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For a bistable potential which is the sum of two hyperbolic cosine functions, the Schrödinger equation for the low-lying states of a homonuclear diatomic molecule can be solved analytically. In this model the potential and the energy eigenvalues depend on three parameters, and the resulting wave functions are continuous and have continuous derivatives everywhere. This last property of the wave functions enables one to generate a family of soluble bistable potentials by applying the theorem of Darboux to the discrete eigenfunctions of the Schrödinger equation.

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