# Event detail

## Matrix Computations and Scientific Computing Seminar: Interpolative Separable Density Fitting Decomposition for Accelerating Hybrid Density Functional Calculations with Applications to Defects in Large Scale Silicon

Seminar | April 5 | 11 a.m.-12 p.m. | 380 Soda Hall

Wei Hu, LBNL

Kohn-Sham density functional theory (KSDFT) is the most powerful tool for performing ab initio electronic structure calculations to study the structural and electronic properties of molecules, solids and liquids in quantum chemistry and materials science. However, as the key ingredient in the KSDFT calculations, the exchange and correlation functional to determine the accuracy must be approximated. There are several choices of the so-called Jacob's ladder in the KSDFT calculations, including local density approximation (LDA), generalized gradient approximation (GGA) and hybrid functionals. However, the widely used semi-local LDA and GGA functionals fail to give accurate electronic structures in molecules and semiconductors due to lack of long-range nonlocal Hartree-Fock exchange interaction in the KSDFT calculations. Fortunately, hybrid density functionals, such as B3LYP, PBE0 and HSE, have already proven to improve the accuracy of electronic structure calculations by introducing a fraction of the Hartree-Fock exchange functional. However, the computational cost of the Hartree-Fock exchange calculations scales as N4 if the exchange operator is constructed explicitly, where N is the number of electrons in the systems. Therefore, hybrid density functional calculations are expensive and challenging even for the systems consisting of hundreds of atoms.

Here, we introduce the interpolative separable density fitting (ISDF) decomposition to accelerate large scale hybrid functional calculations. The high computational cost of hybrid functional plane-wave calculations is often dominated by the cost of applying the exchange operator to N orbitals, which requires solving N2 Poisson-like equations. We demonstrate that the ISDF decomposition can reduce these N2 equations into Or(N) equations with systematically controlled accuracy, and the validity of the method does not depend on the energy gap. In the ISDF decomposition, the interpolation sampling points are selected by using the QR factorization with column pivoting (QRCP) procedure and the interpolation vectors (auxiliary basis) are computed by solving a least squares problem. The overall computational complexity remains Or(N3), but the pre-constant is significantly reduced. Combined with the recently developed adaptively compressed exchange operator formulation (ACE) to reduce the frequency of which the exchange operator needs to be applied, the resulting ACE-ISDF method significantly reduces the computational cost of large scale hybrid functional calculations by nearly two orders of magnitude. We demonstrate that the ACE-ISDF method can obtain accurate energies and forces for insulating and metallic systems, and the converged hybrid functional calculations can be performed for a 1000-atom bulk silicon within 10 minutes using 2000 computational cores. Our method can also scale to 8192 computational cores for a 4096-atom bulk silicon system. Using the ACE-ISDF method, we calculate the defect energy levels for a 1000-atom silicon system with a vacancy using the HSE06 hybrid exchange-correlation functional, which yields quantitative agreement with a recent large scale GW calculation for this large system.