In this work we study flexible Lennard-Jones (LJ) interaction-site chain molecules with fixed bond length L. We obtain "exact" numerical results for the equilibrium configurational and energetic properties of LJ n-mer chains with n=3, 4, and 5 for a range of L (0.8 less than or equal toL/sigma less than or equal to2(1/6)). These results include intramolecular site-site distribution functions and fluctuations in site-site distances, radius of gyration, and average potential energy. We also compute the single-chain specific heat, chemical potential, and theta temperature. With decreasing temperature the chain molecules undergo a collapse transition from an extended disordered state to a compact highly ordered state, exhibiting an associated peak in the specific heat. For n=3 and 4 the collapse occurs in a single stage whereas for the n=5 chains a multistage collapse is possible. In particular, the specific heat peaks associated with the collapse of 5-mer chains with bond lengths in the range 0.85 less than or equal toL/sigma less than or equal to1.0 display either a low temperature shoulder or secondary maximum. We show that this complex specific heat behavior arises when the initial collapsed state consists of a set of distinct and nondegenerate structural isomers. The secondary peak or shoulder is a result of the "freezing out" of all but the ground state isomer. These results are discussed in the context of the similar anomalous specific heat behavior, and low temperature structural transitions observed in recent simulation studies of square-well and LJ chains and atomic clusters.