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We present an analytic theory of the pair correlation function $g(z)$ in a one dimensional (1D) electron liquid. Our approach involves the solution of a zero-energy scattering Schr\"odinger equation for $\sqrt{g(z)}$ where we implemented the Fermi hypernetted-chain approximation including the elementary diagrams corrections.
We present numerical results for $g(z)$ and the static structure factor $S(k)$ and obtain very good agreement with data from difussion Monte Carlo studies of the 1D system. We calculate the total effective electron-electron interaction and the charge excitation spectrum over an extensive range of densities and quantum wire width. Furthermore, we obtain the static correlations in good qualitative agreement with those calculated for the Luttinger liquid model with long-range interactions.
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