# Potential Energy Curve from Hartree-Fock Calculations

I am currently working on some quantum chemical calculations, in particular looking at dissociation energy curves generated from Hartree-Fock, CI, CCSD and MP2 methods for diatomic molecules. Now, I was always under the impression that in order to get a potential energy surface (in the Born-Oppenheimer approximation) one needs the energies for the electronic problem and the nuclear-nuclear interaction: $$E(R) + V_{\mathrm{NN}}(R) = U(R)$$ Where $$E(R)$$ is the energy obtained from the solution of the electronic Schrödinger equation, $$V_{\mathrm{NN}}(R)$$ the internuclear coulomb potential and $$U(R)$$ the potential energy surface. I have now seen multiple publications and books that draw potential energy curves using only $$E(R)$$.

So my question: Are these publications/books wrong, or do I have a misunderstanding of how the PES is computed under the BOA?

Example publication: Dutta, A.; Sherrill, C. D. Full configuration interaction potential energy curves for breaking bonds to hydrogen: An assessment of single-reference correlation methods. J. Chem. Phys. 2003, 118 (4), 1610–1619. DOI: 10.1063/1.1531658.

• In many QM codes, the value that is printed as the Hartree-Fock energy includes the nucleus-nucleus repulsion (VNN) term. Maybe that is what the books and papers reported. – Shoubhik R Maiti Mar 17 at 23:01
• I've added a publication. @ShoubhikRMaiti That would make sense. In that case, would it be sensible to fit a Morse potential to the energies as well(in the diatomic case of course)? – ABCCHEM Mar 17 at 23:09
• Why do you need to fit the energies to a Morse potential? – Shoubhik R Maiti Mar 17 at 23:20
• @ShoubhikRMaiti I'm thinking that using a fit to the energies would make it easier to read off certain quantities of interest, such as eq. bond length, dissociation energy and so on, all of which are parameters of the Morse potential. – ABCCHEM Mar 17 at 23:23
• I guess fitting to a Morse potential could work. I don't know much about this, but parameters of molecular dynamics force fields are derived in this way, so its not unprecedented. You should look at some papers and talk to your supervisor about whether this method is right. One problem is that you can fit the curve only if the energy curve is accurate. For example, doing CCSD on H2 dissociation would give a weird bent curve that won't fit to the eqn of Morse potential. – Shoubhik R Maiti Mar 18 at 9:53

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 "Potential Energy Surface"

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

• I've update the answer to reply to this comment. A Helium atom is used for the zero-bond-lenght case to prevent singular values due to overlapping atomic orbitals. I was also surprised at the importance of the nuclear repulsion to cause the short distance repulsion, hence why I included the whole graph. – user213305 Mar 25 at 15:35
• Notebook to calculate graph colab.research.google.com/drive/… – user213305 Mar 25 at 16:06
• Is it right to use RHF for H2 dissociation? I thought RHF caused H2 to fragment into H+ and H- when forced to dissociate. – Shoubhik R Maiti Mar 25 at 20:14
• Thanks for the clarification. Not a fan of the helium approximation, to be honest. @ShoubhikRMaiti RHF does worse things than this. But it still is a nice showcase. – Martin - マーチン Mar 25 at 23:54