Practical implementation of the instanton theory for the ground-state tunneling splitting

The instanton theory is reformulated with use of the path integral approach and the Wentzel–Kramers–Brillouin approximation to the Schrödinger equation. Both approaches are shown to provide the same results. A new practically useful semiclassical formula is derived for the tunneling splitting of the ground state, which can be implemented for high-dimensional systems. The theory is applicable to systems of arbitrary Riemannian metric and is also supplemented by a practical numerical recipe to evaluate the instanton trajectory, i.e., periodic orbit, in multidimensional space. Numerical examples are presented for three-dimensional (3D) and 21D systems of HO2 and malonaldehyde, respectively.