchange. We apply the tensor hypercontraction (THC) (or interpolative separable density fitting) approximation to periodic hybrid DFT calculations with Gaussian-type orbitals using the Gaussian plane wave approach. This is done to lower the computational scaling with respect to the number of basis functions (N) and k-points (N k ) at a fixed system size. Additionally, we propose an algorithm to fit only occupied orbital products via THC (i.e., a set of points, N ISDF ) to further reduce computation time and memory usage. This algorithm has linear scaling cost with k-points, no explicit dependence of N ISDF on basis set size, and overall cubic scaling with unit cell size. Significant speedups and reduced memory usage may be obtained for moderately sized k-point meshes, with additional gains for large k-point meshes. Adequate accuracy can be obtained using THC-oo-K for self-consistent calculations. We perform illustrative hybrid density function theory calculations on the benzene crystal in the basis set and thermodynamic limits to highlight the utility of this algorithm.

Low-Scaling Algorithms for GW and Constrained Random Phase Approximation Using Symmetry-Adapted Interpolative Separable Density Fitting

“…This factorized representation has demonstrated its efficacy in leading to low-scaling algorithms for GW − and other electronic structure methods ,− in the context of molecules and Γ-point supercells. Its application to crystalline systems with k -point sampling is, as of now, somewhat limited, including hybrid functionals, Hartree–Fock (HF), RPA correlation energy, and the Bethe–Saltpeter equation …”

Tucker Tensor Approach for Accelerating Fock Exchange Computations in a Real-Space Finite-Element Discretization of Generalized Kohn–Sham Density Functional Theory

“…Many algorithms have been proposed to accelerate the Fock exact exchange calculations. Broadly, these algorithms can be classified into two categories: (i) algorithms that construct an approximate Fock exchange operator targeting the occupied orbitals − and (ii) algorithms that reduce the computational complexity of the action of the Fock exchange operator. − Adaptively compressed exchange (ACE) is an example of the former, wherein a low-rank approximation for the Fock exchange operator, which is exact in the space spanned by the occupied orbitals, is constructed. The ACE approach has been shown to drastically reduce the cost of hybrid DFT calculations .…”

“…However, even with ACE, the cost of a hybrid calculation can be substantially higher than using GGA, and the computational cost associated with the action of the Fock exchange operator still remains a bottleneck. Hence, various reduced-order scaling strategies belonging to the later category, such as the interpolative separable density fitting (ISDF) approach, , and linear scaling methods like integral screening approaches and integral evaluation schemes − have been proposed. ISDF approach exploits the rank deficiency in the product of the wave functions that appear in the convolution integrals, while linear-scaling methods utilize the locality of the wave functions to reduce the number of convolution integrals required.…”