# Why should the spherically asymmetric part of the effective potential be small in the central field approximation?

In the central field approximation, each electron is supposed to move in an effective or average potential contributed by its attractive interaction with the nucleus and repulsive interaction with the remaining electrons. It is assumed that a large part of this effective potential is spherically symmetric and the spherically asymmetric residual part is treated as small i.e. as a perturbation. What is the justification?

The justification is that it works. If you do a Hartree-Fock calculation you get reasonably accurate results. Back in the early 1980s I spent a happy few months doing this type of calculation as part of a final year project and the results were pretty good.

However it's not really true to say the potential is split into the central part and an asymmetric part. The deviation from a central field is due to electron correlations and isn't simply treated as a perturbation. Instead we use a technique like configuration interaction to improve the calculation. The self consistent field + CI gives essentially perfect agreement with experiment.

You are talking about the Hartree-Fock calculations for atoms; you are right in saying that calculations for atoms normally use the central-field approximation. Some of the reasons are:

1. It leads to considerable simplifications of the necessary calculations; with a central potential every orbital is in the form $$u(R)Y_{lm}(\theta,\phi)$$, i.e. the angular part can be dealt with analytically with spherical harmonics. Generally speaking it is quite a bit easier to deal with one-dimensional functions like $$u(R)$$ than with three-dimensional ones, as is the case for molecules.
2. Because of the analytical dependence of the angular part of the orbitals, it is easier to enforce on the solution the desired symmetry properties (we want the solutions to be eigenstates of $$\hat{L^2}$$ and $$\hat{L_z}$$).
3. The central-field approximation makes good physical sense, especially for highly charged ions for which the potential due to the central nucleus is dominant.
4. The error introduced by the central-field approximation can be corrected at a later stage, together with a treatment of correlation. Also, for post-Hartree-Fock methods it helps to have orbitals with the analytic spherical harmonic dependence.

If fact, I strongly suspect that the Hartree-Fock potentials of atoms in a closed-shell $$^1S$$ states (e.g., ground states of noble atoms) are exactly spherical symmetric because of symmetry reasons (a related result is known as Unsöld's theorem); the same should be true also in other situations, e.g. for atoms with one one electron or one hole on top of a closed shell.

I performed a Hartree-Fock calculation on the Boron atom (ground $$^2P$$ state) using the quantum chemistry package Molpro, which doesn't use the central-field approximation (because it's designed for molecules). I won't go into details, but the infinite basis set Hartree-Fock energy comes out at -24.52915(1) $$E_h$$. On the other hand the reference central-field Hartree Fock energy is -24.52906 $$E_h$$ [Atomic Data and Nuclear Data Tables 53,113-162 (1993)], so that the central field approximation raises the energy by 0.1 $$mE_h$$. On the other hand, the correlation energy for this atom is about 118 $$mE_h$$, hence 1000 times larger. Similarly, the atomic Hartree-Fock energy for the ground state of the carbon atom ($$^2P$$ term) is 37.68862 $$E_h$$, but using the molecular code in Molpro one obtains 37.68873 $$E_h$$, i.e. a value lower by 0.11 $$mE_h$$; the correlation energy for this atom is about 148 $$mE_h$$.

On top of the considerations above, there is another approximation which is usually done in atomic calculations; in molecular restricted Hartree-Fock calculations electrons are paired up in orbitals which have the same spatial component, so that the number of orbitals to calculate is $$N/2$$, where $$N$$ is the number of electrons (let's consider for now only closed-shell systems). In atoms, all electrons in the same shell (e.g., $$1s$$, $$2p$$...) are usually assigned the same spatial component. This reduces even further the difficulty of the calculation; for example, a generic restricted Hartree-Fock calculation of a 92-electron system such as an uranium atom requires the calculation and storage of 92/2 = 46 three-dimensional functions. The same uranium-atom calculation using the central-field approximation and having all electron in the same shell use the same spatial component reduces the problem to finding 17 one-dimensional functions, which is much easier.