The evaluation of correlation functions by means of approximation of the time evolution operator is discussed. It is shown that different approximations may be obtained depending upon the particular factorization of the equilibrium distribution function in the averages to be computed. With the approximation of free particle dynamics, generalized linear trajectory approximations for correlation functions are obtained. The circumvention in the generalized approximation of the separation of the intermolecular potential employed in the linear trajectory approximation introduced by Helfand is discussed. For low density, it is demonstrated that the constant acceleration approximation introduced by Bloom and Oppenheim is exactly equivalent to a generalized linear trajectory approximation. An explicit expression for the deviation of the constant acceleration approximation result from the exact correlation function expression is obtained. The differences between the constant acceleration and generalized linear trajectory approximations at higher densities are discussed.