How to choose efficiently the size of the Bethe-Salpeter Equation Hamiltonian for accurate exciton calculations on supercells

Abstract

The Bethe-Salpeter Equation (BSE) is the workhorse method to study excitons in materials. The BSE Hamiltonian size, which depends on how many valence-to-conduction band transitions are considered, needs to be chosen to be sufficiently large to converge excitons' energies and wavefunctions but should be minimized to make calculations tractable, as BSE calculations are expensive and scale with the number of atoms as $\mathcal{O}(N_{\mathrm{atoms}}^6)$. In particular, in the case of supercell (SC) calculations composed of $N_{\mathrm{rep}}$ replicas of a primitive cell (PC), a natural choice to build this BSE Hamiltonian is to include all transitions derived from PC calculations by zone folding. However, this leads to a very large BSE Hamiltonian, as the number of matrix elements in it is $(N_k{N}_c{N}_v{)}^2$, where $N_k$ is the number of $k$-points and $N_{c(v)}$ is the number of conduction (valence) states used. When creating a SC, the number of $k$-points decreases by a factor $N_{\mathrm{rep}}$ but both the number of conduction and valence states increase by the same factor, therefore the number of matrix elements in the BSE Hamiltonian increases by a factor $N_{\mathrm{rep}}^2$, making exactly corresponding calculations prohibitive. Here, we provide a workflow to decide how many transitions are necessary to achieve comparable results, based on only PC results. With our method, we show that to converge the first exciton binding energy of a LiF SC composed of 64 PCs, to an energy tolerance of 0.15 eV, we only need 12\% of the valence-to-conduction matrix elements that result from zone folding with a minimal set of bands. As an example, we use the number of bands from our method to obtain the absorption spectrum of LiF with a V${}_k$-like defect. The procedure in our work helps in evaluating excitonic properties in large SC calculations.

References in corpus (15)

• Quantum ESPRESSO: a modular and open-source software project for quantum simulations of materials
• Advanced capabilities for materials modelling with Quantum ESPRESSO
• Quantum ESPRESSO toward the exascale
• Many-body perturbation theory calculations using the yambo code
• The Bethe-Salpeter Equation Formalism: From Physics to Chemistry
• Ab initio theory of polarons: formalism and applications
• Polarons from first principles, without supercells
• Non-uniform sampling schemes of the Brillouin zone for many-electron perturbation-theory calculations in reduced dimensionality
• Hyperspectral imaging of excitons within a moiré unit-cell with a sub-nanometer electron probe
• Excitonic polarons and self-trapped excitons from first-principles exciton-phonon couplings
• Mass enhancement in 3d and s-p perovskites from symmetry breaking
• Theory of excitonic polarons: From models to first-principles calculations
• Solving the Bethe-Salpeter Equation on a Subspace: Approximations and Consequences for Low-dimensional Materials
• Unified Deep Learning Framework for Many-Body Quantum Chemistry via Green's Functions
• Dispersive dark excitons in van der Waals ferromagnet CrI3