Many aspects of computational modelling make it a worthy partner of experimental science. The chemist studying a particular reaction can reach into the computer simulation, alter bond lengths or angles, and then observe the effect of such changes on the process taking place. The geophysicist interested in phase transitions occurring deep inside the earth can model pressures and temperatures which could never be reached in a laboratory. All of this can be achieved with a single piece of apparatus - the computer itself.

Quantum-mechanical calculations stand out because they are by design
* ab initio* i.e. from first-principles, calculations. They do
not depend upon any external parameters except the atomic numbers of
the constituent atoms to be modelled and cannot therefore be biased by
preconceptions about the final result. Such calculations are reliable
and can be used with confidence to predict the behaviour of nature.

Nevertheless, the same complexity which precludes exact analytical solution also results in the highly unfavourable scaling of computational effort and resources required. The computational demands of exact calculations grow exponentially with the size of the system being studied, so that they are too costly to be of significant practical use. Despite the relentless progress of computer technology, this scaling makes this approach inviable for some time yet.

Well-controlled approximations can be employed to enable the equations to be solved much more efficiently without sacrificing the predictive power or parameter-free nature of quantum-mechanical calculations. Much progress has been made in recent years in developing methods which exhibit polynomial rather than exponential scaling. One such method, that of density-functional theory, coupled with a simple description of the quantum-mechanical effects of exchange and correlation and the pseudopotential approximation, has proved to be remarkably successful and is currently applied worldwide by scientists in a wide range of disciplines. Even this method, however, requires a computational effort which scales with the cube of the system-size i.e. is , and so is limited in the scale of simulation which can be realistically attempted.

The
aim of the work described in this dissertation is to develop new
schemes for performing density-functional calculations which lose none
of the accuracy of current approaches, but which require an effort
which scales only * linearly* with system-size i.e. .
A ten-fold increase in computing power then results in a
ten-fold increase in accessible system-size. Therefore these methods
are sought after not only because they increase the range of
applicability of quantum-mechanical calculations now, but also because
they will take full advantage of future improvements in computing
resources.