Peter was awarded the 1999 Dirac Medal and the 2011 Schroedinger Medal of the World Association of Theoretical and Computational Chemists (WATOC), and the 2005 Pople Medal and the 2013 Fukui Medal of the Asia-Pacific Association of Theoretical and Computational Chemists (APATCC). He was also awarded the 2014 Biennial Medal of the Association of Molecular Modellers of Australasia and the 2019 David Craig Medal of the Australian Academy of Science.