Atomic orbital

Atomic orbitals may be defined more precisely in formal quantum mechanical language. They are an approximate solution to the Schrodinger equation for the electrons bound to the atom by the electric field of the atom's nucleus. Specifically, in quantum mechanics, the state of an atom, i.e., an eigenstate of the atomic Hamiltonian, 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 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 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.

3D views of some hydrogen-like atomic orbitals showing probability density and phase (g orbitals and higher are not shown)

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 harmonics Y_ℓm(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 ${\displaystyle e^{-\alpha r^{2}}}$.

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.

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.

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 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.

The Rutherford–Bohr model of the hydrogen atom.

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 atoms, 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.[lower-alpha 1] 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 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 in the hydrogen atom) and remains empty.

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 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.

Orbitals have been given names, which are usually given in the form:

${\displaystyle X\,\mathrm {type} ^{y}\ }$

where X is the energy level corresponding to the principal quantum number n; type is a lower-case letter denoting the shape or subshell of the orbital, corresponding 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 ℓ = 0, denoted as s.

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 n = 1, 2, 3, 4, 5, …, the letters associated with those numbers are K, L, M, N, O, ... respectively.

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 n describes the energy of the electron and is always a positive integer. In fact, it can be any positive integer, but for reasons discussed below, large numbers are seldom encountered. Each atom has, in general, many orbitals associated with each value of n; these orbitals together are sometimes called electron shells.

The azimuthal quantum number ℓ 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 ${\displaystyle 0\leq \ell \leq n_{0}-1}$. For instance, the n = 1 shell has only orbitals with ${\displaystyle \ell =0}$, and the n = 2 shell has only orbitals with ${\displaystyle \ell =0}$, and ${\displaystyle \ell =1}$. The set of orbitals associated with a particular value of ℓ are sometimes collectively called a subshell.

The magnetic quantum number, ${\displaystyle m_{\ell }}$, describes the magnetic moment of an electron in an arbitrary direction, and is also always an integer. Within a subshell where ${\displaystyle \ell }$ is some integer ${\displaystyle \ell _{0}}$, ${\displaystyle m_{\ell }}$ ranges thus: ${\displaystyle -\ell _{0}\leq m_{\ell }\leq \ell _{0}}$.

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

	ℓ = 0	ℓ = 1	ℓ = 2	ℓ = 3	ℓ = 4	...
n = 1	${\displaystyle m_{\ell }=0}$
n = 2	0	−1, 0, 1
n = 3	0	−1, 0, 1	−2, −1, 0, 1, 2
n = 4	0	−1, 0, 1	−2, −1, 0, 1, 2	−3, −2, −1, 0, 1, 2, 3
n = 5	0	−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 ${\displaystyle n}$- and ${\displaystyle \ell }$-values. ${\displaystyle n}$ is represented by its numerical value, but ${\displaystyle \ell }$ 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 ${\displaystyle n=2}$ and ${\displaystyle \ell =0}$ as a '2s subshell'.

Each electron also has a spin quantum number, s, which describes the spin of each electron (spin up or spin down). The number s can be +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]

Animation of continuously varying superpositions between the ${\displaystyle p_{1}}$ and the ${\displaystyle p_{x}}$ orbitals.

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]

${\displaystyle p_{z}=p_{0}}$

${\displaystyle p_{x}={\frac {1}{\sqrt {2}}}\left(p_{1}+p_{-1}\right)}$

${\displaystyle p_{y}={\frac {1}{i{\sqrt {2}}}}\left(p_{1}-p_{-1}\right)}$

where p₀ = R_n 1 Y_1 0, p₁ = R_n 1 Y_1 1, and p₋₁ = R_n 1 Y_1 −1, are the complex orbitals corresponding to ℓ = 1.

The equations for the p_x and p_y orbitals depend on the phase convention used for the spherical harmonics. The above equations suppose that the spherical harmonics are defined by ${\displaystyle Y_{\ell }^{m}(\theta ,\varphi )=Ne^{im\varphi }P_{\ell }^{m}(\cos {\theta })}$. However some quantum physicists[22][23] include a phase factor (−1)^m in these definitions, which has the effect of relating the p_x orbital to a difference of spherical harmonics and the p_y orbital to the corresponding sum. (For more detail, see Spherical harmonics#Conventions).

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 p_z orbital is the same as the p₀ orbital, but the p_x and p_y 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.

	s (ℓ = 0)	p (ℓ = 1)			d (ℓ = 2)					f (ℓ = 3)
	m = 0	m = 0	m = ±1		m = 0	m = ±1		m = ±2		m = 0	m = ±1		m = ±2		m = ±3
	s	pz	px	py	dz2	dxz	dyz	dxy	dx2−y2	fz3	fxz2	fyz2	fxyz	fz(x2−y2)	fx(x2−3y2)	fy(3x2−y2)
n = 1
n = 2
n = 3
n = 4
n = 5										. . .	. . .	. . .	. . .	. . .	. . .	. . .
n = 6					. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .
n = 7		. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .

The shapes of atomic orbitals can be qualitatively understood by considering the analogous case of standing waves on a circular drum.[27] To see the analogy, the mean vibrational displacement of each bit of drum membrane from the equilibrium point over many cycles (a measure of average drum membrane velocity and momentum at that point) must be considered relative to that point's distance from the center of the drum head. If this displacement is taken as being analogous to the probability of finding an electron at a given distance from the nucleus, then it will be seen that the many modes of the vibrating disk form patterns that trace the various shapes of atomic orbitals. The basic reason for this correspondence lies in the fact that the distribution of kinetic energy and momentum in a matter-wave is predictive of where the particle associated with the wave will be. That is, the probability of finding an electron at a given place is also a function of the electron's average momentum at that point, since high electron momentum at a given position tends to "localize" the electron in that position, via the properties of electron wave-packets (see the Heisenberg uncertainty principle for details of the mechanism).

This relationship means that certain key features can be observed in both drum membrane modes and atomic orbitals. For example, in all of the modes analogous to s orbitals (the top row in the animated illustration below), it can be seen that the very center of the drum membrane vibrates most strongly, corresponding to the antinode in all s orbitals in an atom. This antinode means the electron is most likely to be at the physical position of the nucleus (which it passes straight through without scattering or striking it), since it is moving (on average) most rapidly at that point, giving it maximal momentum.

A mental "planetary orbit" picture closest to the behavior of electrons in s orbitals, all of which have no angular momentum, might perhaps be that of a Keplerian orbit with the orbital eccentricity of 1 but a finite major axis, not physically possible (because particles were to collide), but can be imagined as a limit of orbits with equal major axes but increasing eccentricity.

Below, a number of drum membrane vibration modes and the respective wave functions of the hydrogen atom are shown. A correspondence can be considered where the wave functions of a vibrating drum head are for a two-coordinate system ψ(r, θ) and the wave functions for a vibrating sphere are three-coordinate ψ(r, θ, φ).

• s-type drum modes and wave functions
• 
  Drum mode ${\displaystyle u_{01}}$
• 
  Drum mode ${\displaystyle u_{02}}$
• 
  Drum mode ${\displaystyle u_{03}}$
• 
  Wave function of 1s orbital (real part, 2D-cut, ${\displaystyle r_{max}=2a_{0}}$)
• 
  Wave function of 2s orbital (real part, 2D-cut, ${\displaystyle r_{max}=10a_{0}}$)
• 
  Wave function of 3s orbital (real part, 2D-cut, ${\displaystyle r_{max}=20a_{0}}$)

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.

• p-type drum modes and wave functions
• 
  Drum mode ${\displaystyle u_{11}}$
• 
  Drum mode ${\displaystyle u_{12}}$
• 
  Drum mode ${\displaystyle u_{13}}$
• 
  Wave function of 2p orbital (real part, 2D-cut, ${\displaystyle r_{max}=10a_{0}}$)
• 
  Wave function of 3p orbital (real part, 2D-cut, ${\displaystyle r_{max}=20a_{0}}$)
• 
  Wave function of 4p orbital (real part, 2D-cut, ${\displaystyle r_{max}=25a_{0}}$)

• d-type drum modes
• 
  Mode ${\displaystyle u_{21}}$ (3d orbital)
• 
  Mode ${\displaystyle u_{22}}$ (4d orbital)
• 
  Mode ${\displaystyle u_{23}}$ (5d orbital)

For elements with high atomic number Z, 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-Z atoms. This relativistic increase in momentum for high speed electrons causes a corresponding decrease in wavelength and contraction of 6s orbitals relative to 5d orbitals (by comparison to corresponding s and d electrons in lighter elements in the same column of the periodic table); this results in 6s valence electrons becoming lowered in energy.

Examples of significant physical outcomes of this effect include the lowered melting temperature of mercury (which results from 6s electrons not being available for metal bonding) and the golden color of gold and caesium.[29]

In the Bohr Model, an n = 1 electron has a velocity given by ${\displaystyle v=Z\alpha c}$, where Z is the atomic number, ${\displaystyle \alpha }$ is the fine-structure constant, and c 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 ${\displaystyle Z>137}$ 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 (symbol Fy).[30] However, Feynman's approximation fails to predict the exact critical value of Z 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 Z. The critical Z 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 Z 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.[31]

In late period-8 elements a hybrid of 8p_3/2 and 9p_1/2 is expected to exist,[32] where "3/2" and "1/2" refer to the total angular momentum quantum number. This "pp" hybrid may be responsible for the p-block of the period due to properties similar to p subshells in ordinary valence shells. Energy levels of 8p_3/2 and 9p_1/2 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.