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Electronic States in Multielectron Atoms

Monday, March 14, 2022

Atomic Subshells

Looking at the energies for electron orbitals show a pattern where the $1s$ orbital has the least energy, followed by the $2s$ and $2p$ orbitals whose energies are relatively close to one another. This pattern continues with higher energy levels, where different orbitals are almost grouped together with the same energies.

Penetrating Orbits

Two of the $n=3$ orbitals ($3s$ and $3p$) penetrate the inner orbits (meaning they have a large probability density for small values of $r$) significantly. The $3d$ orbital however has negligible penetration, and so has a much higher energy than the other two $n=3$ orbitals.

The same phenomenon occurs with the $n=4$ orbitals, where the $4s$ and $4p$ subshells penetrate so much that their energies almost coincide with the $3d$ orbital.

Although penetrating orbits spend more time closer to the nucleus than non-penetrating ones, they also spend more time farther away, averaging out to approximately the same average radius.

Atomic Shells

The set of orbitals with a certain value of $n$ is called an atomic shell. They are designated by letter, as shown below:

$n$	$1$	$2$	$3$	$4$	$5$
Shell	$K$	$L$	$M$	$N$	$O$

Levels with certain values of $n$ and $l$ are called subshells, such as $3s$ or $4d$. The maximum number of electrons allowed in each of these levels is $2\left(2l+1\right)$. Below is a table with some atomic subshell capacities:

$n$	$l$	Subshell	Capacity
$1$	$0$	$1s$	$2$
$2$	$0$	$2s$	$2$
$2$	$1$	$2p$	$6$
$3$	$0$	$3s$	$2$
$3$	$1$	$3p$	$6$
$4$	$0$	$4s$	$2$
$3$	$2$	$3d$	$10$
$4$	$1$	$4p$	$6$
$5$	$0$	$5s$	$2$
$4$	$2$	$4d$	$10$
$5$	$1$	$5p$	$6$
$6$	$0$	$6s$	$2$
$4$	$3$	$4f$	$14$
$5$	$2$	$5d$	$10$
$6$	$1$	$6p$	$6$
$7$	$0$	$7s$	$2$
$5$	$3$	$5f$	$14$
$6$	$2$	$6d$	$10$

Note: these tables only represent the "outer" or valence electrons since they describe how electrons fill the shells. The properties of electrons change as the $Z$ of an atom changes For instance the nineteenth electron of potassium ($Z=19$) is very different from the nineteenth electron of lead ($Z=82$). Instead, we describe the inner electrons by shells, such as the $K$ shell.

The Periodic Table

The periodic table arranges all the elements by increasing atomic number ($Z$) such that the vertical columns (groups) contain elements with similar properties.

Electron Configurations

We have seen that to fill electronic subshells, there are two rules:

1. The capacity of each subshell is $2\left(2l+1\right)$, called the Pauli exclusion principle
2. The electrons must occupy the lowest energy states available

Electron configurations are written for each element using a notation to identify the subshell and number of electrons it contains. For instance, hydrogen has an electron configuration of $1s^1$, since it has one electron in the $1s$ suborbital.

In general, orbitals are filled in order, however there are some discrepancies such as copper ($Z=29$), which would be expected to fill the $4s$ suborbital before the $3d$ suborbital. However, an electron completing the $3d$ orbital requires slightly less energy, so copper's electron configuration is $1s^2~2s^2~2p^6~3s^2~3p^6~3d^{10}~4s^1$.