We introduce a generalized matrix expansion method for modeling the long-time dynamics of flexible molecules in solution. Applications consider the Brownian dynamics of a united atom model alkane molecule in a white-noise structureless solvent. The time autocorrelation functions for the middle, end, and end-to-end bonds are computed for the alkane molecules from the Brownian dynamics simulations and the matrix expansion method, both using identical interaction potentials. The leading approximation of the matrix expansion method involves neglecting memory functions, but it fares very well (especially for longer chain alkanes) despite the frequent conformational transitions occurring during the Brownian dynamics trajectories. The leading approximation is deficient at short times due to the neglect of the memory terms. We describe the influence of these memory functions by appending basis functions to the generalized matrix expansion method. The added basis functions yield improved agreement between theory and the simulations, and good convergence is obtained towards the simulations for longer times. The Brownian dynamics simulations are also used to test a universal relations for the orientational correlation functions.