The Many Electron Atom

Remember the energy levels of the Bohr atom (one-electron Atom) depend on the number of nodes (n-1), and the charge of the nucleus, Z. E_n = -Z² R_H /n² where R_H has the value 2.180 x 10^-18 J.

The energy of the orbital of a one-electron atom depends on n only, but the shape depends on three quantum numbers: n, l, and m_l. Where

n determines the total number of nodes in the orbital wavefunction. (The number of nodes in n minus one, or n is the number of nodes plus one)
l determines the number of angular nodes (nodes containing the nucleus) that are present in the orbital wavefunction.
m_l determines the number of angular nodes containing the magnetic or quantization axis. This quantum number indicates the orientation of the orbital in space.

For a given value of n, there are n² possible orbitals.
for a given value of n and l, there are (2l+1) different orbitals with distinct values of m_l.
This 'pattern' of allowed quantum numbers can be summarized in the following table:

The energy of the orbitals of a one-electron atom depends only on 1/n², so the energy spectrum looks like this, where all orbitals with the same n quantum number 'shell' are degenerate (i.e. have the same energy).

When describing any element other than Hydrogen, the structure of that atom will by more complicated than the simple Bohr picture can predict.

First, the charge on the nucleus will be larger than +1e. The increased electrostatic attraction of the electrons to the nucleus will have a tendency to draw those electrons closer to the nucleus (and as a result closer to each other). This will lower the total energy of the atom (the Bohr model can handle this effect)
Except for positive ions that contain only a single electron, the electrons of the atom or ion will interact with each other repulsively, since they they have like charges. This will raise the total energy of the atom and 'spread out' the electrons. The Bohr model cannot describe electron-electron repulsion and therefore fails for any multiple electron atom or ion

The description of how most elemental atoms and ions behave is a balance of the two above phenomena.

We will treat the way in which the whole atom behaves as the cumulative behavior of each individual electron. Each individual electron is a 'wave' and has a wavefunction, which may be described approximately with at orbitals of Hydrogen. The energy of each of these wavefunctions (orbitals) is qualitatively different from that of the one-electron atom, however, because of the discriminatory effects of electron-electron repulsion.

Why discriminatory? The electron-electron repulsion will be worse (most destabilizing) for orbitals that have angular nodes rather than radial nodes, so for the same n the orbital with higher l will have higher energy. This splits the degeneracy (the exact equality in energy) of the shell described by n, but leaves only the subshell (described by n and l) degenerate.

Roughly, the order of filling of the orbitals (the order in energy from low to high) can be remembered by the following chart:

We can now 'build' atoms by filling the orbitals expected from a one-electron model 'perturbed' by what we know about electron-electron repulsion. This is called atomic 'Aufbau'. The first elements are easy if we postulate two 'rules':

Each electron has one of two possible 'flavors': up or down which are described by a quantum number called m_s, which has values +1/2 or -1/2, repectively.
Electrons 'exclude' each other in that no two electrons can occupy the same 'state' at the same time. Thus, no two electrons in the same atom can have the same quantum numbers. This is called the Pauli Exclusion Principle.

If it were not for the spin (the 'up' or 'down' flavor) of the electron, many electron atoms would be filled by putting one electron in each hydrogen-like orbital, to satisfy the Pauli exculsion principle. But, as it is, each spatial orbital can two electrons in it, as long as they are spin paired (one up, one down)

Be aware of the notation that explicitly ignores closed inner shells. As these orbitals deep in the atom are significantly more stable than the frontier or valence orbitals, they do not significantly contribute to chemical bonding.

In general we fill orbitals with electrons from the lowest energy (most satble) orbitals up. We put electrons in singly at first when degenerate orbitals are being filled as this minimized electron electron repulsion. Electrons are paired in degenerate orbitals only when there is no room to put any more in unpaired. The energy of an atom with several unpaired electrons in degenerate orbitals is lowered when the unpaired spins are parallel.

Half-filled and completely filled subshells are particularly stable due to a Quantal phenomena called 'exchange'. This phenomenon contributes to the stability of parallel unpaired spins in degenerate orbitals as well as leads to some filling 'exceptions' in the middle and end of the first transition period. Check out Cr and Cu, for example:

You can still see these type of exceptions higher still in atomic number (and thus total number of electrons in the atom

Let's go crazy and list all the elements...

The electron configuation of the elements are what give rise to the shape of the periodic table and the names of the blocks that compose it.

The filling exceptions due to the stability of half filled and completely filled subshells leads to some elements at the middle (Cr) and the end of the transition blocks (Cu) having chemistry somewhat like the alkali metals, but for odd exceptions like this it is not worth rearranging the traditional the Periodic Table.

It is expected that you be able to determine the lowest energy electronic configuration of any neutral atom in the Periodic Table.

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