Energy levels and orbitals help describe the electronic structure of an atom. They designate how electrons are arranged within atoms, and the description of such energies is derived from quantum theory.

Quantum theory postulates that atoms can only exist in certain energy states. If an atom, or an electron by correlation, changes state, it absorbs or emits an amount of energy equal to the energy difference between the states.

The energy emitted or absorbed is quantized; it is energy characterized by definite amounts. These allowed energy states can be described by sets of numbers called quantum numbers.

An electron’s arrangement in an atom can be described by four quantum numbers: n, l, m__l_ and m_s. These relate to energy level, electron subshells, orbital direction and spin, respectively.

The first quantum number is designated by n and is the principal energy level.

The principal energy level definition tells the observer the size of the orbital and determines energy. An increase in n is an increase in energy, and this also means the electron is farther away from the nucleus.

The first quantum number can only take integral values, beginning with 1; n = 1, 2, 3, 4 ... Each energy level corresponds to a letter as well: n = 1 (K), 2 (L), 3 (M), 4 (N) ...

To calculate the amount of orbitals from the principal quantum number, use n². There are n² orbitals for each energy level. For n = 1, there is 1² or one orbital. For n = 2, there are 2² or four orbitals. For n = 3 there are nine orbitals, for n = 4 there are 16 orbitals, for n = 5 there are 5² = 25 orbitals, and so on.

To calculate the maximum number of electrons in each energy level, the formula 2n² can be used, where n is the principal energy level (first quantum number). For example, energy level 1, 2(1)² calculates to two possible electrons that will fit into the first energy level.

The second quantum number denotes sublevels and is designated by the letter l. This quantum number denotes electron subshells and the general shape of the electron cloud.

The first two quantum numbers are related. For any given n, l can take on any integral starting with 0 to a maximum of (n – 1); l = 0, 1, 2, 3 ...

The quantum levels, l = 0, 1, 2, 3 correspond to the electron subshells s, p, d, f, respectively. The shape of s is spherical, p is figure-eight shaped, and the d and f orbitals have a more intricate design, mostly involving clover-shaped orbitals.

Each electron subshell can contain a certain amount of electrons, s = 2, p = 6, d = 10 and f = 14.

The third quantum number m__l_, denotes how the electron cloud is directed in space.

This quantum number can have any integral value, including 0, between l and –l (the second quantum number), or, m__l = _l ... 2, 1, 0, -1, -2 ... -l

For l = 0, there is only 1 m__l value, also 0. This contains only one orbital. For a p orbital, m_l_ = 1, 0, -1. This corresponds to the three p orbitals in three different directions, p_x, p_y, p_z, corresponding to the three-dimensional x,y and z axis.

The fourth quantum number designates clockwise or counterclockwise spin.

An electron is a charged particle spinning on an axis and therefore has magnetic properties. This quantum number is not related to n, l, m_l, and can have only two possible values: +1/2 or -1/2.

The addition of the fourth quantum number allows electrons to fill into orbitals without breaking the Pauli exclusion principle. This states that no two electrons can have the same set of four quantum numbers.

Recall that finding the amount of orbitals in an energy level can be derived by the formula n². For the energy level 3, n = (3)² or nine orbitals.

A more completed calculation can be made using the information from the quantum numbers above. For n = 3, the values of l can be added. For l = 0, there is only one orbital, m_l = 0. For l = 1, there are three values (m_l = −1, 0 or +1). For l = 2, there are five possible values (m_l = −2, −1, 0, +1 or +2). So adding the possibilities gives 1 + 3 + 5 = 9 orbitals in total.