Efficient and Accurate Algorithms for Solving the Bethe-Salpeter Eigenvalue Problem for Crystalline Systems

Abstract Optical properties of materials related to light absorption and scattering are explained by the excitation of electrons. The Bethe–Salpeter equation is the state-of-the-art approach to describe these processes from first principles (ab initio), i.e. without the need for empirical data in the model. To harness the predictive power of the equation, it is mapped to an eigenvalue problem via an appropriate discretization scheme. The eigenpairs of the resulting large, dense, structured matrix can be used to compute dielectric properties of the considered crystalline or molecular system. The matrix always shows a 2 × 2 block structure. Depending on exact circumstances and discretization schemes, one ends up with a matrix structure such as H 1 = A B − B − A ∈ ℂ 2 n × 2 n , A = A H , B = B H , or H 2 = A B − B H − A T ∈ ℂ 2 n × 2 n or R 2 n × 2 n , A = A H , B = B T . H 1 can be acquired for crystalline systems (see Sander et al. (2015)), H 2 is a more general form found e.g. in Shao et al. (2016) and Penke et al. (2020), which can for example be used to study molecules. Additionally, certain definiteness properties may hold. In this work, we compile theoretical results characterizing the structure of H 1 and H 2 in the language of non-standard scalar products. These results enable us to develop a generalized perspective on the currently used direct solution approach for matrices of form H 1 . This new viewpoint is used to develop two alternative methods for solving the eigenvalue problem. Both have advantages over the method currently in use and are well suited for high performance environments and only rely on basic numerical linear algebra building blocks. The results are extended to hold even without the mentioned definiteness property, showing the usefulness of our new perspective.