Energy level

Source: Wikipedia, the free encyclopedia.
(Redirected from
Energy levels
)
Energy levels for an electron in an atom: ground state and excited states. After absorbing energy, an electron may "jump" from the ground state to a higher energy excited state.

A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The term is commonly used for the energy levels of the electrons in atoms, ions, 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.

In

electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" (also called "K shell"), followed by the "2 shell" (or "L shell"), then the "3 shell" (or "M shell"), and so on farther and farther from the nucleus. The shells correspond with the principal quantum numbers (n = 1, 2, 3, 4 ...) or are labeled alphabetically with letters used in the X-ray notation
(K, L, M, N...).

Each shell can contain only a fixed number of electrons: The first shell can hold up to two electrons, the second shell can hold up to eight (2 + 6) electrons, the third shell can hold up to 18 (2 + 6 + 10) and so on. The general formula is that the nth shell can in principle hold up to 2n2 electrons.

Madelung rule for more details.) For an explanation of why electrons exist in these shells see electron configuration.[2]

If the

bound electron states
have negative potential energy.

If an atom, ion, or molecule is at the lowest possible energy level, it and its electrons are said to be in the

degenerate if there is more than one measurable quantum mechanical state
associated with it.

Explanation

atom, showing the probability of finding the electron in the space around the nucleus. Each stationary state defines a specific energy level of the atom.

Quantized energy levels result from the wave behavior of particles, which gives a relationship between a particle's energy and its

wavefunction along a closed path (a path that ends where it started), such as a circular orbit around an atom, where the number of wavelengths gives the type of atomic orbital (0 for s-orbitals, 1 for p-orbitals and so on). Elementary examples that show mathematically how energy levels come about are the particle in a box and the quantum harmonic oscillator
.

Any

collapse of the wavefunction, which results in a new state that consists of just a single energy state. Measurement of the possible energy levels of an object is called spectroscopy
.

History

The first evidence of quantization in atoms was the observation of

Bohr theory of the atom. The modern quantum mechanical theory giving an explanation of these energy levels in terms of the Schrödinger equation was advanced by Erwin Schrödinger and Werner Heisenberg
in 1926.

Atoms

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 n = ∞. When the electron is bound to the atom in any closer value of 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 (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:

(typically between 1 eV and 103 eV),

where R is the

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.

This equation is obtained from combining the Rydberg formula for any hydrogen-like element (shown below) with E = h ν = h c / λ assuming that the principal quantum number n above = n1 in the Rydberg formula and n2 = ∞ (principal quantum number of the energy level the electron descends from, when emitting a photon). The Rydberg formula was derived from empirical spectroscopic emission data.

An equivalent formula can be derived quantum mechanically from the time-independent

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.

In such cases, the orbital types (determined by the

Hund's rule
.

Fine structure splitting

which?
] 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

with

.

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

,

with gS the electron-spin g-factor (about 2), resulting in a total magnetic moment, μ,

.

The interaction energy therefore becomes

.

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. [4] 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:

where Eelectronic is an

.

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

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

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 the 1st, then the 2nd, then the 3rd, etc. of the highest energy electrons, respectively, from the atom originally in the ground state. Energy in corresponding opposite quantities can also be released, sometimes 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

Planck's constant (h) times its frequency (f) and thus is proportional to its frequency, or inversely to its wavelength (λ).[4]

ΔE = h f = h c / λ,

since c, the speed of light, equals to f λ[4]

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*, π → π*, or n → π* meaning excitation of an electron from a σ bonding to a σ 

antibonding orbital, from a π bonding to a π antibonding orbital, or from an n non-bonding to a π antibonding orbital.[4][5]
Reverse electron transitions for all these types of excited molecules are also possible to return to their ground states, which can be designated as σ* → σ, π* → π, or π* → n.

A transition in an energy level of an electron in a molecule may be combined with a

Translational energy levels are practically continuous and can be calculated as kinetic energy using classical mechanics
.

Higher

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 possibly colored glow.

An electron farther from the nucleus has higher potential energy than an electron closer to the nucleus, thus it becomes less bound to the nucleus, since its potential energy is negative and inversely dependent on its distance from the nucleus.[6]

Crystalline materials

conduction band, the Fermi level, the vacuum level, and the energy levels of any defect states
in the crystal.

See also

References

  1. ^ Re: Why do electron shells have set limits ? madsci.org, 17 March 1999, Dan Berger, Faculty Chemistry/Science, Bluffton College
  2. ^ Electron Subshells. Corrosion Source. Retrieved on 1 December 2011.
  3. .
  4. ^ a b c d UV-Visible Absorption Spectra
  5. ^ Theory of Ultraviolet-Visible (UV-Vis) Spectroscopy
  6. ^ "Electron Density and Potential Energy". Archived from the original on 2010-07-18. Retrieved 2010-10-07.