Comparison of two non-primitive methods for path integral simulations: Higher-order corrections versus an effective propagator approach

Two methods are compared that are used in path integral simulations. Both methods aim to achieve faster convergence to the quantum limit than the so-called primitive algorithm (PA). One method, proposed by Takahashi and Imada, is based on a higher-order approximation (HOA) of the quantum-mechanical density operator. The other method is based upon an effective propagator (EPr). This propagator is constructed such that the correct quantum properties are obtained even at finite Trotter numbers P in the limit of small densities. We discuss the conceptual differences between both methods and compare their convergence rate. While the HOA method converges faster than the EPr approach, EPr gives good estimates of thermal quantities already for $P=1.\mathrm{}$ Despite a significant improvement with respect to PA, neither HOA nor EPr overcome the need to increase P linearly with inverse temperature. We also derive the proper estimator for radial distribution functions for HOA based path integral simulations and show that the ${1/P}^4$ convergence in the HOA approach also applies if the interatom repulsion is treated realistically. The case studies include an HOA based virial expansion of ${}^4\mathrm{He}$ and a Lennard-Jones model of solid argon.