Adaptively Compressed Exchange Operator.

The Fock exchange operator plays a central role in modern quantum chemistry. The large computational cost associated with the Fock exchange operator hinders Hartree-Fock calculations and Kohn-Sham density functional theory calculations with hybrid exchange-correlation functionals, even for systems consisting of hundreds of atoms. We develop the adaptively compressed exchange operator (ACE) formulation, which greatly reduces the computational cost associated with the Fock exchange operator without loss of accuracy. The ACE formulation is not dependent on the size of the band gap, and thus can be applied to insulating and semiconducting systems, as well as metallic systems. In an iterative framework for solving Hartree-Fock-like systems, such as that observed in planewave-based methods, the ACE formulation only requires moderate modification of the code. The ACE formulation can also be advantageous for other types of basis sets, especially when the storage cost of the exchange operator is expensive. Numerical results indicate that the ACE formulation can become advantageous, even for small systems with tens of atoms. In particular, the cost of each self-consistent field iteration for the electron density in the ACE formulation is only marginally larger than that of the generalized gradient approximation (GGA) calculation, and thus offers orders-of-magnitude acceleration for Hartree-Fock-like calculations.