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Atomic orbital

In quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function describes the electron's charge distribution around the atom's nucleus, and can be used to calculate the probability of finding an electron in a specific region around the nucleus.

Each orbital in an atom is characterized by a set of values of the three quantum numbers $n$, $ℓ$, and $m_{ℓ}$, which respectively correspond to the electron's energy, its orbital angular momentum, and its orbital angular momentum projected along a chosen axis (magnetic quantum number). The orbitals with a well-defined magnetic quantum number are generally complex-valued. Real-valued orbitals can be formed as linear combinations of $m_{ℓ}$ and $−m_{ℓ}$ orbitals, and are often labeled using the associated harmonic polynomials (e.g., xy, x2 − y2) which describe their angular structure.

An orbital can be occupied by a maximum of two electrons, each with its own projection of spin $$m_s$$. The simple names s orbital, p orbital, d orbital, and f orbital refer to orbitals with angular momentum quantum number $ψ(x, y, z)$ and $ψ(x, y, z)^{2}$ 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 metal spectroscopic lines as sharp, principal, diffuse, and fundamental. Orbitals for $ℓ = 0, 1, 2,$ continue alphabetically (g, h, i, k, ...), omitting j because some languages do not distinguish between the letters "i" and "j".

Atomic orbitals are the basic building blocks of the atomic orbital model (or electron cloud or wave mechanics model), a modern framework for visualizing the submicroscopic behavior of electrons in matter. In this model the electron cloud of an atom may be seen as being built up (in approximation) in an electron configuration that is a product of simpler hydrogen-like atomic orbitals. The repeating periodicity of blocks of 2, 6, 10, and 14 elements within sections of the periodic table arises naturally from the total number of electrons that occupy a complete set of s, p, d, and f orbitals, respectively, though for higher values of quantum number $n$, particularly when the atom bears a positive charge, the energies of certain sub-shells become very similar and so the order in which they are said to be populated by electrons (e.g., Cr = [Ar]4s13d5 and Cr2+ = [Ar]3d4) can be rationalized only somewhat arbitrarily.

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

Wave-like properties:
 * 1) Electrons do not orbit a nucleus in the manner of a planet orbiting a star, 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, 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:
 * 1) The number of electrons orbiting a nucleus can be only an integer.
 * 2) Electrons jump between orbitals like particles. For example, if one photon strikes the electrons, only one electron changes state as a result.
 * 3) 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.

One should remember that these orbital 'states', as described here, are merely eigenstates of an electron in its orbit. An actual electron exists in a superposition of states, which is like a weighted average, but with complex number weights. So, for instance, an electron could be in a state (2, 1, 0), a pure eigenstate, or a mixed state $1⁄2$(2, 1, 0) + $1⁄2$(2, 1, 1), or even the mixed state $2⁄5$(2, 1, 0) + $3⁄5$ $$i$$ (2, 1, 1). For each eigenstate, a property has an eigenvalue. So, for the three states just mentioned, the value of $$n$$ is 2, and the value of $$l$$ is 1. But the value for $$m_l$$ is a superposition of 0 and 1. It is not a fraction like $1⁄2$; it is ambiguous, either 0.0 or 1.0. A superposition of eigenstates (2, 1, 1) and (3, 2, 1) would have an ambiguous $$n$$ and $$l$$, but $$m_l$$ would definitely be 1. Eigenstates make it easier to deal with the math. You can choose a different basis of eigenstates by superimposing eigenstates from any other basis (see Real orbitals below).

Formal quantum mechanical definition
Atomic orbitals may be defined more precisely in formal quantum mechanical language. They are approximate solutions to the Schrödinger equation for the electrons bound to the atom by the electric field of the atom's nucleus. Specifically, in quantum mechanics, the state of an atom, i.e., an eigenstate of the atomic Hamiltonian, 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. (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 many electrons are not in one-electron atoms, and so the one-electron view is an approximation. When thinking about orbitals, we are often given an orbital visualization heavily influenced by the Hartree–Fock approximation, which is one way to reduce the complexities of molecular orbital theory.

Types of orbital
Atomic orbitals can be the hydrogen-like "orbitals" which are exact solutions to the Schrödinger equation for a hydrogen-like "atom" (i.e., atom with one electron). Alternatively, atomic orbitals refer to functions that depend on the coordinates of one electron (i.e., orbitals) but are used as starting points for approximating wave functions that depend on the simultaneous coordinates of all the electrons in an atom or molecule. The coordinate systems chosen for orbitals are usually spherical coordinates $3$ in atoms and Cartesian $ℓ > 3$ 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: $(r, θ, φ)$. The angular factors of atomic orbitals $(x, y, z)$ generate s, p, d, etc. functions as real combinations of spherical harmonics $ψ(r, θ, φ) = R(r) Θ(θ) Φ(φ)$ (where $ℓ$ and $m$ are quantum numbers). There are typically three mathematical forms for the radial functions $Θ(θ) Φ(φ)$ 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 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 $$ e^{-\alpha r} $$.
 * 2) The Slater-type orbital (STO) is a form without radial nodes but decays from the nucleus as does a hydrogen-like orbital.
 * 3) The form of the Gaussian type orbital (Gaussians) has no radial nodes and decays as $$ e^{-\alpha r^2} $$.

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 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 S. 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.

Early models
With J. J. Thomson's discovery of the electron in 1897, it became clear that atoms were not the smallest building blocks of nature, but were rather composite particles. The newly discovered structure within atoms tempted many to imagine how the atom's constituent parts might interact with each other. Thomson theorized that multiple electrons revolve in orbit-like rings within a positively charged jelly-like substance, and between the electron's discovery and 1909, this "plum pudding model" was the most widely accepted explanation of atomic structure.

Shortly after Thomson's discovery, Hantaro Nagaoka 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'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 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; 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. 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 for hydrogen) and remains empty.

Modern conceptions and connections to the Heisenberg uncertainty principle
Immediately after Heisenberg discovered his uncertainty principle, 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. 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, Erwin Schrödinger, Linus Pauling, Mulliken and others noted that the consequence of Heisenberg's relation was that the electron, as a wave packet, could not be considered to have an exact location in its orbital. Max Born suggested that the electron's position needed to be described by a probability distribution which was connected with finding the electron at some point in the wave-function which described its associated wave packet. The new quantum mechanics did not give exact results, but only the probabilities for the occurrence of a variety of possible such results. Heisenberg held that the path of a moving particle has no meaning if we cannot observe it, as we cannot with electrons in an atom.

In the quantum picture of Heisenberg, Schrödinger and others, the Bohr atom number n for each orbital became known as an n-sphere in a three-dimensional atom and was pictured as the most probable energy of the probability cloud of the electron's wave packet which surrounded the atom.

Orbital notation and subshells
Orbitals have been given names, which are usually given in the form:
 * $$X \, \mathrm{type} \ $$

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 momentum quantum number $ℓ$.

For example, the orbital 1s (pronounced as the individual numbers and letters: "'one' 'ess'") is the lowest energy level ($Y_{ℓm}(θ, φ)$) and has an angular quantum number of $R(r)$, denoted as s. Orbitals with $n = 1$ are denoted as p, d and f respectively.

The set of orbitals for a given n and $ℓ$ is called a subshell, denoted
 * $$X \, \mathrm{type}^y \ $$.

The superscript 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 $ℓ$ = 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 $ℓ = 0$, the letters associated with those numbers are K, L, M, N, O, ... respectively.

Hydrogen-like orbitals
The simplest atomic orbitals are those that are calculated for systems with a single electron, such as the hydrogen atom. An atom of any other element ionized down to a single electron is very similar to hydrogen, and the orbitals take the same form. In the Schrödinger equation for this system of one negative and one positive particle, the atomic orbitals are the eigenstates of the Hamiltonian operator for the energy. They can be obtained analytically, meaning that the resulting orbitals are products of a polynomial series, and exponential and trigonometric functions. (see hydrogen atom).

For atoms with two or more electrons, the governing equations can be solved only with the use of methods of iterative approximation. Orbitals of multi-electron atoms are qualitatively similar to those of hydrogen, and in the simplest models, they are taken to have the same form. For more rigorous and precise analysis, numerical approximations must be used.

A given (hydrogen-like) atomic orbital is identified by unique values of three quantum numbers: $n|size=120%$, $ℓ|size=120%$, and $m_{ℓ}|size=120%$. The rules restricting the values of the quantum numbers, and their energies (see below), explain the electron configuration of the atoms and the periodic table.

The stationary states (quantum states) of the hydrogen-like atoms are its atomic orbitals. However, in general, an electron's behavior is not fully described by a single orbital. Electron states are best represented by time-depending "mixtures" (linear combinations) of multiple orbitals. See Linear combination of atomic orbitals molecular orbital method.

The quantum number $n$ 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
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 $n$ 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 $ℓ = 1, 2 and 3$, $ℓ$ ranges across all (integer) values satisfying the relation $$0 \le \ell \le n_0-1$$. For instance, the $n = 1, 2, 3, 4, 5, ...$ shell has only orbitals with $$\ell=0$$, and the $n_{0}$ shell has only orbitals with $$\ell=0$$, and $$\ell=1$$. The set of orbitals associated with a particular value of $ℓ$ are sometimes collectively called a subshell.

The magnetic quantum number, $$m_\ell$$, describes the projection of the orbital angular momentum along a chosen axis. It determines the magnitude of the current circulating around that axis and the orbital contribution to the magnetic moment of an electron via the Ampèrian loop model. Within a subshell $$\ell$$, $$m_\ell$$ obtains the integer values in the range $$-\ell \le m_\ell \le \ell $$.

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 angular momentum in the form of quantum mechanical spin given by spin s = $1⁄2$. Its projection along a specified axis is given by the spin magnetic quantum number, ms, which can be +$1⁄2$ or −$1⁄2$. These values are also called "spin up" or "spin down" respectively.

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 projection ms.

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 $n = 1$ from $n = 2$. As such, the model is most useful when applied to physical systems that share these symmetries. The Stern–Gerlach experimentwhere an atom is exposed to a magnetic fieldprovides one such example.

Real orbitals

Instead of the complex orbitals described above, it is common, especially in the chemistry literature, to use 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 $$\psi_{n,\ell, m}$$ denote a complex orbital with quantum numbers $$n$$, $$l$$, and $$m$$, the real orbitals $$\psi_{n, \ell, m}^{\text{real}}$$ may be defined by $$ \psi_{n,\ell, m}^{\text{real}} = \begin{cases} \sqrt{2} (-1)^m \text{Im}\left\{\psi_{n,\ell,|m|}\right\} &\text{ for } m<0\\ \psi_{n,\ell,|m|} &\text{ for } m=0\\ \sqrt{2} (-1)^m \text{Re}\left\{\psi_{n,\ell,|m|}\right\} &\text{ for } m>0 \end{cases} = \begin{cases} \frac{i}{\sqrt{2}}\left(\psi_{n,\ell, -|m|} - (-1)^m \psi_{n,\ell, |m|}\right) & \text{ for } m<0\\ \psi_{n, \ell, |m|}& \text{ for } m=0\\ \frac{1}{\sqrt{2}}\left(\psi_{n,\ell, -|m|} + (-1)^m \psi_{n,\ell, |m|}\right) & \text{ for } m>0\\ \end{cases} $$

If $$\psi_{n,\ell, m}(r, \theta, \phi) = R_{nl}(r) Y_{\ell}^m(\theta, \phi)$$, with $$R_{nl}(r)$$ the radial part of the orbital, this definition is equivalent to $$\psi_{n,\ell, m}^{\text{real}}(r, \theta, \phi) = R_{nl}(r) Y_{\ell m}(\theta, \phi)$$ where $$Y_{\ell m}$$ is the real spherical harmonic related to either the real or imaginary part of the complex spherical harmonic $$Y_{\ell}^m$$.

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 $$\psi_{n, \ell, m}$$ designation. Orbitals with quantum number $$\ell$$ equal to $$0, 1, 2, 3, 4, 5, 6\ldots$$ are called $$\text{s, p, d, f, g, h, i} \ldots$$ orbitals. With this one can already assign names to complex orbitals such as $$2\text{p}_{\pm 1} = \psi_{2, 1, \pm 1}$$; the first symbol is the $$n$$ quantum number, the second character 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 $$\psi_{n, 1, \pm 1}^{\text{real}}$$. From the table of spherical harmonics, $\psi_{n, 1, \pm1} = R_{n, 1}Y_1^{\pm 1} = \mp R_{n, 1} \sqrt{3/8\pi} \cdot (x\pm i y)/r$ with $$r = \sqrt{x^2+y^2+z^2}$$. Then

\begin{align} \psi_{n, 1, +1}^{\text{real}} =& R_{n, 1} \sqrt{\frac{3}{4\pi}} \cdot \frac{x}{r}\\ \psi_{n, 1, -1}^{\text{real}} =& R_{n, 1} \sqrt{\frac{3}{4\pi}} \cdot \frac{y}{r} \end{align} $$

Likewise $\psi_{n, 1, 0} = R_{n, 1} \sqrt{3/4\pi} \cdot z/r$. As a more complicated example:

\psi_{n, 3, +1}^{\text{real}} = R_{n, 3} \frac{1}{4} \sqrt{\frac{21}{2\pi}} \cdot \frac{x\cdot (5z^2 - r^2)}{r^3} $$

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{align} \psi_{n, 1, -1}^{\text{real}} =& n\text{p}_y = \frac{i}{\sqrt{2}} \left(n\text{p}_{-1} + n\text{p}_{+1}\right)\\ \psi_{n, 1, 0}^{\text{real}} =& n\text{p}_z = 2\text{p}_0\\ \psi_{n, 1, +1}^{\text{real}} =& n\text{p}_x = \frac{1}{\sqrt{2}} \left(n\text{p}_{-1} - n\text{p}_{+1}\right)\\ \psi_{n, 3, +1}^{\text{real}} =& nf_{xz^2} = \frac{1}{\sqrt{2}} \left(nf_{-1} - nf_{+1}\right) \end{align} $$

The expression above all use the Condon–Shortley phase convention which is favored by quantum physicists. Other conventions exist for the phase of the spherical harmonics. Under these different conventions the $$\text{p}_x$$ and $$\text{p}_y$$ orbitals may appear, for example, as the sum and difference of $$\text{p}_{+1}$$ and $$\text{p}_{-1}$$, contrary to what is shown above.

Below is a list of these Cartesian polynomial names for the atomic orbitals. 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 $ℓ = 0 (s)$ has a constant value, chosen so that there is a certain probability (for example 90%) of finding the electron within the contour. Although $ℓ = 1 (p)$ as the square of an absolute value is everywhere non-negative, the sign of the wave function $ℓ = 2 (d)$ is often indicated in each subregion of the orbital picture.

Sometimes the $ℓ = 3 (f)$ function is graphed to show its phases, rather than $ℓ = 4 (g)$ which shows probability density but has no phase (which is lost when taking absolute value, since $n = 1$ is a complex number). $n = 2$ orbital graphs tend to have less spherical, thinner lobes than $n = 3$ graphs, but have the same number of lobes in the same places, and otherwise are recognizable. This article, to show wave function phase, shows mostly $n = 4$ graphs.

The lobes can be seen as standing wave interference patterns between the two counter-rotating, ring-resonant traveling wave $m$ and $n = 5$ modes; the projection of the orbital onto the xy plane has a resonant $m$ wavelength around the circumference. Although rarely shown, the traveling wave solutions can be seen as rotating banded tori; the bands represent phase information. For each $m$ there are two standing wave solutions $m = +1$ and $m = −1$. If $(n = 6, ℓ = 0, m = 0)$, the orbital is vertical, counter rotating information is unknown, and the orbital is z-axis symmetric. If $n > 1$ there are no counter rotating modes. There are only radial modes and the shape is spherically symmetric.

Nodal planes and nodal spheres are surfaces on which the probability density vanishes. The number of nodal surfaces is controlled by the quantum numbers $n$ and $ℓ$. An orbital with azimuthal quantum number $ℓ$ has $ℓ$ radial nodal planes passing through the origin. For example, the s orbitals ($| ψ(r, θ, φ) |^{2}$) are spherically symmetric and have no nodal planes, whereas the p orbitals ($| ψ |^{2}$) have a single nodal plane between the lobes. The number of nodal spheres equals $n-ℓ-1$, consistent with the restriction $ℓ ≤ n-1$ on the quantum numbers. The principal quantum number controls the total number of nodal surfaces which is $n-1$. Loosely speaking, $n$ is energy, $ℓ$ is analogous to eccentricity, and $m$ is orientation.

In general, $n$ determines size and energy of the orbital for a given nucleus; as $n$ increases, the size of the orbital increases. The higher nuclear charge $Z$ 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, $ℓ$ determines an orbital's shape, and $m_{ℓ}$ its orientation. However, since some orbitals are described by equations in complex numbers, the shape sometimes depends on $m_{ℓ}$ 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 ($$\ell=0$$) are shaped like spheres. For $ψ(r, θ, φ)$ it is roughly a solid ball (densest at center and fades outward exponentially), but for $ψ$, 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 $n$ 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 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 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 $| ψ(r, θ, φ) |^{2}$ 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 $m_{ℓ}$. The overall result is a lobe pointing along each direction of the primary axes.

Four of the five d orbitals for $ψ(r, θ, φ)$ 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 center along the x and y axes themselves. The fifth and final d orbital consists of three regions of high probability density: a torus in between two pear-shaped regions placed symmetrically on its z axis. The overall total of 18 directional lobes point in every primary axis direction and between every pair.

There are seven f orbitals, each with shapes more complex than those of the d orbitals.

Additionally, as is the case with the s orbitals, individual p, d, f and g orbitals with $n$ 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 $|ψ(r, θ, φ)|^{2}$ and $ψ(r, θ, φ)$ are the same shape.

Although individual orbitals are most often shown independent of each other, the orbitals coexist around the nucleus at the same time. Also, in 1927, Albrecht Unsöld proved that if one sums the electron density of all orbitals of a particular azimuthal quantum number $ℓ$ of the same shell $n$ (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  $ℓ$) is spherical. This is known as Unsöld's theorem.

Orbitals table
This table shows the real hydrogen-like wave functions for all atomic orbitals up to 7s, and therefore covers the occupied orbitals in the ground state of all elements in the periodic table up to radium and some beyond. "ψ" graphs are shown with − and + wave function phases shown in two different colors (arbitrarily red and blue). The $ψ(r, θ, φ)$ orbital is the same as the $−m$ orbital, but the $⟨m⟩ + ⟨−m⟩$ and $⟨m⟩ − ⟨−m⟩$ are formed by taking linear combinations of the $m = 0$ and $ℓ = 0$ orbitals (which is why they are listed under the $ℓ = 0$ label). Also, the $ℓ = 1$ and $n = 1$ are not the same shape as the $n ≥ 2$, since they are pure spherical harmonics.

* No elements with 6f, 7d or 7f electrons have been discovered yet.

† Elements with 7p electrons have been discovered, but their electronic configurations are only predicted.

‡ For the elements whose highest occupied orbital is a 6d orbital, only some electronic configurations have been confirmed. (Mt, Ds, Rg and Cn are still missing).

These are the real-valued orbitals commonly used in chemistry. Only the $$m = 0$$ orbitals where are eigenstates of the orbital angular momentum operator, $$\hat L_z$$. The columns with $$m = \pm 1, \pm 2,\cdots$$ are combinations of two eigenstates. See comparison in the following picture:

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

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

A mental "planetary orbit" picture closest to the behavior of electrons in s orbitals, all of which have no angular momentum, might perhaps be that of a Keplerian orbit 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 $n = 2$ and the wave functions for a vibrating sphere are three-coordinate $n = 3$.

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 $n$ and another quantum number $j$), 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). The first dictates that no two electrons in an atom may have the same set of values of quantum numbers (this is the Pauli exclusion principle). These quantum numbers include the three that define orbitals, as well as the spin magnetic quantum number $r$. Thus, two electrons may occupy a single orbital, so long as they have different values of $m_{s}$. Because $m_{s}$ takes one of only two values ($m_{s}$ or $1⁄2$), at most two electrons can occupy each orbital.

Additionally, an electron always tends to fall to the lowest possible energy state. It is possible for it to occupy any orbital so long as it does not violate the Pauli exclusion principle, but if lower-energy orbitals are available, this condition is unstable. The electron will eventually lose energy (by releasing a photon) and drop into the lower orbital. Thus, electrons fill orbitals in the order specified by the energy sequence given above.

This behavior is responsible for the structure of the periodic table. The table may be divided into several rows (called 'periods'), numbered starting with 1 at the top. The presently known elements occupy seven periods. If a certain period has number i, it consists of elements whose outermost electrons fall in the ith shell. Niels Bohr was the first to propose (1923) that the periodicity in the properties of the elements might be explained by the periodic filling of the electron energy levels, resulting in the electronic structure of the atom.

The periodic table may also be divided into several numbered rectangular 'blocks'. The elements belonging to a given block have this common feature: their highest-energy electrons all belong to the same $-1⁄2$-state (but the $ℓ$ associated with that $n$-state depends upon the period). For instance, the leftmost two columns constitute the 's-block'. The outermost electrons of Li and Be respectively belong to the 2s subshell, and those of Na and Mg to the 3s subshell.

The following is the order for filling the "subshell" orbitals, which also gives the order of the "blocks" in the periodic table:

 * 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p

The "periodic" nature of the filling of orbitals, as well as emergence of the s, p, d, and f "blocks", is more obvious if this order of filling is given in matrix form, with increasing principal quantum numbers starting the new rows ("periods") in the matrix. Then, each subshell (composed of the first two quantum numbers) is repeated as many times as required for each pair of electrons it may contain. The result is a compressed periodic table, with each entry representing two successive elements:

Although this is the general order of orbital filling according to the Madelung rule, there are exceptions, and the actual electronic energies of each element are also dependent upon additional details of the atoms (see ).

The number of electrons in an electrically neutral atom increases with the atomic number. The electrons in the outermost shell, or valence 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 $ℓ$, 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.

In the Bohr model, an $p_{x}, p_{y},$ electron has a velocity given by $$v = Z \alpha c$$, where $Z$ is the atomic number, $$\alpha$$ is the fine-structure constant, and $p_{z}$ 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 (symbol Fy). 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.

pp hybridization (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 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.

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: By quantum theory, state 1 has a fixed energy of $p_{z}$, and state 2 has a fixed energy of $p_{0}$. 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 $p_{x}$. 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 $p_{y}$ 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.
 * 1) State $p_{+1}$, $p_{−1}$, $m = ±1$ and $p_{+1}$
 * 2) State $p_{−1}$, $p_{0}$, $ℓ = 0$ and $ℓ = 1$

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.

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title: Atomic orbital


text:
Amino acid

Amino acids are organic compounds that contain both amino and carboxylic acid functional groups. Although over 500 amino acids exist in nature, by far the most important are the 22 α-amino acids incorporated into proteins. Only these 22 appear in the genetic code of life.

Amino acids can be classified according to the locations of the core structural functional groups (alpha- (α-), beta- (β-) , gamma- (γ-) amino acids, etc.), other categories relate to polarity, ionization, and side chain group type (aliphatic, acyclic, aromatic, polar, etc.). In the form of proteins, amino acid residues form the second-largest component (water being the largest) of human muscles and other tissues. Beyond their role as residues in proteins, amino acids participate in a number of processes such as neurotransmitter transport and biosynthesis. It is thought that they played a key role in enabling life on Earth and its emergence.

Amino acids are formally named by the IUPAC-IUBMB Joint Commission on Biochemical Nomenclature in terms of the fictitious "neutral" structure shown in the illustration. For example, the systematic name of alanine is 2-aminopropanoic acid, based on the formula CH3\sCH(NH2)\sCOOH. The Commission justified this approach as follows:

"The systematic names and formulas given refer to hypothetical forms in which amino groups are unprotonated and carboxyl groups are undissociated. This convention is useful to avoid various nomenclatural problems but should not be taken to imply that these structures represent an appreciable fraction of the amino-acid molecules."

History
The first few amino acids were discovered in the early 1800s. In 1806, French chemists Louis-Nicolas Vauquelin and Pierre Jean Robiquet isolated a compound from asparagus that was subsequently named asparagine, the first amino acid to be discovered. Cystine was discovered in 1810, although its monomer, cysteine, remained undiscovered until 1884. Glycine and leucine were discovered in 1820. The last of the 20 common amino acids to be discovered was threonine in 1935 by William Cumming Rose, who also determined the essential amino acids and established the minimum daily requirements of all amino acids for optimal growth.

The unity of the chemical category was recognized by Wurtz in 1865, but he gave no particular name to it. The first use of the term "amino acid" in the English language dates from 1898, while the German term, Aminosäure, was used earlier. Proteins were found to yield amino acids after enzymatic digestion or acid hydrolysis. In 1902, Emil Fischer and Franz Hofmeister independently proposed that proteins are formed from many amino acids, whereby bonds are formed between the amino group of one amino acid with the carboxyl group of another, resulting in a linear structure that Fischer termed "peptide".

General structure
2-, alpha-, or α-amino acids have the generic formula H2NCHRCOOH in most cases, where R is an organic substituent known as a "side chain".

Of the many hundreds of described amino acids, 22 are proteinogenic ("protein-building"). It is these 22 compounds that combine to give a vast array of peptides and proteins assembled by ribosomes. Non-proteinogenic or modified amino acids may arise from post-translational modification or during nonribosomal peptide synthesis.

Chirality
The carbon atom next to the carboxyl group is called the α–carbon. In proteinogenic amino acids, it bears the amine and the R group or side chain specific to each amino acid. With four distinct substituents, the α–carbon is stereogenic in all α-amino acids except glycine. All chiral proteogenic amino acids have the L configuration. They are "left-handed" enantiomers, which refers to the stereoisomers of the alpha carbon.

A few D -amino acids ("right-handed") have been found in nature, e.g., in bacterial envelopes, as a neuromodulator ( D -serine), and in some antibiotics. Rarely, D -amino acid residues are found in proteins, and are converted from the L -amino acid as a post-translational modification.

Polar charged side chains
Five amino acids possess a charge at neutral pH. Often these side chains appear at the surfaces on proteins to enable their solubility in water, and side chains with opposite charges form important electrostatic contacts called salt bridges that maintain structures within a single protein or between interfacing proteins. Many proteins bind metal into their structures specifically, and these interactions are commonly mediated by charged side chains such as aspartate, glutamate and histidine. Under certain conditions, each ion-forming group can be charged, forming double salts.

The two negatively charged amino acids at neutral pH are aspartate (Asp, D) and glutamate (Glu, E). The anionic carboxylate groups behave as Brønsted bases in most circumstances. Enzymes in very low pH environments, like the aspartic protease pepsin in mammalian stomachs, may have catalytic aspartate or glutamate residues that act as Brønsted acids.

There are three amino acids with side chains that are cations at neutral pH: arginine (Arg, R), lysine (Lys, K) and histidine (His, H). Arginine has a charged guanidino group and lysine a charged alkyl amino group, and are fully protonated at pH 7. Histidine's imidazole group has a pKa of 6.0, and is only around 10 % protonated at neutral pH. Because histidine is easily found in its basic and conjugate acid forms it often participates in catalytic proton transfers in enzyme reactions.

Polar uncharged side chains
The polar, uncharged amino acids serine (Ser, S), threonine (Thr, T), asparagine (Asn, N) and glutamine (Gln, Q) readily form hydrogen bonds with water and other amino acids. They do not ionize in normal conditions, a prominent exception being the catalytic serine in serine proteases. This is an example of severe perturbation, and is not characteristic of serine residues in general. Threonine has two chiral centers, not only the L (2S) chiral center at the α-carbon shared by all amino acids apart from achiral glycine, but also (3R) at the β-carbon. The full stereochemical specification is (2S,3R)- L -threonine.

Hydrophobic side chains
Nonpolar amino acid interactions are the primary driving force behind the processes that fold proteins into their functional three dimensional structures. None of these amino acids' side chains ionize easily, and therefore do not have pKas, with the exception of tyrosine (Tyr, Y). The hydroxyl of tyrosine can deprotonate at high pH forming the negatively charged phenolate. Because of this one could place tyrosine into the polar, uncharged amino acid category, but its very low solubility in water matches the characteristics of hydrophobic amino acids well.

Special case side chains
Several side chains are not described well by the charged, polar and hydrophobic categories. Glycine (Gly, G) could be considered a polar amino acid since its small size means that its solubility is largely determined by the amino and carboxylate groups. However, the lack of any side chain provides glycine with a unique flexibility among amino acids with large ramifications to protein folding. Cysteine (Cys, C) can also form hydrogen bonds readily, which would place it in the polar amino acid category, though it can often be found in protein structures forming covalent bonds, called disulphide bonds, with other cysteines. These bonds influence the folding and stability of proteins, and are essential in the formation of antibodies. Proline (Pro, P) has an alkyl side chain and could be considered hydrophobic, but because the side chain joins back onto the alpha amino group it becomes particularly inflexible when incorporated into proteins. Similar to glycine this influences protein structure in a way unique among amino acids. Selenocysteine (Sec, U) is a rare amino acid not directly encoded by DNA, but is incorporated into proteins via the ribosome. Selenocysteine has a lower redox potential compared to the similar cysteine, and participates in several unique enzymatic reactions. Pyrrolysine (Pyl, O) is another amino acid not encoded in DNA, but synthesized into protein by ribosomes. It is found in archaeal species where it participates in the catalytic activity of several methyltransferases.

β- and γ-amino acids
Amino acids with the structure NH3+\sCXY\sCXY\sCO2-, such as β-alanine, a component of carnosine and a few other peptides, are β-amino acids. Ones with the structure NH3+\sCXY\sCXY\sCXY\sCO2- are γ-amino acids, and so on, where X and Y are two substituents (one of which is normally H).

Zwitterions
The common natural forms of amino acids have a zwitterionic structure, with \sNH3+ (\sNH2+\s in the case of proline) and \sCO2- functional groups attached to the same C atom, and are thus α-amino acids, and are the only ones found in proteins during translation in the ribosome. In aqueous solution at pH close to neutrality, amino acids exist as zwitterions, i.e. as dipolar ions with both NH3+ and CO2- in charged states, so the overall structure is NH3+\sCHR\sCO2-. At physiological pH the so-called "neutral forms" \sNH2\sCHR\sCO2H are not present to any measurable degree. Although the two charges in the zwitterion structure add up to zero it is misleading to call a species with a net charge of zero "uncharged".

In strongly acidic conditions (pH below 3), the carboxylate group becomes protonated and the structure becomes an ammonio carboxylic acid, NH3+\sCHR\sCO2H. This is relevant for enzymes like pepsin that are active in acidic environments such as the mammalian stomach and lysosomes, but does not significantly apply to intracellular enzymes. In highly basic conditions (pH greater than 10, not normally seen in physiological conditions), the ammonio group is deprotonated to give NH2\sCHR\sCO2-.

Although various definitions of acids and bases are used in chemistry, the only one that is useful for chemistry in aqueous solution is that of Brønsted: an acid is a species that can donate a proton to another species, and a base is one that can accept a proton. This criterion is used to label the groups in the above illustration. The carboxylate side chains of aspartate and glutamate residues are the principal Brønsted bases in proteins. Likewise, lysine, tyrosine and cysteine will typically act as a Brønsted acid. Histidine under these conditions can act both as a Brønsted acid and a base.

Isoelectric point

For amino acids with uncharged side-chains the zwitterion predominates at pH values between the two pKa values, but coexists in equilibrium with small amounts of net negative and net positive ions. At the midpoint between the two pKa values, the trace amount of net negative and trace of net positive ions balance, so that average net charge of all forms present is zero. This pH is known as the isoelectric point pI, so pI = $1⁄2$(pKa1 + pKa2).

For amino acids with charged side chains, the pKa of the side chain is involved. Thus for aspartate or glutamate with negative side chains, the terminal amino group is essentially entirely in the charged form \sNH3+, but this positive charge needs to be balanced by the state with just one C-terminal carboxylate group is negatively charged. This occurs halfway between the two carboxylate pKa values: pI = $1⁄2$(pKa1 + pKa(R)), where pKa(R) is the side chain pKa.

Similar considerations apply to other amino acids with ionizable side-chains, including not only glutamate (similar to aspartate), but also cysteine, histidine, lysine, tyrosine and arginine with positive side chains.

Amino acids have zero mobility in electrophoresis at their isoelectric point, although this behaviour is more usually exploited for peptides and proteins than single amino acids. Zwitterions have minimum solubility at their isoelectric point, and some amino acids (in particular, with nonpolar side chains) can be isolated by precipitation from water by adjusting the pH to the required isoelectric point.

Physicochemical properties
The 20 canonical amino acids can be classified according to their properties. Important factors are charge, hydrophilicity or hydrophobicity, size, and functional groups. These properties influence protein structure and protein–protein interactions. The water-soluble proteins tend to have their hydrophobic residues (Leu, Ile, Val, Phe, and Trp) buried in the middle of the protein, whereas hydrophilic side chains are exposed to the aqueous solvent. (In biochemistry, a residue refers to a specific monomer within the polymeric chain of a polysaccharide, protein or nucleic acid.) The integral membrane proteins tend to have outer rings of exposed hydrophobic amino acids that anchor them in the lipid bilayer. Some peripheral membrane proteins have a patch of hydrophobic amino acids on their surface that sticks to the membrane. In a similar fashion, proteins that have to bind to positively charged molecules have surfaces rich in negatively charged amino acids such as glutamate and aspartate, while proteins binding to negatively charged molecules have surfaces rich in positively charged amino acids like lysine and arginine. For example, lysine and arginine are present in large amounts in the low-complexity regions of nucleic-acid binding proteins. There are various hydrophobicity scales of amino acid residues.

Some amino acids have special properties. Cysteine can form covalent disulfide bonds to other cysteine residues. Proline forms a cycle to the polypeptide backbone, and glycine is more flexible than other amino acids.

Glycine and proline are strongly present within low complexity regions of both eukaryotic and prokaryotic proteins, whereas the opposite is the case with cysteine, phenylalanine, tryptophan, methionine, valine, leucine, isoleucine, which are highly reactive, or complex, or hydrophobic.

Many proteins undergo a range of posttranslational modifications, whereby additional chemical groups are attached to the amino acid residue side chains sometimes producing lipoproteins (that are hydrophobic), or glycoproteins (that are hydrophilic) allowing the protein to attach temporarily to a membrane. For example, a signaling protein can attach and then detach from a cell membrane, because it contains cysteine residues that can have the fatty acid palmitic acid added to them and subsequently removed.

Table of standard amino acid abbreviations and properties
Although one-letter symbols are included in the table, IUPAC–IUBMB recommend that "Use of the one-letter symbols should be restricted to the comparison of long sequences".

The one-letter notation was chosen by IUPAC-IUB based on the following rules:

 * Initial letters are used where there is no ambuiguity: C cysteine, H histidine, I isoleucine, M methionine, S serine, V valine,
 * Where arbitrary assignment is needed, the structurally simpler amino acids are given precedence: A Alanine, G glycine, L leucine, P proline, T threonine,
 * F PHenylalanine and R aRginine are assigned by being phonetically suggestive,
 * W tryptophane is assigned based on the double ring being visually suggestive to the bulky letter W,
 * K lysine and Y tyrosine are assigned as alphabetically nearest to their initials L and T (note that U was avoided for its similarity with V, while X was reserved for undetermined or atypical amino acids); for tyrosine the mnemonic tYrosine was also proposed,
 * D aspartate was assigned arbitrarily, with the proposed mnemonic asparDic acid; E glutamate was assigned in alphabetical sequence being larger by merely one methylene –CH2– group,
 * N asparagine was assigned arbitrarily, with the proposed mnemonic asparagiNe; Q glutamine was assigned in alphabetical sequence of those still available (note again that O was avoided due to similarity with D), with the proposed mnemonic Qlutamine.

Two additional amino acids are in some species coded for by codons that are usually interpreted as stop codons:

In addition to the specific amino acid codes, placeholders are used in cases where chemical or crystallographic analysis of a peptide or protein cannot conclusively determine the identity of a residue. They are also used to summarize conserved protein sequence motifs. The use of single letters to indicate sets of similar residues is similar to the use of abbreviation codes for degenerate bases.

Unk is sometimes used instead of Xaa, but is less standard.

Ter or * (from termination) is used in notation for mutations in proteins when a stop codon occurs. It corresponds to no amino acid at all.

In addition, many nonstandard amino acids have a specific code. For example, several peptide drugs, such as Bortezomib and MG132, are artificially synthesized and retain their protecting groups, which have specific codes. Bortezomib is Pyz–Phe–boroLeu, and MG132 is Z–Leu–Leu–Leu–al. To aid in the analysis of protein structure, photo-reactive amino acid analogs are available. These include photoleucine (pLeu) and photomethionine (pMet).

Proteinogenic amino acids
Amino acids are the precursors to proteins. They join by condensation reactions to form short polymer chains called peptides or longer chains called either polypeptides or proteins. These chains are linear and unbranched, with each amino acid residue within the chain attached to two neighboring amino acids. In nature, the process of making proteins encoded by RNA genetic material is called translation and involves the step-by-step addition of amino acids to a growing protein chain by a ribozyme that is called a ribosome. The order in which the amino acids are added is read through the genetic code from an mRNA template, which is an RNA derived from one of the organism's genes.

Twenty-two amino acids are naturally incorporated into polypeptides and are called proteinogenic or natural amino acids. Of these, 20 are encoded by the universal genetic code. The remaining 2, selenocysteine and pyrrolysine, are incorporated into proteins by unique synthetic mechanisms. Selenocysteine is incorporated when the mRNA being translated includes a SECIS element, which causes the UGA codon to encode selenocysteine instead of a stop codon. Pyrrolysine is used by some methanogenic archaea in enzymes that they use to produce methane. It is coded for with the codon UAG, which is normally a stop codon in other organisms.

Several independent evolutionary studies have suggested that Gly, Ala, Asp, Val, Ser, Pro, Glu, Leu, Thr may belong to a group of amino acids that constituted the early genetic code, whereas Cys, Met, Tyr, Trp, His, Phe may belong to a group of amino acids that constituted later additions of the genetic code.

Standard vs nonstandard amino acids
The 20 amino acids that are encoded directly by the codons of the universal genetic code are called standard or canonical amino acids. A modified form of methionine (N-formylmethionine) is often incorporated in place of methionine as the initial amino acid of proteins in bacteria, mitochondria and plastids (including chloroplasts). Other amino acids are called nonstandard or non-canonical. Most of the nonstandard amino acids are also non-proteinogenic (i.e. they cannot be incorporated into proteins during translation), but two of them are proteinogenic, as they can be incorporated translationally into proteins by exploiting information not encoded in the universal genetic code.

The two nonstandard proteinogenic amino acids are selenocysteine (present in many non-eukaryotes as well as most eukaryotes, but not coded directly by DNA) and pyrrolysine (found only in some archaea and at least one bacterium). The incorporation of these nonstandard amino acids is rare. For example, 25 human proteins include selenocysteine in their primary structure, and the structurally characterized enzymes (selenoenzymes) employ selenocysteine as the catalytic moiety in their active sites. Pyrrolysine and selenocysteine are encoded via variant codons. For example, selenocysteine is encoded by stop codon and SECIS element.

N-formylmethionine (which is often the initial amino acid of proteins in bacteria, mitochondria, and chloroplasts) is generally considered as a form of methionine rather than as a separate proteinogenic amino acid. Codon–tRNA combinations not found in nature can also be used to "expand" the genetic code and form novel proteins known as alloproteins incorporating non-proteinogenic amino acids.

Non-proteinogenic amino acids
Aside from the 22 proteinogenic amino acids, many non-proteinogenic amino acids are known. Those either are not found in proteins (for example carnitine, GABA, levothyroxine) or are not produced directly and in isolation by standard cellular machinery. For example, hydroxyproline, is synthesised from proline. Another example is selenomethionine).

Non-proteinogenic amino acids that are found in proteins are formed by post-translational modification. Such modifications can also determine the localization of the protein, e.g., the addition of long hydrophobic groups can cause a protein to bind to a phospholipid membrane. Examples:
 * the carboxylation of glutamate allows for better binding of calcium cations,
 * Hydroxyproline, generated by hydroxylation of proline, is a major component of the connective tissue collagen.
 * Hypusine in the translation initiation factor EIF5A, contains a modification of lysine.

Some non-proteinogenic amino acids are not found in proteins. Examples include 2-aminoisobutyric acid and the neurotransmitter gamma-aminobutyric acid. Non-proteinogenic amino acids often occur as intermediates in the metabolic pathways for standard amino acids – for example, ornithine and citrulline occur in the urea cycle, part of amino acid catabolism (see below). A rare exception to the dominance of α-amino acids in biology is the β-amino acid beta alanine (3-aminopropanoic acid), which is used in plants and microorganisms in the synthesis of pantothenic acid (vitamin B5), a component of coenzyme A.

In mammalian nutrition

Amino acids are not typical component of food: animals eat proteins. The protein is broken down into amino acids in the process of digestion. They are then used to synthesize new proteins, other biomolecules, or are oxidized to urea and carbon dioxide as a source of energy. The oxidation pathway starts with the removal of the amino group by a transaminase; the amino group is then fed into the urea cycle. The other product of transamidation is a keto acid that enters the citric acid cycle. Glucogenic amino acids can also be converted into glucose, through gluconeogenesis.

Of the 20 standard amino acids, nine (His, Ile, Leu, Lys, Met, Phe, Thr, Trp and Val) are called essential amino acids because the human body cannot synthesize them from other compounds at the level needed for normal growth, so they must be obtained from food.

Semi-essential and conditionally essential amino acids, and juvenile requirements
In addition, cysteine, tyrosine, and arginine are considered semiessential amino acids, and taurine a semi-essential aminosulfonic acid in children. Some amino acids are conditionally essential for certain ages or medical conditions. Essential amino acids may also vary from species to species. The metabolic pathways that synthesize these monomers are not fully developed.

Non-protein functions
Many proteinogenic and non-proteinogenic amino acids have biological functions beyond being precursors to proteins and peptides.In humans, amino acids also have important roles in diverse biosynthetic pathways. Defenses against herbivores in plants sometimes employ amino acids. Examples:

Standard amino acids

 * Tryptophan is a precursor of the neurotransmitter serotonin.
 * Tyrosine (and its precursor phenylalanine) are precursors of the catecholamine neurotransmitters dopamine, epinephrine and norepinephrine and various trace amines.
 * Phenylalanine is a precursor of phenethylamine and tyrosine in humans. In plants, it is a precursor of various phenylpropanoids, which are important in plant metabolism.
 * Glycine is a precursor of porphyrins such as heme.
 * Arginine is a precursor of nitric oxide.
 * Ornithine and S-adenosylmethionine are precursors of polyamines.
 * Aspartate, glycine, and glutamine are precursors of nucleotides. However, not all of the functions of other abundant nonstandard amino acids are known.

Roles for nonstandard amino acids

 * Carnitine is used in lipid transport.
 * gamma-aminobutyric acid is a neurotransmitter.
 * 5-HTP (5-hydroxytryptophan) is used for experimental treatment of depression.
 * L -DOPA ( L -dihydroxyphenylalanine) for Parkinson's treatment,
 * Eflornithine inhibits ornithine decarboxylase and used in the treatment of sleeping sickness.
 * Canavanine, an analogue of arginine found in many legumes is an antifeedant, protecting the plant from predators.
 * Mimosine found in some legumes, is another possible antifeedant. This compound is an analogue of tyrosine and can poison animals that graze on these plants.

Animal feed
Amino acids are sometimes added to animal feed because some of the components of these feeds, such as soybeans, have low levels of some of the essential amino acids, especially of lysine, methionine, threonine, and tryptophan. Likewise amino acids are used to chelate metal cations in order to improve the absorption of minerals from feed supplements.

Food
The food industry is a major consumer of amino acids, especially glutamic acid, which is used as a flavor enhancer, and aspartame (aspartylphenylalanine 1-methyl ester), which is used as an artificial sweetener. Amino acids are sometimes added to food by manufacturers to alleviate symptoms of mineral deficiencies, such as anemia, by improving mineral absorption and reducing negative side effects from inorganic mineral supplementation.

Chemical building blocks
Amino acids are low-cost feedstocks used in chiral pool synthesis as enantiomerically pure building blocks.

Amino acids are used in the synthesis of some cosmetics.

Fertilizer
The chelating ability of amino acids is sometimes used in fertilizers to facilitate the delivery of minerals to plants in order to correct mineral deficiencies, such as iron chlorosis. These fertilizers are also used to prevent deficiencies from occurring and to improve the overall health of the plants.

Biodegradable plastics
Amino acids have been considered as components of biodegradable polymers, which have applications as environmentally friendly packaging and in medicine in drug delivery and the construction of prosthetic implants. An interesting example of such materials is polyaspartate, a water-soluble biodegradable polymer that may have applications in disposable diapers and agriculture. Due to its solubility and ability to chelate metal ions, polyaspartate is also being used as a biodegradable antiscaling agent and a corrosion inhibitor.

Chemical synthesis
The commercial production of amino acids usually relies on mutant bacteria that overproduce individual amino acids using glucose as a carbon source. Some amino acids are produced by enzymatic conversions of synthetic intermediates. 2-Aminothiazoline-4-carboxylic acid is an intermediate in one industrial synthesis of L -cysteine for example. Aspartic acid is produced by the addition of ammonia to fumarate using a lyase.

Biosynthesis
In plants, nitrogen is first assimilated into organic compounds in the form of glutamate, formed from alpha-ketoglutarate and ammonia in the mitochondrion. For other amino acids, plants use transaminases to move the amino group from glutamate to another alpha-keto acid. For example, aspartate aminotransferase converts glutamate and oxaloacetate to alpha-ketoglutarate and aspartate. Other organisms use transaminases for amino acid synthesis, too.

Nonstandard amino acids are usually formed through modifications to standard amino acids. For example, homocysteine is formed through the transsulfuration pathway or by the demethylation of methionine via the intermediate metabolite S-adenosylmethionine, while hydroxyproline is made by a post translational modification of proline.

Microorganisms and plants synthesize many uncommon amino acids. For example, some microbes make 2-aminoisobutyric acid and lanthionine, which is a sulfide-bridged derivative of alanine. Both of these amino acids are found in peptidic lantibiotics such as alamethicin. However, in plants, 1-aminocyclopropane-1-carboxylic acid is a small disubstituted cyclic amino acid that is an intermediate in the production of the plant hormone ethylene.

Primordial synthesis
The formation of amino acids and peptides are assumed to precede and perhaps induce the emergence of life on earth. Amino acids can form from simple precursors under various conditions. Surface-based chemical metabolism of amino acids and very small compounds may have led to the build-up of amino acids, coenzymes and phosphate-based small carbon molecules. Amino acids and similar building blocks could have been elaborated into proto-peptides, with peptides being considered key players in the origin of life.

In the famous Urey-Miller experiment, the passage of an electric arc through a mixture of methane, hydrogen, and ammonia produces a large number of amino acids. Since then, scientists have discovered a range of ways and components by which the potentially prebiotic formation and chemical evolution of peptides may have occurred, such as condensing agents, the design of self-replicating peptides and a number of non-enzymatic mechanisms by which amino acids could have emerged and elaborated into peptides. Several hypotheses invoke the Strecker synthesis whereby hydrogen cyanide, simple aldehydes, ammonia, and water produce amino acids.

According to a review, amino acids, and even peptides, "turn up fairly regularly in the various experimental broths that have been allowed to be cooked from simple chemicals. This is because nucleotides are far more difficult to synthesize chemically than amino acids." For a chronological order, it suggests that there must have been a 'protein world' or at least a 'polypeptide world', possibly later followed by the 'RNA world' and the 'DNA world'. Codon–amino acids mappings may be the biological information system at the primordial origin of life on Earth. While amino acids and consequently simple peptides must have formed under different experimentally probed geochemical scenarios, the transition from an abiotic world to the first life forms is to a large extent still unresolved.

Reactions
Amino acids undergo the reactions expected of the constituent functional groups.

Peptide bond formation

As both the amine and carboxylic acid groups of amino acids can react to form amide bonds, one amino acid molecule can react with another and become joined through an amide linkage. This polymerization of amino acids is what creates proteins. This condensation reaction yields the newly formed peptide bond and a molecule of water. In cells, this reaction does not occur directly; instead, the amino acid is first activated by attachment to a transfer RNA molecule through an ester bond. This aminoacyl-tRNA is produced in an ATP-dependent reaction carried out by an aminoacyl tRNA synthetase. This aminoacyl-tRNA is then a substrate for the ribosome, which catalyzes the attack of the amino group of the elongating protein chain on the ester bond. As a result of this mechanism, all proteins made by ribosomes are synthesized starting at their N-terminus and moving toward their C-terminus.

However, not all peptide bonds are formed in this way. In a few cases, peptides are synthesized by specific enzymes. For example, the tripeptide glutathione is an essential part of the defenses of cells against oxidative stress. This peptide is synthesized in two steps from free amino acids. In the first step, gamma-glutamylcysteine synthetase condenses cysteine and glutamate through a peptide bond formed between the side chain carboxyl of the glutamate (the gamma carbon of this side chain) and the amino group of the cysteine. This dipeptide is then condensed with glycine by glutathione synthetase to form glutathione.

In chemistry, peptides are synthesized by a variety of reactions. One of the most-used in solid-phase peptide synthesis uses the aromatic oxime derivatives of amino acids as activated units. These are added in sequence onto the growing peptide chain, which is attached to a solid resin support. Libraries of peptides are used in drug discovery through high-throughput screening.

The combination of functional groups allow amino acids to be effective polydentate ligands for metal–amino acid chelates. The multiple side chains of amino acids can also undergo chemical reactions.

Catabolism

Degradation of an amino acid often involves deamination by moving its amino group to α-ketoglutarate, forming glutamate. This process involves transaminases, often the same as those used in amination during synthesis. In many vertebrates, the amino group is then removed through the urea cycle and is excreted in the form of urea. However, amino acid degradation can produce uric acid or ammonia instead. For example, serine dehydratase converts serine to pyruvate and ammonia. After removal of one or more amino groups, the remainder of the molecule can sometimes be used to synthesize new amino acids, or it can be used for energy by entering glycolysis or the citric acid cycle, as detailed in image at right.

Complexation
Amino acids are bidentate ligands, forming transition metal amino acid complexes.

Chemical analysis
The total nitrogen content of organic matter is mainly formed by the amino groups in proteins. The Total Kjeldahl Nitrogen (TKN) is a measure of nitrogen widely used in the analysis of (waste) water, soil, food, feed and organic matter in general. As the name suggests, the Kjeldahl method is applied. More sensitive methods are available.

title: Amino acid