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

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.

💡 Research Summary

The paper introduces a rigorous and computationally efficient method for unfolding electronic band structures obtained from supercell calculations that employ atom‑centered, non‑orthogonal basis functions. Traditional unfolding techniques have largely relied on plane‑wave representations, which benefit from orthogonality and independence from atomic positions. However, when using linear combinations of atomic orbitals (LCAO) or numeric atom‑centered orbitals (NAOs) as in many all‑electron codes, two major complications arise: (i) the overlap matrix S is non‑diagonal, and (ii) the basis functions move with the atoms, making the translational symmetry less straightforward to handle. Existing approaches circumvent these issues by Wannierization or by projecting the atomic orbitals onto a plane‑wave basis, but such strategies become cumbersome for core and semi‑core states that require a large number of plane waves, and they add considerable computational overhead.

The authors develop a formalism that works directly in the AO basis. They start by defining the integer transformation matrix M that relates the primitive‑cell lattice vectors a to the supercell vectors A (A = a·M). This yields the reciprocal‑space relationship B = M⁻¹·b, where b and B are the primitive‑cell and supercell reciprocal‑lattice matrices, respectively. Using these relations, the mapping between primitive‑cell k‑points and supercell K‑points is expressed analytically (F_K = f_k·M), allowing automatic identification of all k‑points that fold onto a given K‑point.

Next, the translational operator in the AO basis, denoted \tilde{T}, is introduced. Its eigenfunctions are Bloch‑type atomic orbitals |χ_{KJ}⟩ = (1/√L_tot) Σ_L e^{iK·L} |φ_{LJ}⟩, where |φ_{LJ}⟩ are the numeric atomic orbitals centered on atom L with orbital index J. Because the AO set is non‑orthogonal, the Hamiltonian and overlap matrices in K‑space, H_K and S_K, are constructed, and the electronic states are obtained from the generalized eigenvalue problem H_K C_{KN} = E_{KN} S_K C_{KN}. The coefficients C_{KN} constitute the electron wavefunction in the Bloch‑type AO basis.

Unfolding is performed by projecting the supercell eigenstates onto the primitive‑cell subspace using the projector P_k = Σ_j |k j⟩⟨k j| (or equivalently onto the primitive‑cell eigenstates |ψ_{kn}⟩). The unfolding weight for a given supercell state |Ψ_{KN}⟩ is W_{kKN} = |⟨ψ_{kn}|Ψ_{KN}⟩|², which reduces to 0 or 1 for a perfect supercell composed of identical primitive cells, but becomes fractional when the supercell is perturbed (defects, phonons, disorder). This fractional weight enables the definition of a k‑resolved spectral function A(k,E) = Σ_{KN} W_{kKN} δ(E−E_{KN}), which can be used to compute temperature‑dependent electronic spectra.

The methodology is implemented in the all‑electron code FHI‑aims, which uses numeric atom‑centered orbitals. Implementation details include (i) on‑the‑fly construction of the Bloch‑type AO basis without recourse to FFTs, (ii) Lӧwdin orthogonalization of the overlap matrix to ensure numerical stability, and (iii) efficient handling of large supercells by exploiting the sparsity of S and H. The authors demonstrate scalability by unfolding the band structure of a 4 096‑atom zinc‑blende GaN supercell (≈100 000 basis functions), showing that the unfolded bands reproduce the primitive‑cell dispersion with negligible error.

Finally, the method is applied to CuI, a material known for strong anharmonicity at room temperature. Molecular‑dynamics snapshots of large supercells are generated, and for each snapshot the electronic structure is computed and unfolded. By averaging over snapshots, a temperature‑dependent spectral function is obtained, revealing band broadening and shifts that are absent in static calculations but agree with ARPES measurements. Importantly, the approach handles conduction, valence, and core states uniformly, something that plane‑wave‑based unfolding struggles with for core levels.

In summary, the paper provides (1) a general theoretical framework for band unfolding in non‑orthogonal, atom‑centered bases, (2) a practical, scalable implementation in an all‑electron code, and (3) compelling applications that illustrate its capability to treat large, anharmonic systems and to compute temperature‑dependent electronic spectra. This work opens the door for accurate unfolding analyses in a wide range of materials where plane‑wave methods are impractical, including systems with defects, strong electron‑phonon coupling, and core‑level spectroscopy.