Editing Electron configuration (section)

== Shells and subshells ==
{{Main|Electron shell}}
{| class="wikitable" align=right
|-
!
! s  ({{mvar|l}} = 0)
! colspan="3" |p ({{mvar|l}} = 1)
|-
!
! ''m'' = 0
! ''m'' = 0
! colspan="2" |''m'' = ±1
|-
!
! s
! p<sub>''z''</sub>
! p<sub>''x''</sub>
! p<sub>''y''</sub>
|- align=center
!''n'' = 1
| [[File:Atomic-orbital-cloud n1 l0 m0.png|32px]]
|
|
|
|- align=center
!''n'' = 2
| [[File:Atomic-orbital-cloud n2 l0 m0.png|48px]]
| [[File:Atomic-orbital-cloud n2 l1 m0.png|48px]]
| [[File:Atomic-orbital-cloud n2 px.png|48px]]
| [[File:Atomic-orbital-cloud n2 py.png|48px]]
|}

Electron configuration was first conceived under the [[Bohr model]] of the [[atom]], and it is still common to speak of [[Electron shell|shells and subshells]] despite the advances in understanding of the [[Quantum mechanics|quantum-mechanical]] nature of [[Electron|electrons]].

An [[electron shell]] is the [[Set (mathematics)|set]] of [[Quantum state|allowed states]] that share the same [[principal quantum number]], ''n'', that electrons may occupy. In each [[Term (logic)|term]] of an electron configuration, ''n'' is the [[Natural number|positive integer]] that precedes each [[Atomic orbital#Shapes of orbitals|orbital letter]] ([[helium]]'s electron configuration is 1s<sup>2</sup>, therefore ''n'' = 1, and the orbital contains two electrons). An atom's ''n''th electron shell can accommodate 2''n''<sup>2</sup> electrons. For example, the first shell can accommodate two&nbsp;electrons, the second shell eight&nbsp;electrons, the third shell eighteen, and so on. The factor of two arises because the number of allowed states doubles with each successive shell due to [[Spin quantum number|electron spin]]—each atomic orbital admits up to two otherwise identical electrons with opposite spin, one with a spin +{{1/2}} (usually denoted by an up-arrow) and one with a spin of −{{1/2}} (with a down-arrow).

A [[Electron shell#Subshells|subshell]] is the set of states defined by a common [[azimuthal quantum number]], {{mvar|l}}, within a shell. The value of {{mvar|l}} is in the range from 0 to ''n''&nbsp;−&nbsp;1. The values {{mvar|l}}&nbsp;= 0, 1, 2, 3 correspond to the s, p, d, and f labels, respectively. For example, the 3d subshell has ''n''&nbsp;=&nbsp;3 and {{mvar|l}}&nbsp;=&nbsp;2. The maximum number of electrons that can be placed in a subshell is given by 2(2{{mvar|l}} + 1). This gives two electrons in an s&nbsp;subshell, six electrons in a p&nbsp;subshell, ten electrons in a d&nbsp;subshell and fourteen electrons in an f&nbsp;subshell.

The numbers of electrons that can occupy each shell and each subshell arise from the equations of quantum mechanics,{{efn|name="SchrodNote"|In formal terms, the [[quantum number]]s ''n'', {{mvar|l}} and ''m''{{sub|{{mvar|l}}}} arise from the fact that the solutions to the time-independent [[Schrödinger equation]] for [[hydrogen-like atom]]s are based on [[spherical harmonics]].}} in particular the [[Pauli exclusion principle]], which states that no two electrons in the same atom can have the same values of the four [[quantum number]]s.<ref>{{GoldBookRef|file=PT07089|title=Pauli exclusion principle}}</ref>

Exhaustive technical details about the complete quantum mechanical theory of atomic spectra and structure can be found and studied in the basic book of Robert D. Cowan.<ref>{{Cite book |last=Cowan |first=Robert D. |title=The Theory of Atomic Structure and Spectra |date=2020 |publisher=University of California Press |isbn=9780520906150}}</ref>