Quantum Numbers and Atomic Orbitals (s, p, d, f)
Quantum numbers are four numbers (n, l, ml and ms) that describe the energy, shape, orientation and spin of an electron in an atom, and an atomic orbital is the region around the nucleus where an electron is most likely to be found, designated s, p, d or f according to its shape. In brief: n gives the shell, l gives the subshell (0 = s, 1 = p, 2 = d, 3 = f), ml gives the orbital within the subshell and ms is the spin, either +½ or −½; no two electrons in an atom can have the same set of four numbers.
Quantum numbers come from the solution of the Schrödinger equation for the hydrogen atom, and they organize all of the electrons in all atoms. This article explains each number and its allowed values, describes the shapes of s, p, d and f orbitals, shows how many orbitals and electrons each subshell holds, works through examples of valid and invalid quantum numbers and gives the history of the notation.
What are the four quantum numbers?
- Principal quantum number, n: the shell. n = 1, 2, 3, and so on. A larger n means a higher energy and a larger average distance from the nucleus.
- Angular momentum (azimuthal) quantum number, l: the subshell and the shape of the orbital. l = 0, 1, 2, 3 correspond to s, p, d, f.
- Magnetic quantum number, ml: the orientation of the orbital in space. It distinguishes the orbitals within a subshell.
- Spin quantum number, ms: the spin of the electron, +½ (“spin up”) or −½ (“spin down”).
What values are allowed?
- n is a positive whole number: 1, 2, 3, …
- l can be any whole number from 0 to n − 1.
- ml can be any whole number from −l to +l, including 0. There are 2l + 1 values.
- ms is +½ or −½.
For n = 2, l can be 0 or 1 (2s and 2p). For l = 1, ml can be −1, 0 or +1, which gives three p orbitals. Each orbital can hold two electrons with opposite spins, which is the Pauli exclusion principle.
What do s, p, d and f orbitals look like?
- s orbitals (l = 0): spherical. There is one s orbital per shell, and the 2s orbital is larger than the 1s orbital with a spherical node inside it.
- p orbitals (l = 1): dumbbell shaped, with two lobes on opposite sides of the nucleus and a node at the nucleus. The three p orbitals (px, py, pz) point along the three perpendicular axes.
- d orbitals (l = 2): mostly cloverleaf shaped with four lobes; there are five d orbitals, one of which has a different shape (a dumbbell with a ring).
- f orbitals (l = 3): more complex shapes with eight lobes in many of the seven f orbitals.
An orbital is not a path. The picture shows the region in which the electron is found with a high probability, usually 90%. The number of nodes (regions of zero probability) in an orbital is n − 1: the radial nodes number n − l − 1 and the angular nodes number l.
Shells, subshells, orbitals and electrons
| Shell n | Subshells | Orbitals per subshell | Orbitals in the shell (n2) | Maximum electrons (2n2) |
|---|---|---|---|---|
| 1 | 1s | 1 | 1 | 2 |
| 2 | 2s, 2p | 1, 3 | 4 | 8 |
| 3 | 3s, 3p, 3d | 1, 3, 5 | 9 | 18 |
| 4 | 4s, 4p, 4d, 4f | 1, 3, 5, 7 | 16 | 32 |
| Letter | l | ml values | Orbitals | Maximum electrons |
|---|---|---|---|---|
| s | 0 | 0 | 1 | 2 |
| p | 1 | −1, 0, +1 | 3 | 6 |
| d | 2 | −2, −1, 0, +1, +2 | 5 | 10 |
| f | 3 | −3 to +3 | 7 | 14 |
How are orbital energies ordered?
In hydrogen, all subshells in the same shell have the same energy. In atoms with more than one electron, electron-electron repulsion and shielding make the energy depend also on l, with s lower than p, p lower than d and d lower than f in the same shell. As a result the 4s orbital is generally filled before the 3d. The order of filling is described in the Aufbau principle, covered in the next article.
Worked examples
- Valid set? (n, l, ml, ms) = (3, 2, −2, +½). n = 3 allows l = 0, 1, 2, so l = 2 is valid. For l = 2, ml goes from −2 to +2, so −2 is valid. ms = +½ is valid. This is a 3d electron.
- Invalid set 1. (2, 2, 0, +½). For n = 2, l can only be 0 or 1. l = 2 is not allowed.
- Invalid set 2. (3, 1, 2, −½). For l = 1, ml must be −1, 0 or +1, so 2 is not allowed.
- Naming an orbital. (4, 3, −3, −½) has n = 4 and l = 3 (f). It is a 4f electron.
- Counting orbitals. The 4d subshell has 2l + 1 = 5 orbitals, and holds a maximum of 10 electrons. The whole fourth shell has 42 = 16 orbitals and can hold 32 electrons.
- All allowed sets for n = 2. There are 2 × 22 = 8 sets: 2s has (2, 0, 0, ±½) = 2 sets; 2p has (2, 1, −1, ±½), (2, 1, 0, ±½) and (2, 1, +1, ±½) = 6 sets.
- Nodes in 3p. n = 3, l = 1. Radial nodes = 3 − 1 − 1 = 1, angular nodes = 1, and the total is n − 1 = 2.
How were the quantum numbers developed?
The letters s, p, d and f come from early spectroscopy, where lines were described as sharp, principal, diffuse and fundamental. Niels Bohr introduced the principal quantum number in 1913. Arnold Sommerfeld added a second number in 1916 to describe elliptical orbits, and the magnetic quantum number was added to explain the splitting of spectral lines in magnetic fields. In 1925, Samuel Goudsmit and George Uhlenbeck proposed electron spin, and Wolfgang Pauli stated his exclusion principle. Schrödinger’s wave equation in 1926 gave the quantum numbers a natural basis, since they arise from the mathematics of the solutions.
Which mistakes are common?
Mistake 1: Allowing l equal to n
l goes up to n − 1. There is no 2d or 1p orbital.
Mistake 2: Thinking that an orbital is an electron’s path
It is a region of probability.
Mistake 3: Mixing up subshell and orbital
A subshell, such as 2p, contains several orbitals (three for p).
Mistake 4: Forgetting that each orbital holds two electrons
The two electrons have opposite spins, ms = +½ and −½.