Editing Energy level (section)

=== Intrinsic energy levels ===
In the formulas for energy of electrons at various levels given below in an atom, the zero point for energy is set when the electron in question has completely left the atom; i.e. when the electron's principal quantum number {{math|1=''n'' = ∞}}. When the electron is bound to the atom in any closer value of {{mvar|n}}, the electron's energy is lower and is considered negative.

==== Orbital state energy level: atom/ion with nucleus + one electron ====
Assume there is one electron in a given atomic orbital in a [[Hydrogen-like atom|hydrogen-like atom (ion)]]. The energy of its state is mainly determined by the electrostatic interaction of the (negative) electron with the (positive) nucleus. The energy levels of an electron around a nucleus are given by: 
: <math>E_n = - h c R_{\infty} \frac{Z^2}{n^2}</math>
(typically between 1&nbsp;[[electronvolt|eV]] and 10<sup>3</sup>&nbsp;eV), where {{math|''R''<sub>∞</sub>}} is the [[Rydberg constant]], {{mvar|Z}} is the [[atomic number]], {{mvar|n}} is the principal quantum number, {{math|''h''}} is the [[Planck constant]], and {{math|''c''}} is the [[speed of light]]. For hydrogen-like atoms (ions) only, the Rydberg levels depend only on the principal quantum number {{mvar|n}}.

This equation is obtained from combining the [[Rydberg formula#Rydberg formula for any hydrogen-like element|Rydberg formula for any hydrogen-like element]] (shown below) with {{math|1=''E'' = ''h&nu;'' = ''hc'' / ''&lambda;''}} assuming that the principal quantum number {{mvar|n}} above = {{math|''n''<sub>1</sub>}} in the Rydberg formula and {{math|1=''n''<sub>2</sub> = ∞}} (principal quantum number of the energy level the electron descends from, when emitting a [[photon]]). The [[Rydberg formula]] was derived from empirical [[Emission spectrum|spectroscopic emission]] data.
: <math>\frac{1}{\lambda} = RZ^2 \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)</math>

An equivalent formula can be derived quantum mechanically from the time-independent [[Schrödinger equation]] with a kinetic energy [[Hamiltonian operator]] using a [[wave function]] as an [[eigenfunction]] to obtain the energy levels as [[Eigenvalue#Schrödinger equation|eigenvalues]], but the Rydberg constant would be replaced by other fundamental physics constants.

==== Electron–electron interactions in atoms ====
If there is more than one electron around the atom, electron–electron interactions raise the energy level. These interactions are often neglected if the spatial overlap of the electron wavefunctions is low.

For multi-electron atoms, interactions between electrons cause the preceding equation to be no longer accurate as stated simply with {{mvar|Z}} as the [[atomic number]]. A simple (though not complete<!--
**Note** what is hinted at here is that screening is only a mean-field effect. Electron-electron interactions also lead to dynamic correlation-exchange energy shifts. If strong enough, correlation-exchange can prevent us from being able to look at the atom in terms of orbitals at all, leaving only the consideration of many-body states. However, in the case of atoms the correlation-exchange seems to be a small perturbation (usually).
-->) way to understand this is as a [[shielding effect]], where the outer electrons see an effective nucleus of reduced charge, since the inner electrons are bound tightly to the nucleus and partially cancel its charge. This leads to an approximate correction where {{mvar|Z}} is substituted with an [[effective nuclear charge]] symbolized as {{math|''Z''<sub>eff</sub>}} that depends strongly on the principal quantum number.
<math display="block">E_{n,\ell} = - h c R_{\infty} \frac{{Z_{\rm eff}}^2}{n^2}</math>

In such cases, the orbital types (determined by the [[azimuthal quantum number]] {{mvar|ℓ}}) as well as their levels within the molecule affect {{math|''Z''<sub>eff</sub>}} and therefore also affect the various atomic electron energy levels. The [[Aufbau principle]] of filling an atom with electrons for an [[electron configuration]] takes these differing energy levels into account. For filling an atom with electrons in the ground state, the lowest energy levels are filled first and consistent with the [[Pauli exclusion principle]], the [[Aufbau principle]], and [[Hund's rule]].

==== Fine structure splitting ====
[[Fine structure]] arises from relativistic kinetic energy corrections, [[spin–orbit coupling]] (an electrodynamic interaction between the electron's [[spin (physics)|spin]] and motion and the nucleus's electric field) and the Darwin term (contact term interaction of {{serif|s}} shell{{which|reason=of which principal q.n.?|date=January 2014}} electrons inside the nucleus). These affect the levels by a typical order of magnitude of 10<sup>−3</sup>&nbsp;eV.

==== Hyperfine structure ====
{{Main|Hyperfine structure}} 
This even finer structure is due to electron–nucleus [[Angular momentum coupling#Spin–spin coupling|spin–spin interaction]],<!--might be better to link to [[J-coupling]]. Will check later--> resulting in a typical change in the energy levels by a typical order of magnitude of 10<sup>−4</sup>&nbsp;eV.