The real Coulomb interaction between electrons is given by

${\hat {V}}_{ee}={\frac {1}{2}}{\frac {e_{0}^{2}}{4\pi \epsilon _{0}}}\sum _{i\neq j}{\frac {1}{\left|{\vec {r}}_{i}-{\vec {r}}_{j}\right|}}$

and the full hamiltonian is

${\hat {H}}=\sum _{i}{\frac {-\hbar ^{2}}{2m}}\nabla _{i}^{2}-{\frac {Ze_{0}^{2}}{4\pi \epsilon _{0}}}\sum _{i}{\frac {1}{r_{i}}}+V_{ee}$

Instead, Hartree hamiltonian is

${\hat {H}}_{H}=\sum _{i}{\frac {-\hbar ^{2}}{2m}}\nabla _{i}^{2}-{\frac {Ze_{0}^{2}}{4\pi \epsilon _{0}}}\sum _{i}{\frac {1}{r_{i}}}+\sum _{i}V_{H}\left({\vec {r_{i}}}\right)$

Taking the difference between

${\hat {H}}$ and

${\hat {H}}_{H}$ we get

${\hat {H}}_{1}={\hat {V}}_{ee}-\sum _{i}{\hat {V}}_{H}\left({\vec {r_{i}}}\right)$

We can consider

${\hat {H}}_{1}$ as a perturbation with respect to

${\hat {H}}_{H}$, representing the difference between the actual coulomb interaction between electrons and the avarage electron repulsion.
The eigenstates of

${\hat {H}}_{H}$ are labeled with the

*good quantum numbers* $\left|n_{i},l_{i},m_{l_{i}},m_{s_{i}}\right\rangle$ and are degenerate in

$m_{l}$ and

$m_{s}$ since energies depend only on

$n$ and

$l$.
In order to deal with simpler calculations we'd like to find a basis that makes the perturbation

${\hat {H}}_{1}$ diagonal together with

${\hat {H}}_{H.F}$.

If ${\hat {H}}_{1}$ is the most relevant perturbation ^{[1]} and thus we do not have to deal with other terms in the hamiltonian, we can proceed as follows.

Since ${\hat {H}}_{1}$ does not contain spin-orbit energy terms it commutes not only with ${\hat {\vec {J}}}=\sum _{i}{\hat {\vec {L_{i}}}}+{\hat {\vec {S_{i}}}}$ but also with ${\hat {\vec {L}}}=\sum _{i}{\hat {\vec {L_{i}}}}$ and ${\hat {\vec {S}}}=\sum _{i}{\hat {\vec {S_{i}}}}$. Furthermore, $\left[{\hat {H}}_{1},{\hat {L^{2}}}_{i}\right]=0$ so that $l_{i}$ is still a good quantum number together with $n_{i}$.

The right way of labeling states is thus: $\left|\{n_{i},l_{i}\}L,S,M_{L},M_{S}\right\rangle$

Using Clebsch-Gordan coefficients we can easily change basis:

$\left|\{n_{i},l_{i}\}L,S,M_{L},M_{S}\right\rangle =\sum _{m_{l_{i}},m_{s_{i}}}\underbrace {\left(\dots \right)} _{\mbox{ Clebsch-Gordan}}\left|n_{i},l_{i},m_{l_{i}},m_{s_{i}}\right\rangle$

The degeneracy of energy levels is partially removed by ${\hat {H}}_{1}$:

$\left\langle {\hat {H}}_{1}\right\rangle =E^{\left(1\right)}\left(\{n_{i},l_{i}\},L,S\right)$

States with different

$L,S$ are no longer degenerate.
This is the reason why when we take into account non-spherical corrections we label energy levels no longer with

$[n]s,[n]p,\dots$ but rather with

**atomic terms in LS scheme**:

$^{2S+1}L_{J}$

where the capital letter

$L$ has to be replaced with

$S,P,D,F,\dots$ for

$L=0,1,2,3,\dots$
A state

$1s^{2}2s^{2}2p^{2}$, which represents 15 degenerate states, is now split in

$^{1}D_{2}\,(5{\mbox{ states}})\quad ^{3}P_{0}\,^{3}P_{1}\,^{3}P_{2}(9{\mbox{ states}})\quad ^{1}S_{\frac {1}{2}}(1{\mbox{ state}})$

and each of these terms with a different capital letter

$L$ has different energy.

^{[2]}
After having taken into account non-spherical corrections, we can consider other less relevant perturbations such has spin-orbit coupling: as we'll see in the next chapter the effect of this perturbation is to split the terms according to their $J$, since

$\left\langle \{n_{i},l_{i}\}L,S,J,M_{J}\left|{\hat {H}}_{SO}\right|\{n_{i},l_{i}\}L,S,J,M_{J}\right\rangle =A\left(\{n_{i},l_{i}\}\right){\frac {\hbar ^{2}}{2}}\left(J(J+1)-L(L+1)-S(S+1)\right)$

This leads to a further removal of degeneracy, since states such as

$^{3}P_{0}\,^{3}P_{1}\,^{3}P_{2}$ no longer have the same energy.

The physical fact partially hidden by all the maths is that spins interact giving a total spin ${\hat {\vec {S}}}$ and so do the orbital angular momenta ${\hat {\vec {L}}}_{i}$, which give rise to a total angular momentum ${\hat {\vec {L}}}$. Then ${\hat {\vec {L}}}$ and ${\hat {\vec {S}}}$ couple into a ${\hat {\vec {J}}}$ (*LS coupling*).

In order to find the term with the lowest energy the following empirical rules hold:

- The term with the maximum spin has the lowest energy
- If there still are more terms, choose the one with the highest angular momentum
- if the shell is less than half-full the lowest energy term is the one with the lowest value of J, otherwise the one with the highest value of J

- ↑ in particular, if spin-orbit effects are small
- ↑ For a detailed discussion about how to find atomic terms read LS scheme