The basic characters of the electronic correlation effect from weak to strong in the three dimensional electron gas

In this paper, we present the rigorous expression of the ground state energy and study the phase transition of the three-dimensional homogeneous electron gas by the eigenfunctional theory. The ground state energy is decided completely by the pair distribution function $g(r)$, which is, by definition, strictly a positive function. But when the density decreases, the electronic correction effect becomes strong from weak, the previous theories basing on one-particle approximation, such as: RPA, Hubbard and STLS ,\cite{4,9} can't insure $g(r)$ positive always, which implies that they are becoming invalid and overestimate the ground state energy. The eigenfunctional theory has a significant improvement over them, under the linear approximation in solving the equation of the phase field, the $g(r)$ obtained by the eigenfunctional theory is always positive and can well satisfy the normalization integral ($n_{0}\int d^3r[g(r)-1]=-1$) which is one of the important features of $g(r)$. After obtaining $g(r)$ and calculating the ground state energy of the electron gas both in paramagnetic and ferromagnetic by $g(r)$, we observe a continuous phase transition from paramagnetic to the ferromagnetic occurring at $r_s=19.9\pm0.8$. This can be indirectly supported by the observation of a ferromagnetic state in doped hexaboride $(Ca_{1-x}La_xB_b)$,\cite{42} and is close to the result of G.Ortiz et.al ($r_s=20\pm5$).\cite{12}