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Magnetic Quantum Number (m): Gives the orientation of the orbital in space; in other words, the value of m describes whether an orbital lies along the x-, y-, or z-axis on a three-dimensional graph, with the nucleus of the atom at the origin. m can take on any value from -l to l. For our purposes, it is only important that this quantum number tells us that for each value of n there may be up to one s-orbital, three p-orbitals, five d- orbitals, and so on: The s orbital (l = 0) has one orbital, since m can only equal 0. That orbital is spherically symmetrical about the nucleus.

Figure %: s orbital
The p orbital (l = 1) has three orbitals, since m = -1, 0, and 1. These three orbitals lie along the x-, y-, and z-axes.
Figure %: p orbitals
The d orbital (l = 2) has five orbitals, since m = -2, -1, 0, 1, and 2. It is far more difficult to describe the orientation of d orbitals, as you can see:
Figure %: d orbitals


Spin Quantum Number (s): Tells whether a given electron is spin up (+1/2) or spin down (-1/2). Because the Pauli Exclusion Principle tells us that no two electrons of an atom can have the same set of quantum numbers, each orbital is limited to holding two electrons at most.

Orbital Energy Diagrams

It is often convenient to depict orbitals in an orbital energy diagram, as seen below in . Such diagrams show the orbitals and their electron occupancies, as well as any orbital interactions that exist. In this case we have the orbitals of the hydrogen atom with electrons omitted. The first electron shell (n = 1) contains just the 1s orbital. The second shell (n = 2) holds a 2s orbital and three 2p orbitals. The third shell (n = 3) holds one 3s orbital, three 3p orbitals, and five 3d orbitals, and so forth. Note that the relative spacing between orbitals becomes smaller for larger n. In fact, as n gets large the spacing becomes infinitesimally small.

Figure %: Energy diagram of the unoccupied atomic orbitals of hydrogen. Potential energy is on the y-axis.

You will see such energy diagrams quite often in your continuing study of organic chemistry. Notice that all orbitals with the same n have the same energy. Orbitals with identical energies are said to be degenerate (not in the moral sense!). Electrons in higher level orbitals have more potential energy and are more reactive, i.e. more likely to undergo chemical reactions.

Multi-electron atoms

When an atom only contains a single electron, its orbital energies depend only on the principle quantum numbers: a 2s orbital would be degenerate with a 2p orbital. However, this degeneracy is broken when an atom has more than one electron. This is due to the fact that the attractive nuclear force any electron feels is shielded by the other electrons. s-orbitals tend to be closer to the nucleus than p-orbitals and don't get as much shielding, and hence become lower in energy. This process of breaking degeneracies within a shell is known as splitting. In general s orbitals become lowest in energy, followed by p orbitals, d orbitals, and so forth.

Figure %: Splitting of orbital energies in multi-electron systems
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