# Obtaining data for plotting Fermi and Coulomb holes of a hydrogen molecule

I need to visualize the exchange (Fermi) and correlation (Coulomb) hole of a hydrogen molecule to qualitatively illustrate the compensation of of both holes to form the total hole that is localized around the reference electron at the dissociation limit. I have found some resources which contain the corresponding plots (see F. Jensen: Introduction to Computational Chemistry, 2. ed, John Wiley & Ltd, 2009, p. 244). Is there a way to create these plots with actual data for e. g. the PBE functional ? Which calculations do I have to carry out ? (I want to create them myself because of aesthetic reasons and I'm just curious.)

## 1 Answer

I am not sure I fully understood your question, but I will give it a try.

The hydrogen molecule in its ground state only contains two electrons of opposite spin, which means that the exchange (Fermi) contribution to the Hamiltonian (and therefore, the wavefunction, which generates the density) of either electron has to exactly cancel the (unphysical) Coulomb contribution to the Hamiltonian generated by that same electron.

That is, you can just take either converged orbital and square it to get the difference in probability density of the Fermi hole, which in turn equals the alpha- or beta- electron density.

As for the Coulomb hole, you can take the density difference between the solution of the correlated method and the Hartree-Fock solution, which only treats electron potentials generated by opposite spin electrons as an average.

Depending on how precise you want to be I would choose Full-CI instead of DFT because in F-CI there is a one-to-one correspondence with HF of each Hamiltonian term, whereas DFT approximates and then corrects some of these operators in ways that break the mathematical correspondence. Full-CI for a two-electron system only scales as O(N^4) with the size of the basis set, but even then most calculations for a linear molecule can be cancelled out by symmetry.

You can use practically any QC program, as most of them output the AO basis weights for the solution orbitals or densities, and then use your favorite plotting tool (GNUplot, Mathematica, etc.) to visualize it. You might also need the weights and exponents used for the basis set, which you can often get from the program, but can also be found here:

https://www.basissetexchange.org/