Order-$N$ implementation of exact exchange in extended insulating systems

Exact (Hartree-Fock) exchange is needed to overcome some of the limitations of local and semilocal approximations of density-functional theory. So far, however, computational cost has limited the use of exact exchange in plane-wave calculations for extended systems. We show that this difficulty can be overcome by performing a unitary transformation from Bloch to maximally localized Wannier functions in combination with an efficient technique to compute real-space Coulomb integrals. The resulting scheme scales linearly with system size. We validate the scheme with representative applications.

Figure 1

(Color online) Overlap between a tagged Wannier orbital (red at the center) and its nearest neighboring Wannier orbitals (blue) in the 64-atom Si supercell. Si atoms are denoted by the green spheres.

Figure 2

Computational cost of exact exchange per iteration of AIMD dynamics with different supercells. The computational cost is given by the time (s) necessary to compute exact exchange on a 16-CPU 3.2 GHz Intel Xeon computer cluster.

Figure 3

(Color online) Potential energy per atom $E^{\text{PBE}0}$ (red line), kinetic energy per atom $K$ (black line), and internal energy per atom $U$ (blue line) vs AIMD time steps. The average $U$ sets the zero of the energy scale. A time step $\Delta =8\text{a}\text{.u}\text{.}$ and a fictitious electronic mass of 800 a.u. were used.