Consider a single proton and add an electron orbiting around it, the electron will be termed to have negative energy. This is numerically the same amount of energy you need to put in to take the electron out of the orbital. An orbital with "lower energy" requires more energy to take an election out.
When describing atomic orbitals scientists prefer to consider an atom with one electron, either hydrogen, or a "hydrogenic atom" which can have multiple protons so long as it has just one electron. This is because a system with multiple electrons has electron-electron repulsion changing the energy levels of atomic orbitals, and working out this effect can be really hard.
Orbitals are described by physicists using wave functions (usually denoted as Ψ) which contain their properties, such as their shapes and energy levels.
A consequence of using wave functions is that the orbitals of a hydrogenic atom can be described by three quantum numbers:
1. The principle quantum number, denoted as n. This is an integer which can range from 1 to infinity. The energy level of the orbital increases as n increases.
This number also indicates what shell the orbital is in. In A-level chemistry you are taught that atoms are made more stable by completing a shell, either 2 electrons (n = 1), 8 electrons (n = 2) or 18 electrions (n =3).
Shells with a higher principle quantum number are also larger and more diffuse.
2. The angular momentum quantum number (also called the azimuthal quantum number), denoted as ℓ. "ℓ" is just a lower case "L". This exists at the values 0 ≤ ℓ ≤ n − 1
This describes the shape of the orbital. It also denotes the subshell of n which the orbital is in. In A-level chemistry you learn these as S, P and D subshells, or visually as sphere, dumbbell, and double-dumbbell. The S subshell corresponds to ℓ=0, P is ℓ=1, and D is ℓ=2. In multiple-electron systems these are split into different energy levels, in that case the lower values are usually lower in energy level (i.e. S becomes lower than P).
So far we have:
N = 1, ℓ = 0
N = 2, ℓ = 0, 1
N = 3, ℓ = 0, 1, 2
This should make sense compared to A-level, where the first shell only had a S subshell, the second had S and P, and the third had S, P and D.
3. The magnetic quantum number, denoted as mℓ. This can take the values −ℓ ≤ mℓ ≤ ℓ. This describes the direction of the orbital. Now we have:
N = 1, ℓ = 0, mℓ = 0
N = 2, ℓ = 0 mℓ = 0
ℓ = 1 mℓ = 1, 0, -1
N = 3 ℓ = 0 mℓ = 0
ℓ = 1 mℓ = 1, 0, -1
ℓ = 2 mℓ = 2, 1, 0, -1, -2
So the ℓ = 0, or S subshell, can only point one direction. The P subshell can point three directions, and the D subshell can point in five directions, as we already knew.
The Pauli exclusion principle tell us:
No two fermions (which includes electrons) can occupy the same space with the same spin.
Electrons can have a spin of + 1/2 or -1/2, so a maximum of two electrons can fit into an orbital. If you apply that to the above table and work out the number of possible electrons you could fit into each shell, you get 2 electrons at n = 1, 8 electrons at n = 2, and 18 electrons at n = 3. So this system of notation fits perfectly with filling up the shells in A-level chemistry.
Electrons should also be assumed to fall down to orbitals with lower energy levels, provided they are not prevented by the Pauli exclusion principle. This is summed up in the Aufbau principle.
The orbitals of lower energy are filled in first with the electrons and only then the orbitals of high energy are filled.
The German word "Aufbau" means "building up, construction".
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