# What function is used to graph the potential curve of two hydrogens with parallel spins?

TL; DR

• an analytic potential curve function for the triplet state of $$\ce{H2}$$ (preferred) with experimental parameters, or
• experimental data (enough points to realise a smooth-looking curve).

Also see solid red curve pictured below.

To characterise the energy of a two-atom system, Morse potential is applied as a second approximation after the harmonic oscillator model. This is an analytic function of the form $$E=D_\mathrm{e}\left\{1-\exp[-a(r-r_\mathrm{c})]\right\}$$

where the parameters $$D_e$$, $$a$$ and $$r_c$$ depend on the atoms in play. Surely enough, better results are achievable. For examples on simple molecules, I found the paper by Hulburt and Hirschfelder, 'Potential Energy Functions for Diatomic Molecules', J. Phys. Chem., ($$1940$$) to be rather useful. Have a look at equations $$(14)$$ and ($$15$$) specifically (available more easily here). The one I used was

$$E_{\mathrm{P}}=\mathrm{D_e}[(1-\mathrm{e}^{-z})^2+\mathrm{c}x^3\mathrm{e}^{-2z}(1+\mathrm{b}z)],$$

where

$$z=\frac{\omega_\mathrm{e}}{{2}\sqrt{\mathrm{B_eD_e}}}\left(\frac{x-\mathrm{r_0}}{\mathrm{r_0}}\right)$$

This is for the case where spins are antiparallel.

If atoms of parallel spin approach one another, the qualitative picture is quite different. A smaller distance will equal higher energy regardless of separation, i.e., no stable equilibrium state exists. Unfortunately, finding a nice function for the antibonding situation proved less fruitful. An example of what I'm after is the solid red curve found in this Wolfram demonstration.

• I did contact a representative of Wolfram Research Europe, but (for some reason) failed to receive an answer. Furthermore, I was initally reluctant to post this question due to the fact that it is essentially a data question... but here it is anyway. If it's off-topic, closing will not offend. Jul 31, 2017 at 17:34

I believe that you wont be able to find a closed analytic expression for $\mathrm{H_2}$, since no closed analytic expresion is possible for a many-body problem (more than 2 bodies). That said you can calculate as many points on the triplet curve as you like by using a QM program able of doing unrestricted Hartree-Fock (UHF). Here you should be aware that UHF does not give very good potential curves, because the Hartree-Fock approximation lacks the electron correleation energy. $\mathrm{H_2}$ is a quite small system, so you will be able to make a potential curve using unrestricted MP2 or unrectriced CCSD. I do believe that PySCF is capable of doing such calculations. Regarding the experimental data, I am not aware of any databases for such kind of data. I would suggest finding a paper where a curve is presented and then use engauge digitizer to extract the points you need.

• Thank you for the insight! Apologies for the delay, I was swamped. Didn't even think of the engauge digitiser approach. Will give it a go in the coming days, and depending on the outcome (probably) will accept the answer :). Aug 13, 2017 at 23:45

Potentiology is a field that began in the early days of quantum mechanics:

• 1924: John Lennard-Jones publishes the Lennard-Jones pontial
• 1929: Philip M. Morse publishes the Morse potenital.

The most recent reference you gave was a potential from:

• 1940: by Hugh Hulburt and Joseph Hirschfelder

You seem to be hoping that in the last 80 years, someone might have come up with something better.
The answer is the 2009 Morse/Long-range, which is the most flexible potential energy function known. It has the theoretically correct long-range behaviour built into it. You can easily do the anti-bonding potential, and also double-well potentials, potentials with a shelf, potentials with a shoulder, potentials with a barrier, etc.