# Non-spherical corrections and LS scheme

The real Coulomb interaction between electrons is given by

${\displaystyle {\hat {V}}_{ee}={\frac {1}{2}}{\frac {e_{0}^{2}}{4\pi \epsilon _{0}}}\sum _{i\neq j}{\frac {1}{\left|{\vec {r}}_{i}-{\vec {r}}_{j}\right|}}}$
and the full hamiltonian is
${\displaystyle {\hat {H}}=\sum _{i}{\frac {-\hbar ^{2}}{2m}}\nabla _{i}^{2}-{\frac {Ze_{0}^{2}}{4\pi \epsilon _{0}}}\sum _{i}{\frac {1}{r_{i}}}+V_{ee}}$
Instead, Hartree hamiltonian is
${\displaystyle {\hat {H}}_{H}=\sum _{i}{\frac {-\hbar ^{2}}{2m}}\nabla _{i}^{2}-{\frac {Ze_{0}^{2}}{4\pi \epsilon _{0}}}\sum _{i}{\frac {1}{r_{i}}}+\sum _{i}V_{H}\left({\vec {r_{i}}}\right)}$
Taking the difference between ${\displaystyle {\hat {H}}}$ and ${\displaystyle {\hat {H}}_{H}}$ we get
${\displaystyle {\hat {H}}_{1}={\hat {V}}_{ee}-\sum _{i}{\hat {V}}_{H}\left({\vec {r_{i}}}\right)}$
We can consider ${\displaystyle {\hat {H}}_{1}}$ as a perturbation with respect to ${\displaystyle {\hat {H}}_{H}}$, representing the difference between the actual coulomb interaction between electrons and the avarage electron repulsion. The eigenstates of ${\displaystyle {\hat {H}}_{H}}$ are labeled with the good quantum numbers ${\displaystyle \left|n_{i},l_{i},m_{l_{i}},m_{s_{i}}\right\rangle }$ and are degenerate in ${\displaystyle m_{l}}$ and ${\displaystyle m_{s}}$ since energies depend only on ${\displaystyle n}$ and ${\displaystyle l}$. In order to deal with simpler calculations we'd like to find a basis that makes the perturbation ${\displaystyle {\hat {H}}_{1}}$ diagonal together with ${\displaystyle {\hat {H}}_{H.F}}$.

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

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

Using Clebsch-Gordan coefficients we can easily change basis:

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

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

${\displaystyle \left\langle {\hat {H}}_{1}\right\rangle =E^{\left(1\right)}\left(\{n_{i},l_{i}\},L,S\right)}$
States with different ${\displaystyle L,S}$ are no longer degenerate. This is the reason why when we take into account non-spherical corrections we label energy levels no longer with ${\displaystyle [n]s,[n]p,\dots }$ but rather with atomic terms in LS scheme:
${\displaystyle ^{2S+1}L_{J}}$
where the capital letter ${\displaystyle L}$ has to be replaced with ${\displaystyle S,P,D,F,\dots }$ for ${\displaystyle L=0,1,2,3,\dots }$ A state ${\displaystyle 1s^{2}2s^{2}2p^{2}}$, which represents 15 degenerate states, is now split in
${\displaystyle ^{1}D_{2}\,(5{\mbox{ states}})\quad ^{3}P_{0}\,^{3}P_{1}\,^{3}P_{2}(9{\mbox{ states}})\quad ^{1}S_{\frac {1}{2}}(1{\mbox{ state}})}$
and each of these terms with a different capital letter ${\displaystyle L}$ 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 ${\displaystyle J}$, since

${\displaystyle \left\langle \{n_{i},l_{i}\}L,S,J,M_{J}\left|{\hat {H}}_{SO}\right|\{n_{i},l_{i}\}L,S,J,M_{J}\right\rangle =A\left(\{n_{i},l_{i}\}\right){\frac {\hbar ^{2}}{2}}\left(J(J+1)-L(L+1)-S(S+1)\right)}$
This leads to a further removal of degeneracy, since states such as ${\displaystyle ^{3}P_{0}\,^{3}P_{1}\,^{3}P_{2}}$ no longer have the same energy.

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

## Hund's rules

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

1. The term with the maximum spin has the lowest energy
2. If there still are more terms, choose the one with the highest angular momentum
3. 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

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