If one of the two electrons fo a hydrogen molecule is removed, we get a hydrogen molecule ion `(H_(2)^(+))`. In the ground state of `H_(2)^(+)`, the two protons are separated roughly by `1.5 Å` and electron is roughly `1 Å` from each proton. Determine the potential energy of the system. Specify your choice of zero of potential energy.

Explanation

To determine the potential energy of the hydrogen molecule ion \( H_2^+ \) in its ground state, we need to consider the interactions between the two protons and the single electron. The potential energy \( U \) of the system can be calculated using the formula for electrostatic potential energy.

### Step-by-Step Solution:

1. **Identify the Charges and Distances**:
- The charges involved are:
- Two protons, each with charge \( +e \) (where \( e = 1.6 \times 10^{-19} \, C \)).
- One electron with charge \( -e \).
...