Approximate Euler-Lagrange Equations for the Ground-State Electron Gas

The ground-state wave function for a quantum electron gas is approximated by $\psi =\Phi D$, where $D$ is the ground-state ideal-gas wave function and $\Phi =\exp [-(\frac{1}{2}\left)\Sigma {i<j}^{}u\left(r_{\mathrm{ij}}\right)\right]$. Three-body correlation functions in the energy are approximated so that the energy is a functional of only pair-correlation functions. The variational principle is applied by utilizing the functional derivative of the pair-correlation function with respect to $u\left(r\right)\mathrm{}$. The three- and four-body correlation functions in the functional derivative are approximated in a consistent manner. Use of the Kirkwood superposition approximation for the many-body correlation functions in the functional derivative gives unphysical results. A new approximation to the functional derivative is obtained by using the convolution approximation for the many-body terms. This gives a reasonable result for the asymptotic form of the wave function. The hypernetted-chain functional derivative combined with the convolution approximation for the energy gives the same asymptotic form. When the random-phase approximation to the energy is used with the Gaskell-Broyles-Sahlin-Carley functional derivative, the equation may be solved analytically, giving the same result obtained by Gaskell. The asymptotic form is the same as the two previous methods, namely, $u\sim \left(\frac{e}{\hslash }\right)\frac{{\left(\frac{m}{\pi \rho }\right)}^{\frac{1}{2}}}{r}$.