Harmonic-oscillator-referenced ring-polymer molecular dynamics. I. Practical computation of correlation functions

We present a practical formulation of harmonic-oscillator-referenced ring-polymer molecular dynamics (RPMD) for the calculation of linear and nonlinear real-time correlation functions under quantum statistical conditions. The method is formulated as a direct extension of standard RPMD but incorporates three coordinated modifications aimed at observables for which a direct ring-polymer average is either inconvenient or strongly contaminated by internal-mode artifacts. First, the imaginary-time distribution is built from an exact harmonic oscillator reference rather than the primitive free particle Trotter factorization so that the exact harmonic canonical variance is already recovered at a finite bead number. Second, the real-time dynamics is centroid-anchored, ensuring that the slow collective mode evolves on the physical timescale while the internal modes remain statistically organized by the reference quadratic form. Third, nonlinear observables are reconstructed from the anchored linear correlation and sampled static moments through a Gaussian moment reconstruction. In the present work, this reconstruction is used primarily for canonical/Wigner-type correlation functions, while a Kubo mixed-moment reconstruction is retained as a special harmonic benchmark. The resulting scheme preserves the familiar ring-polymer workflow, is straightforward to incorporate into existing RPMD implementations, and provides a practical route to higher moments such as x(0)x(t)3 and related mixed correlations. We present the working formulation, implementation strategy, and benchmark structure for harmonic, mildly anharmonic, and strongly anharmonic quartic model systems.