Using LCAO method, we can determine the ground state of the hydrogen molecular ion. It is the spatially symmetric state given by $$\psi_+=\frac{1}{\sqrt{2}}\left[\psi_{1s}(r_A)+\psi_{1s}(r_B)\right]$$ where $r_A,r_B$ are the distances of the electron from the nuclei A and B. This is called the bonding state. It turns out that the spatially antisymmetric state given by $$\psi_-=\frac{1}{\sqrt{2}}\left[\psi_{1s}(r_A)-\psi_{1s}(r_B)\right]$$ has positive energy. Therefore, I think, the H-molecular ion in this state is not a bound system. In this state, the hydrogen molecular ion is disintegrated. If so, then H molecular ion has only one bound state - the bonding state.
Why is this the case? I mean, H-atom has many bound states. But H molecular ion has only one bound state. Why is this so?
