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A One-Dimensional Model of Hydrogen Molecular Ion

We present a study on a one-dimensional hydrogen molecular ion under the Born-Oppenheimer approximation. A canonical transformation produces the classical system directly to be a pendulum. The quantum Schrödinger equation is solved analytically and the electronic energy curves show that the bound states of this 1D model differ from the 2D and 3D . The vibration spectroscopy is also obtained by employing the Morse's eigen wavefunctions as basis vectors to diagonalize the Hamiltonian for R. The semiclassical quantization yields electronic energies in agreement with the quantum ones reasonably.