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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.
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Note that in the Lanczos approach the energy range includes always the lowest energies. In principle it is possible to choose a different energy range, not necessarily containing the lowest energy, using the modification proposed by G. Grosso, L. Martinelli and G. Pastori Parravini, Phys. Rev. B 51
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Because of the 4-point nature of the kernel, within BS the reformulation in terms of the eigenproblem for H RPA H^{\text{RPA}} is the only feasible alternative. To the contrary, within TD-DFT the direct solution of the Dyson-like equation is affordable and even the most convenient in the case of bulk periodic systems. On the other hand, for finite or sparse systems even within TDDFT the solution of the eigenproblem for 𝐇 RPA {\bf H}^{\text{RPA}} can be more efficient than the direct solution especially when one is interested only to a frequency range where the excitations are well separated
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A. Marinopoulos and M. Grüning, in preparation
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In particular, at each iteration both the computational cost and the storage are increasing, since all the vectors in the basis are needed to determine the new one
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In principle also the Hermitian algorithm can break down. In fact after many iterations, because of the numerical error, the vectors in the Lanczos basis may stop to be orthonormal
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