Research overview

Quantum Monte Carlo



The term «quantum Monte Carlo» (QMC) refers to a class of numerical algorithms aimed at solving the Schrödinger equation for a quantum-mechanical system in a stochastic manner. Its best-known flavours are the variational and diffusion Monte Carlo methods (VMC and DMC):


The advantages of QMC over other methods are:

Thus QMC provides more accuracy than faster methods (e.g. DFT), while it features a more favourable scaling with system size than the more accurate ones (e.g. Coupled Cluster).


VMC is strictly driven by the trial wave function, and it is therefore limited by how good a trial wave function we can construct. DMC should in principle be free from this constraint, but due to the fermion sign problem the method needs to be reformulated into a fixed-node (FN) version, which is only exact when the nodes (zero-valued regions) of the trial wave function coincide with those of the exact ground-state wave function.

There are two main lines of research aimed at solving (or alleviating the effects of) the fermion sign problem:

The former has seen promising, gradual developments in the past couple of years, whereas the second is still to yield a satisfactory solution to the problem.

Backflow transformations in QMC



The standard QMC wave function for fermionic systems is the Slater-Jastrow wave function, consisting of a Slater determinant times a positive Jastrow factor,

Ψ(R)=e J(R) D(R)

The Slater determinant is constructed using one-particle orbitals, usually obtained from a cheaper method, and accounts for exchange. The Jastrow factor, which usually contains optimizable parameters, depends explicitly on the inter-particle distances so it is capable of describing correlation effects. It is also used to enforce the Kato cusp conditions, which cancel out the divergencies of the Coulomb potential and give a more stable and accurate calculation.

The Jastrow factor gives the functional form a great deal of flexibility, and it can typically recover about 60%–80% of the correlation energy of the system at the VMC level. However it cannot modify the nodes of the wave function, since e J ≥0, therefore it does not affect the DMC energy. Note that this statement can be reversed to give an useful conclusion: DMC is equivalent to VMC with a perfect Jastrow.

Pairing wave functions in QMC



Technical improvements to QMC methods

Application of QMC