# Multi-electron atoms and term symbols

Multi-electron atoms are studied thanks to several approximations and assumptions. In particular, remember that:

• We assume to work in the particular reference frame of one electron;
• This electron feels the coulomb potential of the others, thus we introduce a charge distribution per unit volume:

${\displaystyle \rho ({\overrightarrow {x}})=\sum _{i}\left|\psi _{i}({\overrightarrow {x}})\right|^{2}}$

• The potential introduced by the presence of this charge is called Hartree potential, and it's defined as:

${\displaystyle V_{H}({\overrightarrow {x}})={\frac {e^{2}}{4\pi \epsilon _{0}}}\int d^{3}{\overrightarrow {x\prime }}{\frac {\rho ({\overrightarrow {x^{\prime }}})}{\left|{\overrightarrow {x}}-{\overrightarrow {x^{\prime }}}\right|}}}$

• A more precise result can be obtained taking into account another potential (actually it's an operator), called Fock potential:

${\displaystyle {\hat {V}}_{F}\psi _{i}\left({\overrightarrow {x}}\right)=\sum _{j}(-){\frac {e^{2}}{4\pi \epsilon _{0}}}\int d^{3}{\overrightarrow {x\prime }}\psi _{j}^{\star }({\overrightarrow {x\prime }}){\frac {\psi _{j}({\overrightarrow {x}})\psi _{i}({\overrightarrow {x^{\prime }}})}{\left|{\overrightarrow {x}}-{\overrightarrow {x^{\prime }}}\right|}}}$

Light atoms ${\displaystyle (Z<30/40)}$ are weakly affected from Spin-Orbit correction, so ${\displaystyle L_{tot}}$ and ${\displaystyle S_{tot}}$ are good quantum numbers that can be used to describe the state of these atoms. All these informations are given in the term symbol:

${\displaystyle ^{2S+1}L}$

Where ${\displaystyle S}$ stands for the total spin, while the letter ${\displaystyle L}$ is fixed by the value of the total angular momentum according to these rules:

 Ltot L 0 S 1 P 2 D 3 F

Where the values of total spin and angular momentum have to be chosen, starting from the electronic configuration, according to selection rules ${\displaystyle -L\leq M_{L}\leq L}$ and ${\displaystyle M_{S}\leq S\leq M_{S}}$, Pauli's exclusion principle and Hund's rules:

1. Electrons take place in orbitals in such a way to minimize their total energy. Maximum multiplicity ${\displaystyle 2S+1}$ (i.e. maximum spin) corresponds to minimum energy.
1. Being the multiplicity maximum, the term with maximum angular momentum has the minimum energy.
1. For a given term, the level with lower energy corresponds to: the minimum value of ${\displaystyle J=L+S}$ for atoms with outermost subshell half-filled or less, the maximum value of ${\displaystyle J}$ for atoms with outermost shell more than half filled.

As a consequence of the third Hund's rule, the term symbol has to be written as:

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

Note that term symbols have a particular recurrence: all the term symbols of the ${\displaystyle p^{n^{th}}}$ level are the same of the symbols of the ${\displaystyle p^{(6-n)^{th}}}$ level. When we construct the term symbol in this way we are working with the LS coupling (or scheme). In heavier atoms, the spin orbit correction has to be considered since it has an important effect on energy levels. Hence, each electron gives a contribute of ${\displaystyle j_{i}}$ having its own angular momentum ${\displaystyle l_{i}}$ and its own spin ${\displaystyle s_{i}}$. Now, all the ${\displaystyle j_{i}}$ will couple up to give a generalized angular momentum ${\displaystyle J=\sum _{i}j_{i}}$. These informations are used to construct the term symbol indicating the split of the orbitals and the total value of ${\displaystyle J}$, as it follows:

${\displaystyle \left([\cdot ]n_{1}p_{j_{i}}^{\alpha }n_{1}p_{j_{k}}^{\beta }\right)_{J}}$

Clearly this is just an example, since we can have more splitted levels. Anyway the total number of electrons ${\displaystyle \alpha +\beta }$ has to sum up to the number of electrons in the starting orbital, and ${\displaystyle J}$ has to be decided according to the third Hund's rule. The above scheme is called jj coupling.