Efficient Band Structure Unfolding with Atom-centered Orbitals: General Theory and Application

TL;DR

This work addresses the challenge of interpreting band structures from large, symmetry-broken supercells by projecting supercell states onto primitive-cell symmetry within a non-orthogonal atom-centered orbital (AO) basis. It introduces a Löwdin-based simplification and an analytical expression for primitive-cell translation eigenvectors, enabling efficient computation of unfolding weights $W^{\mathbf{k}}_{\mathbf{K}N}$ directly in the supercell basis without requiring explicit primitive-cell wavefunctions. The method is implemented in the all-electron NAO code FHI-aims and demonstrated on a 4,096-atom GaN supercell, as well as for CuI where non-perturbative temperature-dependent spectral functions $A(\mathbf{k},E)$ reveal strong anharmonic electron–vibrational coupling. Together, these developments provide a scalable, accurate tool for interpreting complex band structures and predicting vibrationally renormalized spectra in materials under realistic conditions, including defects and finite-temperature effects.

Abstract

Band structure unfolding is a key technique for analyzing and simplifying the electronic band structure of large, internally distorted supercells that break the primitive cell's translational symmetry. In this work, we present an efficient band unfolding method for atomic orbital (AO) basis sets that explicitly accounts for both the non-orthogonality of atomic orbitals and their atom-centered nature. Unlike existing approaches that typically rely on a plane-wave representation of the (semi-)valence states, we here derive analytical expressions that recasts the primitive cell translational operator and the associated Bloch-functions in the supercell AO basis. In turn, this enables the accurate and efficient unfolding of conduction, valence, and core states in all-electron codes, as demonstrated by our implementation in the all-electron ab initio simulation package FHI-aims, which employs numeric atom-centered orbitals. We explicitly demonstrate the capability of running large-scale unfolding calculations for systems with thousands of atoms and showcase the importance of this technique for computing temperature-dependent spectral functions in strongly anharmonic materials using CuI as example.

Figures (7)

• Figure 1: Atomic structure (a) and the first Brillouin zone (b) of a one-atom cubic hydrogen cell with lattice vector $1.5$Å and a rotated, eight-atom SC. Blue square indicate the PC, red square indicate the SC and grey squares indicate replicas. (c) Band structure of the PC. (d) Band structure of the SC.
• Figure 2: Illustration of the action of the translational operator in a non-orthogonal LCAO basis. Red and purple curves represent the AOs $I$ and $J$.
• Figure 3: (a) Illustration of how the translational operators acts on each AO in real space. Here we use the black lattice to represent our PC and the two AOs in the PC are represented by yellow and green circles. The SC consists of two PCs is represented by the blue lattice. (b) Illustration of the matrix elements of the translational operator in K-space. We labeled the corresponding bloch-type AO belonging to two different PCs in our SC by $I_1$ and $I_2$. $D_I$ is the complex permutation matrix that represent the full path of orbital $I$ under the PC translational operator, as shown in Eq.(\ref{['matrix:D']}).
• Figure 4: (a) The perfect PC with 1 atomic orbital (b) A distorted SC with displaced AOs. (c) The "extended perfect" SC constructed by adding placeholder orbitals $\varphi'$ in the original distorted SC.
• Figure 5: Schematic code flow of the band unfolding implementation in FHI-aims.