  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 does not
depend on the size of the band gap, and thus can be applied to insulating,
semiconducting as well as metallic systems. In an iterative framework for
solving Hartree-Fock-like systems, the ACE formulation only requires moderate
modification of the code, and can be potentially beneficial for all electronic
structure software packages involving exchange calculations. 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 speedup for Hartree-Fock-like
calculations.
