Quantum tunneling between symmetry-related molecular structures produces characteristic splittings that sensitively probe rovibrational dynamics and the underlying potential energy surface. In this talk, I will present two recent advances in exact path-integral methods for such splittings. First, I will show how the previous symmetrized path-integral molecular dynamics (PIMD) framework for the rotational ground state^[1] can be extended to rotationally excited states by projecting onto selected J manifolds and symmetry subspaces with an Eckart spring, allowing rigorous rotationally resolved splittings for multiple J values to be extracted from a single set of simulations.^[2]

In the second part, I will introduce path-integral hybrid Monte Carlo with enveloping bridging potentials (PIHMC-EBP), a new path-integral sampling method based on the same rotational-projection formalism.^[3] In this approach, an EBP is constructed for direct free-energy sampling, and two tailored nonlocal updates are designed to enhance the sampling of slow collective motions. Compared with PIMD based on thermodynamic integration, this approach removes the need for quadrature and time-step convergence checks, greatly reducing manual effort in the analysis. Applications to malonaldehyde and the HCl dimer yield the most precise reported splittings while reducing the computational cost by several-fold and by roughly three orders of magnitude, respectively. Application to the water dimer provides the first numerically exact path-integral calculations for the ground-state splittings on three different PESs, all obtained simultaneously from a single set of trajectories by reweighting.

[1] G. Trenins, l. Meuser, H. Bertschi, O. Vavourakis, R. Flütsch, J. O. Richardson. J. Chem. Phys., 2023, 159, 034108.

[2] L. Zupan, Y.-C. Wang, J. O. Richardson. 2026, arXiv:2604.12638.

[3] Y.-C. Wang, J. O. Richardson. 2026, arXiv:2604.12639.