Using density functional theory (DFT) combined with the first-principles nonequilibrium Green's function (NEGF), we investigated the electron-transport properties and rectifying behaviors of several molecular junctions based on the bis-2-(5-ethynylthienyl)ethyne (BETE) molecule. To examine the roles of different rectification factors, asymmetric electrode molecule contacts and donor-acceptor substituent groups were introduced into the BETE-based molecular junction. The asymmetric current-voltage characteristics were obtained for the molecular junctions containing asymmetric contacts and donor-acceptor groups. In our models, the computed rectification ratios show that the mode of electrode-molecule contacts plays a crucial role in rectification and that the rectifying effect is not enhanced significantly by introducing the additional donor-acceptor components for the molecular rectifier with asymmetric electrode molecule contacts. The current-voltage characteristics and rectifying behaviors are discussed in terms of transmission spectra, molecular projected self-consistent Hamiltonian (MPSH) states, and energy levels of MPSH states.