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atomic theory Atomic theory is the scientific theory that matter is composed of particles called atoms. Atomic theory traces its origins to an ancient philosophical tradition known as atomism. According to this idea, if one were to take a lump of matter ...
and
quantum mechanics 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, ...
, an atomic orbital is a function describing the location and wave-like behavior of an
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
in an
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, a ...
. This function can be used to calculate the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
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 number In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can ...
s , , and , which respectively correspond to the electron's
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
,
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, and an angular momentum vector component ( magnetic quantum number). 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 Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
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 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 ato ...
s of atoms. They are derived from the description by early spectroscopists of certain series of
alkali metal The alkali metals consist of the chemical elements lithium (Li), sodium (Na), potassium (K),The symbols Na and K for sodium and potassium are derived from their Latin names, ''natrium'' and ''kalium''; these are still the origins of the names ...
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 The periodic table, also known as the periodic table of the (chemical) elements, is a rows and columns arrangement of the chemical elements. It is widely used in chemistry, physics, and other sciences, and is generally seen as an icon of ch ...
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 Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
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 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, ...
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 Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical physics, classical concepts "particle" or "wave" to fu ...
. 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 wave In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
s. Thus the lowest possible energy an electron can take is similar to the
fundamental frequency The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. I ...
of a wave on a string. Higher energy states are similar to harmonics 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. 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 A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
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 language. They are approximate solutions to the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
for the electrons bound to the atom by the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
of the atom's
nucleus Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: * Atomic nucleus, the very dense central region of an atom *Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA Nucl ...
. Specifically, in quantum mechanics, the state of an atom, i.e., an
eigenstate In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
of the atomic Hamiltonian, is approximated by an expansion (see configuration interaction expansion and basis set) into linear combinations of anti-symmetrized products (
Slater determinant In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electro ...
s) of one-electron functions. The spatial components of these one-electron functions are called atomic orbitals. (When one considers also their
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
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 Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...
, the atomic spectral lines correspond to transitions ( quantum leaps) between
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
s of an atom. These states are labeled by a set of
quantum number In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can ...
s summarized in the
term symbol In quantum mechanics, the term symbol is an abbreviated description of the (total) angular momentum quantum numbers in a multi-electron atom (however, even a single electron can be described by a term symbol). Each energy level of an atom with a giv ...
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 Neon is a chemical element with the symbol Ne and atomic number 10. It is a noble gas. Neon is a colorless, odorless, inert monatomic gas under standard conditions, with about two-thirds the density of air. It was discovered (along with krypt ...
-term symbol: 1S0). This notation means that the corresponding Slater determinants have a clear higher weight in the configuration interaction 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
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s ruled by the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
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 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 In chemistry, molecular orbital theory (MO theory or MOT) is a method for describing the electronic structure of molecules using quantum mechanics. It was proposed early in the 20th century. In molecular orbital theory, electrons in a molec ...
.


Types of orbital

Atomic orbitals can be the hydrogen-like "orbitals" which are exact solutions to the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
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 system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
s chosen for orbitals are usually spherical coordinates 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 In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
(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. The part of the function that depends on distance ''r'' from the nucleus has radial nodes and decays as . # The
Slater-type orbital Slater-type orbitals (STOs) are functions used as atomic orbitals in the linear combination of atomic orbitals molecular orbital method. They are named after the physicist John C. Slater, who introduced them in 1930. They possess exponential deca ...
(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 in 1932 as short for ''one-electron orbital wave function''. Niels Bohr 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, that of the electron moving around a nucleus. Japanese physicist Hantaro Nagaoka 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 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, ...
.


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 The plum pudding model is one of several historical scientific models of the atom. First proposed by J. J. Thomson in 1904 soon after the discovery of the electron, but before the discovery of the atomic nucleus, the model tried to explain two ...
" was the most widely accepted explanation of atomic structure. Shortly after Thomson's discovery, Hantaro Nagaoka 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 turned out to have more in common with modern theory than any of its contemporaries.


Bohr atom

In 1909, Ernest Rutherford 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, 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 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 Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...
's explanation of the photoelectric effect 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 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 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 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, ...
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 The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
treatment of hydrogen-like atoms, 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-to ...
. 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 Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic ta ...
(two electrons), neon (10 electrons), and
argon Argon is a chemical element with the symbol Ar and atomic number 18. It is in group 18 of the periodic table and is a noble gas. Argon is the third-most abundant gas in Earth's atmosphere, at 0.934% (9340 ppmv). It is more than twice a ...
(18 electrons) exhibit similar chemical inertness. Modern
quantum mechanics 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, ...
explains this in terms of electron shells and subshells which can each hold a number of electrons determined by the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
. 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 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. B ...
noted that the existence of any sort of wave packet 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 Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a ...
suggested that the electron's position needed to be described by a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
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 ; type is a lower-case letter denoting the shape or subshell of the orbital, corresponding to the
angular momentum 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 th ...
 . 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, 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 A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen cons ...
. 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 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 A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen cons ...
). 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 periodic table, also known as the periodic table of the (chemical) elements, is a rows and columns arrangement of the chemical elements. It is widely used in chemistry, physics, and other sciences, and is generally seen as an icon of ch ...
. The stationary states (
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
s) 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 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". 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 or complex 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 describes the energy of the electron and is always a
positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
. 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 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" ( ...
''. The azimuthal quantum number 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, 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 In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe t ...
, ''s'', which describes the spin of each electron (spin up or spin down). The number ''s'' can be + or −. The
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
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 — 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 This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree \ell = 10. Some of these formulas are expressed in terms of the Cartesian expansion of the spherical harmonics into polynomials in ''x'', ''y'' ...
, \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 In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
is everywhere non-negative, the sign of the wave function 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 In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
). 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 In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
interference patterns between the two counter rotating, ring resonant traveling wave 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, 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 number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, 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
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as th ...
s with a point of tangency at the
nucleus Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: * Atomic nucleus, the very dense central region of an atom *Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA Nucl ...
(the two-lobed shape is sometimes referred to as a " dumbbell"—there are two lobes pointing in opposite directions from each other). The three p-orbitals in each shell 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 geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does n ...
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 In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
. 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 Albrecht Otto Johannes Unsöld (20 April 1905 – 23 September 1995) was a German astrophysicist known for his contributions to spectroscopic analysis of stellar atmospheres. Career Albrecht Unsöld was born in Bolheim, Württemberg, Germany. ...
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 Radium is a chemical element with the symbol Ra and atomic number 88. It is the sixth element in group 2 of the periodic table, also known as the alkaline earth metals. Pure radium is silvery-white, but it readily reacts with nitrogen (rat ...
and some beyond. "ψ" graphs are shown with − and + wave function 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 In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
. * ''No elements with 6f, 7d or 7f electrons have been discovered yet.'' † ''Elements with 7p electrons have been discovered, but their electronic configurations 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 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 ...
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 A node is a point along a standing wave where the wave has minimum amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effect ...
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 with the
orbital eccentricity In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values bet ...
of 1 but a finite major axis, not physically possible (because
particle In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from ...
s were to collide), but can be imagined as a
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
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), 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 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 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 levels 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 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 ato ...
''). 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 In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
). These quantum numbers include the three that define orbitals, as well as , or
spin quantum number In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe t ...
. 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 A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
) 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 periodic table, also known as the periodic table of the (chemical) elements, is a rows and columns arrangement of the chemical elements. It is widely used in chemistry, physics, and other sciences, and is generally seen as an icon of ch ...
. 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 was the first to propose (1923) that the
periodicity Periodicity or periodic may refer to: Mathematics * Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups * Periodic function, a function whose output contains values tha ...
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 atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of ever ...
. The electrons in the outermost shell, or '' valence electrons'', 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 Gold is a chemical element with the symbol Au (from la, aurum) and atomic number 79. This makes it one of the higher atomic number elements that occur naturally. It is a bright, slightly orange-yellow, dense, soft, malleable, and ductile ...
and
caesium Caesium (IUPAC spelling) (or cesium in American English) is a chemical element with the symbol Cs and atomic number 55. It is a soft, silvery-golden alkali metal with a melting point of , which makes it one of only five elemental metals that a ...
. In the Bohr model, an  electron has a velocity given by v = Z \alpha c, where is the atomic number, \alpha is the fine-structure constant, 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, 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. Element 137 is sometimes informally called
feynmanium An extended periodic table theorises about chemical elements beyond those currently known in the periodic table and proven. , the element with the highest atomic number known is oganesson (''Z'' = 118), which completes the seventh period (row) ...
(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. This "pp" hybrid may be responsible for the
p-block A block of the periodic table is a set of elements unified by the atomic orbitals their valence electrons or vacancies lie in. The term appears to have been first used by Charles Janet. Each block is named after its characteristic orbital: s-blo ...
of the period due to properties similar to p subshells in ordinary valence shells. 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 through
xenon Xenon is a chemical element with the symbol Xe and atomic number 54. It is a dense, colorless, odorless noble gas found in Earth's atmosphere in trace amounts. Although generally unreactive, it can undergo a few chemical reactions such as the ...
.


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 The aufbau principle , from the German ''Aufbauprinzip'' (building-up principle), also called the aufbau rule, states that in the ground state of an atom or ion, electrons fill subshells of the lowest available energy, then they fill subshells ...
* Condensed matter physics *
Electron 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 ato ...
* Energy level *
Hund's rules In atomic physics, Hund's rules refers to a set of rules that German physicist Friedrich Hund formulated around 1927, which are used to determine the term symbol that corresponds to the ground state of a multi- electron atom. The first rule is ...
*
Molecular orbital In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of find ...
*
Quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
* Quantum chemistry computer programs *
Solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how th ...
* Wave function collapse


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