Hall was educated at Queen's University Belfast and St John's College, Cambridge, where he studied for a PhD under the supervision of John Lennard-Jones. For his work on the Roothaan-Hall equations, Hall was awarded a doctorate by the University of Cambridge in 1950.[2] He then lectured at Cambridge as a Research Assistant in theoretical chemistry, before being elected to a Fellowship at St John's College in 1953. From 1955 to 1962 he lectured in Mathematics at Imperial College London. In 1957–58 he spent a year with Per-Olov Löwdin in Uppsala, Sweden. He became Professor of Mathematics at the University of Nottingham in 1962. In 1982 he took early retirement from Nottingham and was appointed an emeritus professor. He moved in 1983 to Kyoto University, Japan, returning to Nottingham in 1988. He has collaborated with (inter alia) A.T. Amos, K. Collard, and D. Rees. He was Emeritus Professor and Senior Research Fellow in the Shell Centre for Mathematical Education at the University of Nottingham.