Jae Gil Kim^a, Yoon-Hyuck Sean Choi^a, and Eok Kyun Lee^a and Soonchil Lee^b

^a Department of Chemistry, Korea Advanced Institute of Science and Technology, Taejon, Republic of Korea
^b Department of Physics, Korea Advanced Institute of Science and Technology, Taejon, Republic of Korea

The analytical forms of free-electron susceptibilities _d(q) and their range functions _d(r) are derived at nonzero temperature starting from the Green's function representation by properly evaluating the contributions from the poles of free-electron temperature Green's function for each dimension d = 1, 2, and 3. The present formalism produces not only _d(q) and _d(r) which show more accurate temperature-dependent behavior than our previous results for d = 1 and 3, but also temperature-dependent two-dimensional ₂(q) and ₂(r) for a wide range of temperature. Our analytical results show that irrespective of dimension, the singular behavior of _d(q) at q = ±2k_F becomes suppressed at nonzero temperature as the singular points transit to complex wave vectors 2k, and this transition causes _d(r) to be exponentially damped with common damping factor e^{–2₀ sin _rk_Fr} ~ e^–Tk_Fr for low enough temperature, where the exponent of the damping factor corresponds to an imaginary part of wave vectors 2k. We also show that the causality relation of the response function is essential in understanding the correct behavior of _d(q) and _d(r) for all dimensions.