SCDM-k: Localized orbitals for solids via selected columns of the density matrix

The recently developed selected columns of the density matrix (SCDM) method [J. Chem. Theory Comput. 11, 1463, 2015] is a simple, robust, efficient and highly parallelizable method for constructing localized orbitals from a set of delocalized Kohn-Sham orbitals for insulators and semiconductors with $\Gamma$ point sampling of the Brillouin zone. In this work we generalize the SCDM method to Kohn-Sham density functional theory calculations with k-point sampling of the Brillouin zone, which is needed for more general electronic structure calculations for solids. We demonstrate that our new method, called SCDM-k, is by construction gauge independent and is a natural way to describe localized orbitals. SCDM-k computes localized orbitals without the use of an optimization procedure, and thus does not suffer from the possibility of being trapped in a local minimum. Furthermore, the computational complexity of using SCDM-k to construct orthogonal and localized orbitals scales as O(N log N ) where N is the total number of k-points in the Brillouin zone. SCDM-k is therefore efficient even when a large number of k-points are used for Brillouin zone sampling. We demonstrate the numerical performance of SCDM-k using systems with model potentials in two and three dimensions.

💡 Research Summary

The paper introduces SCDM‑k, an extension of the Selected Columns of the Density Matrix (SCDM) technique to Kohn‑Sham density functional theory (DFT) calculations that employ k‑point sampling of the Brillouin zone. Traditional SCDM works only for Γ‑point (single‑k) calculations, limiting its use for realistic solid‑state problems where many k‑points are required to capture periodicity and electronic dispersion. SCDM‑k overcomes this limitation by exploiting the gauge‑invariant nature of the one‑particle density matrix. For each k‑point, the density matrix P_k = Ψ_k Ψ_k† (with Ψ_k containing the occupied Bloch states) is constructed; the sum over all k‑points yields the total density matrix P, which remains unchanged under arbitrary phase factors or unitary rotations of the Bloch states. Because the density matrix decays exponentially in real space (the nearsightedness principle), its columns are intrinsically localized. The algorithm therefore selects a small set of representative columns using a column‑pivoted QR factorization performed on a local supercell Ω_ℓ (a modest collection of adjacent unit cells). The indices of the selected columns define a set of non‑orthogonal localized functions that span the occupied subspace exactly.

A crucial step is the orthogonalization of these functions. By employing fast Fourier transforms (FFT) to carry out convolutions across the full supercell, the orthogonalization can be achieved with O(N log N) computational cost, where N is the total number of k‑points. The overall procedure consists of (1) O(N) work to build the density matrices and perform the column selection, and (2) O(N log N) work for the global orthogonalization, making the method scalable to tens of thousands of k‑points. Importantly, because the density matrix is gauge‑independent, SCDM‑k does not require any prior gauge‑smoothening, a step that is mandatory for maximally localized Wannier functions (MLWF) and can be ambiguous. Consequently, SCDM‑k avoids the non‑linear optimization and possible trapping in local minima that plague MLWF constructions.

The authors validate SCDM‑k on two‑ and three‑dimensional model potentials, demonstrating that the resulting localized orbitals have spreads and symmetry properties comparable to those of traditional Wannier functions. Numerical experiments confirm the predicted O(N log N) scaling and show that the quality of the orbitals is relatively insensitive to the size of the local supercell, which is the only tunable parameter.

In summary, SCDM‑k provides a robust, efficient, and gauge‑free route to generate orthogonal, localized orbitals for periodic systems with arbitrary k‑point meshes. Its linear‑logarithmic scaling, lack of iterative optimization, and straightforward implementation make it an attractive tool for large‑scale electronic‑structure workflows, including hybrid functional calculations, band‑structure interpolation, Berry‑phase analyses, and linear‑scaling DFT. Future work may focus on integrating SCDM‑k into mainstream plane‑wave codes, extending it to metallic systems, and coupling it with electron‑phonon or many‑body perturbation theory frameworks.