Selectively localized Wannier functions

Since the seminal work of Marzari and Vanderbilt, maximally localized Wannier functions have become widely used as a real-space representation of the electronic structure of periodic materials. In this paper we introduce selectively localized Wannier functions which extend the method of Marzari and Vanderbilt in two important ways. First, our method allows us to focus on localizing a subset of orbitals of interest. Second, our method allows us to fix centers of these orbitals, and ensure the preservation of the point-group symmetry. These characteristics are important when Wannier functions are used in methodologies that go beyond density functional theory by treating a local subspace of the Hamiltonian more effectively. Application of our method to GaAs, ${\mathrm{SrMnO}}_3$, and Co demonstrates that selectively localized Wannier functions can offer improvements over the maximally localized Wannier function technique.

Figure 1

Wannier functions for the 1-d chain of $\delta$-function potentials. Large tick marks denote the $\delta$ function, while small tick marks denote the midpoint. (a) MLWF for $J=J^{\prime }=4$. ${\Omega }_1={\Omega }_4=0.2853a^2,{\Omega }_2={\Omega }_3=0.2983a^2$. (b) SLWF for $J=4,J^{\prime }=2$. ${\Omega }_1=0.0162a^2$, ${\Omega }_2=0.0162a^2$. (c) SLWF+C for $J=4,J^{\prime }=2$. Both are located at $x=0.5a$; ${\Omega }_1=0.0154a^2$, ${\Omega }_2=0.0605a^2$. (d) SLWF+CS for $J=4,J^{\prime }=2$. Both are symmetric about $0.5a$; ${\Omega }_1=0.0154a^2,{\Omega }_2=0.1387a^2$.

Figure 2

Wannier functions for GaAs obtained from different methods. In panels (a)–(c), Ga ions are indicated by purple spheres on the lattice corners; As ions are blocked from view by the Wannier function isosurfaces. The absolute value of the isosurfaces is $0.5/\sqrt{V}$, where V is the unit cell volume. Isosurfaces with positive amplitudes are colored red; those with negative amplitudes are colored blue. The Wannier functions are all real valued. Panel (d) shows a slice of each Wannier function along the Ga-As bond. The OWF function is plotted for the SLWF and SLWF+C methods.

Figure 3

Comparison between the MLWF and SLWF method for ${\mathrm{SrMnO}}_3$. Sr is the light blue sphere in the center, Mn is the blue sphere on the corner, O is the red sphere. In both panels, the 5 $d$-like Wannier functions are on the order of $3z^2-r^2$, $xz$, $yz$, $x^2-y^2$, $xy$, from left to right. The absolute value of the isosurfaces is $0.1/\sqrt{V}$.

Figure 4

Wannier function along the $z$ axis for the $3z^2-r^2$-like Wannier function generated from MLWF and SLWF in ${\mathrm{SrMnO}}_3$.

Figure 5

Comparison between the MLWF and SLWF method for Co. Co atoms are indicated by blue spheres on the lattice corners. In both panels, the 5 $d$-like Wannier functions are in the order of $3z^2-r^2$, $xz$, $yz$, $x^2-y^2$, $xy$, from left to right. The absolute value of the isosurfaces is $0.2/\sqrt{V}$.

Figure 6

Wannier function along the center axis of each lobe for the $xz$-like Wannier function generated from MLWF and SLWF in Co. The light (green) lines represent MLWF while the dark (red) lines represent SLWF. For a given method, the solid lines are along the lobes with positive phase while the dashed lines are along the ones with negative phase. The horizontal axis is in units of the lattice constant.

Figure 7

Sliced band structures of ${\mathrm{SrMnO}}_3$. In both panels, dashed gray lines represent the DFT band structure; solid dark (red) lines represent bands for the block $H_d$; solid light (green) lines represent bands for the block $H_p$.

Figure 8

Sliced band structures of Co. In both panels, dashed gray lines represent the DFT band structure; solid dark (red) lines represent bands for the block $H_d$; solid light (green) lines represent bands for the block $H_s$.

Figure 9

Wannier functions for the 1-d chain of negative $\delta$-function potentials. Large tick marks denote the $\delta$ function, while small tick marks denote the midpoint. Panels (a) and (b) use 100 $k$ points while panel (c) uses 20 $k$ points. (a) Wannier functions obtained for $J=2,J^{\prime }=2$ (MLWF), and $J=2,J^{\prime }=1$ (SLWF). The spreads are $0.2713a^2$ and $0.0506a^2$, respectively. (b) OWF with centers fixed at $0.5a$ and $0.3a$ (SLWF+C), in the case of $J=2,J^{\prime }=1$. The spreads are $0.0596a^2$ and $0.0637a^2$, respectively. (c) OWF with centers and symmetries controlled (SLWF+SC) in the case of $J=7,J^{\prime }=1$. The spreads are $0.0050a^2$ and $0.0205a^2$ for symmetric OWF and antisymmetric OWF, respectively.