  We study the low-lying eigenvalues of the semiclassical Robin Laplacian in a
smooth planar domain symmetric with respect to an axis. In the case when the
curvature of the boundary of the domain attains its maximum at exactly two
points away from the axis of symmetry, we establish an explicit asymptotic
formula for the splitting of the first two eigenvalues. This is a rigorous
derivation of the semiclassical tunneling effect induced by the domain's
geometry. Our approach is close to the Born-Oppenheimer one and yields, as a
byproduct, a Weyl formula of independent interest.
