Harmonic-oscillator-referenced ring-polymer molecular dynamics. II. Theoretical foundations of reference-modified...

The Paper I [H. Wang, J. Chem. Phys. 165, ■ (2026)] introduced harmonic-oscillator-referenced ring-polymer molecular dynamics (HO-RPMD) as a practical framework for approximately computing linear and nonlinear real-time quantum correlation functions. The present study develops the corresponding theoretical foundation in greater detail. The method is organized around three coordinated ingredients: exact harmonic-reference sampling in imaginary time, centroid-anchored real-time propagation, and Gaussian reconstruction of nonlinear observables from an anchored linear kernel together with sampled static information. We show that these elements are structurally linked. Once the quantum Boltzmann operator is represented by a harmonic reference kernel rather than by the primitive free-particle Trotter factorization, the centroid component of the real-time dynamical generator cannot generally be left unchanged without shifting the harmonic oscillation frequency. Likewise, beyond linear centroid observables, a direct nonlinear ring-polymer correlation function is no longer naturally protected from contamination by the internal fluctuation modes, and an explicit reconstruction step is therefore needed. We formulate these points at the operator, path integral, normal mode, and observable reconstruction levels and derive the finite-P reference distribution, the centroid-anchoring condition, and the Gaussian reconstruction logic. The resulting theory clarifies the relation of HO-RPMD to standard RPMD, centroid molecular dynamics, symmetrized/Wigner-type correlation-function formulations, and Gaussian moment-reconstruction ideas. It also makes clear that, beyond the harmonic limit, the natural practical nonlinear target of the present framework is the canonical hybrid correlation function, while nonlinear Kubo quantities remain primarily useful as harmonic benchmarks.