There is considerable experimental evidence of covalent dimerization of aromatic compounds occurring under shock conditions. Because of their endothermicity, these reactions could play a large role in the shock initiation process of aromatic molecular explosives such as 2,4,6-trinitrotoluene and 1,3,5-triamino-2,4,6-trinitrobenzene by withdrawing energy from the shock compression. Very little is known about the energetics, however, and this knowledge is crucial for the design of empirical force fields that can treat shock-induced chemistry. We have employed ab initio electronic structure and density functional methods to study the Diels-Alder (DA) dimerizations of benzene and anthracene. The enthalpy of reaction for DA benzene dimerization is predicted to be +35.9 kcal/mol. The stepwise pathway to this dimer involves formation of a stable triplet intermediate that requires 71.8 kcal/mol of energy. Transition states along both the concerted and stepwise pathways were optimized and the energetics of the reaction pathways are detailed. The former is found to be the energetically preferred mechanism. Nine DA dimers of anthracene were found, with six predicted to have dimerization DeltaH(rxn)'s of 24-55 kcal/mol, two with dimerization energies near zero and one that is formed through an exothermic reaction. Twelve triplet dimers of anthracene, with DeltaH(rxn)'s ranging from 33-50 kcal/mol, are also described. Finally, the potential importance of these reactions in the context of shock compression of these materials is discussed.