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Aquantum mechanical system orparticle that isbound—that is, confined spatially—can only take on certain discrete values of energy, calledenergy levels. This contrasts withclassical particles, which can have any amount of energy. The term is commonly used for the energy levels of the electrons in atoms,ions, ormolecules, which are bound by the electric field of thenucleus, but can also refer to energy levels of nuclei orvibrational orrotational energy levels in molecules. The energy spectrum of a system with such discrete energy levels is said to bequantized.
Inchemistry andatomic physics, an electron shell, or principal energy level, may be thought of as theorbit of one or moreelectrons around anatom'snucleus. 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 further and further from the nucleus. The shells correspond with theprincipal quantum numbers (n = 1, 2, 3, 4, ...) or are labeled alphabetically with letters used in theX-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 thenth shell can in principle hold up to 2n2 electrons.[1] Since electrons areelectrically attracted to the nucleus, an atom's electrons will generally occupy outer shells only if the more inner shells have already been completely filled by other electrons. However, this is not a strict requirement: atoms may have two or even three incomplete outer shells. (SeeMadelung rule for more details.) For an explanation of why electrons exist in these shells seeelectron configuration.[2]
If thepotential energy is set to zero atinfinite distance from the atomic nucleus or molecule, the usual convention, thenbound 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 theground state. If it is at a higher energy level, it is said to beexcited, or any electrons that have higher energy than the ground state areexcited. An energy level is regarded asdegenerate if there is more than one measurable quantum mechanicalstate associated with it.
Quantized energy levels result from the wave behavior of particles, which gives a relationship between a particle's energy and itswavelength. For a confined particle such as anelectron in an atom, thewave functions that have well defined energies have the form of astanding wave.[3] States having well-defined energies are calledstationary states because they are the states that do not change in time. Informally, these states correspond to a whole number of wavelengths of thewavefunction 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 ofatomic orbital (0 for s-orbitals, 1 for p-orbitals and so on). Elementary examples that show mathematically how energy levels come about are theparticle in a box and thequantum harmonic oscillator.
Anysuperposition (linear combination) of energy states is also a quantum state, but such states change with time and do not have well-defined energies. A measurement of the energy results in thecollapse 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 calledspectroscopy.
The first evidence of quantization in atoms was the observation ofspectral lines in light from the sun in the early 1800s byJoseph von Fraunhofer andWilliam Hyde Wollaston. The notion of energy levels was proposed in 1913 by Danish physicistNiels Bohr in theBohr theory of the atom. The modern quantum mechanical theory giving an explanation of these energy levels in terms of theSchrödinger equation was advanced byErwin Schrödinger andWerner Heisenberg in 1926.
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 numbern = ∞. When the electron is bound to the atom in any closer value ofn, the electron's energy is lower and is considered negative.
Assume there is one electron in a given atomic orbital in ahydrogen-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), whereR∞ is theRydberg constant,Z is theatomic number,n is the principal quantum number,h is thePlanck constant, andc is thespeed of light. For hydrogen-like atoms (ions) only, the Rydberg levels depend only on the principal quantum numbern.
This equation is obtained from combining theRydberg formula for any hydrogen-like element (shown below) withE =hν =hc / λ assuming that the principal quantum numbern above =n1 in the Rydberg formula andn2 = ∞ (principal quantum number of the energy level the electron descends from, when emitting aphoton). TheRydberg formula was derived from empiricalspectroscopic emission data.
An equivalent formula can be derived quantum mechanically from the time-independentSchrödinger equation with a kinetic energyHamiltonian operator using awave function as aneigenfunction to obtain the energy levels aseigenvalues, but the Rydberg constant would be replaced by other fundamental physics constants.
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 withZ as theatomic number. A simple (though not complete) way to understand this is as ashielding 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 whereZ is substituted with aneffective nuclear charge symbolized asZeff that depends strongly on the principal quantum number.
In such cases, the orbital types (determined by theazimuthal quantum numberℓ) as well as their levels within the molecule affectZeff and therefore also affect the various atomic electron energy levels. TheAufbau principle of filling an atom with electrons for anelectron 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 thePauli exclusion principle, theAufbau principle, andHund's rule.
Fine structure arises from relativistic kinetic energy corrections,spin–orbit coupling (an electrodynamic interaction between the electron'sspin and motion and the nucleus's electric field) and the Darwin term (contact term interaction ofs shell[which?] electrons inside the nucleus). These affect the levels by a typical order of magnitude of 10−3 eV.
This even finer structure is due to electron–nucleusspin–spin interaction, resulting in a typical change in the energy levels by a typical order of magnitude of 10−4 eV.
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
withgS the electron-sping-factor (about 2), resulting in a total magnetic moment,μ,
The interaction energy therefore becomes
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 tocovalently bond, their orbitals affect each other's energy levels to form bonding and antibondingmolecular orbitals. The energy level of thebonding orbitals is lower, and the energy level of theantibonding 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. Anon-bonding orbital in a molecule is an orbital with electrons in outershells which do not participate in bonding and its energy level is the same as that of the constituent atom. Such orbitals can be designated asn orbitals. The electrons in an n orbital are typicallylone pairs.[4] In polyatomic molecules, different vibrational and rotational energy levels are also involved.
Roughly speaking, a molecular energy state (i.e., aneigenstate of themolecular Hamiltonian) is the sum of the electronic, vibrational, rotational, nuclear, and translational components, such that:whereEelectronic is aneigenvalue of theelectronic molecular Hamiltonian (the value of thepotential energy surface) at theequilibrium geometry of the molecule.
The molecular energy levels are labelled by themolecular term symbols. The specific energies of these components vary with the specific energy state and the substance.
There are various types of energy level diagrams for bonds between atoms in a molecule.
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Electrons in atoms and molecules can change (maketransitions in) energy levels by emitting or absorbing aphoton (ofelectromagnetic 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, orion. Complete removal of an electron from an atom can be a form ofionization, which is effectively moving the electron out to anorbital with an infiniteprincipal 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 theground state. Energy in corresponding opposite quantities can also be released, sometimes in the form ofphoton energy, when electrons are added to positively charged ions or sometimes atoms. Molecules can also undergo transitions in theirvibrational 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 theground state. If it is at a higher energy level, it is said to beexcited, or any electrons that have higher energy than the ground state areexcited. Such a species can be excited to a higher energy level byabsorbing 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 the Planck constant (h) times itsfrequency (f) and thus is proportional to its frequency, or inversely to itswavelength (λ).[4]
sincec, the speed of light, equals tofλ[4]
Correspondingly, many kinds ofspectroscopy are based on detecting the frequency or wavelength of the emitted orabsorbed photons to provide information on the material analyzed, including information on the energy levels and electronic structure of materials obtained by analyzing thespectrum.
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 avibrational transition and called avibronic transition. A vibrational androtational transition may be combined byrovibrational coupling. Inrovibronic 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 asX-ray,ultraviolet,visible light,infrared, ormicrowave 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 usingclassical mechanics.
Highertemperature causes fluid atoms and molecules to move faster increasing their translational energy, and thermally excites molecules to higher average amplitudes of vibrational and rotational modes (excites the molecules to higher internal energy levels). This means that as temperature rises, translational, vibrational, and rotational contributions to molecularheat capacity let molecules absorb heat and hold moreinternal energy.Conduction of heat typically occurs as molecules or atoms collidetransferring 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 coloured glow.
An electron further 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 solids are found to haveenergy bands, instead of or in addition to energy levels. Electrons can take on any energy within an unfilled band. At first this appears to be an exception to the requirement for energy levels. However, as shown inband theory, energy bands are actually made up of many discrete energy levels which are too close together to resolve. Within a band the number of levels is of the order of the number of atoms in the crystal, so although electrons are actually restricted to these energies, they appear to be able to take on a continuum of values. The important energy levels in a crystal are the top of thevalence band, the bottom of theconduction band, theFermi level, thevacuum level, and the energy levels of anydefect states in the crystal.
