Semi-automated estimation of hydrogenic initial states for localized Wannier functions

TL;DR

This work tackles the challenge of generating meaningful initial guesses for Wannier function construction by introducing a semi-automated pipeline that derives hydrogenic projections from Bloch functions at the $\Gamma$-point using a neural-network encoder to predict spherical-harmonic and radial coefficients. The method also employs a clustering-based determination of projection centers, enabling automatic treatment of interstitial and molecular states. Applied to SrVO$_{3}$, FeSe, Si, Na$_8$Al$_6$Si$_6$O$_{24}$, and $(\text{TMTTF})_2\text{PF}_6$, the approach yields Wannier functions with comparable or improved localization and interpretability relative to SCDM, while maintaining accurate band reproduction and reasonable gauge-invariant spreads. Integrated with cif2qewan, this framework supports efficient, interpretable, and potentially high-throughput Wannierization workflows for complex materials.

Abstract

We present a semi-automated method for obtaining an initial estimate of Wannier functions, designed to facilitate the construction of Wannier functions for describing low-energy effective models of solids, particularly those relevant to strongly correlated electron systems. Our approach automatically determines the hydrogenic projections orbitals and the center of the Wannier functions from information on Bloch wavefunctions at the $Γ$ point. This method is integrated into cif2qewan, enabling seamless generation of input files for Quantum ESPRESSO and Wannier90. We validate our method through applications to both inorganic and organic compounds, such as Si, SrVO$_3$, FeSe, Na$_8$Al$_6$Si$_6$O$_{24}$, and (TMTTF)$_2$PF$_6$. The obtained results demonstrate that our semi-automated projections give a good initial estimate of the Wannier functions. We also show the comparisons with other methods for estimating the initial states of the Wannier functions, such as the Selected Columns of the Density Matrix (SCDM). Our methodology shows an efficient way to construct Wannier functions, paving the way for high-throughput calculations in the study of complex materials.

Figures (8)

• Figure 1: Schematic workflow of the proposed semi-automated method for constructing Wannier functions. The details are given in Sec. \ref{['sec:method']} of the main text.
• Figure 2: (a) Band structure of SrVO$_{3}$ with Wannier interpolated bands. Circles are obtained by DFT calculations. The red line and blue dashed line are obtained by Wannier interpolation using our method and the SCDM method, respectively. The black thin line denotes the Fermi energy level. (b) Bloch functions at $\Gamma$ point. Green, blue and red spheres are Sr atoms, V atoms and O atoms, respectively. The yellow (sky blue) region indicates the isosurface of the positive (negative) intensity of the Bloch functions. (c-d) Wannier functions obtained by the SCDM method (c) and our method (d). Yellow and sky-blue regions mean the isosurface of the positive and negative intensity of the Wannier functions, respectively. Other notations are the same as those in panel (b).
• Figure 3: Clustering of a Bloch wavefunction of FeSe around the Fermi energy. Brown and deep red spheres are Fe atoms and Se atoms, respectively. The Bloch wavefunction (top) is clustered into four box clusters with dashed lines (bottom). The center of each cluster corresponds to $\tilde{\bm{r}}_\nu^{(n)}$. Bottom panels on the left side show components of the Bloch wavefunction within each cluster including one Se atom, and on the right side show around each cluster including one Fe atom. The maximum value of $|c_{lm}^{(n)}|$ of each cluster and its corresponding $l$ and $m$ are shown.
• Figure 4: (a) Band structure of FeSe with Wannier interpolated bands. (b-c) Wannier functions obtained by the SCDM method (b) and our method (c). Brown and deep red spheres are Fe atoms and Se atoms, respectively. The other notations are the same as in Fig. \ref{['fig:svo']}.
• Figure 5: (a) Band structure of Si with Wannier interpolated bands. Green dotted line is obtained by the SCDM method without the SMV disentanglement procedure. The other notations are the same as in Fig. \ref{['fig:svo']}. (b) Total spread $\Omega_{\rm tot}$ during the MLWF procedure. Red, blue and green lines are obtained by our method, the SCDM method with and without the SMV disentanglement procedure, respectively. The inset represents a Wannier function obtained by our method. Blue spheres are Si atoms.