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In atomic theory and quantum mechanics, an atomic orbital is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus. The term ''atomic orbital'' may also refer to the physical region or space where the electron can be calculated to be present, as predicted by the particular mathematical form of the orbital. Each orbital in an atom is characterized by a set of values of the three quantum numbers , , and , which respectively correspond to the electron's energy, angular momentum, and an angular momentum
vector component In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors a ...
(
magnetic quantum number In atomic physics, the magnetic quantum number () is one of the four quantum numbers (the other three being the principal, azimuthal, and spin) which describe the unique quantum state of an electron. The magnetic quantum number distinguishes the ...
). Alternative to the magnetic quantum number, the orbitals are often labeled by the associated harmonic polynomials (e.g., ''xy'', ). Each such orbital can be occupied by a maximum of two electrons, each with its own projection of spin m_s. The simple names s orbital, p orbital, d orbital, and f orbital refer to orbitals with angular momentum quantum number and respectively. These names, together with the value of , are used to describe the electron configurations of atoms. They are derived from the description by early spectroscopists of certain series of alkali metal spectroscopic lines as sharp, principal, diffuse, and fundamental. Orbitals for > 3 continue alphabetically (g, h, i, k, ...), omitting j because some languages do not distinguish between the letters "i" and "j". Atomic orbitals are the basic building blocks of the atomic orbital model (or electron cloud or wave mechanics model), a modern framework for visualizing the submicroscopic behavior of electrons in matter. In this model the electron cloud of an atom may be seen as being built up (in approximation) in an electron configuration that is a product of simpler hydrogen-like atomic orbitals. The repeating ''periodicity'' of blocks of 2, 6, 10, and 14 elements within sections of the periodic table arises naturally from the total number of electrons that occupy a complete set of s, p, d, and f orbitals, respectively, though for higher values of quantum number , particularly when the atom bears a positive charge, the energies of certain sub-shells become very similar and so the order in which they are said to be populated by electrons (e.g., Cr = rs13d5 and Cr2+ = rd4) can be rationalized only somewhat arbitrarily.


Electron properties

With the development of quantum mechanics and experimental findings (such as the two slit diffraction of electrons), it was found that the electrons orbiting a nucleus could not be fully described as particles, but needed to be explained by wave–particle duality. In this sense, electrons have the following properties: Wave-like properties: # Electrons do not orbit a nucleus in the manner of a planet orbiting the Sun, but instead exist as standing waves. Thus the lowest possible energy an electron can take is similar to the fundamental frequency of a wave on a string. Higher energy states are similar to
harmonics A harmonic is a wave with a frequency that is a positive integer multiple of the '' fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', ...
of that fundamental frequency. # The electrons are never in a single point location, though the probability of interacting with the electron at a single point can be found from the electron's wave function. The electron's charge acts like it is smeared out in space in a continuous distribution, proportional at any point to the squared magnitude of the electron's
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
. Particle-like properties: # The number of electrons orbiting a nucleus can be only an integer. # Electrons jump between orbitals like particles. For example, if one photon strikes the electrons, only one electron changes state as a result. # Electrons retain particle-like properties such as: each wave state has the same electric charge as its electron particle. Each wave state has a single discrete spin (spin up or spin down) depending on its superposition. Thus, electrons cannot be described simply as solid particles. An analogy might be that of a large and often oddly shaped "atmosphere" (the electron), distributed around a relatively tiny planet (the nucleus). Atomic orbitals exactly describe the shape of this "atmosphere" only when one electron is present. When more electrons are added, the additional electrons tend to more evenly fill in a volume of space around the nucleus so that the resulting collection ("electron cloud") tends toward a generally spherical zone of probability describing the electron's location, because of the
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
.


Formal quantum mechanical definition

Atomic orbitals may be defined more precisely in formal
quantum mechanical Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
language. They are approximate solutions to the Schrödinger equation for the electrons bound to the atom by the electric field of the atom's nucleus. Specifically, in quantum mechanics, the state of an atom, i.e., an eigenstate of the atomic
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
, is approximated by an expansion (see
configuration interaction Configuration interaction (CI) is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematical ...
expansion and basis set) into linear combinations of anti-symmetrized products ( Slater determinants) of one-electron functions. The spatial components of these one-electron functions are called atomic orbitals. (When one considers also their spin component, one speaks of atomic spin orbitals.) A state is actually a function of the coordinates of all the electrons, so that their motion is correlated, but this is often approximated by this independent-particle model of products of single electron wave functions. (The London dispersion force, for example, depends on the correlations of the motion of the electrons.) In atomic physics, the
atomic spectral line Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter w ...
s correspond to transitions ( quantum leaps) between quantum states of an atom. These states are labeled by a set of quantum numbers summarized in the term symbol and usually associated with particular electron configurations, i.e., by occupation schemes of atomic orbitals (for example, 1s2 2s2 2p6 for the ground state of neon-term symbol: 1S0). This notation means that the corresponding Slater determinants have a clear higher weight in the
configuration interaction Configuration interaction (CI) is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematical ...
expansion. The atomic orbital concept is therefore a key concept for visualizing the excitation process associated with a given transition. For example, one can say for a given transition that it corresponds to the excitation of an electron from an occupied orbital to a given unoccupied orbital. Nevertheless, one has to keep in mind that electrons are fermions ruled by the Pauli exclusion principle and cannot be distinguished from each other. Moreover, it sometimes happens that the configuration interaction expansion converges very slowly and that one cannot speak about simple one-determinant wave function at all. This is the case when
electron correlation Electronic correlation is the interaction between electrons in the electronic structure of a quantum system. The correlation energy is a measure of how much the movement of one electron is influenced by the presence of all other electrons. Ato ...
is large. Fundamentally, an atomic orbital is a one-electron wave function, even though many electrons are not in one-electron atoms, and so the one-electron view is an approximation. When thinking about orbitals, we are often given an orbital visualization heavily influenced by the Hartree–Fock approximation, which is one way to reduce the complexities of molecular orbital theory.


Types of orbital

Atomic orbitals can be the hydrogen-like "orbitals" which are exact solutions to the Schrödinger equation for a hydrogen-like "atom" (i.e., atom with one electron). Alternatively, atomic orbitals refer to functions that depend on the coordinates of one electron (i.e., orbitals) but are used as starting points for approximating wave functions that depend on the simultaneous coordinates of all the electrons in an atom or molecule. The coordinate systems chosen for orbitals are usually
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
in atoms and Cartesian in polyatomic molecules. The advantage of spherical coordinates here is that an orbital wave function is a product of three factors each dependent on a single coordinate: . The angular factors of atomic orbitals generate s, p, d, etc. functions as real combinations of spherical harmonics (where and are quantum numbers). There are typically three mathematical forms for the radial functions  which can be chosen as a starting point for the calculation of the properties of atoms and molecules with many electrons: # The ''hydrogen-like orbitals'' are derived from the exact solutions of the Schrödinger Equation for one electron and a nucleus, for a
hydrogen-like atom A hydrogen-like atom (or hydrogenic atom) is any atom or ion with a single valence electron. These atoms are isoelectronic with hydrogen. Examples of hydrogen-like atoms include, but are not limited to, hydrogen itself, all alkali metals such a ...
. The part of the function that depends on distance ''r'' from the nucleus has radial nodes and decays as . # The Slater-type orbital (STO) is a form without radial nodes but decays from the nucleus as does a hydrogen-like orbital. # The form of the Gaussian type orbital (Gaussians) has no radial nodes and decays as e^ . Although hydrogen-like orbitals are still used as pedagogical tools, the advent of computers has made STOs preferable for atoms and diatomic molecules since combinations of STOs can replace the nodes in hydrogen-like orbitals. Gaussians are typically used in molecules with three or more atoms. Although not as accurate by themselves as STOs, combinations of many Gaussians can attain the accuracy of hydrogen-like orbitals.


History

The term "orbital" was coined by
Robert Mulliken Robert Sanderson Mulliken Note Longuet-Higgins' amusing title for reference B238 1965 on page 354 of this Biographical Memoir. The title should be "Selected papers of Robert S Mulliken." (June 7, 1896 – October 31, 1986) was an American ph ...
in 1932 as short for ''one-electron orbital wave function''.
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
explained around 1913 that electrons might revolve around a compact nucleus with definite angular momentum. Bohr's model was an improvement on the 1911 explanations of
Ernest Rutherford Ernest Rutherford, 1st Baron Rutherford of Nelson, (30 August 1871 – 19 October 1937) was a New Zealand physicist who came to be known as the father of nuclear physics. ''Encyclopædia Britannica'' considers him to be the greatest ...
, that of the electron moving around a nucleus. Japanese physicist
Hantaro Nagaoka was a Japanese physicist and a pioneer of Japanese physics during the Meiji period. Life Nagaoka was born in Nagasaki, Japan on August 19, 1865 and educated at the University of Tokyo. After graduating with a degree in physics in 1887, Naga ...
published an orbit-based hypothesis for electron behavior as early as 1904. These theories were each built upon new observations starting with simple understanding and becoming more correct and complex. Explaining the behavior of these electron "orbits" was one of the driving forces behind the development of quantum mechanics.


Early models

With J. J. Thomson's discovery of the electron in 1897, it became clear that atoms were not the smallest building blocks of nature, but were rather composite particles. The newly discovered structure within atoms tempted many to imagine how the atom's constituent parts might interact with each other. Thomson theorized that multiple electrons revolve in orbit-like rings within a positively charged jelly-like substance, and between the electron's discovery and 1909, this " plum pudding model" was the most widely accepted explanation of atomic structure. Shortly after Thomson's discovery,
Hantaro Nagaoka was a Japanese physicist and a pioneer of Japanese physics during the Meiji period. Life Nagaoka was born in Nagasaki, Japan on August 19, 1865 and educated at the University of Tokyo. After graduating with a degree in physics in 1887, Naga ...
predicted a different model for electronic structure. Unlike the plum pudding model, the positive charge in Nagaoka's "Saturnian Model" was concentrated into a central core, pulling the electrons into circular orbits reminiscent of Saturn's rings. Few people took notice of Nagaoka's work at the time, and Nagaoka himself recognized a fundamental defect in the theory even at its conception, namely that a classical charged object cannot sustain orbital motion because it is accelerating and therefore loses energy due to electromagnetic radiation. Nevertheless, the
Saturnian model was a Japanese physicist and a pioneer of Japanese physics during the Meiji period. Life Nagaoka was born in Nagasaki, Japan on August 19, 1865 and educated at the University of Tokyo. After graduating with a degree in physics in 1887, Naga ...
turned out to have more in common with modern theory than any of its contemporaries.


Bohr atom

In 1909,
Ernest Rutherford Ernest Rutherford, 1st Baron Rutherford of Nelson, (30 August 1871 – 19 October 1937) was a New Zealand physicist who came to be known as the father of nuclear physics. ''Encyclopædia Britannica'' considers him to be the greatest ...
discovered that the bulk of the atomic mass was tightly condensed into a nucleus, which was also found to be positively charged. It became clear from his analysis in 1911 that the plum pudding model could not explain atomic structure. In 1913, Rutherford's post-doctoral student,
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
, proposed a new model of the atom, wherein electrons orbited the nucleus with classical periods, but were permitted to have only discrete values of angular momentum, quantized in units ħ. This constraint automatically allowed only certain electron energies. The
Bohr model In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar Syst ...
of the atom fixed the problem of energy loss from radiation from a ground state (by declaring that there was no state below this), and more importantly explained the origin of spectral lines. After Bohr's use of Einstein's explanation of the
photoelectric effect The photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material. Electrons emitted in this manner are called photoelectrons. The phenomenon is studied in condensed matter physics, and solid sta ...
to relate energy levels in atoms with the wavelength of emitted light, the connection between the structure of electrons in atoms and the emission and
absorption spectra Absorption spectroscopy refers to spectroscopic techniques that measure the absorption of radiation, as a function of frequency or wavelength, due to its interaction with a sample. The sample absorbs energy, i.e., photons, from the radiating ...
of atoms became an increasingly useful tool in the understanding of electrons in atoms. The most prominent feature of emission and absorption spectra (known experimentally since the middle of the 19th century), was that these atomic spectra contained discrete lines. The significance of the Bohr model was that it related the lines in emission and absorption spectra to the energy differences between the orbits that electrons could take around an atom. This was, however, ''not'' achieved by Bohr through giving the electrons some kind of wave-like properties, since the idea that electrons could behave as
matter waves Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wav ...
was not suggested until eleven years later. Still, the Bohr model's use of quantized angular momenta and therefore quantized energy levels was a significant step toward the understanding of electrons in atoms, and also a significant step towards the development of quantum mechanics in suggesting that quantized restraints must account for all discontinuous energy levels and spectra in atoms. With de Broglie's suggestion of the existence of electron matter waves in 1924, and for a short time before the full 1926 Schrödinger equation treatment of
hydrogen-like atom A hydrogen-like atom (or hydrogenic atom) is any atom or ion with a single valence electron. These atoms are isoelectronic with hydrogen. Examples of hydrogen-like atoms include, but are not limited to, hydrogen itself, all alkali metals such a ...
s, a Bohr electron "wavelength" could be seen to be a function of its momentum; so a Bohr orbiting electron was seen to orbit in a circle at a multiple of its half-wavelength. The Bohr model for a short time could be seen as a classical model with an additional constraint provided by the 'wavelength' argument. However, this period was immediately superseded by the full three-dimensional wave mechanics of 1926. In our current understanding of physics, the Bohr model is called a semi-classical model because of its quantization of angular momentum, not primarily because of its relationship with electron wavelength, which appeared in hindsight a dozen years after the Bohr model was proposed. The Bohr model was able to explain the emission and absorption spectra of
hydrogen Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic ...
. The energies of electrons in the ''n'' = 1, 2, 3, etc. states in the Bohr model match those of current physics. However, this did not explain similarities between different atoms, as expressed by the periodic table, such as the fact that helium (two electrons), neon (10 electrons), and argon (18 electrons) exhibit similar chemical inertness. Modern quantum mechanics explains this in terms of
electron shell In chemistry and atomic physics, an electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" (also called the "K shell"), followed by the "2 shell" (or ...
s and subshells which can each hold a number of electrons determined by the Pauli exclusion principle. Thus the ''n'' = 1 state can hold one or two electrons, while the ''n'' = 2 state can hold up to eight electrons in 2s and 2p subshells. In helium, all ''n'' = 1 states are fully occupied; the same is true for ''n'' = 1 and ''n'' = 2 in neon. In argon, the 3s and 3p subshells are similarly fully occupied by eight electrons; quantum mechanics also allows a 3d subshell but this is at higher energy than the 3s and 3p in argon (contrary to the situation for hydrogen) and remains empty.


Modern conceptions and connections to the Heisenberg uncertainty principle

Immediately after Heisenberg discovered his
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
, Bohr noted that the existence of any sort of
wave packet In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of diff ...
implies uncertainty in the wave frequency and wavelength, since a spread of frequencies is needed to create the packet itself. In quantum mechanics, where all particle momenta are associated with waves, it is the formation of such a wave packet which localizes the wave, and thus the particle, in space. In states where a quantum mechanical particle is bound, it must be localized as a wave packet, and the existence of the packet and its minimum size implies a spread and minimal value in particle wavelength, and thus also momentum and energy. In quantum mechanics, as a particle is localized to a smaller region in space, the associated compressed wave packet requires a larger and larger range of momenta, and thus larger kinetic energy. Thus the binding energy to contain or trap a particle in a smaller region of space increases without bound as the region of space grows smaller. Particles cannot be restricted to a geometric point in space, since this would require infinite particle momentum. In chemistry, Schrödinger, Pauling, Mulliken and others noted that the consequence of Heisenberg's relation was that the electron, as a wave packet, could not be considered to have an exact location in its orbital. Max Born suggested that the electron's position needed to be described by a probability distribution which was connected with finding the electron at some point in the wave-function which described its associated wave packet. The new quantum mechanics did not give exact results, but only the probabilities for the occurrence of a variety of possible such results. Heisenberg held that the path of a moving particle has no meaning if we cannot observe it, as we cannot with electrons in an atom. In the quantum picture of Heisenberg, Schrödinger and others, the Bohr atom number ''n'' for each orbital became known as an ''n-sphere'' in a three-dimensional atom and was pictured as the most probable energy of the probability cloud of the electron's wave packet which surrounded the atom.


Orbital names


Orbital notation and subshells

Orbitals have been given names, which are usually given in the form: :X \, \mathrm \ where ''X'' is the energy level corresponding to the
principal quantum number In quantum mechanics, the principal quantum number (symbolized ''n'') is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. Its values are natural numbers (from 1) making it a discrete variable. A ...
; type is a lower-case letter denoting the shape or subshell of the orbital, corresponding to the angular momentum quantum number . For example, the orbital 1s (pronounced as the individual numbers and letters: "'one' 'ess'") is the lowest energy level () and has an angular quantum number of , denoted as s. Orbitals with are denoted as p, d and f respectively. The set of orbitals for a given n and is called a ''subshell'', denoted :X \, \mathrm^y \ . The exponent y shows the number of electrons in the subshell. For example, the notation 2p4 indicates that the 2p subshell of an atom contains 4 electrons. This subshell has 3 orbitals, each with n = 2 and = 1.


X-ray notation

There is also another, less common system still used in X-ray science known as
X-ray notation X-ray notation is a method of labeling atomic orbitals that grew out of X-ray science. Also known as IUPAC notation, it was adopted by the International Union of Pure and Applied Chemistry in 1991 as a simplification of the older Siegbahn notation. ...
, which is a continuation of the notations used before orbital theory was well understood. In this system, the principal quantum number is given a letter associated with it. For , the letters associated with those numbers are K, L, M, N, O, ... respectively.


Hydrogen-like orbitals

The simplest atomic orbitals are those that are calculated for systems with a single electron, such as the hydrogen atom. An atom of any other element ionized down to a single electron is very similar to hydrogen, and the orbitals take the same form. In the Schrödinger equation for this system of one negative and one positive particle, the atomic orbitals are the eigenstates of the
Hamiltonian operator Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonia ...
for the energy. They can be obtained analytically, meaning that the resulting orbitals are products of a polynomial series, and exponential and trigonometric functions. (see hydrogen atom). For atoms with two or more electrons, the governing equations can be solved only with the use of methods of iterative approximation. Orbitals of multi-electron atoms are ''qualitatively'' similar to those of hydrogen, and in the simplest models, they are taken to have the same form. For more rigorous and precise analysis, numerical approximations must be used. A given (hydrogen-like) atomic orbital is identified by unique values of three quantum numbers: , , and . The rules restricting the values of the quantum numbers, and their energies (see below), explain the electron configuration of the atoms and the periodic table. The stationary states ( quantum states) of the hydrogen-like atoms are its atomic orbitals. However, in general, an electron's behavior is not fully described by a single orbital. Electron states are best represented by time-depending "mixtures" ( linear combinations) of multiple orbitals. See Linear combination of atomic orbitals molecular orbital method. The quantum number first appeared in the
Bohr model In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar Syst ...
where it determines the radius of each circular electron orbit. In modern quantum mechanics however, determines the mean distance of the electron from the nucleus; all electrons with the same value of ''n'' lie at the same average distance. For this reason, orbitals with the same value of ''n'' are said to comprise a "
shell Shell may refer to: Architecture and design * Shell (structure), a thin structure ** Concrete shell, a thin shell of concrete, usually with no interior columns or exterior buttresses ** Thin-shell structure Science Biology * Seashell, a hard o ...
". Orbitals with the same value of ''n'' and also the same value of  are even more closely related, and are said to comprise a " subshell".


Quantum numbers

Because of the quantum mechanical nature of the electrons around a nucleus, atomic orbitals can be uniquely defined by a set of integers known as quantum numbers. These quantum numbers occur only in certain combinations of values, and their physical interpretation changes depending on whether
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
versions of the atomic orbitals are employed.


Complex orbitals

In physics, the most common orbital descriptions are based on the solutions to the hydrogen atom, where orbitals are given by the product between a radial function and a pure spherical harmonic. The quantum numbers, together with the rules governing their possible values, are as follows: The
principal quantum number In quantum mechanics, the principal quantum number (symbolized ''n'') is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. Its values are natural numbers (from 1) making it a discrete variable. A ...
describes the energy of the electron and is always a positive integer. In fact, it can be any positive integer, but for reasons discussed below, large numbers are seldom encountered. Each atom has, in general, many orbitals associated with each value of ''n''; these orbitals together are sometimes called '' electron shells''. The
azimuthal quantum number The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe ...
describes the orbital angular momentum of each electron and is a non-negative integer. Within a shell where is some integer , ranges across all (integer) values satisfying the relation 0 \le \ell \le n_0-1. For instance, the  shell has only orbitals with \ell=0, and the  shell has only orbitals with \ell=0, and \ell=1. The set of orbitals associated with a particular value of  are sometimes collectively called a ''subshell''. The
magnetic quantum number In atomic physics, the magnetic quantum number () is one of the four quantum numbers (the other three being the principal, azimuthal, and spin) which describe the unique quantum state of an electron. The magnetic quantum number distinguishes the ...
, m_\ell, describes the magnetic moment of an electron in an arbitrary direction, and is also always an integer. Within a subshell where \ell is some integer \ell_0, m_\ell ranges thus: -\ell_0 \le m_\ell \le \ell_0. The above results may be summarized in the following table. Each cell represents a subshell, and lists the values of m_\ell available in that subshell. Empty cells represent subshells that do not exist. Subshells are usually identified by their n- and \ell-values. n is represented by its numerical value, but \ell is represented by a letter as follows: 0 is represented by 's', 1 by 'p', 2 by 'd', 3 by 'f', and 4 by 'g'. For instance, one may speak of the subshell with n=2 and \ell=0 as a '2s subshell'. Each electron also has a spin quantum number, ''s'', which describes the spin of each electron (spin up or spin down). The number ''s'' can be + or −. The Pauli exclusion principle states that no two electrons in an atom can have the same values of all four quantum numbers. If there are two electrons in an orbital with given values for three quantum numbers, (, , ), these two electrons must differ in their spin. The above conventions imply a preferred axis (for example, the ''z'' direction in Cartesian coordinates), and they also imply a preferred direction along this preferred axis. Otherwise there would be no sense in distinguishing from . As such, the model is most useful when applied to physical systems that share these symmetries. The
Stern–Gerlach experiment The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent throug ...
— where an atom is exposed to a magnetic field — provides one such example.


Real orbitals

In addition to the complex orbitals described above, it is common, especially in the chemistry literature, to utilize ''real'' atomic orbitals. These real orbitals arise from simple linear combinations of complex orbitals. Using the Condon–Shortley phase convention, real orbitals are related to complex orbitals in the same way that the real spherical harmonics are related to complex spherical harmonics. Letting \psi_ denote a complex orbital with quantum numbers n, l, and m, the real orbitals \psi_^ may be defined by : \psi_^ = \begin \sqrt (-1)^m \text\left\ &\text m<0\\ \psi_ &\text m=0\\ \sqrt (-1)^m \text\left\ &\text m>0 \end = \begin \frac\left(\psi_ - (-1)^m \psi_\right) & \text m<0\\ \psi_& \text m=0\\ \frac\left(\psi_ + (-1)^m \psi_\right) & \text m<0\\ \end If \psi_(r, \theta, \phi) = R_(r) Y_^m(\theta, \phi), with R_(r) the radial part of the orbital, this definition is equivalent to \psi_^(r, \theta, \phi) = R_(r) Y_(\theta, \phi) where Y_ is the real spherical harmonic related to either the real or imaginary part of the complex spherical harmonic Y_^m. Real spherical harmonics are physically relevant when an atom is embedded in a crystalline solid, in which case there are multiple preferred symmetry axes but no single preferred direction. Real atomic orbitals are also more frequently encountered in introductory chemistry textbooks and shown in common orbital visualizations. In real hydrogen-like orbitals, quantum numbers n and \ell have the same interpretation and significance as their complex counterparts, but m is no longer a good quantum number (but its absolute value is). Some real orbitals are given specific names beyond the simple \psi_ designation. Orbitals with quantum number \ell equal to 0, 1, 2, 3, 4, 5, 6\ldots are called s, p, d, f, g, h, \ldots orbitals. With this one can already assign names to complex orbitals such as 2p_ = \psi_; the first symbol is the n quantum number, the second number is the symbol for that particular \ell quantum number and the subscript is the m quantum number. As an example of how the full orbital names are generated for real orbitals, one may calculate \psi_^. From the table of spherical harmonics, \psi_ = R_Y_1^ = \mp R_ \sqrt \cdot (x\pm i y)/r with r = \sqrt. Then : \begin \psi_^ =& R_ \sqrt \cdot \frac\\ \psi_^ =& R_ \sqrt \cdot \frac \end Likewise \psi_ = R_ \sqrt \cdot z/r. As a more complicated example: : \psi_^ = R_ \frac \sqrt \cdot \frac In all these cases we generate a Cartesian label for the orbital by examining, and abbreviating, the polynomial in x, y, and z appearing in the numerator. We ignore any terms in the z, r polynomial except for the term with the highest exponent in z. We then use the abbreviated polynomial as a subscript label for the atomic state, using the same nomenclature as above to indicate the n and \ell quantum numbers. : \begin \psi_^ =& np_y = \frac \left(np_ + np_\right)\\ \psi_^ =& np_z = 2p_0\\ \psi_^ =& np_x = \frac \left(np_ - np_\right)\\ \psi_^ =& nf_ = \frac \left(nf_ - nf_\right) \end The expression above all use the Condon–Shortley phase convention which is favored by quantum physicists. Other conventions for the phase of the spherical harmonics exists. Under these different conventions the p_x and p_y orbitals may appear, for example, as the sum and difference of p_ and p_, contrary to what is shown above. Below is a list of these Cartesian polynomial names for the atomic orbitals. Note that there does not seem to be reference in the literature as to how to abbreviate the long Cartesian spherical harmonic polynomials for \ell>3 so there does not seem be consensus on the naming of g orbitals or higher according to this nomenclature.


Shapes of orbitals

Simple pictures showing orbital shapes are intended to describe the angular forms of regions in space where the electrons occupying the orbital are likely to be found. The diagrams cannot show the entire region where an electron can be found, since according to quantum mechanics there is a non-zero probability of finding the electron (almost) anywhere in space. Instead the diagrams are approximate representations of boundary or contour surfaces where the probability density has a constant value, chosen so that there is a certain probability (for example 90%) of finding the electron within the contour. Although as the square of an absolute value is everywhere non-negative, the sign of the
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
is often indicated in each subregion of the orbital picture. Sometimes the function is graphed to show its phases, rather than which shows probability density but has no phase (which is lost when taking absolute value, since is a complex number). orbital graphs tend to have less spherical, thinner lobes than graphs, but have the same number of lobes in the same places, and otherwise are recognizable. This article, in order to show wave function phase, shows mostly graphs. The lobes can be seen as standing wave interference patterns between the two counter rotating, ring resonant
traveling wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
and modes; the projection of the orbital onto the xy plane has a resonant wavelength around the circumference. Alhough rarely shown, the traveling wave solutions can be seen as rotating banded tori; the bands represent phase information. For each there are two standing wave solutions and . If , the orbital is vertical, counter rotating information is unknown, and the orbital is ''z''-axis symmetric. If there are no counter rotating modes. There are only radial modes and the shape is spherically symmetric. For any given , the smaller is, the more radial nodes there are. For any given , the smaller is, the fewer radial nodes there are (zero for whichever first has that orbital). Loosely speaking, is energy, is analogous to
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
, and is orientation. In the classical case, a ring resonant traveling wave, for example in a circular transmission line, unless actively forced, spontaneously decays into a ring resonant standing wave because reflections build up over time at even the smallest imperfection or discontinuity. In general, determines size and energy of the orbital for a given nucleus; as increases, the size of the orbital increases. The higher nuclear charge of heavier elements causes their orbitals to contract by comparison to lighter ones, so that the size of the atom remains very roughly constant, even as the number of electrons increases. Also in general terms, determines an orbital's shape, and its orientation. However, since some orbitals are described by equations in complex numbers, the shape sometimes depends on also. Together, the whole set of orbitals for a given and fill space as symmetrically as possible, though with increasingly complex sets of lobes and nodes. The single s-orbitals (\ell=0) are shaped like spheres. For it is roughly a solid ball (densest at center and fades outward exponentially), but for , each single s-orbital is made of spherically symmetric surfaces which are nested shells (i.e., the "wave-structure" is radial, following a sinusoidal radial component as well). See illustration of a cross-section of these nested shells, at right. The s-orbitals for all numbers are the only orbitals with an anti-node (a region of high wave function density) at the center of the nucleus. All other orbitals (p, d, f, etc.) have angular momentum, and thus avoid the nucleus (having a wave node ''at'' the nucleus). Recently, there has been an effort to experimentally image the 1s and 2p orbitals in a SrTiO3 crystal using scanning transmission electron microscopy with energy dispersive x-ray spectroscopy. Because the imaging was conducted using an electron beam, Coulombic beam-orbital interaction that is often termed as the impact parameter effect is included in the final outcome (see the figure at right). The shapes of p, d and f-orbitals are described verbally here and shown graphically in the ''Orbitals table'' below. The three p-orbitals for have the form of two ellipsoids with a point of tangency at the nucleus (the two-lobed shape is sometimes referred to as a "
dumbbell The dumbbell, a type of free weight, is a piece of equipment used in weight training. It can be used individually or in pairs, with one in each hand. History The forerunner of the dumbbell, halteres, were used in ancient Greece as lifting ...
"—there are two lobes pointing in opposite directions from each other). The three p-orbitals in each
shell Shell may refer to: Architecture and design * Shell (structure), a thin structure ** Concrete shell, a thin shell of concrete, usually with no interior columns or exterior buttresses ** Thin-shell structure Science Biology * Seashell, a hard o ...
are oriented at right angles to each other, as determined by their respective linear combination of values of . The overall result is a lobe pointing along each direction of the primary axes. Four of the five d-orbitals for look similar, each with four pear-shaped lobes, each lobe tangent at right angles to two others, and the centers of all four lying in one plane. Three of these planes are the xy-, xz-, and yz-planes—the lobes are between the pairs of primary axes—and the fourth has the centre along the x and y axes themselves. The fifth and final d-orbital consists of three regions of high probability density: a torus in between two pear-shaped regions placed symmetrically on its z axis. The overall total of 18 directional lobes point in every primary axis direction and between every pair. There are seven f-orbitals, each with shapes more complex than those of the d-orbitals. Additionally, as is the case with the s orbitals, individual p, d, f and g orbitals with values higher than the lowest possible value, exhibit an additional radial node structure which is reminiscent of harmonic waves of the same type, as compared with the lowest (or fundamental) mode of the wave. As with s orbitals, this phenomenon provides p, d, f, and g orbitals at the next higher possible value of (for example, 3p orbitals vs. the fundamental 2p), an additional node in each lobe. Still higher values of further increase the number of radial nodes, for each type of orbital. The shapes of atomic orbitals in one-electron atom are related to 3-dimensional spherical harmonics. These shapes are not unique, and any linear combination is valid, like a transformation to cubic harmonics, in fact it is possible to generate sets where all the d's are the same shape, just like the and are the same shape. Although individual orbitals are most often shown independent of each other, the orbitals coexist around the nucleus at the same time. Also, in 1927, Albrecht Unsöld proved that if one sums the electron density of all orbitals of a particular azimuthal quantum number of the same shell (e.g., all three 2p orbitals, or all five 3d orbitals) where each orbital is occupied by an electron or each is occupied by an electron pair, then all angular dependence disappears; that is, the resulting total density of all the atomic orbitals in that subshell (those with the same ) is spherical. This is known as Unsöld's theorem.


Orbitals table

This table shows the real hydrogen-like wave functions for all atomic orbitals up to 7s, and therefore covers the occupied orbitals in the ground state of all elements in the periodic table up to radium and some beyond. "ψ" graphs are shown with − and +
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
phases shown in two different colors (arbitrarily red and blue). The orbital is the same as the orbital, but the and are formed by taking linear combinations of the and orbitals (which is why they are listed under the label). Also, the and are not the same shape as the , since they are pure spherical harmonics. * ''No elements with 6f, 7d or 7f electrons have been discovered yet.'' † ''Elements with 7p electrons have been discovered, but their
electronic configuration In atomic physics and quantum chemistry, the electron configuration is the distribution of electrons of an atom or molecule (or other physical structure) in atomic or molecular orbitals. For example, the electron configuration of the neon atom ...
s are only predicted.'' ‡ ''For the elements whose highest occupied orbital is a 6d orbital, only some electronic configurations have been confirmed.'' ( Ds, Rg and Cn are still missing). These are the real-valued orbitals commonly used in chemistry. Only the m = 0 orbitals where are eigenstates of the orbital angular momentum operator, \hat L_z. The columns with m = \pm 1, \pm 2,\cdots are combinations of two eigenstates. See comparison in the following picture:


Qualitative understanding of shapes

The shapes of atomic orbitals can be qualitatively understood by considering the analogous case of standing waves on a circular drum. To see the analogy, the mean vibrational displacement of each bit of drum membrane from the equilibrium point over many cycles (a measure of average drum membrane velocity and momentum at that point) must be considered relative to that point's distance from the center of the drum head. If this displacement is taken as being analogous to the probability of finding an electron at a given distance from the nucleus, then it will be seen that the many modes of the vibrating disk form patterns that trace the various shapes of atomic orbitals. The basic reason for this correspondence lies in the fact that the distribution of kinetic energy and momentum in a matter-wave is predictive of where the particle associated with the wave will be. That is, the probability of finding an electron at a given place is also a function of the electron's average momentum at that point, since high electron momentum at a given position tends to "localize" the electron in that position, via the properties of electron wave-packets (see the Heisenberg uncertainty principle for details of the mechanism). This relationship means that certain key features can be observed in both drum membrane modes and atomic orbitals. For example, in all of the modes analogous to s orbitals (the top row in the animated illustration below), it can be seen that the very center of the drum membrane vibrates most strongly, corresponding to the antinode in all s orbitals in an atom. This antinode means the electron is most likely to be at the physical position of the nucleus (which it passes straight through without scattering or striking it), since it is moving (on average) most rapidly at that point, giving it maximal momentum. A mental "planetary orbit" picture closest to the behavior of electrons in s orbitals, all of which have no angular momentum, might perhaps be that of a
Keplerian orbit Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
with the orbital eccentricity of 1 but a finite major axis, not physically possible (because particles were to collide), but can be imagined as a limit of orbits with equal major axes but increasing eccentricity. Below, a number of drum membrane vibration modes and the respective wave functions of the hydrogen atom are shown. A correspondence can be considered where the wave functions of a vibrating drum head are for a two-coordinate system and the wave functions for a vibrating sphere are three-coordinate . File:Drum vibration mode01.gif, Drum mode u_ File:Drum vibration mode02.gif, Drum mode u_ File:Drum vibration mode03.gif, Drum mode u_ File:Phi 1s.gif, Wave function of 1s orbital (real part, 2D-cut, r_=2 a_0) File:Phi 2s.gif, Wave function of 2s orbital (real part, 2D-cut, r_=10 a_0) File:Phi 3s.gif, Wave function of 3s orbital (real part, 2D-cut, r_=20 a_0) None of the other sets of modes in a drum membrane have a central antinode, and in all of them the center of the drum does not move. These correspond to a node at the nucleus for all non-s orbitals in an atom. These orbitals all have some angular momentum, and in the planetary model, they correspond to particles in orbit with eccentricity less than 1.0, so that they do not pass straight through the center of the primary body, but keep somewhat away from it. In addition, the drum modes analogous to p and d modes in an atom show spatial irregularity along the different radial directions from the center of the drum, whereas all of the modes analogous to s modes are perfectly symmetrical in radial direction. The non radial-symmetry properties of non-s orbitals are necessary to localize a particle with angular momentum and a wave nature in an orbital where it must tend to stay away from the central attraction force, since any particle localized at the point of central attraction could have no angular momentum. For these modes, waves in the drum head tend to avoid the central point. Such features again emphasize that the shapes of atomic orbitals are a direct consequence of the wave nature of electrons. File:Drum vibration mode11.gif, Drum mode u_ File:Drum vibration mode12.gif, Drum mode u_ File:Drum vibration mode13.gif, Drum mode u_ File:Phi 2p.gif, Wave function of 2p orbital (real part, 2D-cut, r_=10 a_0) File:Phi 3p.gif, Wave function of 3p orbital (real part, 2D-cut, r_=20 a_0) File:Phi 4p.gif, Wave function of 4p orbital (real part, 2D-cut, r_=25 a_0) File:Drum vibration mode21.gif, Drum mode u_ File:Drum vibration mode22.gif, Drum mode u_ File:Drum vibration mode23.gif, Drum mode u_


Orbital energy

In atoms with one electron (
hydrogen-like atom A hydrogen-like atom (or hydrogenic atom) is any atom or ion with a single valence electron. These atoms are isoelectronic with hydrogen. Examples of hydrogen-like atoms include, but are not limited to, hydrogen itself, all alkali metals such a ...
), the energy of an orbital (and, consequently, any electron in the orbital) is determined mainly by n. The n=1 orbital has the lowest possible energy in the atom. Each successively higher value of n has a higher energy, but the difference decreases as n increases. For high n, the energy becomes so high that the electron can easily escape the atom. In single electron atoms, all levels with different \ell within a given n are degenerate in the Schrödinger approximation, and have the same energy. This approximation is broken slightly in the solution to the Dirac equation (where energy depends on and another quantum number ), and by the effect of the magnetic field of the nucleus and
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
effects. The latter induce tiny binding energy differences especially for s electrons that go nearer the nucleus, since these feel a very slightly different nuclear charge, even in one-electron atoms; see
Lamb shift In physics, the Lamb shift, named after Willis Lamb, is a difference in energy between two energy levels 2''S''1/2 and 2''P''1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which th ...
. In atoms with multiple electrons, the energy of an electron depends not only on its orbital, but also on its interactions with other electrons. These interactions depend on the detail of its spatial probability distribution, and so the
energy level 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 t ...
s of orbitals depend not only on n but also on \ell. Higher values of \ell are associated with higher values of energy; for instance, the 2p state is higher than the 2s state. When \ell = 2, the increase in energy of the orbital becomes so large as to push the energy of orbital above the energy of the s-orbital in the next higher shell; when \ell = 3 the energy is pushed into the shell two steps higher. The filling of the 3d orbitals does not occur until the 4s orbitals have been filled. The increase in energy for subshells of increasing angular momentum in larger atoms is due to electron–electron interaction effects, and it is specifically related to the ability of low angular momentum electrons to penetrate more effectively toward the nucleus, where they are subject to less screening from the charge of intervening electrons. Thus, in atoms with higher atomic number, the \ell of electrons becomes more and more of a determining factor in their energy, and the principal quantum numbers n of electrons becomes less and less important in their energy placement. The energy sequence of the first 35 subshells (e.g., 1s, 2p, 3d, etc.) is given in the following table. Each cell represents a subshell with n and \ell given by its row and column indices, respectively. The number in the cell is the subshell's position in the sequence. For a linear listing of the subshells in terms of increasing energies in multielectron atoms, see the section below. ''Note: empty cells indicate non-existent sublevels, while numbers in italics indicate sublevels that could (potentially) exist, but which do not hold electrons in any element currently known.''


Electron placement and the periodic table

Several rules govern the placement of electrons in orbitals ('' electron configuration''). The first dictates that no two electrons in an atom may have the same set of values of quantum numbers (this is the Pauli exclusion principle). These quantum numbers include the three that define orbitals, as well as , or spin quantum number. Thus, two electrons may occupy a single orbital, so long as they have different values of . However, ''only'' two electrons, because of their spin, can be associated with each orbital. Additionally, an electron always tends to fall to the lowest possible energy state. It is possible for it to occupy any orbital so long as it does not violate the Pauli exclusion principle, but if lower-energy orbitals are available, this condition is unstable. The electron will eventually lose energy (by releasing a photon) and drop into the lower orbital. Thus, electrons fill orbitals in the order specified by the energy sequence given above. This behavior is responsible for the structure of the periodic table. The table may be divided into several rows (called 'periods'), numbered starting with 1 at the top. The presently known elements occupy seven periods. If a certain period has number ''i'', it consists of elements whose outermost electrons fall in the ''i''th shell.
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
was the first to propose (1923) that the periodicity in the properties of the elements might be explained by the periodic filling of the electron energy levels, resulting in the electronic structure of the atom. The periodic table may also be divided into several numbered rectangular ' blocks'. The elements belonging to a given block have this common feature: their highest-energy electrons all belong to the same -state (but the associated with that -state depends upon the period). For instance, the leftmost two columns constitute the 's-block'. The outermost electrons of Li and Be respectively belong to the 2s subshell, and those of Na and Mg to the 3s subshell. The following is the order for filling the "subshell" orbitals, which also gives the order of the "blocks" in the periodic table: :1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p The "periodic" nature of the filling of orbitals, as well as emergence of the s, p, d, and f "blocks", is more obvious if this order of filling is given in matrix form, with increasing principal quantum numbers starting the new rows ("periods") in the matrix. Then, each subshell (composed of the first two quantum numbers) is repeated as many times as required for each pair of electrons it may contain. The result is a compressed periodic table, with each entry representing two successive elements: Although this is the general order of orbital filling according to the Madelung rule, there are exceptions, and the actual electronic energies of each element are also dependent upon additional details of the atoms (see ). The number of electrons in an electrically neutral atom increases with the atomic number. The electrons in the outermost shell, or ''
valence electron In chemistry and physics, a valence electron is an electron in the outer shell associated with an atom, and that can participate in the formation of a chemical bond if the outer shell is not closed. In a single covalent bond, a shared pair form ...
s'', tend to be responsible for an element's chemical behavior. Elements that contain the same number of valence electrons can be grouped together and display similar chemical properties.


Relativistic effects

For elements with high atomic number , the effects of relativity become more pronounced, and especially so for s electrons, which move at relativistic velocities as they penetrate the screening electrons near the core of high- atoms. This relativistic increase in momentum for high speed electrons causes a corresponding decrease in wavelength and contraction of 6s orbitals relative to 5d orbitals (by comparison to corresponding s and d electrons in lighter elements in the same column of the periodic table); this results in 6s valence electrons becoming lowered in energy. Examples of significant physical outcomes of this effect include the lowered melting temperature of mercury (which results from 6s electrons not being available for metal bonding) and the golden color of gold and caesium. In the
Bohr model In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar Syst ...
, an  electron has a velocity given by v = Z \alpha c, where is the atomic number, \alpha is the
fine-structure constant In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between el ...
, and is the speed of light. In non-relativistic quantum mechanics, therefore, any atom with an atomic number greater than 137 would require its 1s electrons to be traveling faster than the speed of light. Even in the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
, which accounts for relativistic effects, the wave function of the electron for atoms with Z > 137 is oscillatory and unbounded. The significance of element 137, also known as untriseptium, was first pointed out by the physicist
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfl ...
. Element 137 is sometimes informally called feynmanium (symbol Fy). However, Feynman's approximation fails to predict the exact critical value of  due to the non-point-charge nature of the nucleus and very small orbital radius of inner electrons, resulting in a potential seen by inner electrons which is effectively less than . The critical  value, which makes the atom unstable with regard to high-field breakdown of the vacuum and production of electron-positron pairs, does not occur until is about 173. These conditions are not seen except transiently in collisions of very heavy nuclei such as lead or uranium in accelerators, where such electron-positron production from these effects has been claimed to be observed. There are no nodes in relativistic orbital densities, although individual components of the wave function will have nodes.


pp hybridisation (conjectured)

In late period 8 elements, a hybrid of 8p3/2 and 9p1/2 is expected to exist, where "3/2" and "1/2" refer to the
total angular momentum quantum number In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin). If s is the particle's sp ...
. This "pp" hybrid may be responsible for the p-block of the period due to properties similar to p subshells in ordinary
valence shell In chemistry and physics, a valence electron is an electron in the outer shell associated with an atom, and that can participate in the formation of a chemical bond if the outer shell is not closed. In a single covalent bond, a shared pair forms ...
s. Energy levels of 8p3/2 and 9p1/2 come close due to relativistic spin–orbit effects; the 9s subshell should also participate, as these elements are expected to be analogous to the respective 5p elements
indium Indium is a chemical element with the symbol In and atomic number 49. Indium is the softest metal that is not an alkali metal. It is a silvery-white metal that resembles tin in appearance. It is a post-transition metal that makes up 0.21 parts ...
through xenon.


Transitions between orbitals

Bound quantum states have discrete energy levels. When applied to atomic orbitals, this means that the energy differences between states are also discrete. A transition between these states (i.e., an electron absorbing or emitting a photon) can thus happen only if the photon has an energy corresponding with the exact energy difference between said states. Consider two states of the hydrogen atom: # State , , and # State , , and By quantum theory, state 1 has a fixed energy of , and state 2 has a fixed energy of . Now, what would happen if an electron in state 1 were to move to state 2? For this to happen, the electron would need to gain an energy of exactly . If the electron receives energy that is less than or greater than this value, it cannot jump from state 1 to state 2. Now, suppose we irradiate the atom with a broad-spectrum of light. Photons that reach the atom that have an energy of exactly will be absorbed by the electron in state 1, and that electron will jump to state 2. However, photons that are greater or lower in energy cannot be absorbed by the electron, because the electron can jump only to one of the orbitals, it cannot jump to a state between orbitals. The result is that only photons of a specific frequency will be absorbed by the atom. This creates a line in the spectrum, known as an absorption line, which corresponds to the energy difference between states 1 and 2. The atomic orbital model thus predicts line spectra, which are observed experimentally. This is one of the main validations of the atomic orbital model. The atomic orbital model is nevertheless an approximation to the full quantum theory, which only recognizes many electron states. The predictions of line spectra are qualitatively useful but are not quantitatively accurate for atoms and ions other than those containing only one electron.


See also

* Atomic electron configuration table * Wiswesser's rule * Condensed matter physics * Electron configuration *
Energy level 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 t ...
* Hund's rules * Molecular orbital * Quantum chemistry *
Quantum chemistry computer programs Quantum chemistry computer programs are used in computational chemistry to implement the methods of quantum chemistry. Most include the Hartree–Fock (HF) and some post-Hartree–Fock methods. They may also include density functional theory (DF ...
* Solid-state physics *
Wave function collapse In quantum mechanics, wave function collapse occurs when a wave function—initially in a quantum superposition, superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is ...


Notes


References

* * * * * *


External links

* 3D hydrogen orbitals on Wikimedia Commons
Guide to atomic orbitals



Animation of the time evolution of an hydrogenic orbital

3D representation of hydrogenic orbitals

The Orbitron
a visualization of all common and uncommon atomic orbitals, from 1s to 7g

Still images of many orbitals {{Authority control Atomic physics Chemical bonding Electron states Quantum chemistry Articles containing video clips