Newton's Principia Mathematica was the climax of a revolution in man's perception of the Universe, resulting in the acceptance of mathematical physics as a reliable and powerful tool for describing nature. The same laws which accurately predicted the motion of planets around the sun also accounted for the trajectories of terrestrial projectiles, including that legendary windfall apple. So remarkable was the enormous range of scales over which the laws were observed to work, that many believed that they applied universally. Two centuries later, however, a second revolution took place in which classical Newtonian mechanics was found to be inadequate for explaining phenomena on the atomic scale, and a new theory was required. This theory was quantum mechanics.
Despite the philosophical questions of interpretation [1] which arise from the new theory, few question the astounding accuracy with which quantum mechanics describes the world around us. The favourite example cited is that of relativistic quantum field theory's prediction of the gyromagnetic ratio of the electron [2], which agrees with experiment [3] to better than one part in a million. Today there is little doubt that quantum theory applied to electrons and atomic nuclei provides the foundation for all of low-energy physics, chemistry and biology, and that if we wish to describe complex processes occurring in real materials precisely, we should attempt to solve the equations of quantum mechanics.
Unfortunately, the equations are too complicated to be solved analytically for all but the simplest (and hence most trivial) of systems. The only hope of bringing the power of quantum mechanics to bear on real phenomena of genuine interest to contemporary scientists, and of relevance to our society in general, is to solve the equations numerically by modelling the processes of interest computationally.