# Polyelectronic atoms : Quantum mechanics

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- Written by: Germán Fernández
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We write the Hamiltonian of an atom with N electrons, under the approximation of an immobile punctual nucleus, with $m_N/m_e\rightarrow \infty$. \begin{equation} \hat{H}=\frac{-\hbar^2}{2m}\sum_{i=1}^{N}\nabla_{i}^2+\sum_{i=1}^ {N}\frac{Ze^2}{r_i}+\sum_{i=1}^{N}\sum_{j>i}^{N}\frac{e^2}{r_{ij}} \end{equation}

The first term contains the kinetic energy for the N electrons.

The second term is the potential energy of electron-nucleus interaction.

The third addend takes into account the interelectronic interactions (repulsions between electrons). The constraint i>j prevents adding the same interaction twice, $\frac{e^2}{r_{12}}=\frac{e^2}{r_{21}}$.

Read more: Non-relativistic Hamiltonian of a many-electron atom

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For a many-electron atom, the angular momentum operators (orbital and spin) of the individual electrons do not commute with the Hamiltonian, but their sum does.

Consider an atom with two electrons, which we will call (1) and (2):

$\hat{\vec{l}}_1$, is the operator of the orbital angular momentum of the electron (1).

$\hat{\vec{l}}_2$, is the operator of the orbital angular momentum of the electron (2).

$\hat{\vec{L}}=\hat{\vec{l}}_1+\hat{\vec{l}}_2$, is the operator of the total angular momentum of the atom.

Read more: Addition of angular momenta in many electron atoms

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The total orbital angular momentum (L) of an atom of N electrons, $\vec{L}=\sum_{i=1}^N\vec{l}_i$, is the vector sum of the orbital angular momentum of the electrons individual. The operators $\hat{L}^2$ and $\hat{L}_z$ commute with each other and with $\hat{H}$, their eigenvalues being: \begin{equation} \hat{L}^2 \;\;\rightarrow \hbar^2L(L+1) \end{equation} \begin{equation} \hat{L}_z\;\;\;\rightarrow M_L\hbar \end{equation} with $M_L $ taking values between -L,........,+L. Orbital angular momentum is designated by an uppercase letter, while lowercase is for the orbital angular momentum of individual electrons.

L | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

letter | S | P | D | F | G | h | I | k |

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The total electronic spin angular momentum (S) of an atom of N electrons, $\vec{S}=\sum_{i=1}^N\vec{s}_i$, is the vector sum of the angular momenta of spin of individual electrons. The operators $\hat{S}^2$ and $\hat{S}_z$ commute with each other, with the operators of the total orbital angular momentum and with the Hamiltonian of the many-electron atom, which allows to know simultaneously the observables associated with these operators in each quantum state of the polyelectronic atom. Let's see what are the eigenvalues of the operators $\hat{S}^2$ and $\hat{S}_z$: \begin{equation} \hat{S}^2\;\;\rightarrow\;\; \hbar^2 S(S+1) \end{equation} \begin{equation} \hat{S}_z\;\;\rightarrow\;\;M_s\hbar \end{equation} with $M_s$ taking values between -S,........,+S

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Russell-Saunders (RS) terms classify the states of the many-electron atom according to their energy. That is, each RS term corresponds to an energy level. The RS terms have the following form: \begin{equation} ^{2s+1}L \end{equation} where L represents the orbital angular momentum expressed by the letter corresponding to its value (S,P,D,F.. ..), and 2S+1 is the multiplicity of the term. If 2S+1=1, we speak of a spin singlet; 2S+1=2, is a spin doublet; 2S+1=3, spin triplet; 2S+1=4, spin quartet, etc. 3P, read triplet P; $^{1}D$, read singlet D.

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The spin and orbital angular momenta are coupled giving rise to a total angular momentum that we represent by J, $\vec{J}=\vec{L}+\vec{S}$. The total angular momentum quantum number is obtained by summing the total orbital and spin angular momentum quantum numbers: \begin{equation} J=L\otimes S= L+S, L+S-1, L+S-2 ,......,|LS| \end{equation} The operators compatible with the fine structure Hamiltonian are $J^2$ and $J_z$, whose eigenvalues are $\hbar^2 J(J+1)$ and $M_J\hbar$, respectively. A fine structure level is given by: $^{2S+1}L_J$ Fine structure states have the form: $\left|^{2S+1}L_J\;M_J\right\rangle$. The degeneracy of each fine structure level is $2J+1$ The spin-orbit coupling produces the splitting of the RS terms into fine structure levels, which have a slightly different energy. For example, carbon in its ground state $1s^22s^22p^2$, has three RS terms $^1S,^3P$ and $^1D$. Considering the spin-orbit coupling, the term $^3P$ is split into three levels $^3P_2, ^3P_1$ and $^3P_0$.