Research data supporting "Effective classical potential for quantum statistical averages"

Main Text Figures FIG1: Quartic Oscillator The data in this figure represents a comparison between the exact quantum position distribution and several classical-like effective potentials for a one-dimensional quartic oscillator. It shows the potential energy curves and the resulting position distribution functions (PDFs) at three different temperatures (high, intermediate, and low). The data demonstrates how the proposed Local Harmonic (LH) potential accurately captures quantum effects like zero-point energy and tunneling, whereas the standard classical potential fails at lower temperatures. FIG2: Morse Potential This figure contains data for a Morse potential, which is often used to model chemical bonds. The data includes the potential energy and position distributions calculated at different temperatures. It highlights the robustness of the LH effective potential in dealing with potentials that have asymmetric shapes and a finite dissociation limit, comparing it against exact quantum results. FIG3: Double Well The data here compares the proposed LH potential against established methods like the Feynman-Hibbs and Feynman-Kleinert potentials. It focuses on the position distribution function and illustrates that while centroid-based methods are excellent for partition functions, the LH approach provides a more direct and accurate estimation of the actual position-dependent observables. Supplementary Information (SI) Figures FIG1_SI. FIG2_SI and FIG3_SI: Comparison of LH Potential Variants (Quartic) This figure displays data comparing the "bare" unrenormalized local harmonic potential, the renormalized version, and the final refined version for the quartic oscillator (at two different parameters) and Morse oscillator. It shows how specific mathematical refinements (like harmonic mapping) improve the accuracy of the potential energy and position distribution curves at various temperatures. It also documents how the refined potential avoids unphysical localization (narrowing) of the distribution that occurs in simpler approximations at very low temperatures. FIG4_SI: Detailed Centroid vs. Position Distribution Comparison The data here provides a side-by-side comparison of the position distributions generated by the Feynman-Kleinert and Feynman-Hibbs methods against the exact quantum distribution. It highlights the tendency of centroid-based methods to remain more "classical-like" and localized compared to the true quantum distribution. FIG5_SI, FIG6_SI and FIG7_SI: Absolute Zero Comparison (T=0 K) This figure presents data for several potentials (Quartic, Morse, and Double-well) at absolute zero temperature. The data shows that even at the limit of zero temperature, the local harmonic PDF remains in good agreement with the exact quantum mechanics results.