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\section{Linear response in the electron-hole basis}  \label{sec:casida}  An alternate approach to linear response is not to solve for the response function but rather for its poles (the excitation energies $\omega_k$) and residues (\textit{e.g.} electric dipole matrix elements $\vec{d}_k$) \cite{Strubbe2012}. The polarizability is given by  \begin{align}