- © The Mineralogical Society Of America
In this brief article, various types of quantum Monte Carlo (QMC) methods are introduced, in particular, those that are applicable to systems in extreme regimes of temperature and pressure. References to longer articles have been given where detailed discussion of applications and algorithms appear.
One does QMC for the same reason as one does classical simulations; there is no other method able to treat exactly the quantum many-body problem aside from the direct simulation method where electrons and ions are directly represented as particles, instead of as a “fluid” as is done in mean-field based methods. However, quantum systems are more difficult than classical systems because one does not know the distribution to be sampled, it must be solved for. In fact, it is not known today which quantum problems can be “solved” with simulation on a classical computer in a reasonable amount of computer time. One knows that certain systems, such as most quantum many-body systems in 1D and most bosonic systems are amenable to solution with Monte Carlo methods, however, the “sign problem” prevents making the same statement for systems with electrons in 3D. Some limitations or approximations are needed in practice. On the other hand, in contrast to simulation of classical systems, one does know the Hamiltonian exactly: namely charged particles interacting with a Coulomb potential. Given sufficient computer resources, the results can be of quite high quality and for systems where there is little reliable experimental data. For this reason, QMC methods, though more expensive, are useful to benchmark and validate results from other methods.
The two main applications discussed in this review are the “electronic structure problem”; computing the energy of the interacting systems of electrons and fixed ions, and the problem of quantum effects of the nuclei.
RANDOM WALK AND MARKOV CHAINS
Most QMC algorithms are based on random …