Linear-scaling DFT + $U$ with full local orbital optimization

We present an approach to the DFT + $U$ method (density functional theory $+$ Hubbard model) within which the computational effort for calculation of ground-state energies and forces scales linearly with system size. We employ a formulation of the Hubbard model using nonorthogonal projector functions to define the localized subspaces, and we apply it to a local orbital DFT method including in situ orbital optimization. The resulting approach thus combines linear-scaling and systematic variational convergence. We demonstrate the scaling of the method by applying it to nickel-oxide nanoclusters with sizes exceeding 7000 atoms.

Figure 1

Scaling of the energy-minimization algorithm for NiO nanoclusters of increasing size. Timings are for three density kernel optimization steps and one orbital optimization step, comparing DFT and DFT + $U$ calculations. Simulations were performed on 300 Intel Westmere $2.67$ GHz cores connected using quad data rate Infiniband.

Figure 2

Computational time spent in subroutines associated with the DFT + $U$ functionality in the tests shown in Fig. 1. Specifically, timings shown are for computing the DFT + $U$ energy of Eq. (4), the Hamiltonian matrix of Eq. (21) in its Hubbard projector and local orbital representations, and the ionic forces given by Eq. (C6).

Figure 3

Nanocluster size dependence of filling factors of principal matrices affecting the computational cost of linear-scaling DFT and DFT + $U$. Namely, these are the overlap between orbitals, the density kernel, the Hamiltonian matrices conventional for DFT and for DFT + $U$, and the overlap matrices between the local orbitals and the nonlocal pseudopotential projectors and Hubbard projectors.

Figure 4

Fractional change in time expended on energy-minimization operations when the DFT + $U$ functionality is activated. Namely, these are calculation of the local orbital (NGWF) gradient, expansion of the Hamiltonian in the basis, kernel optimization using the LNV method, and calculation of the matrix elements of the local potential.