Two-point correlation function in systems with van der Waals type interaction

The behavior of the bulk two-point correlation function $G({\bf r};T|d)$ in $d$-dimensional system with van der Waals type interactions is investigated and its consequences on the finite-size scaling properties of the susceptibility in such finite systems with periodic boundary conditions is discussed within mean-spherical model which is an example of Ornstein and Zernike type theory. The interaction is supposed to decay at large distances $r$ as $r^{-(d+\sigma)}$, with $2<d<4$, $2<\sigma<4$ and $d+\sigma \le 6$. It is shown that $G({\bf r};T|d)$ decays as $r^{-(d-2)}$ for $1\ll r\ll \xi$, exponentially for $\xi\ll r \ll r^*$, where $r^*=(\sigma-2)\xi \ln \xi$, and again in a power law as $r^{-(d+\sigma)}$ for $r\gg r^*$. The analytical form of the leading-order scaling function of $G({\bf r};T|d)$ in any of these regimes is derived.