Physics:Wiswesser's rule

From HandWiki

The Wiswesser rule gives a simple method to determine the energetic sequence of the atomic subshells [math]\displaystyle{ (n,l) }[/math]. n is the principal quantum number and l is the azimuthal quantum number. The energetic sequence of the subshells characterized by the quantum numbers [math]\displaystyle{ (n,l) }[/math] is the sequence that leads to a monotonically increasing row of function values for the Wiswesser function.

[math]\displaystyle{ W(n,l) = n + l - \left( \frac{l}{l + 1} \right) }[/math]

For example: if [math]\displaystyle{ n = 2 }[/math] and [math]\displaystyle{ l = 1 }[/math], this corresponds to a 2p-orbital.

If the results of this function for each shell and subshell are ordered, the resulting sequence is: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p...

To illustrate this, the corresponding values of each of these shells can be found in the table below:

Order in which orbitals are arranged by increasing energy according to the Madelung rule. Each diagonal red arrow corresponds to a different value of [math]\displaystyle{ n + l }[/math].
[math]\displaystyle{ n }[/math] [math]\displaystyle{ l }[/math] [math]\displaystyle{ W(n,l) }[/math]
2 1 2.5
2 2 3.33
3 1 3.5
3 2 4.33
4 1 4.5
3 3 5.25
4 2 5.33
5 1 5.5
4 3 6.25
5 2 6.33
6 1 6.5
4 4 7.2
5 3 7.25
6 2 7.33
7 1 7.5
5 4 8.2
6 3 8.25
7 2 8.33

We can now clearly see that the Aufbau principle and Wiswesser's Rule come to the same order of filling electron shells.

See also

References