Variational problem in Jastrow theory

The ground-state wave function of an infinite system of fermions is approximated by the state-independent Jastrow ansatz. In order to optimize the pair correlation function, the Euler-Lagrange equations of the variational problem for the energy expectation value are derived. The short- and long-range behaviors of the optimum pair correlation function are discussed. Application of graphical techniques and use of rigorous results on the connection between the slope of the static structure function as $k\to 0$+ and long-range Jastrow correlations allow one to prove that the optimum pair correlation function behaves like $1+O\left(r^{-2\mathrm{}}\right)$ as $r\to \infty$. A connection is derived between the weight of the long-range correlations and the Landau parameters. As simple examples the limit of Bose statistics and the electron-gas problem are considered. The consequences of these investigations on numerical calculations and their relation to alternative expansion methods are investigated.