Barnett spent most of the World War II years near
Fleetwood in Lancashire. He attended
Baines' Grammar School in
Poulton-le-Fylde, then went to
King's College, London in 1945, where he received a BSc in chemistry in 1948, a PhD for work in the theoretical physics department with
Charles Coulson in 1952, that he continued on a one-year post-doctoral fellowship. His assigned project was to determine if
electrostatic forces could account for the energy needed to make two parts of an
ethane molecule rotate around the bond that joins them. This work required the evaluation of certain mathematical objects – molecular integrals over
Slater orbitals. Barnett extended some earlier work by
Charles Coulson by discovering some
recurrence formulas, that are part of a method of analysis and computation frequently referred to as the Barnett-Coulson expansion. Molecular integrals remain a significant problem in quantum chemistry and continued to be one of Barnett's main interests. Two years after Barnett started this work, he was invited to be one of the twenty-five participants in a conference that was organised by
Robert Mulliken, sponsored by the
National Academy of Sciences and known, from its venue, as the
Shelter Island Conference on Quantum Mechanics in Valence Theory. Barnett's attendance was enabled by the
British Rayon Research Association, which supported his post-graduate work. At the
Royal Radar Establishment, Barnett held a Senior Government Fellowship. He worked on aspects of theoretical
solid state physics, that included the properties of
organic semiconductors. As part of his work at IBM United Kingdom, he directed an
IBM model 650 computer centre. He directed and participated in numerous projects that included (1) calculating
DNA structures from crystallographic data, and (2) simulations to plan the location and operation of dams and reservoirs on the
River Nile, working with Humphry Morrice, the hydrological advisor to the Government of the Sudan, and his predecessor, Nimmo Allen. In 1957, Barnett accepted an invitation from
Joseph Hirschfelder, in the Chemistry department of the
University of Wisconsin–Madison, to work on mathematical theories of
combustion and
detonation. ==Activities at MIT==