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In this note we derive the displacement charge of a site which is coupled to a noninteracting tight binding semi-infinite chain.   First we derive the Green function for a semi-infinite chain and after we coupled such chain to an impurity with a energy level $\epsilon_d$.  \begin{equation}  H=\sum_{\sigma} \epsilon_0 c^\dagger_{i} c_{i} + t \sum_{n=1}^{n=\infty} \left(c^\dagger_{i}c_{i+1}+h.c\right)  \end{equation}  \section{Semi-infinite chain Green function}  We start with the following wavefunction  \begin{equation}  |\Psi\rangle =\sum_{\beta=1}^{\infty} \sin(k\beta)|\beta\rangle 
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This is the surface (boundary) DOS, which never diverges in contrast with the bulk DOS