Back-rotation of the wave function in the complex scaling method

The complex resonance eigenvalue can easily be obtained by scaling the internal coordinates of the Hamiltonian by a complex factor since then the resonance eigenfunction ${\mathrm{\Psi }}_{\mathrm{\theta }}$ is square integrable. The back-transformation of ${\mathrm{\Psi }}_{\mathrm{\theta }}$, however, yields the exact eigenfunction of the original Hamiltonian only when ${\mathrm{\Psi }}_{\mathrm{\theta }}$ is given in closed form. We show here that in the case when a basis-set method is invoked, even if the ${\lim }_{\mathit{n}\to \mathrm{\infty }}$ ${\mathrm{\Psi }}_{\mathrm{\theta }}^{\mathit{n}}$ is the exact solution of the transformed problem, the n→∞ limit of the back-rotated ${\mathrm{\Psi }}_{\mathrm{\theta }}^{\mathit{n}}$ may produce a divergent wave function. Our numerical examples suggest that the quality of approximation in the back-rotated wave function strongly depends on the basis set.