Exact tunneling splittings of rotationally excited states from symmetrized path-integral molecular dynamics

We extend our previous symmetrized path-integral molecular dynamics approach to calculate tunneling splittings of molecules in rotationally excited states. In this new formalism, the system is rigorously projected onto selected rotational manifolds and states of a chosen symmetry through an Eckart spring, which connects the two end beads of the ring polymer via a permutation-inversion-rotation operation. In this way, the tunneling splittings for the lowest vibrational states in the projected subspace can be obtained by taking the low-temperature limit. This method is numerically exact within statistical uncertainty once convergence with respect to all simulation parameters has been achieved. Importantly, it enables the simultaneous extraction of tunneling splittings for multiple total angular-momentum quantum numbers J from a single set of simulations, without additional computational cost relative to the original approach. After validating the formalism by computing the rotational levels of water (beyond the rigid-rotor approximation), we apply it to ammonia and obtain rotationally resolved tunneling splittings in excellent agreement with exact variational benchmarks. Except for small errors due to the underlying potential energy surface, the results capture the experimentally observed trend that the tunneling splitting decreases with J.