Efficient and accurate algorithms for solving the Bethe–Salpeter eigenvalue problem for crystalline systems

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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 H1=ABBA2n×2n,A=AH,B=BH,orH2=ABBHAT2n×2norR2n×2n,A=AH,B=BT. H1 can be acquired for crystalline systems (see Sander et al. (2015)), H2 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 H1 and H2 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 H1. 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.

Section snippets

Introduction and preliminaries

The accurate and efficient computation of optical properties of molecules and condensed matter has been an objective actively pursued in recent years [1], [2], [3], [4]. In particular, the increasing importance of renewable energies reinforces the interest in the in silico prediction of optical properties of novel composite materials and nanostructures.

New theoretical and algorithmic developments need to go hand in hand with the ever advancing computer technology. In view of the ongoing massive

Results on the spectral structure of BSE matrices

Non-definite scalar products introduced in (4) provide a language to describe the structure of the BSE matrices in a more concise way, not relying on the matrix block structure. The following two matrices, and the scalar products induced by them, play a central role: Jn=0InIn0,Kn=In00In.We drop the index when the dimension is clear from its context. The identities J1=J and K1=K are regularly used in the following. The results compiled in this section are partly known and can be proven

Ways to solve the definite product eigenvalue problem

We have seen in Theorem 6 that the BSE problem of form I with size 2n×2n can be interpreted as a product eigenvalue problem with two Hermitian factors of size n×n. In this section we assume that the definiteness property (5) holds. Then the factors of the product eigenvalue problem are Hermitian positive definite.

In practice, the complete set of eigenvectors provides additional insight into excitonic effects. To compute them, left and right eigenvectors of the smaller product eigenvalue problem

Numerical experiments

We implemented and compared serial versions of algorithms presented in Table 1 in MATLAB. They compute positive eigenvalues and associated eigenvectors of a BSE matrix H2n×2n of form I (2), which fulfills the definiteness property KH>0. The eigenvalues are given as a diagonal matrix DRn×n. The eigenvectors V2n×n are scaled s.t. K-orthogonality holds, i.e. VHKV=In. The K-orthogonality is an important property in the application. It is exploited in order to construct the polarizability

Towards structure preserving methods for non-definite problems

The presented algorithms assume (5) to hold, i.e. KH is assumed to be positive definite. However, as pointed out in [8], [13], complex eigenvalues can occur in certain contexts and are related to finite lifetimes of particles. In this section, we point out how our framework can be used to gain insight into this case. In contrast to the definite case, the stable computation of required quantities can be more challenging and routines are not as readily available. This section serves as a

Conclusions and discussion

We presented a unifying framework for solving the Bethe–Salpeter eigenvalue problem as it appears in the computation of optical properties of crystalline systems. Two presented methods are superior to the one often used in current implementations, which is based on the computation of a matrix square root. Computing the matrix square root constitutes a high computational effort for non-diagonal matrices. Our first proposed method substitutes the matrix square root with a Cholesky factorization

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