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Orbitale Atomique Explication Essay

Pour les articles homonymes, voir orbitale.

En mécanique quantique, une orbitale atomique est une fonction mathématique qui décrit le comportement ondulatoire d'un électron ou d'une paire d'électrons dans un atome. Cette fonction donne la probabilité de présence d'un électron d'un atome dans une région donnée de cet atome. On la représente ainsi souvent à l'aide d'isosurfaces, qui délimitent la région à l'intérieur duquel la probabilité de présence de l'électron est supérieure à un seuil donné, par exemple 90 %. De telles régions ne sont pas nécessairement connexes et peuvent présenter des formes complexes issues des harmoniques sphériques.

Chaque orbitale atomique est définie par un triplet(n, ℓ, m) unique de nombres quantiques qui représentent respectivement l'énergie de l'électron, son moment angulaire et la projection de ce moment angulaire sur un axe donné. Chacune de ces orbitales peut être occupée par au plus deux électrons différant l'un de l'autre par leur nombre quantique magnétique de spinms. On parle d'orbitales s, p, d et f pour désigner les orbitales définies par un moment angulaire ℓ égal respectivement à 0, 1, 2 et 3. Ces noms proviennent d'anciennes dénominations des raies spectrales des métaux alcalins décrites comme sharp, principal, diffuse et fine ou fundamental ; les orbitales correspondant à ℓ > 3 sont ensuite nommées alphabétiquement g, h, i, k, etc.[1],[2],[3], en omettant la lettre j car certaines langues ne la distinguent pas de la lettre i[4].

Les orbitales atomiques sont les constituants élémentaires du nuage électronique, qui permet de modéliser le comportement des électrons dans la matière. Dans ce modèle, le nuage électronique d'un atome à plusieurs électrons peut être approché comme une configuration électronique formée du produit de plusieurs orbitales hydrogénoïdes. La structure du tableau périodique des éléments en blocs comprenant, sur chaque période, un total de 2, 6, 10 ou 14 éléments, est une conséquence directe du nombre maximum d'électrons pouvant occuper des orbitales atomiques s, p, d et f.

Propriétés de l'électron[modifier | modifier le code]

Nature quantique[modifier | modifier le code]

Le développement de la mécanique quantique et les observations expérimentales telles que la diffraction d'un faisceau d'électrons à travers des fentes de Young ont établi la dualité onde-corpuscule pour décrire les particules élémentaires. Ainsi, les électrons ne gravitent pas autour des noyaux atomiques sur des orbites définies comme le font les planètes autour du Soleil. En effet, ils ne peuvent être décrits comme de petites sphères solides définies par une position et une vitesse autour du noyau. Au contraire, ils doivent être vus comme des ondes stationnaires occupant un volume à l'intérieur duquel ils ont des propriétés quantiques définies et ils sont susceptibles d'interagir avec d'autres particules.

  • Nature corpusculaire :
  1. il ne peut y avoir qu'un nombre entier d'électrons dans un atome ;
  2. un échange d'énergie entre une particule incidente, par exemple un photon, et les électrons d'un atome ne concerne toujours qu'un seul électron à l'exclusion de tous les autres, quand bien même plusieurs électrons ont une probabilité non nulle de se trouver en même temps à l'endroit où l'un d'entre eux interagit avec ce photon ;
  3. un électron est pourvu d'une charge électrique définie et constante, et présente un spin dont la projection sur un axe de quantification peut valoir + 1/2 (spin up) ou – 1/2 (spin down).
  • Nature ondulatoire :
  1. l'énergie minimum d'un électron correspond à un état comparable à la fréquence fondamentale de la vibration d'une corde, dit état fondamental ; les niveaux d'énergie supérieurs de l'électron peuvent être vus comme les harmoniques de cette fréquence fondamentale ;
  2. les électrons ne sont jamais localisés en un point précis de l'espace mais se manifestent dans un volume à l'intérieur duquel la probabilité d'interagir avec eux en un point donné est déterminée par leur fonction d'onde.

Expression mathématique[modifier | modifier le code]

Les orbitales atomiques peuvent être définies plus précisément à travers le formalisme mathématique de la mécanique quantique. Dans ce cadre, l'état quantique d'un électron est la fonction d'ondeΨ qui satisfait l'équation aux valeurs propres de l'hamiltonienH, appelée aussi « équation de Schrödinger indépendante du temps », ou encore « équation des états stationnaires » : H Ψ = E Ψ, où E est l'énergie associée à cette fonction d'onde. La configuration électronique d'un atome multi-électronique est approchée par combinaison linéaire (interaction de configuration, bases) de produits de fonctions mono-électroniques (déterminants de Slater). La composante spatiale de ces fonctions mono-électroniques sont les orbitales atomiques, et la prise en compte de la composante de spin définit les spinorbitales.

L'orbitale atomique est une amplitude de probabilité de présence d'un électron autour du noyau d'un atome isolé. Cette densité de probabilité dépend du carré du module de la fonction d'ondeΨ. Celle-ci est déterminée par l'équation de Schrödinger en utilisant l'approximation orbitale, qui consiste à ignorer les corrélations entre électrons et à calculer la configuration électronique d'un atome comme produits de fonctions d'onde mono-électroniques[5]. Il s'agit cependant d'une approximation, car la distribution des différents électrons dans leurs orbitales est en réalité corrélée, les forces de London étant une manifestation de cette corrélation.

Calcul d'orbitales[modifier | modifier le code]

Les orbitales atomiques peuvent être des orbitales mono-électroniques qui sont des solutions exactes de l'équation de Schrödinger pour un atomehydrogénoïde (c'est-à-dire à un seul électron). Elles peuvent aussi être à la base du calcul des fonctions d'onde décrivant les différents électrons d'un atome ou d'une molécule. Le système de coordonnées choisi est généralement celui de coordonnées sphériques(r, θ, φ) dans les atomes et de coordonnées cartésiennes(x, y, z) dans les molécules polyatomiques. L'avantage des coordonnées sphériques est que la fonction d'onde d'une orbitale est le produit de trois fonctions ne dépendant chacune que d'une seule des trois coordonnées : ψ(r, θ, φ) = R(r) Θ(θ) Φ(φ).

La fonction radiale R(r) peut être généralement modélisée à travers trois formes mathématiques couramment employées :

  • Les orbitales atomiques hydrogénoïdes sont dérivées de solutions exactes de l'équation de Schrödinger pour un électron et un noyau atomique. La partie de la fonction qui dépend de la distance radiale présente des nœuds radiaux et décroît selon e-a r, a étant une constante.
  • L'orbitale de type Slater (STO) est une forme d'orbitale dépourvue de nœuds radiaux mais décroît selon r comme une orbitale hydrogénoïde.
  • L'orbitale gaussienne (GTO) est dépourvue de nœuds radiaux et décroît selon e-a r2.

Les orbitales hydrogénoïdes sont utilisées dans les outils pédagogiques mais ce sont les orbitales de type Slater qui sont préférentiellement utilisées pour modéliser les atomes et les molécules diatomiques en chimie numérique. Les molécules polyatomiques à trois atomes ou plus sont généralement modélisées à l'aide d'orbitales gaussiennes, moins précises que les orbitales de Slater mais dont la combinaison en grand nombre permet d'approcher la précision des orbitales hydrogénoïdes.

Le facteur angulaire Θ(θ) Φ(φ) génère des fonctions qui sont des combinaisons linéaires réelles d'harmoniques sphériquesYm
ℓ(θ, φ), où ℓ et m sont respectivement le nombre quantique azimutal et le nombre quantique magnétique.

Orbitales d'un atome hydrogénoïde[modifier | modifier le code]

Les orbitales sont calculées comme des nombres complexes, de sorte qu'on parle d’orbitales complexes, mais on utilise le plus souvent des combinaisons linéaires d'harmoniques sphériques choisies de telle sorte que les parties imaginaires s'annulent : les orbitales deviennent ainsi des nombres réels, et on parle d’orbitales réelles.

Orbitales complexes[modifier | modifier le code]

Chaque orbitale atomique est définie par un triplet de nombres quantiques(n, ℓ, m) et peut contenir au plus deux électrons différant chacun par leur nombre quantique magnétique de spin, qui ne peut être que up ou down ; le principe d'exclusion de Pauli interdit en effet à deux électrons d'un même atome de partager le même état quantique :

Le nombre quantique principal n définit la couche électronique tandis que le nombre quantique azimutal ℓ définit le type de sous-couche électronique de l'électron. Selon que ℓ vaut 0, 1, 2, 3, 4, 5, 6 ou davantage, le type de ces sous-couches est noté par les lettres s, p, d, f, g, h, i, etc.

Les sous-couches elles-mêmes sont notées en associant le nombre quantique n avec la lettre représentant le nombre quantique ℓ ; ainsi, la sous-couche correspondant à (n, ℓ) = (2, 1) est notée 2p.

La configuration électronique des atomes est notée en listant les sous-couches électroniques avec, en exposant, le nombre d'électrons sur cette sous-couche. Par construction, le nombre de sous-couches par couche électronique est égal à n, tandis que le nombre d'orbitales par sous-couche électronique s, p, d, f vaut 1, 3, 5, 7, etc. Chacune de ces orbitales pouvant contenir au plus deux électrons, le nombre maximum d'électrons par type de sous-couches s, p, d, f vaut 2, 6, 10, 14.

Table des premières orbitales complexes d'un atomehydrogénoïde

Les couleurs des surfaces représentées ci-dessus indiquent le signe des parties réelle et imaginaire de la fonction d'onde (ces couleurs sont arbitraires et ne reflètent pas une convention) :

 

Ces calculs impliquent le choix d'un axe privilégié — par exemple l'axe z en coordonnées cartésiennes — et d'un sens privilégié sur cet axe : c'est ce qui permet de définir le signe du nombre quantique magnétiquem. Ce modèle est par conséquent utile avec les systèmes qui partagent cette symétrie, comme dans le cas de l'expérience de Stern et Gerlach, où des atomes d'argent sont soumis à un champ magnétique non uniforme vertical.

Orbitales réelles[modifier | modifier le code]

Un atome situé dans un solide cristallin est soumis à plusieurs axes préférentiels, mais à aucune orientation préférentielle sur ces axes. Dans ces conditions, au lieu de construire les orbitales atomiques d'un tel atome à partir de fonctions radiales et d'une harmonique sphérique unique, on utilise généralement des combinaisons linéaires d'harmoniques sphériques choisies de telle sorte que leur partie imaginaire s'annule, ce qui donne des harmoniques réelles. Ce sont ces orbitales réelles qui sont généralement utilisées pour visualiser les orbitales atomiques.

Dans les orbitales hydrogénoïdes réelles, par exemple, n et ℓ ont la même signification sur leur contrepartie complexe, mais m n'est plus un nombre quantique valable, bien que son module le soit. Les orbitales p hydrogénoïdes réelles, par exemple, sont données par[6],[7] :

, et .

Les équations pour les orbitales px et py dépendent de la convention de phase choisie pour les harmoniques sphériques. Les équations précédentes supposent que les harmoniques sphériques sont définies par Ym
ℓ(θ, φ) = N eimφ Pm
ℓ(cosθ). On inclut cependant parfois[8],[9] un facteur de phase (–1)m qui a pour effet de faire correspondre l'orbitale px à une différence d'harmoniques, et l'orbitale py à une somme d'harmoniques.

Représentation des nuages de probabilité de présence de l'électron (en haut) et des isosurfaces à 90 % (en bas) pour les orbitales 1s, 2s et 2p. Dans le cas des orbitales 2p, les trois isosurfaces 2px, 2py et 2pz représentées correspondent à ℓ = –1, ℓ = 0 et ℓ = +1. Les couleurs indiquent la phase de la fonction d'onde : positive en bleu, négative en rouge.

In quantum mechanics, an atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom.[1] 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 defined by the particular mathematical form of the orbital.[2]

Each orbital in an atom is characterized by a unique set of values of the three quantum numbersn, ℓ, and m, which respectively correspond to the electron's energy, angular momentum, and an angular momentum vector component (the magnetic quantum number). Each such orbital can be occupied by a maximum of two electrons, each with its own spin quantum numbers. The simple names s orbital, p orbital, d orbital and f orbital refer to orbitals with angular momentum quantum number = 0, 1, 2 and 3 respectively. These names, together with the value of n, are used to describe the electron configurations of atoms. They are derived from the description by early spectroscopists of certain series of alkali metalspectroscopic lines as sharp, principal, diffuse, and fundamental. Orbitals for ℓ > 3 continue alphabetically, omitting j (g, h, i, k, …)[3][4][5] because some languages do not distinguish between the letters "i" and "j".[6]

Atomic orbitals are the basic building blocks of the atomic orbital model (alternatively known as the 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 a multi-electron 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 the 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 atomic orbitals, respectively, although for higher values of the quantum number n, particularly when the atom in question 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 = [Ar]4s13d5 and Cr2+ = [Ar]3d4) can only be rationalized somewhat arbitrarily.

Electron properties[edit]

With the development of quantum mechanics and experimental findings (such as the two slits diffraction of electrons), it was found that the orbiting electrons around a nucleus could not be fully described as particles, but needed to be explained by the wave-particle duality. In this sense, the electrons have the following properties:

Wave-like properties:

  1. The electrons do not orbit the 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 of that fundamental frequency.
  2. The electrons are never in a single point location, although the probability of interacting with the electron at a single point can be found from the wave function of the electron. The charge on the electron 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:

  1. The number of electrons orbiting the nucleus can only be an integer.
  2. Electrons jump between orbitals like particles. For example, if a single photon strikes the electrons, only a single electron changes states in response to the photon.
  3. The electrons retain particle-like properties such as: each wave state has the same electrical charge as its electron particle. Each wave state has a single discrete spin (spin up or spin down) depending on its superposition.

Thus, despite the popular analogy to planets revolving around the Sun, electrons cannot be described simply as solid particles. In addition, atomic orbitals do not closely resemble a planet's elliptical path in ordinary atoms. A more accurate analogy might be that of a large and often oddly shaped "atmosphere" (the electron), distributed around a relatively tiny planet (the atomic nucleus). Atomic orbitals exactly describe the shape of this "atmosphere" only when a single electron is present in an atom. When more electrons are added to a single atom, the additional electrons tend to more evenly fill in a volume of space around the nucleus so that the resulting collection (sometimes termed the atom's "electron cloud"[7]) tends toward a generally spherical zone of probability describing the electron's location, because of the uncertainty principle.

Formal quantum mechanical definition[edit]

Atomic orbitals may be defined more precisely in formal quantum mechanical language. Specifically, in quantum mechanics, the state of an atom, i.e., an eigenstate of the atomic Hamiltonian, is approximated by an expansion (see configuration interaction 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.[8] (The London dispersion force, for example, depends on the correlations of the motion of the electrons.)

In atomic physics, the atomic spectral lines 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 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 the other electrons in the atom. 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 most electrons do not exist 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 orbitals[edit]

Atomic orbitals can be the hydrogen-like "orbitals" which are exact solutions to the Schrödinger equation for a hydrogen-like "atom" (i.e., an 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 atomic orbitals are usually spherical coordinates(r, θ, φ) in atoms and cartesians(x, y, z) in polyatomic molecules. The advantage of spherical coordinates (for atoms) is that an orbital wave function is a product of three factors each dependent on a single coordinate: ψ(r, θ, φ) = R(r) Θ(θ) Φ(φ). The angular factors of atomic orbitals Θ(θ) Φ(φ) generate s, p, d, etc. functions as real combinations of spherical harmonicsYℓm(θ, φ) (where ℓ and m are quantum numbers). There are typically three mathematical forms for the radial functions R(r) which can be chosen as a starting point for the calculation of the properties of atoms and molecules with many electrons:

  1. The hydrogen-like atomic orbitals are derived from the exact solution of the Schrödinger Equation for one electron and a nucleus, for a hydrogen-like atom. The part of the function that depends on the distance r from the nucleus has nodes (radial nodes) and decays as e−(constant × distance).
  2. The Slater-type orbital (STO) is a form without radial nodes but decays from the nucleus as does the hydrogen-like orbital.
  3. The form of the Gaussian type orbital (Gaussians) has no radial nodes and decays as .

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 atomic orbital. 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[edit]

Main article: Atomic theory

The term "orbital" was coined by Robert Mulliken in 1932 as an abbreviation for one-electron orbital wave function.[9] However, the idea that electrons might revolve around a compact nucleus with definite angular momentum was convincingly argued at least 19 years earlier by Niels Bohr,[10] and the Japanese physicist Hantaro Nagaoka published an orbit-based hypothesis for electronic behavior as early as 1904.[11] Explaining the behavior of these electron "orbits" was one of the driving forces behind the development of quantum mechanics.[12]

Early models[edit]

With J. J. Thomson's discovery of the electron in 1897,[13] 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 revolved in orbit-like rings within a positively charged jelly-like substance,[14] 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 predicted a different model for electronic structure.[11] 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,[15] 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.[16] Nevertheless, the Saturnian model turned out to have more in common with modern theory than any of its contemporaries.

Bohr atom[edit]

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 as Rutherford's post-doctoral student, Niels Bohr proposed a new model of the atom, wherein electrons orbited the nucleus with classical periods, but were only permitted to have discrete values of angular momentum, quantized in units h/2π.[10] This constraint automatically permitted only certain values of 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'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 towards 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 Bohr electron "wavelength" could be seen to be a function of its momentum, and thus a Bohr orbiting electron was seen to orbit in a circle at a multiple of its half-wavelength (this physically incorrect Bohr model is still often taught to beginning students). 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. 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 shells 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 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 in the hydrogen atom) and remains empty.

Modern conceptions and connections to the Heisenberg uncertainty principle[edit]

Immediately after Heisenberg discovered his uncertainty principle,[17]Bohr 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.[18] 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 an 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[citation needed] in a three dimensional atom and was pictured as the mean energy of the probability cloud of the electron's wave packet which surrounded the atom.

Orbital names[edit]

Orbitals are given names in the form:

where X is the energy level corresponding to the principal quantum numbern, type is a lower-case letter denoting the shape or subshell of the orbital and it corresponds to the angular quantum number ℓ, and y is the number of electrons in that orbital.

For example, the orbital 1s2 (pronounced as the individual numbers and letters: "one ess two") has two electrons and is the lowest energy level (n = 1) and has an angular quantum number of = 0. In X-ray notation, the principal quantum number is given a letter associated with it. For n = 1, 2, 3, 4, 5, …, the letters associated with those numbers are K, L, M, N, O, … respectively.

Hydrogen-like orbitals[edit]

Main article: Hydrogen-like atom

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 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 only be solved 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, the numerical approximations must be used.

A given (hydrogen-like) atomic orbital is identified by unique values of three quantum numbers: n, ℓ, and m. 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.[clarification needed] 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 n first appeared in the Bohr model where it determines the radius of each circular electron orbit. In modern quantum mechanics however, n 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 ℓ are even more closely related, and are said to comprise a "subshell".

Quantum numbers[edit]

Main article: Quantum number

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 only occur 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[edit]

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 numbern 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 numberℓ describes the orbital angular momentum of each electron and is a non-negative integer. Within a shell where n is some integer n0, ℓ ranges across all (integer) values satisfying the relation . For instance, the n = 1 shell has only orbitals with , and the n = 2 shell has only orbitals with , and . The set of orbitals associated with a particular value of ℓ are sometimes collectively called a subshell.

The magnetic quantum number, , describes the magnetic moment of an electron in an arbitrary direction, and is also always an integer. Within a subshell where is some integer , ranges thus: .

The above results may be summarized in the following table. Each cell represents a subshell, and lists the values of available in that subshell. Empty cells represent subshells that do not exist.

= 0 = 1 = 2 = 3 = 4
n = 1
n = 20−1, 0, 1
n = 30−1, 0, 1−2, −1, 0, 1, 2
n = 40−1, 0, 1−2, −1, 0, 1, 2−3, −2, −1, 0, 1, 2, 3
n = 50−1, 0, 1−2, −1, 0, 1, 2−3, −2, −1, 0, 1, 2, 3−4, −3, −2, −1, 0, 1, 2, 3, 4

Subshells are usually identified by their - and -values. is represented by its numerical value, but 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 and 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 +1/2 or −1/2.

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, (n, l, m), 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 m = +1 from m = −1. 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.[19]

Real orbitals[edit]

An atom that is embedded in a crystalline solid feels multiple preferred axes, but often no preferred direction. Instead of building atomic orbitals out of the product of radial functions and a single spherical harmonic, linear combinations of spherical harmonics are typically used, designed so that the imaginary part of the spherical harmonics cancel out. These real orbitals are the building blocks most commonly shown in orbital visualizations.

In the real hydrogen-like orbitals, for example, n and ℓ have the same interpretation and significance as their complex counterparts, but m is no longer a good quantum number (though its absolute value is). The orbitals are given new names based on their shape with respect to a standardized Cartesian basis. The real hydrogen-like p orbitals are given by the following[20][21]

where p0 = Rn 1Y1 0, p1 = Rn 1Y1 1, and p−1 = Rn 1Y1 −1, are the complex orbitals corresponding to = 1.

The equations for the px and py orbitals depend on the phase convention used for the spherical harmonics. The above equations suppose that the spherical harmonics are defined by . However some quantum physicists[22][23] include a phase factor (-1)m in these definitions, which has the effect of relating the px orbital to a difference of spherical harmonics and the py orbital to the corresponding sum. (For more detail, see Spherical harmonics#Conventions).

Shapes of orbitals[edit]

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 | ψ(r, θ, φ) |2 has a constant value, chosen so that there is a certain probability (for example 90%) of finding the electron within the contour. Although | ψ |2 as the square of an absolute value is everywhere non-negative, the sign of the wave functionψ(r, θ, φ) is often indicated in each subregion of the orbital picture.

Sometimes the ψ function will be graphed to show its phases, rather than the | ψ(r, θ, φ) |2 which shows probability density but has no phases (which have been lost in the process of taking the absolute value, since ψ(r, θ, φ) is a complex number). | ψ(r, θ, φ) |2 orbital graphs tend to have less spherical, thinner lobes than ψ(r, θ, φ) graphs, but have the same number of lobes in the same places, and otherwise are recognizable. This article, in order to show wave function phases, shows mostly ψ(r, θ, φ) graphs.

The lobes can be viewed as standing waveinterference patterns between the two counter rotating, ring resonant travelling wave "m" and "−m" modes, with the projection of the orbital onto the xy plane having a resonant "m" wavelengths around the circumference. Though rarely depicted the travelling wave solutions can be viewed as rotating banded tori, with the bands representing phase information. For each m there are two standing wave solutions ⟨m⟩+⟨−m⟩ and ⟨m⟩−⟨−m⟩. For the case where m = 0 the orbital is vertical, counter rotating information is unknown, and the orbital is z-axis symmetric. For the case where = 0 there are no counter rotating modes. There are only radial modes and the shape is spherically symmetric. For any given n, the smaller ℓ is, the more radial nodes there are. Loosely speaking n is energy, ℓ is analogous to eccentricity, and m is orientation. In the classical case, a ring resonant travelling wave, for example in a circular transmission line, unless actively forced, will spontaneously decay into a ring resonant standing wave because reflections will build up over time at even the smallest imperfection or discontinuity.

Generally speaking, the number n determines the size and energy of the orbital for a given nucleus: as n increases, the size of the orbital increases. When comparing different elements, the higher nuclear charge Z of heavier elements causes their orbitals to contract by comparison to lighter ones, so that the overall size of the whole atom remains very roughly constant, even as the number of electrons in heavier elements (higher Z) increases.

Also in general terms, ℓ determines an orbital's shape, and m its orientation. However, since some orbitals are described by equations in complex numbers, the shape sometimes depends on m also. Together, the whole set of orbitals for a given ℓ and n fill space as symmetrically as possible, though with increasingly complex sets of lobes and nodes.

The single s-orbitals () are shaped like spheres. For n = 1 it is roughly a solid ball (it is most dense at the center and fades exponentially outwardly), but for n = 2 or more, each single s-orbital is composed 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 n 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 orbitials in a SrTiO3 crystal using scanning transmission electron microscopy with energy dispersive x-ray spectroscopy.[24] 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 n = 2 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"—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 m. The overall result is a lobe pointing along each direction of the primary axes.

Four of the five d-orbitals for n = 3 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 centres along the x and y axes themselves. The fifth and final d-orbital consists of three regions of high probability density: a torus with 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 n 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 n (for example, 3p orbitals vs. the fundamental 2p), an additional node in each lobe. Still higher values of n 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 px, py, and pz are the same shape.[25][26]

Although individual orbitals are most often shown independent of each other, the orbitals coexist around the nucleus at the same time.

Orbitals table[edit]

This table shows all orbital configurations for the real hydrogen-like wave functions up to 7s, and therefore covers the simple electronic configuration for all elements in the periodic table up to radium. "ψ" graphs are shown with and +wave function phases shown in two different colors (arbitrarily red and blue). The pz orbital is the same as the p0 orbital, but the px and py are formed by taking linear combinations of the p+1 and p−1 orbitals (which is why they are listed under the m = ±1 label). Also, the p+1 and p−1 are not the same shape as the p0, since they are pure spherical harmonics.

The shapes of the first five atomic orbitals are: 1s, 2s, 2px, 2py, and 2pz. The two colors show the phase or sign of the wave function in each region. These are graphs of ψ(x, y, z) functions which depend on the coordinates of one electron. To see the elongated shape of ψ(x, y, z)2 functions that show probability density more directly, see the graphs of d-orbitals below.
Atomic orbitals of the electron in a hydrogen atom at different energy levels. The probability of finding the electron is given by the color, as shown in the key at upper right.
Heat maps of some hydrogen-like atomic orbitals showing the probability density (f orbitals and higher are not shown)
Cross-section of computed hydrogen atom orbital (ψ(r, θ, φ)2) for the 6s (n = 6, = 0, m = 0) orbital. The s orbitals, though spherically symmetrical, have radially placed wave-nodes for n > 1. Only s orbitals invariably have a center anti-node; the other types never do.
Experimentally imaged 1s and 2p core-electron orbitals of Sr, including the effects of atomic thermal vibrations and excitation broadening, retrieved from energy dispersive x-ray spectroscopy (EDX) in scanning transmission electron microscopy (STEM).[24]
The 1s, 2s, & 2p orbitals of a sodium atom.