Dirac's equation, quantum electrodynamics, atomic and molecular structure

This essay was written to give readers who are not familiar with quantum electrodynamics or atomic and molecular theory a brief history of the progress made on the physical interpretation of Dirac's celebrated wave equation. We offer a somewhat biased, personal review as it describes primarily British contributions since the 1950's by the first author with the help of several students and researchers. First, some of the main physical and mathematical challenges posed by Dirac's relativistic quantum mechanics are introduced. The first contribution in the 1950's was to formulate the solution in terms of the 4-spinor wave function. The first breakthrough occurred in the 1980's with the introduction of kinetic matching. The second occurred in 2021, when we realised with the increase in computing power that we could solve Dirac's equation with an external potential. This is achieved with CKG-spinor primitive basis sets and Gaussian-based 4-spinors with charge conjugation symmetry and kinetic matching to construct wave functions for electrons and positrons. This mathematical framework solves some of the puzzles of the early work by Dirac (‘the hole theory’) and avoids the complexities of QED theory, still obtaining in a relatively simple way the QED corrections.