The **Aufbau principle** (building up principle) states that an atom in the ground state has electrons filling its orbitals in the order of increasing energy. It was proposed by Niels Bohr and Wolfgang Pauli in the 1920s and is based on the observation that the lower the energy of a system is, the more stable it is.

Specifically, the principle adopts the rule, which was first suggested by Charles Janet in 1928, in his attempt to construct a version of the **periodic table**. It was later adopted by Erwin Madelung in 1936, as a rule on how atomic sub-shells are filled.

The empirical rule states that electrons fill sub-shells in the order of increasing value of where is the principal quantum number and is the angular quantum number. It further mentions that electrons fill sub-shells in the order of increasing value of for sub-shells with identical values of . For example,

Subshell | Order | |||

1s |
1 | 0 | 1 | 1 |

2s |
2 | 0 | 2 | 2 |

2p |
2 | 1 | 3 | 3 |

3s | 3 | 0 | 3 |
4 |

3p | 3 | 1 | 4 |
5 |

3d | 3 | 2 | 5 |
7 |

4s | 4 | 0 | 4 |
6 |

The order of fill is represented by the diagram above. So, the ground state electron configuration (distribution of electrons) for calcium is 1*s ^{2}* 2

*s*2

^{2}*p*3

^{6}*s*3

^{2}*p*4

^{6}*s*or [

^{2}*Ar*]4

*s*, where [

^{2}*Ar*] is the electron configuration of argon. The Aufbau principle works well for elements with atomic number but must be applied with a better understanding of orbital energy and electron repulsion for .

For elements with *Z* between 1 and 6, calculations show that the energy of the 4s sub-shell is higher than the 3d sub-shell (see diagram above). For *Z* between 7 and 20, the reverse is true, as a result of the interplay between increasing nuclear charge and increasing electron repulsion. The relative energy of the *s* and *d* sub-shells again changes for , where the 3*d* sub-shell has a lower energy than the 4*s* sub-shell. This is because electrons in 3*d* orbitals do not shield each other well from nuclear forces, leading to the lowering of their energies.

With that in mind, one may conclude that the electron configurations of scandium and titanium are [*Ar*]3*d ^{3 }*and [

*Ar*]3

*d*respectively. However, they are [

^{4}*Ar*]3

*d*4

^{1}*s*and [

^{2 }*Ar*]3

*d*4

^{2}*s*. Numerical solutions of the Schrodinger equation for scandium and titanium not only show that the 3

^{2}*d*orbitals have lower energies than the 4

*s*orbitals, but also reveal that the 3

*d*orbitals are smaller in size compared to the 4

*s*orbitals. Electrons occupying 3

*d*orbitals therefore experience greater repulsions than electrons residing in 4

*s*orbitals, with the order of increasing repulsion being:

where *V* is the potential energy due to repulsion.

To determine the stability of an atom in the ground state, we need to consider the net effect of the relative energies of 4*s*/3*d* orbitals and the repulsion of electrons. In fact, calculations for the overall energies of scandium are as follows:

Consequently, when a transition metal undergoes ionisation, the electron is removed from the 4*s* orbital rather than the 3*d* sub-shell. Despite 3*d* being lower in energy than 4s for the first row of transition metals, the rule applies. However, the rule breaks down for chromium and copper, where the ground state electronic configuration of chromium is [*Ar*]3*d*^{5}4*s*^{1 }instead of [*Ar*]3*d*^{4}4*s*^{2} and that of copper is [*Ar*]3*d*^{10}4*s*^{1} instead of [*Ar*]3*d*^{9}4*s*^{2}. This is attributed to Hund’s rule.