  A multidimensional semiclassical method for calculating tunneling splittings
in vibrationally excited states of molecules using Cartesian coordinates is
developed. It is an extension of the theory by Mil'nikov and Nakamura
[$\textit{ J. Chem. Phys.}$ $\textbf{122}$, $124311$ $(2005)$] to asymmetric
paths that are necessary for calculating tunneling splitting patterns in
multi-well systems, such as water clusters. Additionally, new terms are
introduced in the description of the semiclassical wavefunction that
drastically improve the splitting estimates for certain systems. The method is
based on the instanton theory and builds the semiclassical wavefunction of the
vibrationally excited states from the ground-state instanton wavefunction along
the minimum action path and its harmonic neighborhood. The splittings of
excited states are thus obtained at a negligible added numerical effort. The
cost is concentrated, as for the ground-state splittings, in the instanton path
optimization and the hessian evaluation along the path. The method can thus be
applied without modification to many mid-sized molecules in full dimensionality
and in combination with on-the-fly evaluation of electronic potentials. The
tests were performed on several model potentials and on the water dimer.
