  Monte Carlo simulations are a powerful tool to investigate the thermodynamic
properties of atomic systems. In practice however, sampling of the complete
configuration space is often hindered by high energy barriers between different
regions of configuration space which can make ergodic sampling completely
infeasible within accessible simulation times. Although several extensions to
the conventional Monte Carlo scheme have been developed, that enable the
treatment of such systems, these extensions often entail substantial
computational cost or rely on the harmonic approximation. In this work we
propose an exact method called Funnel Hopping Monte Carlo (FHMC) that is
inspired by the the ideas of smart darting but is more efficient. Gaussian
mixtures are used to approximate the Boltzmann distribution around local energy
minima which are then used to propose high quality Monte Carlo moves that
enable the Monte Carlo simulation to directly jump between different funnels.
To fit the Gaussian mixtures we developed an extended version of the
expectation-maximization algorithm that is able to take advantage of the high
symmetry present in many low energy configurations. We demonstrate the methods
performance on the example of the 38 as well as the 75 atom Lennard-Jones
clusters which are well known for their double funnel energy landscapes that
prevent ergodic sampling with conventional Monte Carlo simulations. By
integrating FHMC into the parallel tempering scheme we were able to reduce the
number of steps required until convergence of the simulation significantly.
