4.2 History and background

At the time of writing, the uptake and use of QMC methods for electronic structure is quite limited. The reasons for the limited uptake provide a useful insight into the technical challenges of QMC methods in their present form.

Since Ceperley and Alder's QMC calculations for the Homogeneous Electron Gas (HEG)[13] in 1980 there has been a gradual increase in the use of QMC methods. [24] Successful application of QMC methods to inhomogeneous systems - those including atoms - is considerably more difficult than for the HEG.

The computational cost of QMC methods scale with the third power of number of electrons in the system. However, in practice it has been established that the prefactor for realistic calculations is very large. Hence, the barrier to performing calculations has been the large prefactor or intrinsic variance of the wavefunctions used in the calculation as well as the number of particles (electrons). The development of improved, more efficient wavefunction forms and methods for their optimisation has led to a considerable reduction in the prefactor.

A hidden scaling of QMC methods is their scaling with the atomic number, $Z$, of the atoms included in the simulation. It has been argued [23,35,36] that the scaling of computer time, $T$, with Z is,

$$T \sim Z^{5.5-6.5} \;\;\;.$$ (4.1)

This scaling is severe, and results from the small size of the atomic cores requiring a different sampling (such as a different timestep) than valence regions. A further limitation results from the large kinetic energies and strong potentials in core regions. These can result in large fluctuations in the local energy and it is difficult to design trial functions that significantly decrease the fluctuations. The wavefunctions in section 4.3 are able to reduce but not eliminate the fluctuations.

Fortunately, for most properties, the atomic cores are almost unaffected by the valence (outermost) electrons and their environment. This observation can be used to remove the core electrons from the simulation, greatly reducing the problems of time and energy scales. The removal is achieved by the incorporation of a pseudopotential. This procedure, which is also common in DFT and quantum chemical calculations,^4.2 is described in section 4.5.