Bloch-state-based interpolation: An efficient generalization of the Shirley approach to interpolating electronic structure

We present an efficient generalization of the $k$-space interpolation scheme for electronic structure presented by Shirley [Phys. Rev. B 54, 16464 (1996)]. The method permits the construction of a compact $k$-dependent Hamiltonian using a numerically optimal basis derived from a coarse-grained set of effective single-particle electronic-structure calculations (based on density-functional theory in this work). We provide some generalizations of the initial approach which reduce the number of required initial electronic-structure calculations, enabling accurate interpolation over the entire Brillouin zone based on calculations at the zone center only for large systems. We also generalize the representation of nonlocal Hamiltonians, leading to a more efficient implementation which permits the use of both norm-conserving and ultrasoft pseudopotentials in the input calculations. Numerically interpolated electronic eigenvalues with accuracy that is within 0.01 eV can be produced at very little computational cost. Furthermore, accurate eigenfunctions—expressed in the optimal basis—provide easy access to useful matrix elements for simulating spectroscopy and we provide details for computing optical transition amplitudes. The approach is also applicable to other theoretical frameworks such as the Dyson equation for quasiparticle excitations or the Bethe-Salpeter equation for optical responses.

Figure 1

The steps involved in building the compact $k$-dependent Hamiltonian in the optimal basis, beginning (top) with a self-consistent charge density, from which states are generated for a range of bands and $k$ points. An overlap matrix is constructed from the periodic parts of these Bloch states and diagonalized. The optimal basis is chosen as those eigenvectors of the overlap matrix (ordered by eigenvalue magnitude) which span a user-defined fraction of the space defined by the input states. The $k$-dependent Hamiltonian is constructed in this basis from its various parts: kinetic energy, local potential, and nonlocal potential.

Figure 2

Accuracy in reproducing the band structure of bcc Na with respect to choice of input $k$-point grid. Band energies are reported in electron volt with respect to the Fermi level and $k$ points follow the $\Delta$ line from the zone center $\left(\Gamma \right)$ to one of the zone boundaries $\left(H\right)$ and beyond to a periodic image of the zone center $\left(2H\right)$. DFT (Shirley interpolated) band structure is indicated by light solid (dark dashed) lines. The size of the input $k$-point grid is varied: (a) $\Gamma$ only; (b) $\Gamma$ and $H$; (c) $\Gamma$, $H$, and $2H$; and (d) $\Gamma$ together with its seven periodic images from the corners of the unit cube ${\left[0,1\right]}^3$.

Figure 3

(Color online) The atomic structure of $\gamma$ brass $\left({\text{Cu}}_5{\text{Zn}}_8\right)$.

Figure 4

The electron band structure of $\gamma$ brass $\left({\text{Cu}}_5{\text{Zn}}_8\right)$ generated using plane-wave DFT calculations (dots) and using Shirley interpolation based solely on DFT wave functions generated at the $\Gamma$ point (lines). The root-mean-square deviation of the Shirley interpolated values is 2 meV. The light (heavy) horizontal line indicates the Fermi level resulting from a $4\times 4\times 4$ $k$-point plane-wave DFT calculation ($10\times 10\times 10$ $k$-point Shirley interpolation).

Figure 5

Details of the band structure of $\gamma$ brass in three energy regions: (a) the $\text{Zn}d$ bands; (b) the $\text{Cu}d$ bands; and (c) high in the conduction bands. DFT plane-wave calculations are indicated by dots. Shirley interpolated bands are shown as lines. Together with Fig. 4 this indicates the ability of Shirley interpolation to accurately reproduce the bands of metallic states of varying character with minimal effort ($\Gamma$-point calculations only).

Figure 6

(Color online) A subset of the optimal basis functions of fcc Cu, determined using eight input $k$ points ($\Gamma$ plus seven images) and displayed in real space in a $2\times 2\times 2$ supercell. Copper atom positions are indicated by copper-colored spheres. Outside (inside) of basis function isosurfaces indicated in purple (green). Numbers in parentheses indicate the ordering in terms of overlap matrix eigenvalue magnitude corresponding to the following coverage of the entire space: (1) 7.6%; (2) 7.3%; (3) 6.4%; (4) 6.0%; (5) 5.7%; (6) 5.4%; (18) 1.8%; and (36) 0.78%.