Efficient and accurate algorithms for solving the Bethe–Salpeter eigenvalue problem for crystalline systems
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: We drop the index when the dimension is clear from its context. The identities and 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 can be interpreted as a product eigenvalue problem with two Hermitian factors of size . 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 of form I (2), which fulfills the definiteness property . The eigenvalues are given as a diagonal matrix . The eigenvectors are scaled s.t. -orthogonality holds, i.e. . The -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. 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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