# Energy level

Energy level

A quantum mechanical system or particle that is bound -- that is, confined spatially—can only take on certain discrete values of energy. This contrasts with classical particles, which can have any energy. These discrete values are called energy levels. The term is commonly used for the energy levels of electrons in atoms or molecules, which are bound by the electric field of the nucleus, but can also refer to energy levels of nuclei or vibrational or rotational energy levels in molecules. The energy spectrum of a system with such discrete energy levels is said to be quantized.

If the potential energy is set to zero at infinite distance from the atomic nucleus or molecule, the usual convention, then bound electron states have negative potential energy.

If more than one quantum mechanical state is at the same energy, the energy levels are " degenerate ". They are then called degenerate energy levels.

## Explanation

Quantized energy levels result from the relation between a particle's energy and its wavelength. For a confined particle such as an electron in an atom, the wave function has the form of standing waves. Only stationary states with energies corresponding to integral numbers of wavelengths can exist; for other states the waves interfere destructively, resulting in zero probability density. Elementary examples that show mathematically how energy levels come about are the particle in a box and the quantum harmonic oscillator.

## Atoms

### Intrinsic energy levels

#### Orbital state energy level

Assume there is one electron in a given atomic orbital in a 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 :

$E_n = - h c R_{\infty} \frac{Z^2}{n^2} \$

(typically between 1 eV and 103 eV), where $R_{\infty} \$ is the Rydberg constant , Z is the Atomic number, n is the principal quantum number, h is Planck's constant, and c is the speed of light. For hydrogen-like atoms (ions) only, the Rydberg levels depend only on the principal quantum number n.

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. Instead an approximate correction may be used where Z is substituted with an effective nuclear charge symbolized as Zeff.

$E_{n,l} = - h c R_{\infty} \frac{{Z_{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). Typical magnitude 10 − 3 eV.

#### Hyperfine structure

Spin-nuclear-spin coupling. Typical magnitude 10 − 4 eV.

#### Electrostatic interaction of an electron with other electrons

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.

### 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

$U = -\boldsymbol{\mu}_L\cdot\mathbf{B}$

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## 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 others energy levels to form bonding and anti-bonding molecular orbitals. The energy level of the bonding orbitals is lower, and the energy level of the anti-bonding 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. 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:

$E = E_\mathrm{electronic}+E_\mathrm{vibrational}+E_\mathrm{rotational}+E_\mathrm{nuclear}+E_\mathrm{translational}\,$

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.

In molecular physics and quantum chemistry, an energy level is a quantized energy of a bound quantum mechanical state.

### Energy level diagrams

There are various types of energy level diagrams for bonds between atoms in a molecule. Examples include ''molecular orbital diagram''s, ''Jablonski diagram''s, and ''Franck-Condon'' diagrams.

## Energy level transitions

An increase in energy level from E1 to E2 resulting from absorption of a photon represented by the red squiggly arrow, and whose energy = h ν
A decrease in energy level from E2 to E1 resulting in emission of a photon represented by the red squiggly arrow, and whose energy = h ν

Electrons in atoms and molecules can change (make transitions in) energy levels by emitting or absorbing a photon (of electromagnetic radiation) whose energy must be exactly equal to the energy difference between the two levels. Electrons can also be completely removed from a chemical species such as an atom, molecule, or ion. Complete removal of an electron from an atom can be a form of ionization, which is effectively moving the electron out to an orbital with an infinite principal quantum number, in effect so far away so as to have practically no more effect on the remaining atom (ion). For various types of atoms, there are 1st, 2nd, 3rd, etc. ionization energies for removing 1, 2, 3, etc. of the highest energy electrons from the atom in a ground state. Energy in corresponding opposite quantities can also be released, often in the form of photon energy, when electrons are added to positively-charged ions or sometimes atoms. Molecules can also undergo transitions in their vibrational or rotational energy levels. Energy level transitions can also be nonradiative, meaning emission or absorption of a photon is not involved.

If an atom, ion, or molecule is at the lowest possible energy level, it and its electrons are said to be in the ground state. If it is at a higher energy level, it is said to be excited, or any electrons that have higher energy than the ground state are excited. Such a species can be excited to a higher energy level by absorbing a photon whose energy is equal to the energy difference between the levels. Conversely, an excited species can go to a lower energy level by spontaneously emitting a photon equal to the energy difference. A photon's energy is equal to Planck's constant (h) times its frequency (ν) and thus is proportional to its frequency, or inversely to its wavelength. Correspondingly, many kinds of spectroscopy are based on detecting the frequency or wavelength of the emitted or absorbed photons to provide information on the material analyzed, including information on the energy levels and electronic structure of materials obtained by analyzing the spectrum.

An asterisk is commonly used to designate an excited state. An electron transition in a molecule's bond from a ground state to an excited state may have a designation such as σ→σ* or π→π*, meaning excitation of an electron from a σ bonding to a σ antibonding orbital or from a π bonding to a π antibonding orbital.

A transition in an energy level of an electron in a molecule may be combined with a vibrational transition and called a vibronic transition. A vibrational and rotational transition may be combined by rovibrational coupling. In rovibronic coupling, electron transitions are simultaneously combined with both vibrational and rotational transitions. Photons involved in transitions may have energy of various ranges in the electromagnetic spectrum, such as X-ray, ultraviolet, visible light, infrared, or microwave radiation, depending on the type of transition. In a very general way, energy level differences between electronic states are larger, differences between vibrational levels are intermediate, and differences between rotational levels are smaller, although there can be overlap. Translational energy levels are practically continuous and can be calculated as kinetic energy using classical mechanics.

Higher temperature causes fluid atoms and molecules to move faster increasing their translational energy and can thermally excite (nonradiatively) polyatomic molecules to a higher average distribution of vibrational and rotational energy levels. This means as temperature rises, translational, vibrational, and rotational contributions to molecular heat capacity let molecules absorb heat and hold more internal energy. Conduction of heat typically occurs as molecules or atoms collide transferring the heat between each other. At even higher temperatures, electrons can be thermally excited to higher energy orbitals in atoms or molecules. A subsequent drop of an electron to a lower energy level can release a photon, causing a possible colored glow.

An electron farther from the nucleus has higher potential energy than an electron closer to the nucleus.[1]

## Crystalline materials

Crystalline solids are found to have energy bands, instead of or in addition to energy levels. Electrons can take on any energy within an unfilled band.

## References

Wikimedia Foundation. 2010.

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