A self-consistent field (SCF) theoretic approach, using a general excess Helmholtz energy density functional that includes a square gradient term, is derived for polymer melt surfaces and implemented for linear polyethylene films over a variety of temperatures and chain lengths. The formulation of the SCF plus square gradient approximation (SGA) developed is generic and can be applied with any equation of state (EoS) suitable for the estimation of the excess Helmholtz energy. As a case study, the approach is combined with the Sanchez Lacombe (SL) EoS to predict reduced density profiles, chain conformational properties, and interfacial free energies, yielding very favorable agreement with atomistic simulation results and noticeable improvement relative to simpler SCF and SGA approaches. The reduced influence parameter invoked in the SGA to achieve accurate density profiles and interfacial free energies is consistent with the definition of Poser and Sanchez, J. Colloid Interface Sci. 1979, 69, 539-548. The new SCF_SL + SGA approach is used to quantify the dominance of chain end segments compared to that of middle segments at free polyethylene surfaces. Schemes are developed to distinguish surface adsorbed from free chains and to decompose the surface density profiles into contributions from trains, loops, and tails; the results for molten polyethylene are compared with the observables of atomistic simulations. Reduced chain shape profiles indicate flattening of the chains in the surface region as compared to the bulk chains. The range of this transitional region is approximately 1.6 radii of gyration (R-g). The inclusion of chain conformational entropy effects, as described by the modified Edwards diffusion equation of the SCF, in addition to the square gradient term in density, provides more accurate predictions of the surface tension, in good match with experimental measurements on a variety of polymer melts and with atomistic simulation findings.