Potential Energy Curve from Hartree-Fock Calculations
The energy calculated by a quantum chemistry program is essential always the total potential energy $U(R)$.  For example Fig 1 and Fig 4 of Dutta and Sherrill referenced in the question both plot the total sum of electronic and nuclear energies $U(R)=E(R)+V_{NN}(R)$.
For an example, have a look at a PES I have generated for $\ce{H2}$, using Restricted Hartree-Fock and the STO-3G basis set using PySCF.  The characteristic PES, with a single minimum is actually the total sum of energies $U(R)$, labelled the &quot;Potential Energy Surface&quot;

N.B. The zero bond-length electronic energy is calculated using a helium nucleus as the program doesn't allow superposition of atoms.