  We consider quantum tunnelling in asymmetric double-well systems for which
the local minima in the two wells have the same energy, but the frequencies
differ slightly. We derive a generalization of instanton theory for these
asymmetric systems, leading to a semiclassical expression for the tunnelling
matrix element and hence the energy level splitting. We benchmark the method
using a set of one- and two-dimensional models, for which the results compare
favourably with numerically exact quantum calculations. Using the ring-polymer
instanton approach, we apply the method to compute the level splittings in
various isotopic-substituted versions of malonaldehyde in full dimensionality
and analyse the relative contributions from the zero-point energy difference
and tunnelling effects.
