Collisional energy transfer remains an important area of uncertainty in master equation simulations. Quasi-classical trajectory (QCT) calculations were used to examine the energy transfer probability density distribution (energy transfer kernel), which depends on translational temperature, on the nature of the collision partners, and on the initial and final total internal energies and angular momenta: P(E, J; E', J'). For this purpose, model potential energy functions were taken from the literature or were formulated for pyrazine + Ar and for ethane + Ar collisions. For each collision pair, batches of 10(5) trajectories were computed with three selected initial vibrational energies and five selected values for initial total angular momentum. Most trajectories were carried out with relative translational energy distributions at 300 K, but some were carried out at 1000 or 1200 K. In addition, some trajectories were computed for artificially "heavy" ethane, in which the H-atoms were assigned masses of 20 amu. The results were binned according to (Delta E, Delta J), and a least-squares analysis was carried out by omitting the quasi-elastic trajectories from consideration. By trial-and-error, an empirical function was identified that fitted all 45 batches of trajectories with moderate accuracy. The results reveal significant correlations between initial and final energies and angular momenta. In particular, a strong correlation between Delta E and Delta J depends on the smallest rotational constant in the excited polyatomic. These results show that the final rotational energy distribution is not independent of the initial distribution, showing that the plausible simplifying assumption described by Smith and Gilbert [Int. J. Chem. Kinet. 1988, 20, 307-329] and extended by Miller, Klippenstein, and Rally [J. Phys. Chem. A 2002, 106, 4904-4913] is invalid for the systems studied.