LCT, UMR7616, University Paris-Sorbonne
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Solving the Schroedinger equation can be recast as computing large dimensional integrals, or large sums. This is true for variational methods, perturbative methods, or even in principle for exact methods (using Feynman-Kac integrals). Monte Carlo (MC) methods rely on a probabilistic interpretation of integrals, and compute them using a statistical sampling, drawn through a stochastic process. They have a modest numerical scaling, to compute large dimensional integrals. For this reason they are a natural tool to
tackle quantum physics or quantum chemistry, where they are called Quantum Monte Carlo methods (QMC). Because, they avoid to the constraint to compute integrals analytically they allow a large variational flexibility, leading to high accuracies to obtain groundstate energies. This accuracy can be achieved for a modest computational cost if we compare to other methods based on the description of the wavefunction: post-Hartree Fock methods, like configuration of interaction (CI) or coupled cluster (CC) methods. QMC methods are benchmark methods but up to now, they are far to be the most used methods in quantum chemistry. This is because of a numerical bottleneck, the statistical fluctuations are large when small energy differences and derivatives are involved, they grow like O(N1−2 ) with the number N of particles. This is at the origin of large numerical cost O(N4−5 ) to compute excited states, or to optimize the wavefunction or the geometry of a large molecule. As a result, if the scaling of these methods is usually lower than post-Hartree Fock methods it’s much larger than methods based on the description of the density: Density Functional Theory (DFT).
We introduce Monte Carlo methods, alongside with some basic notions on probability theory and Markov chains. We draw a perspective on Quantum Monte Carlo methods, focusing mainly on recent developments in Monte Carlo method in real space, to the scaling for optimizing the wavefunction, and the geometry of a molecule [1-2]. Some perspective will be also given tocompute excited states, and on current developments which may lower the scaling of QMC techniques close to Density Functional Theory standards.
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