Partly occupied Wannier functions: Construction and applications

We have developed a practical scheme to construct partly occupied, maximally localized Wannier functions (WFs) for a wide range of systems. We explain and demonstrate how the inclusion of selected unoccupied states in the definition of the WFs can improve both their localization and symmetry properties. A systematic selection of the relevant unoccupied states is achieved by minimizing the spread of the resulting WFs. The method is applied to a silicon cluster, a copper crystal, and a Cu(100) surface with nitrogen adsorbed. In all cases we demonstrate the existence of a set of WFs with particularly good localization and symmetry properties, and we show that this set of WFs is characterized by a maximal average localization.

Figure 1

Schematic of the bonding-antibonding closure for a hydrogen molecule. The construction of well-localized atomic $s$ orbitals involves a matching of bonding and antibonding orbitals, independent of their occupation. The sign of the wave functions is indicated by the shading.

Figure 2

Relation between the first BZ of the unit cell, defined by the reciprocal basis vectors ${\mathbf{b}}_1,{\mathbf{b}}_2,{\mathbf{b}}_3$ (light gray), and the first BZ of the repeated unit cell (dark gray). In this case $N_1$ and $N_2$ from Eq. (21) both equals 3. The relation between $\mathbf{k}$ and ${\mathbf{k}}^{\prime }$, given in Eq. (29), is indicated.

Figure 3

(Color online) (a) Geometry of the ${\mathrm{Si}}_5$ cluster. (b) Contour plots of the WFs corresponding to $L=4$. The position of the WF centers are indicated by black spheres.

Figure 4

Average spread of the WFs of the ${\mathrm{Si}}_5$ cluster for different values of $N_b$ and $L$.

Figure 5

Band structure of Cu(fcc). The full lines are the original DFT bands and the dots are the approximate bands computed from a set of six WFs $(N_w=6)$. The WFs have been constructed using $11\times 11\times 11\mathbf{k}$ points and keeping all occupied states in the localization space, i.e., $E_0=0.0\mathrm{eV}$ relative to the Fermi level.

Figure 6

Like Fig. 5 except that the WFs have been generated with $E_0=3.0\mathrm{eV}$.

Figure 7

(Color online) Two $d$-like WFs for Cu(fcc). The orbitals are centered at the atoms (not shown).

Figure 8

(Color online) (a) Two tetrahedral interstitial sites in the fcc crystal. (b)–(d) Contour plots of the $s$-like WF obtained for Cu (fcc). The WFs in (b) and (c) have been generated with $N_w=6$, $E_0=0.0\mathrm{eV}$ and $N_w=6,E_0=3.0\mathrm{eV}$, respectively. Both WFs are located in one of the interstitial sites. The WF in (d) corresponds to $N_w=7$ and $E_0=0.0\mathrm{eV}$. In this case there is an equivalent WF located in the other interstitial site. The same contour value has been used for all plots.

Figure 9

Like Fig. 5 except that the WFs have been generated with $N_w=7$.

Figure 10

Topographic view on the Cu(100) surface with adsorbed nitrogen. The white spheres are nitrogen, while the light gray spheres represent the Cu surface layer. A supercell is indicated.

Figure 11

Average localization of the WFs of the nitrogen covered Cu surface for different values of $N_b$ and $N_w$.

Figure 12

Average localization of the $d$-, $s{p}^3$-, and $s$-like WFs considered separately for different values of $N_b$ and $N_w$.

Figure 13

(Color online) Orientation of the four $s{p}^3$-like WFs belonging to the N atom. Seen from above the orbitals point to the bridge sites of the Cu atoms in the surface. Each orbital points either up from or down into the surface. An example of each type is shown in the lower panel.

Figure 14

(Color online) Top: PDOS for the atomic $p$ orbitals obtained by diagonalizing the Hamiltonian in the subspace spanned by the $s{p}^3$ WFs of the N atom. Bottom: PDOS for the group orbitals corresponding to each of the atomic $p$ orbitals.