基于Lanczos算法的随机相位近似特征值问题的实现与测试
Implementation and testing of Lanczos-based algorithms for Random-Phase Approximation eigenproblems
摘要：
在不同的框架中遇到的随机相位近似哈密顿量的处理，如含时密度泛函理论或Bethe-Salpeter方程，由于它们的非厄米性而变得复杂。与厄米哈密顿量相比，处理非厄米哈密顿量的计算方法通常效率较低，稳定性较差，有时会导致方法的崩溃。最近[Grüning et al. Nano Lett. { \bf 8}，2820（2009）]，我们已经确定这样的哈密顿算子通常是伪厄米算子。利用这一属性，我们已经实现了一个算法的Lanczos型随机相位近似哈密顿，受益于相同的稳定性和计算量作为其厄米对应，并将其应用到碳纳米管的光学响应的研究。我们在这里介绍了相关的理论基础和技术细节，并研究了在Bethe-Salpeter方程框架内计算分子光吸收的算法的性能。
英文摘要：
The treatment of the Random-Phase Approximation Hamiltonians, encountered in different frameworks, like Time-Dependent Density Functional Theory or Bethe-Salpeter equation, is complicated by their non-Hermicity. Compared to their Hermitian Hamiltonian counterparts, computational methods for the treatment of non-Hermitian Hamiltonians are often less efficient and less stable, sometimes leading to the breakdown of the method. Recently [Grüning et al. Nano Lett. {\bf 8}, 2820 (2009)], we have identified that such Hamiltonians are usually pseudo-Hermitian. Exploiting this property, we have implemented an algorithm of the Lanczos type for random-Phase Approximation Hamiltonians that benefits from the same stability and computational load as its Hermitian counterpart, and applied it to the study of the optical response of carbon nanotubes. We present here the related theoretical grounds and technical details, and study the performance of the algorithm for the calculation of the optical absorption of a molecule within the Bethe-Salpeter equation framework.
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