A new computational scheme for calculating vibrational energy relaxation rate constants is presented that is based on applying the linearized semiclassical approximation to the symmetrized force-force correlation function. Unlike the previous scheme of Geva and Shi [Shi, Q.; Geva, E. J. Phys. Chem A 2003, 107, 9059], which was based on applying the linearized semiclassical approximation to the standard force-force correlation function, the new scheme does not involve a power expansion of the initial force in terms of the Wigner transform integration variable Delta and as a result is more accurate and does not require force derivatives as input. The main disadvantages of the new scheme are its slower convergence rate in comparison to the Shi-Geva scheme and its ambiguity with respect to the choice of configuration around which the local harmonic approximation is performed. The computational feasibility and accuracy of the new approach is demonstrated on a number of benchmark models, including the highly challenging case of nonpolar diatomic liquids. It is observed that performing the local harmonic approximation around the configuration used for computing the initial force yields the best agreement with experiment.