9.3 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Energy level | 2/3 | https://en.wikipedia.org/wiki/Energy_level | reference | science, encyclopedia | 2026-05-05T10:52:25.529347+00:00 | kb-cron |
1
λ
=
R
Z
2
(
1
n
1
2
−
1
n
2
2
)
{\displaystyle {\frac {1}{\lambda }}=RZ^{2}\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)}
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 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 Z as the atomic number. A simple (though not complete) 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 Z is substituted with an effective nuclear charge symbolized as Zeff that depends strongly on the principal quantum number.
E
n
,
ℓ
=
−
h
c
R
∞
Z
e
f
f
2
n
2
{\displaystyle E_{n,\ell }=-hcR_{\infty }{\frac {{Z_{\rm {eff}}}^{2}}{n^{2}}}}
In such cases, the orbital types (determined by the azimuthal quantum number ℓ) as well as their levels within the molecule affect Zeff 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 and motion and the nucleus's electric field) and the Darwin term (contact term interaction of s shell electrons inside the nucleus). These affect the levels by a typical order of magnitude of 10−3 eV.
==== Hyperfine structure ====
This even finer structure is due to electron–nucleus spin–spin interaction, resulting in a typical change in the energy levels by a typical order of magnitude of 10−4 eV.
=== Energy levels due to external fields ===
==== Zeeman effect ====
There is an interaction energy associated with the magnetic dipole moment, μL, arising from the electronic orbital angular momentum, L, given by
U
=
−
μ
L
⋅
B
{\displaystyle U=-{\boldsymbol {\mu }}_{L}\cdot \mathbf {B} }
with
−
μ
L
=
e
ℏ
2
m
L
=
μ
B
L
{\displaystyle -{\boldsymbol {\mu }}_{L}={\dfrac {e\hbar }{2m}}\mathbf {L} =\mu _{B}\mathbf {L} }
. Additionally taking into account the magnetic momentum arising from the electron spin. Due to relativistic effects (Dirac equation), there is a magnetic momentum, μS, arising from the electron spin
−
μ
S
=
−
μ
B
g
S
S
{\displaystyle -{\boldsymbol {\mu }}_{S}=-\mu _{\text{B}}g_{S}\mathbf {S} }
, with gS the electron-spin g-factor (about 2), resulting in a total magnetic moment, μ,
μ
=
μ
L
+
μ
S
{\displaystyle {\boldsymbol {\mu }}={\boldsymbol {\mu }}_{L}+{\boldsymbol {\mu }}_{S}}
. The interaction energy therefore becomes
U
B
=
−
μ
⋅
B
=
μ
B
B
(
M
L
+
g
S
M
S
)
{\displaystyle U_{B}=-{\boldsymbol {\mu }}\cdot \mathbf {B} =\mu _{\text{B}}B(M_{L}+g_{S}M_{S})}
.
==== Stark effect ====
== Molecules == Chemical bonds between atoms in a molecule form because they make the situation more stable for the involved atoms, which generally means the sum energy level for the involved atoms in the molecule is lower than if the atoms were not so bonded. As separate atoms approach each other to covalently bond, their orbitals affect each other's energy levels to form bonding and antibonding molecular orbitals. The energy level of the bonding orbitals is lower, and the energy level of the antibonding orbitals is higher. For the bond in the molecule to be stable, the covalent bonding electrons occupy the lower energy bonding orbital, which may be signified by such symbols as σ or π depending on the situation. Corresponding anti-bonding orbitals can be signified by adding an asterisk to get σ* or π* orbitals. A non-bonding orbital in a molecule is an orbital with electrons in outer shells which do not participate in bonding and its energy level is the same as that of the constituent atom. Such orbitals can be designated as n orbitals. The electrons in an n orbital are typically lone pairs. In polyatomic molecules, different vibrational and rotational energy levels are also involved. Roughly speaking, a molecular energy state (i.e., an eigenstate of the molecular Hamiltonian) is the sum of the electronic, vibrational, rotational, nuclear, and translational components, such that:
E
=
E
electronic
+
E
vibrational
+
E
rotational
+
E
nuclear
+
E
translational
{\displaystyle E=E_{\text{electronic}}+E_{\text{vibrational}}+E_{\text{rotational}}+E_{\text{nuclear}}+E_{\text{translational}}}
where Eelectronic is an eigenvalue of the electronic molecular Hamiltonian (the value of the potential energy surface) at the equilibrium geometry of the molecule. The molecular energy levels are labelled by the molecular term symbols. The specific energies of these components vary with the specific energy state and the substance.
=== Energy level diagrams === There are various types of energy level diagrams for bonds between atoms in a molecule.
Examples Molecular orbital diagrams, Jablonski diagrams, and Franck–Condon diagrams.
== Energy level transitions ==