Effective and accurate representation of extended Bloch states on finite Hilbert spaces

We present a straightforward, noniterative projection scheme that can represent the electronic ground state of a periodic system on a finite atomic-orbital-like basis, up to a predictable number of electronic states and with controllable accuracy. By cofiltering the projections of plane-wave Bloch states with high-kinetic-energy components, the richness of the finite space and thus the number of exactly-reproduced bands can be selectively increased at a negligible computational cost, an essential requirement for the design of efficient algorithms for electronic structure simulations of realistic material systems and massive high-throughput investigations.

Figure 1

Si band structures from projected LCAO Hamiltonian matrices $H^{\mathbf{k}}(\kappa ,N)$. Eigenenergies are represented by circles, triangles, or horizontal solid lines (flat bands). The electronic structure is expanded in an $sp$ (a,b) and an $spd$ (c,d) finite space. (a,c) Direct-projection scheme (no filtering, $N=M$; no shifting, $\kappa =0$). (b,d) Filtered+shifted projections. Degenerate flat bands (3- and 10-fold) are rigidly shifted by $\kappa$ (4.1 and 9.6 eV) and plotted as horizontal solid lines. Reference PW bands are shown in black (${\mathcal{P}}_n\ge 0.9$) or gray (${\mathcal{P}}_n<0.9$).

Figure 2

Band structure of Mo bcc under space ${\Omega }_3$. (a) Direct-projection scheme ($N=13$, $\kappa =0$ eV). (b) Filtered+shifted projection ($N=7$, $\kappa =10$ eV). (c) Unfiltered+shifted projection ($N=M=13$, $\kappa =12.5$ eV). The eigenenergies of the LCAO Hamiltonian $H^{\mathbf{k}}(\kappa ,N)$ are represented by circles. PW bands of high (${\mathcal{P}}_n\ge 0.9$) projectability are shown by solid black lines; those of low (${\mathcal{P}}_n<0.9$) projectability, by solid gray lines. The four low-lying semicore bands ($4s4p$) are not shown.

Figure 3

Fermi surface of Mo bcc bulk. (a) Colored circles identify the same band crossing the Fermi energy (horizontal dashed line). Orange, band 1; green, band 2; red, band 3. (b–d) Individual-band contributions to the Fermi surface. (e) Total Fermi surface.

Figure 4

Band structure and quantum transmittance spectrum of a gold nanowire. (a) Direct-projection and (b) filtered+shifted ($N=62$, $\kappa =3$ eV) band structures are represented by circles; PW bands, by solid lines. (c) Quantized transmittance of the nanowire.

Figure 5

Effect of the nearest principal-layer approximation on the interpolated band structure (left) and quantized conductance (right). The principal layer is approximated with (a) one and (b) two unit cells ($=$4.71 Å). The (blue) circles are the eigenvalues of $H^{\mathbf{k}}$ and the underlying solid lines are the corresponding interpolated bands.

Figure 6

Schematic representation of the quadratic eigenvalue equations (A2) and (A3) as a one-dimensional tight-binding model.