Partly Occupied Wannier Functions

We introduce a scheme for constructing partly occupied, maximally localized Wannier functions (WFs) for both molecular and periodic systems. Compared to the traditional occupied WFs the partly occupied WFs possess improved symmetry and localization properties achieved through a bonding-antibonding closing procedure. We demonstrate the equivalence between bonding-antibonding closure and the minimization of the average spread of the WFs in the case of a benzene molecule and a linear chain of Pt atoms. The general applicability of the method is demonstrated through the calculation of WFs for a metallic system with an impurity: a Pt wire with a hydrogen molecular bridge.

Figure 1

Occupied and partly occupied Wannier functions for benzene ${\mathrm{C}}_6{\mathrm{H}}_6$. The traditional (occupied) WFs are seen to mix the $\sigma$ and $\pi$ orbitals. Inclusion of the three antibonding $\pi$ orbitals leads to the partly occupied WFs which are more localized and separate the orbitals with different mirror symmetry.

Figure 2

Average localization per Wannier function as a function of the number of included unoccupied orbitals for the three systems studied. For the benzene molecule the maximum appears at $L=3$ corresponding to the inclusion of the three antibonding $\pi$ orbitals. For the Pt chain the maximum at $L=6$ corresponds to completion of the valence bands. (The values of $⟨\Omega ⟩$ do not compare from one system to another due to dependence on the size of the supercell.)

Figure 3

The norm of the projection of eigenstates onto the subspace spanned by the partly occupied WFs for the three systems studied. The eigenstates are listed in increasing order according to their energy. States below the dashed line are occupied. It is seen that in the case of the benzene molecule and the Pt chain the algorithm selects particular unoccupied states necessary for the bonding-antibonding closing. In the case of the hydrogen molecule in the Pt chain, one of the WFs has weight on an unoccupied resonance distributed over several eigenstates.

Figure 4

Partly occupied Wannier functions for a periodically repeated, linear string of six Pt atoms without and with an inserted hydrogen bridge. The WFs have been generated with $L=6$ (${\mathrm{P}\mathrm{t}}_6$) and $L=7$ (${\mathrm{P}\mathrm{t}}_6\mathrm{\text{−}}{\mathrm{H}}_2$). The WFs shown for the ${\mathrm{P}\mathrm{t}}_6\mathrm{\text{−}}{\mathrm{H}}_2$ system correspond to H-H and H-Pt $\sigma$ bonds.