  Convergence of path integral simulations requires a substantial number of
beads when quantum effects are significant. Traditional Trotter scaling
approaches estimate the continuum limit through extrapolation, however they are
restricted to the asymptotic behavior near this limit. We introduce an
efficient extrapolation approach for thermodynamic properties of quantum
oscillators and crystals from primitive path integral simulations. The method
utilizes a fitting function inspired by the analytic solution of the harmonic
oscillator (HO), or the Einstein crystal for solids. The formulation is for
first derivative properties, such as energy and pressure; however, extension to
second derivative properties, such as elastic constants, is straightforward. We
apply the method to a one-dimensional HO and anharmonic oscillator (AO), as
well as a three-dimensional Lennard-Jones crystal. Configurations are sampled
using path integral molecular dynamics simulations in the canonical ensemble,
at a low temperature of $T=0.1$ (simulation units). Compared to Trotter
extrapolation approaches, the new method demonstrates substantial accuracy in
estimating the continuum limit, using only a few simulations of relatively
smaller system sizes. This capability significantly reduces computational cost,
providing a powerful tool to facilitate computations for more complex and
challenging systems, such as molecules and real crystals.
