Introduction

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):

• VMC is a simple application of the Monte Carlo integration technique to obtain the expectation value of the Hamiltonian of the system with respect to a given trial wave function. This expectation value is guaranteed to be an upper bound of the ground-state energy of the system, as per the variational principle. In addition, one can optimize the parameters of the wave function in order to minimize the variational energy, thus improving the accurary of the description of the system.
• DMC is a more sophisticated method which uses the imaginary-time-dependent Schrödinger equation to project out the higher-energy components of the trial wave function to give, in principle, the exact ground state energy of the system.

Advantages

The advantages of QMC over other methods are:

• Because of using wave functions, QMC is a truly many-body method. This is in contrast with Density Functional Theory, whose density-based description of the system removes many-body effects. These must be re-introduced approximately by choosing an appropriate exchange-correlation functional, which is arbitrary in any case.
• Because of being a numerical integration technique, QMC can use any wave function to describe the system. In particular, physically-sensible constraints like the Kato cusp conditions can be enforced directly without prohibitively increasing the cost of the calculation. This is in contrast with some quantum Chemistry methods which rely on being able to calculate integrals analytically, and therefore require large expansions of simple terms to achieve a good accuracy.

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).

Limitations

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:

• Understanding the nodal properties of trial wave functions and designing better functional forms.
• Designing new QMC methods.

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.

Current applicability

Some ten years ago very few people used QMC for real applications. But presently the number of groups using QMC is increasing at an accelerated pace. One reason is that now QMC is more reliable as a production method than it was then. Another is that computers are getting faster, and the almost-perfect parallelization attainable in QMC implementations can exploit the capabilities of computer clusters better than most other techniques.

Introduction

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.

Shaping landscapes

Just to exercise our visualization skills, let us imagine a landscape, with mountains and valleys, and fill it with water up to, say, half the maximum height, thus adding some lakes to the picture. Now focus on three elements: land, water and shorelines.

Clearly the parallelism is that the value of the wave function corresponds with the height of each point in our landscape. Land corresponds to positive nodal regions, and water corresponds to negative ones. The shoreline, at which the height is zero, is the nodal surface (in this simple 2D example it is the nodal curve).

It is easy to see that the Jastrow factor, a position-dependent positive scale factor, could indeed modify the Slater landscape much at will. It can build and destroy mountains, it can create sharp summits (as it indeed does when cusp conditions are applied) and dig holes. The only thing it cannot do is turn land into water or vice versa, the Jastrow factor leaves the shoreline intact. And that turns out to be rather important: getting the shoreline wrong means moving DMC away from the ground state.

So the Jastrow factor works by scaling the local height of the underlying wave function. If we want to modify shorelines, we need to seek a different landscape transformation function. Consider this:

Ψ[X(R)]= e ^J(R) D[X(R)]

X is a transformed set of coordinates, which we can write as X=R+Δ(R), where Δ is a position-dependent displacement.

Back in our landscape example, we can see why this is useful. Imagine that we sit at a point R in the Slater land near the shoreline. With this transformation, instead of evaluating the wave function at R we will evaluate it at R+Δ, and if Δ is chosen well, R+Δ may be in the water. So we are changing the wave function at R from a positive value D(R) to a negative one D[X(R)]. And provided Δ is a continuous function of R, this means we are changing the shoreline.

This transformation is called the backflow transformation. Notice that this transformation complements that effected by the Jastrow factor, in the sense that backflow exerts a sort of transversal shift of the Slater landscape, and then the Jastrow factor applies a vertical rescaling. The combination of the two transformations is potentially extremely powerful in reshaping Slater wave functions.

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• Optimal sampling
• The zero variance limit

• Phase diagram of the electron-hole system