Pauli and The Hunds Rule

The structure of periodical system of elements is essentially described by the Pauli extraction principle (Wolfgang Pauli) which we have already discussed earlier. It sounds in the following way as applied to an atom:

"Electrons in an atom must be represented by different sets of quantum numbers (n, l, m_l, m_s)."

This rule is a particular case of more general statement:
 

We would like to use mathematical consequences of the Pauli principle in order to determine the electronic configuration of atoms (The short introduction to many-electron systems is given here).

We have already found out that there are values l = 0,1,2,...,n for each n value of hydrogen atom. In its turn, for each l value there are 2l + 1 values of m_l and for each m_l there are two values of m_s (= ±½ ;  Spin up and Spin down), since s = ½.
 

The configurations of first ten elements are given in the diagram below:

The s-subshell has a block for two electrons having opposite directions of spins; the p-subshell consists of 3 blocks (m_l = −1, 0, +1) having maximum two spin states, correspondingly. The spin orientation and J-values for the total kinetic moment can be found according to the following empirical rules (Friedrich Hund):
 

Hunds Rules:

1. The resultant spin of the atomic ground state takes the highest value, that is also the Pauli extraction principle.
    
2. If shell is filled in less than half of its complete capacity then the lowest state will have the minimum J (J = |L − S|). And if shell is filled in more than half of its complete capacity then the lowest state will have the maximum J (J = L + S).

One can find periodic table description here: Periodic Table.  (a 1st class, a must have by each chemist!)

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