We investigate the local structure factor S(z; q) at a free liquid-gas interface in systems with
short-ranged intermolecular forces and determine the corrections to the leading-order, capillarywave-like,
Goldstone mode divergence of S(z; q) known to occur for parallel (i.e., measured along
the interface) wavevectors q → 0. We show from explicit solution of the inhomogeneous OrnsteinZernike
equation that for distances z far from the interface, where the profile decays exponentially,
S(z; q) splits unambiguously into bulk and interfacial contributions. On each side of the interface,
the interfacial contributions can be characterised by distinct liquid and gas wavevector dependent
surface tensions, σl(q) and σg(q), which are determined solely by the bulk two-body and three-body
direct correlation functions. At high temperatures, the wavevector dependence simplifies and is
determined almost entirely by the appropriate bulk structure factor, leading to positive rigidity
coefficients. Our predictions are confirmed by explicit calculation of S(z; q) within square-gradient
theory and the Sullivan model. The results for the latter predict a striking temperature dependence
for σl(q) and σg(q), and have implications for fluctuation effects. Our results account quantitatively
for the findings of a recent very extensive simulation study by H¨ofling and Dietrich of the total
structure factor in the interfacial region, in a system with a cut-off Lennard-Jones potential, in
sharp contrast to extended Capillary-Wave models which failed completely to describe the simulation
results.