Overview of Electronic Structure Theory. Uncovering and predicting the fundamental quantum processes which lead to macro- scopic physical properties in real materials is the motivation behind electronic struc- ture theory. Uniquely, the quantum-mechanical governing equations are known [10–14] but an analytic solution is only possible for the simplest quantum mechanical systems [15–17] or systems with a large number of exploitable symmetries [18]. Numerical solution generally requires approximation too, owing to the extremely large dimen- sion of the many-body Xxxxxxx space for molecules [19] and solids [20]. In certain physical situations, the full quantum-mechanical formalism can be approximated to remove unnecessary features. As an example, in techniques based on the numerical renormalisation-group procedure of Xxxxxx [21], only many-electron quantum states belonging to a variational submanifold are considered [22, 23]. These methods pro- vide an extremely accurate solution to the quantum many-body ground state but, as a result, are typically restricted to either low-dimensional lattice models [24, 25] or embedded within molecules [26]. At a similar level of accuracy are the quantum Monte Carlo (QMC) methods (see Fig.(1.1)) which stochastically integrate expectation values given some wavefunction [20, 27]. There are many flavours of QMC, of particular importance are diffusion Monte Carlo [28–30] and variational Monte Carlo [31, 32] as they have been applied to the cohesive energy of real solid-state systems such as the semiconductors Si [33] and Ge [34], and also solid H [35–39]. QMC has also been used to study the homogenous electron gas [32, 40]; these results play a central role in more approximate methods. There are computational advantages to QMC; its computational scaling is O (N 3) in time and certain flavours are embarrassingly parallel [27]. However, just like classical Monte Carlo [41], the stochastic error scales with M−1/2 where M is the number of simulation steps. The cubic scaling of QMC (with the number of electrons N ) combined with a huge prefactor puts accurate QMC out of reach for most real materials at present. The next set of electronic structure methods down in accuracy are the mean- or
