# Hartree-Fock Hydrogen Basis Functions

I'm interested in running Hartree-Fock calculations on single atoms for a project, and I was trying to find more information about how to actually run them instead of just derivations of equations. Most of what I've found involves simulating molecules, but since I just want to do single atoms, can I use the hydrogen wavefunctions as basis functions? They seem nice because they're all orthogonal, and calculating the overlap integrals will be easy. Is there a reason not to use them?

Short answer: For a close-to-optimal representation, instead of Coulomb-wave functions, use spherical harmonics for the angular part and a grid-like representation in radial direction. ("grid-like" means finite-differences, collocation methods, discrete-variable-representations, or similar methods).

Long answer: The usual approach to solving the Hartree-Fock equations is implemented in terms of the electron integrals, i.e. the matrix representation of the single-particle Hamiltonian $h_{ij}$ and the two-particle Coulomb-interaction $g_{ijkl}$ (all indices $i,j,k,l$ run from $1$ up to the sp-basis size $N_b$).

In particular, the Coulomb-interaction tensor is required to calculate the Fock (or mean-field) matrix and thus determines a good part of the total run-time of your computation (together with the linear algebra task of solving the eigenvalue problem). It is therefore advantageous to choose a basis that leads to an as sparse representation of the two-electron integrals as possible. (In this regard, nothing beats a grid-like basis which reduces the size of the two-electron integrals to $N_b^2$).

On the other hand, one wants a basis that well represents the problem. For atoms, and particular for ground-state calculations, the Hamiltonian has spherical symmetry and therefore spherical harmonics are ideal. The sp-basis is represented as

$$\phi_{klm} = \sum_{klm} R_{klm}(r) Y_{lm}(\theta,\varphi)$$

Note that the index $i$ becomes a multi-index $(klm)$.

This leaves the radial part $R_{klm}(r)$ to determine. Coulomb functions make a specific choice here (that only depends on $kl$ but not on $m$). As you said, this leads to orthogonal basis functions, which is ok, but for the crucial two-particle integrals, nothing is gained.

A grid like representation for $R_{klm}(r)$ does much better here, as it leads to two-electron integrals which depend only quadratically on the size of the grid (instead of quartically as for the index $k$ in the Coulomb wave functions). This is important, as for ground-state calculations, the angular basis can be kept quite small, so the radial part dominates. Moreover, for large grids, as they're required for time-dependent calculations, this gives a huge advantage. Moreover, the flexibility to freely adjust the grid can be used to obtain a higher accuracy.

So, this is basically the answer to the question what would be best. Anyways, one usually starts by implementing the general case, and once you're there, just go on and use the Coulomb basis. But then, as a next step, maybe consider to go on to grid-like representations. It's a nice example of how the single-particle basis affects the many-particle solution method.

Finally, a good reference for Hartree-Fock calculations (and also their implementation) is the book Molecular Electronic Structure Theory by Helgaker et al. Have a look.

• I'm looking into the grid approach. I found a paper from Sandia Labs that goes into it. Thanks! – HiddenBabel Sep 19 '18 at 22:18
• @HiddenBabel: you can also consider the book of Froese-Fischer. And feel free to ask further questions, I've done that a lot some years ago. – davidhigh Sep 20 '18 at 5:25