Quantum Monte Carlo

Quantum Monte-Carlo (QMC) methods are now mentioned briefly because of the accurate exchange-correlation data they can generate for use in functional development, such as total energies, energy densities and holes. An extensive and recent review of QMC methods is given by Foulkes  et al. in Ref. [88].

Whereas Hartree-Fock theory and DFT are mean-field theories in that they invoke a single-particle description which replaces the real forces of interaction between electrons with an averaged or mean-field, QMC takes the alternative approach and computes the actual many-electron wavefunction $\Psi$ for the system. There are two principle QMC methods - variational Monte-Carlo (VMC) and diffusion Monte-Carlo (DMC). VMC is the cheapest and less accurate of the two methods, but exchange-correlation data is more readily available from this method. A VMC simulation proceeds by first choosing a trial wavefunction $\Psi_{\hbox{{T}}}$, which contains functions representing the electron-nuclear and electron-electron correlations. These functions are then adjusted so as to minimise the variance of the total energy according to the variational principle. So the accuracy of the VMC method depends on how well the form of $\Psi_{\hbox{{T}}}$ can represent the particular system.

The DMC method [89,90,91] is the most accurate groundstate electronic structure method, at least for extended systems. The only approximation in DMC is the location of the nodes of the wavefunction, i.e. where $\Psi$ equals zero and changes sign. These are fixed throughout a simulation and the wavefunction is optimised between the nodes. This is commonly called the fixed-node approximation [90]. Usually the nodes from VMC wavefunctions are used as the input. The DMC method involves solving the imaginary time many-electron Schrödinger equation using a population of ``walkers'' that randomly sample the $3N$-dimensional vector space - the groundstate wavefunction can be obtained from the population density of the walkers after a sufficient amount of imaginary time has elapsed.

Probably the most important DMC simulations were performed for the homogeneous electron gas by Ceperley and Alder in $1980$ [51], since this lead to the accurate determination of the parameters in the correlation part of the LDA. As a consequence, QMC essentially made DFT practical. However, a symbiotic relationship exists between DFT and the QMC methods nowadays: although the DMC method effectively brought about the LDA - which forms the basis of all functionals - the majority of QMC calculations currently performed, use densities and even pseudopotentials generated from DFT - usually with the LDA.