# The central field approximation and the quantum number $n$

Under the central field approximation the Hamiltonian of a multi-electron atom is approximated as: $$\hat H= \sum_i \left( -\frac{\hbar^2}{2m} \nabla_i^2 +V(r_i) \right)$$ for some central potential $V(r_i)$. This can be solved separately for each electron, thus reducing the problem to solving: $$-\frac{\hbar^2}{2m} \nabla_i^2\psi_i +V(r_i)\psi_i=E\psi_i$$ This can be further solved by separation, with the angular part been given (as usual) by the spherical harmonics $Y^m_l(\theta, \phi)$ and the radial part been given by $R(r)$. Now it is commonly accepted that $R(r)$ depends on two quantum numbers $n$ and $l$ analogous to hydrogen. Clearly it must depend on $l$ due to the presence of the spherical harmonics, but given that $V(r_i)$ is a rather arbitrary potential how can we be sure that $R(r)$ depends on one and only one further quantum number $n$?

The inter-electron repulsion contains a large spherically symmetric component. So, it is possible to construct a potential energy function $V(r_i)$ which is spherically symmetric. The Hamiltonian for such a potential is written as follows using perturbation theory.

$H$ = $H^*$ + $H'$

where $H^*$ = $\hat H= \sum_i \left( -\frac{\hbar^2}{2m} \nabla_i^2 +V(r_i) \right)$

$V(r_i)$ = $-\sum_i ^N\frac{Ze^2}{r_i}$ + $\langle \sum_{i<j}\frac{e^2}{r_{ij}} \rangle$ (Not arbitrary; spherically symmetric; still a function of $\frac{1}{r}$ only)

and $H'$ = $\sum_i ^N\frac{e^2}{r_{ij}}$ - $\langle \sum_{i<j}\frac{e^2}{r_{ij}} \rangle$ (Clearly, no spherical part)

The $\langle \sum_{i<j}\frac{e^2}{r_{ij}} \rangle$ term, which is added and subtracted is the average over a sphere of the electron repulsion. In $V(r_i)$, it is added to mean that the attraction experienced by the $i^{th}$ electron is reduced by this amount , and in the perturbation term, it is subtracted to nullify its addition in $V(r_i)$. Dont be troubled too much with this term since it is just a number (not an operator)

Hence, $H'$ is independent of the angular coordinates; then $H'$ becomes the Hamiltonian which contains the non spherical part of the electron repulsions, whereas $H^*$ contains the K.E, P.E in the field of the nucleus, and the spherical average electron repulsion energy. It is assumed that $H^*$ contains most of the inter electron repulsion such that the remaining term $H'$ is small enough to be treated as a perturbation.

Hence there is no reason to think that the solution to this hamiltonian will yield a radial part of the wavefunction $R(r)$ that depends on other quantum numbers besides $n$ and $l$