In this study a simple relation has been derived for the influence parameter in the gradient theory of van der Waals in terms of simple accessible quantities like temperature, the equilibrium densities, and the equilibrium isothermal compressibilities, Application of this relation leads to a substantially better agreement between interfacial tensions computed from the gradient theory and tensions obtained from experiment and simulation. The basis for this novel relation is an expression that connects the influence parameter to the second moment of the direct correlation function of pure fluids at states within the binodal region. The direct correlation functions for this study have been obtained from solving of the Ornstein-Zernike equation and the Percus-Yevick (PY) or hypernetted chain (HNC) closure relations for a Lennard-Jones (LJ) fluid. Special attention was paid to the behavior of solutions in the vicinity of the nonsolution region. It was shown for the PY closure that at the low density side the isothermal compressibility remains finite at the boundary of the nonsolution region. Along the isotherms and isochors the isothermal compressibility terminates at this boundary in so-called square root branch points. The isothermal compressibility diverges on the high density side although the correct location of the spinodal locus could not be found because of numerical inaccuracies. Diverging compressibilities are never encountered as the solution boundary is reached using the HNC closure. In all cases the isothermal compressibility terminates in square root branch points along the isotherms and isochors. In addition, computations show that at this boundary, the second moment of the direct correlation function seems to diverge for both the PY and the HNC closure. Comparison of the tensions obtained from the gradient theory with those obtained from a partial summation of the gradient expansion shows that at low temperatures the former results are similar to 50% higher. Comparison of the results obtained from the latter model with experiments and simulations shows good agreement.