[8][9][10] Dewar's reputation for providing original solutions to vexing puzzles first developed when he was still a postdoctoral fellow at the University of Oxford.

[11] The discovery of the tropolone structure launched the field of non-benzenoid aromaticity, which witnessed feverish activity for several decades and greatly expanded the chemists' understanding of cyclic π-electron systems.

[8] Following Woodward and Hoffmann's suggestion of selection rules for pericyclic reactions, Dewar championed (concurrently with Howard Zimmerman) an alternative approach (which he erroneously felt had been pioneered by M. G. Evans) to understanding pericyclic reactivity based on aromatic and antiaromatic transition states.

He is known most famously for the development in the 1970s and 1980s of the Semi-empirical quantum chemistry methods, MINDO, MNDO,[21] AM1 and PM3 that are in the MOPAC computer program, and which for the first time enabled the quantitative study of the structure and mechanism of reaction (transition state) of many real (i.e. large) systems.

[8] This was illustrated in 1974 by computing (using the technique of energy minimisation) the structure of a molecule as large as LSD (with 49 atoms) at a quantum mechanical level (the calculation taking several days of the then state-of-the-art supercomputer time, a CDC 6600).