The two lowest eigenvalues E0(λ), E1(λ) of a symmetric double well tunnelling problem −Δ + λ2V as λ → ∞ are considered and they are compared to the two lowest eigenvalues Ẽ0(λ), Ẽ1(λ) of −Δ + λ2(V + W), where W is supported away from the well-bottoms of V. We determine the leading exponential splitting of various differences of the four numbers E0, E1, Ẽ0, Ẽ1. Related problems are discussed.