In the framework of second-order Moller-Plesset linear-r(12) (MP2-R12) perturbation theory, a method is developed and implemented that uses an auxiliary basis set for the resolution-of-the-identity (RI) approximation for the three- and four-electron integrals. In contrast to previous work, the two-electron integrals that must be evaluated never involve more than one auxiliary basis function. The new method therefore scales linearly with the number of auxiliary basis functions and is much more efficient than the previous one, which scaled quadratically. A general formulation of MP2-R12 theory is presented for various ansatze, approximations, and orbitals (canonical or localized). The new method is assessed by computations of the valence-shell second-order Moller-Plesset correlation energy of a few small closed-shell systems. The preliminary calculations indicate that the difference between the new and previous methods is about one order of magnitude smaller than the errors that occur due to basis-set truncations and RI approximations and under the assumptions of generalized and extended Brillouin conditions. (C) 2004 American Institute of Physics.