Total and direct correlations in the liquid–vapor interface and the capillary-wave theory

The two-point total correlation function ${\text{H̃(1,2)=〈δ}}_{\mathrm{\rho -k}}{\mathrm{\delta \rho }}_k{\mathrm{〉=H(z}}_1{\mathrm{,z}}_2\mathrm{,k)}$ in the interfacial zone between a Lennard-Jones liquid and its coexisting vapor is determined in a molecular dynamics simulation at a low temperature $\text{T=0.75.}$ New terms absent in the standard capillary wave theory are discovered. Eigenvector analysis and matrix partitioning of $H$ and of the direct correlation function $C$ revealed a strong $k^2$ dependence at $\text{k→0}$ in a set of terms proportional to ${\rho }^{\prime }{\rho }^{\prime },$ ${\rho }^{\mathrm{\prime \prime }}{\rho }^{\mathrm{\prime \prime }},$ ${\rho }^{\mathrm{\prime \prime \prime }}{\rho }^{\mathrm{\prime \prime \prime }}\mathrm{,\dots \hspace{0.17em}.}$ The bending coefficient is found nonexistent. Projections of $\mathrm{H,}$ $\mathrm{C,}$ and susceptibilities are discussed in the context of removal of all capillary wave contributions to obtain intrinsic quantities.