= Solution
For clamped nuclei and SI units, the many-electron electronic <Hamiltonian> is
$$
\boxed{\hat H_e=
-\frac{\hbar^2}{2m_e}(\nabla_1^2+\nabla_2^2)
-\frac{e^2}{4\pi\epsilon_0}\sum_{i=1}^2\sum_{A=1}^2\frac1{|\mathbf r_i-\mathbf R_A|}
+\frac{e^2}{4\pi\epsilon_0|\mathbf r_1-\mathbf r_2|}.}
$$
The terms are electron kinetic energy, electron–nucleus attraction, and electron–electron repulsion. The nuclear kinetic operator is omitted in the <Born-Oppenheimer approximation>. The nuclear repulsion
$$
V_{NN}=\frac{e^2}{4\pi\epsilon_0|\mathbf R_1-\mathbf R_2|}
$$
is a constant at fixed separation and is added to electronic energies to form the total potential curve. Some conventions include it in $\hat H_e$ as a constant; the electronic-energy convention used below excludes it so that the two one-electron energies can be compared without double-counting nuclear repulsion.
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