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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 1s1s orbital has the least energy, followed by the 2s2s and 2p2p 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=3n=3 orbitals (3s3s and 3p3p) penetrate the inner orbits (meaning they have a large probability density for small values of rr) significantly. The 3d3d orbital however has negligible penetration, and so has a much higher energy than the other two n=3n=3 orbitals.

The same phenomenon occurs with the n=4n=4 orbitals, where the 4s4s and 4p4p subshells penetrate so much that their energies almost coincide with the 3d3d 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 nn is called an atomic shell. They are designated by letter, as shown below:

nn 11 22 33 44 55

Levels with certain values of nn and ll are called subshells, such as 3s3s or 4d4d. The maximum number of electrons allowed in each of these levels is 2(2l+1)2\left(2l+1\right). Below is a table with some atomic subshell capacities:

nn ll Subshell Capacity
11 00 1s1s 22
22 00 2s2s 22
22 11 2p2p 66
33 00 3s3s 22
33 11 3p3p 66
44 00 4s4s 22
33 22 3d3d 1010
44 11 4p4p 66
55 00 5s5s 22
44 22 4d4d 1010
55 11 5p5p 66
66 00 6s6s 22
44 33 4f4f 1414
55 22 5d5d 1010
66 11 6p6p 66
77 00 7s7s 22
55 33 5f5f 1414
66 22 6d6d 1010

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 ZZ of an atom changes For instance the nineteenth electron of potassium (Z=19Z=19) is very different from the nineteenth electron of lead (Z=82Z=82). Instead, we describe the inner electrons by shells, such as the KK shell.

The Periodic Table

Periodic table

The periodic table arranges all the elements by increasing atomic number (ZZ) 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(2l+1)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 1s11s^1, since it has one electron in the 1s1s suborbital.

In general, orbitals are filled in order, however there are some discrepancies such as copper (Z=29Z=29), which would be expected to fill the 4s4s suborbital before the 3d3d suborbital. However, an electron completing the 3d3d orbital requires slightly less energy, so copper's electron configuration is 1s2 2s2 2p6 3s2 3p6 3d10 4s11s^2~2s^2~2p^6~3s^2~3p^6~3d^{10}~4s^1.