Atomic orbital

# Atomic orbital

Overview
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

or a pair of electrons in an atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

. The term may also refer to the physical region where the electron can be calculated to be, as defined by the particular mathematical form of the orbital.

Atomic orbitals are mathematical functions that depend on the coordinates of only one electron.
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Encyclopedia
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

or a pair of electrons in an atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

. The term may also refer to the physical region where the electron can be calculated to be, as defined by the particular mathematical form of the orbital.

Atomic orbitals are mathematical functions that depend on the coordinates of only one electron. They describe the wave-like nature of this electron in any energy state. They can be the hydrogen-like "orbitals" which are exact solutions to the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

for a hydrogen-like "atom"
Hydrogen-like atom
A hydrogen-like ion is any atomic nucleus with one electron and thus is isoelectronic with hydrogen. Except for the hydrogen atom itself , these ions carry the positive charge e, where Z is the atomic number of the atom. Examples of hydrogen-like ions are He+, Li2+, Be3+ and B4+...

(i.e., the hydrogen atom or any ion formed by one electron and a nucleus). Alternatively, atomic orbitals refer to functions that depend on the coordinates of one electron (i.e. orbitals) but are used as starting points for approximating wave functions that depend on the simultaneous coordinates of all the electrons in an atom or molecule. The coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

s for atomic orbitals are usually spherical coordinates (r,θ,φ) in atoms and cartesians
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

(x,y,z) in poly-atomic molecules.

Within a visual context, atomic orbitals are the basic building blocks of the introductory pedagogical electron cloud model (derived from the wave mechanics model or atomic orbital model, but using particle concepts in order to visualize the mathematical procedures used to approximate wave functions for atoms with many electrons). Thus, (with particle concepts in italics) this model provides a framework for describing the placement of electrons in an atom. In this model, the atom consists of a nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

surrounded by orbiting electrons. These electrons exist in so called atomic orbitals, which are a set of quantum
Quantum
In physics, a quantum is the minimum amount of any physical entity involved in an interaction. Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization". This means that the magnitude can take on only certain discrete...

states of the negatively charged electrons trapped in the electrical field generated by the positively-charged nucleus (which may be weakened by the effect of other electrons, but still remains attractive in sum). The electron cloud model can ultimately be described by quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, in which the electrons are more accurately described as standing waves surrounding the nucleus.

Atomic orbitals are typically categorized by n, l, and m quantum numbers, which correspond to the electron's energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

, angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

, and an angular momentum vector component, respectively. Each orbital is defined by a different set of quantum numbers and contains a maximum of two electrons. The simple names s orbital, p orbital, d orbital and f orbital refer to orbitals with angular momentum quantum number l = 0, 1, 2 and 3 respectively. These names indicate the orbital shape and are used to describe the electron configuration
Electron configuration
In atomic physics and quantum chemistry, electron configuration is the arrangement of electrons of an atom, a molecule, or other physical structure...

s. They are derived from the characteristics of their spectroscopic lines: sharp, principal, diffuse, and fundamental, the rest being named in alphabetical order (omitting j).

The wave function for the electron cloud of a multi-electron atom may be seen as being built up (in approximation) in an electron configuration
Electron configuration
In atomic physics and quantum chemistry, electron configuration is the arrangement of electrons of an atom, a molecule, or other physical structure...

that is a product of simpler hydrogen-like atomic orbitals. ΨHe(x1,y1,z1,x2,y2,z2) ≈ 1s(x1,y1,z1) • 1s(x2,y2,z2) = 1s2. (This is read as, "The exact wave function depending on the simultaneous coordinates of the two electrons in the helium atom is approximated as a product of two one-s atomic orbitals each of which depends on the coordinates of only one electron.") In such a configuration, pairs of electrons are arranged in simple repeating patterns of increasing odd numbers (1,3,5,7..), each of which represents a set of electron pairs with a given energy and angular momentum. The repeating periodicity of the blocks of 2, 6, 10, and 14 elements within sections of the periodic table
Periodic table
The periodic table of the chemical elements is a tabular display of the 118 known chemical elements organized by selected properties of their atomic structures. Elements are presented by increasing atomic number, the number of protons in an atom's atomic nucleus...

arises naturally from the total number of electrons which occupy a complete set of s, p, d and f atomic orbitals, respectively.

There are typically three mathematical forms for atomic orbitals which can be chosen as a starting point for the calculation of the properties of atoms and molecules with many electrons. They all have forms that generate s, p, d, etc. functions. Their differences are: 1) the hydrogen-like atomic orbitals are derived from the exact solution of the Schrödinger Equation for one electron and a nucleus. The part of the function that depends on the distance from the nucleus has nodes (radial nodes) and decays as e(-distance) from the nucleus. 2) The Slater-type orbital
STO
STO may refer to:*Seoul Tourism Organization The Seoul Tourism Organization*Service du travail obligatoire, a forced labour programme introduced by the Vichy French government during World War II...

(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
Gaussian orbital
In computational chemistry and molecular physics, Gaussian orbitals are functions used as atomic orbitals in the LCAO method for the computation of electron orbitals in molecules and numerous properties that depend on these.- Rationale :The principal reason for the use of Gaussian basis functions...

(Gaussians) has no radial nodes and decays as e(-distance squared). 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.

## Introduction

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

### Wave-like properties

The electrons do not orbit the nucleus in the sense of a planet orbiting the sun, but instead exist as standing waves. The lowest possible energy an electron can take is therefore analogous to the fundamental frequency of a wave on a string. Higher energy states are then similar to harmonics of the fundamental frequency
Fundamental frequency
The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...

.

The electrons are never in a single point location, although the probability of interacting with the electron at a single point can be found from the wave function of the electron.

### Particle-like properties

There is always an integer number of electrons orbiting the nucleus.

Electrons jump between orbitals in a particle-like fashion. For example, if a single photon strikes the electrons, only a single electron changes states in response to the photon.

The electrons retain particle like-properties such as: each wave state has the same electrical charge as the electron particle. Each wave state has a single discrete spin (spin up or spin down).

### Visualizing atomic orbitals intuitively

Despite the obvious analogy to planets revolving around the Sun, electrons cannot be described as solid particles. In addition, atomic orbitals do not closely resemble a planet's elliptical path in ordinary atoms. A more accurate analogy might be that of a large and often oddly-shaped "atmosphere" (the electron), distributed around a relatively tiny planet (the atomic nucleus). One difference is that some of an atom's electrons, those in s orbitals, have zero angular momentum, so they cannot in any sense be thought of as moving "around" the nucleus, as a planet does. (A planet would need to fall vertically into the Sun and oscillate up and down through it, to be in an orbit with no angular momentum). Other electrons do have varying amounts of angular momentum.

Atomic orbitals exactly describe the shape of this "atmosphere" only when a single electron is present in an atom. When more electrons are added to a single atom, the additional electrons tend to more evenly fill in a volume of space around the nucleus so that the resulting collection (sometimes termed the atom’s “electron cloud” ) tends toward a generally spherical zone of probability describing where the atom’s electrons will be found.

## History

The term "orbital" was coined by Robert Mulliken in 1932. However, the idea that electrons might revolve around a compact nucleus with definite angular momentum was convincingly argued at least 19 years earlier by Niels Bohr
Niels Bohr
Niels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...

, and the Japanese physicist Hantaro Nagaoka published an orbit-based hypothesis for electronic behavior as early as 1904.
Explaining the behavior of these electron "orbits" was one of the driving forces behind the development of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

.

### 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 revolved in orbit-like rings within a positively-charged jelly-like substance, and between the electron's discovery and 1909, this "plum pudding model
Plum pudding model
The plum pudding model of the atom by J. J. Thomson, who discovered the electron in 1897, was proposed in 1904 before the discovery of the atomic nucleus. In this model, the atom is composed of electrons The plum pudding model of the atom by J. J. Thomson, who discovered the electron in 1897, was...

" was the most widely-accepted explanation of atomic structure.

Shortly after Thomson's discovery, Hantaro Nagaoka, a Japanese physicist, 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.

### The Bohr atom

In 1909 Ernest Rutherford
Ernest Rutherford
Ernest Rutherford, 1st Baron Rutherford of Nelson OM, FRS was a New Zealand-born British chemist and physicist who became known as the father of nuclear physics...

discovered that the positive half of atoms was tightly condensed into a nucleus,
and it became clear from his analysis in 1911 that the plum pudding model could not explain atomic structure. Shortly after, in 1913, Rutherford's postdoctoral student Niels Bohr
Niels Bohr
Niels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...

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. This constraint automatically permitted only certain values of electron energies. The Bohr model
Bohr model
In atomic physics, the Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing attraction,...

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
Photoelectric effect
In the photoelectric effect, electrons are emitted from matter as a consequence of their absorption of energy from electromagnetic radiation of very short wavelength, such as visible or ultraviolet light. Electrons emitted in this manner may be referred to as photoelectrons...

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
Bohr model
In atomic physics, the Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing attraction,...

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 twelve 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
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

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
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

treatment of hydrogen like atom, a Bohr electron "wavelength" could be seen to be a function of its momentum, and thus a Bohr orbiting electron was seen to orbit in a circle at a multiple of its half-wavelength (this historically incorrect Bohr model is still occasionally taught to students). The Bohr model for a short time could be seen as a classical model with an additional constraint provided by the 'wavelength' argument. However, this period was immediately superseded by the full three-dimensional wave mechanics of 1926. In our current understanding of physics, the Bohr model is called a semi-classical model because of its quantization of angular momentum, not primarily because of its relationship with electron wavelength, which appeared in hindsight a dozen years after the Bohr model was proposed.

The Bohr model was able to explain the emission and absorption spectra of hydrogen
Hydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...

. 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 (2 electrons), neon (10 electrons), and argon (18 electrons) exhibit similar chemical behavior. Modern physics explains this by noting that the n=1 state holds 2 electrons, the n=2 state holds 8 electrons, and the n=3 state holds 8 electrons (in argon). In the end, this was solved by the discovery of modern quantum mechanics and the Pauli Exclusion Principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

.

### Modern conceptions and connections to the Heisenberg Uncertainty Principle

Immediately after Heisenberg discovered his uncertainty relation
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

,
it was noted by Bohr
Niels Bohr
Niels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...

that the existence of any sort of wave packet
Wave packet
In physics, a wave packet is a short "burst" or "envelope" of wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere...

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 an infinite particle momentum.

In chemistry, Schrödinger, Pauling
Pauling
Pauling is a surname. People with this surname include:*Linus Pauling**Paulingite**Pauling's rules**4674 Pauling**Linus Pauling Institute**Linus Pauling Library**Linus Pauling Award**Pauling Field*Ava Helen Pauling, wife of Linus*Tom Pauling...

, Mulliken
Robert S. Mulliken
Robert Sanderson Mulliken was an American physicist and chemist, primarily responsible for the early development of molecular orbital theory, i.e. the elaboration of the molecular orbital method of computing the structure of molecules. Dr. Mulliken received the Nobel Prize for chemistry in 1966...

and others noted that the consequence of Heisenberg's relation was that the electron, as a wave packet, could not be considered to have an exact location in its orbital. Max Born
Max Born
Max Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...

suggested that the electron's position needed to be described by a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

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 mean energy of the probability cloud of the electron's wave packet which surrounded the atom.

## Orbital names

Orbitals are given names in the form:
where X is the energy level corresponding to the principal quantum number
Principal quantum number
In atomic physics, the principal quantum symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The principal quantum number can only have positive integer values...

n, type is a lower-case letter denoting the shape or subshell of the orbital and it corresponds to the angular quantum number l, and y is the number of electrons in that orbital.

For example, the orbital 1s2 has two electrons and is the lowest energy level (n = 1) and has an angular quantum number of l = 0. In X-ray notation
X-ray notation
X-ray notation is a method of labeling atomic orbitals that grew out of X-ray science. It is still traditionally used with most x-ray spectroscopy techniques including AES and XPS. In X-ray notation, every principal quantum number is given a letter associated with it.-Conversion:-Use:*X-ray sources...

, the principal quantum number is given a letter associated with it. For , the letters associated with those numbers are K, L, M, N, O, ..., respectively.

## Formal quantum mechanical definition

In quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, the state of an atom, i.e. the eigenstates of the atomic Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

, is expanded (see configuration interaction
Configuration interaction
Configuration interaction is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematically, configuration simply describes the linear combination...

expansion and basis set
Basis set (chemistry)
A basis set in chemistry is a set of functions used to create the molecular orbitals, which are expanded as a linear combination of such functions with the weights or coefficients to be determined. Usually these functions are atomic orbitals, in that they are centered on atoms. Otherwise, the...

) into linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...

s of anti-symmetrized products (Slater determinant
Slater determinant
In quantum mechanics, a Slater determinant is an expression that describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry requirements and consequently the Pauli exclusion principle by changing sign upon exchange of fermions . It is named for its discoverer, John C...

s) of one-electron functions. The spatial components of these one-electron functions are called atomic orbitals. (When one considers also their spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

component, one speaks of atomic spin orbitals.)

In atomic physics
Atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...

, the atomic spectral line
Atomic spectral line
In physics, atomic spectral lines are of two types:* An emission line is formed when an electron makes a transition from a particular discrete energy level of an atom, to a lower energy state, emitting a photon of a particular energy and wavelength...

s correspond to transitions (quantum leaps) between quantum states of an atom. These states are labeled by a set of quantum number
Quantum number
Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...

s summarized in the term symbol
Term symbol
In quantum mechanics, the Russell-Saunders term symbol is an abbreviated description of the angular momentum quantum numbers in a multi-electron atom. It is related with the energy level of a given electron configuration. LS coupling is assumed...

and usually associated with particular electron configurations, i.e., by occupation schemes of atomic orbitals (e.g., 1s2 2s2 2p6 for the ground state of neon
Neon
Neon is the chemical element that has the symbol Ne and an atomic number of 10. Although a very common element in the universe, it is rare on Earth. A colorless, inert noble gas under standard conditions, neon gives a distinct reddish-orange glow when used in either low-voltage neon glow lamps or...

-- term symbol: 1S0).

This notation means that the corresponding Slater determinants have a clear higher weight in the configuration interaction
Configuration interaction
Configuration interaction is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematically, configuration simply describes the linear combination...

expansion. The atomic orbital concept is therefore a key concept for visualizing the excitation process associated with a given transition. For example, one can say for a given transition that it corresponds to the excitation of an electron from an occupied orbital to a given unoccupied orbital. Nevertheless, one has to keep in mind that electrons are fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s ruled by the Pauli exclusion principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

and cannot be distinguished from the other electrons in the atom. Moreover, it sometimes happens that the configuration interaction expansion converges very slowly and that one cannot speak about simple one-determinant wave function at all. This is the case when electron correlation is large.

Fundamentally, an atomic orbital is a one-electron wave function, even though most electrons do not exist in one-electron atoms, and so the one-electron view is an approximation. When thinking about orbitals, we are often given an orbital vision which (even if it is not spelled out) is heavily influenced by this Hartree–Fock approximation, which is one way to reduce the complexities of molecular orbital theory
Molecular orbital theory
In chemistry, molecular orbital theory is a method for determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule...

.

## Hydrogen-like atoms

The simplest atomic orbitals are those that are calculated for systems with a single electron, such as the hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

. An atom of any other element ion
Ion
An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. The name was given by physicist Michael Faraday for the substances that allow a current to pass between electrodes in a...

ized 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
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

).

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

A given (hydrogen-like) atomic orbital is identified by unique values of three quantum numbers: n
Principal quantum number
In atomic physics, the principal quantum symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The principal quantum number can only have positive integer values...

, l
Azimuthal quantum number
The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital...

, and ml
Magnetic quantum number
In atomic physics, the magnetic quantum number is the third of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter m...

. The rules restricting the values of the quantum numbers, and their energies (see below), explain the electron configuration of the atoms and the periodic table
Periodic table
The periodic table of the chemical elements is a tabular display of the 118 known chemical elements organized by selected properties of their atomic structures. Elements are presented by increasing atomic number, the number of protons in an atom's atomic nucleus...

.

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 combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...

s) of multiple orbitals. See Linear combination of atomic orbitals molecular orbital method
Linear combination of atomic orbitals molecular orbital method
A linear combination of atomic orbitals or LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry. In quantum mechanics, electron configurations of atoms are described as wavefunctions...

.

The quantum number n first appeared in the Bohr model
Bohr model
In atomic physics, the Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing attraction,...

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
Electron shell
An electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" , followed by the "2 shell" , then the "3 shell" , and so on further and further from the nucleus. The shell letters K,L,M,.....

". 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, they cannot be described by a location and momentum. Instead, they are described by a set of quantum numbers that encompasses both the particle-like nature and the wave-like nature of the electrons. An atomic orbital is uniquely identified by the values of the three quantum numbers, and each set of the three quantum numbers corresponds to exactly one orbital, but the quantum numbers only occur in certain combinations of values. The quantum numbers, together with the rules governing their possible values, are as follows:

The principal quantum number
Principal quantum number
In atomic physics, the principal quantum symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The principal quantum number can only have positive integer values...

, 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
Azimuthal quantum number
The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital...

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

The magnetic quantum number
Magnetic quantum number
In atomic physics, the magnetic quantum number is the third of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter m...

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

The above results may be summarized in the following table. Each cell represents a subshell, and lists the values of available in that subshell. Empty cells represent subshells that do not exist.
l=0 l=1 l=2 l=3 l=4 ...
n=1
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 - and -values. is represented by its numerical value, but is represented by a letter as follows: 0 is represented by 's', 1 by 'p', 2 by 'd', 3 by 'f', and 4 by 'g'. For instance, one may speak of the subshell with and as a '2s subshell'.

Each electron also has a spin quantum number
Spin quantum number
In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given particle...

, s, which describes the spin of each electron (spin up or spin down). The number s can be + or -.

The Pauli exclusion principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

states that no two electrons can occupy the same quantum state: every electron in an atom must have a unique combination of quantum numbers.

## The shapes of orbitals

Any discussion of the shapes of electron orbitals is necessarily arbitrary since a number representing a certain probability of finding the electron must be chosen (usually from zero up to one, or from 0 up to 100%). According to the uncertainty principle
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

and the hypotheses of Quantum Mechanics
Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...

a given electron has one and only one probability of being found at any moment at a particular point (x,y,z) in space. For an atom, it is easier to calculate the probability using a distance r from the nucleus and a direction (θ,φ) using polar coordinates. If an atomic orbital is used to describe this electron, then the probability of finding an electron at point (r,θ,φ) is obtained from this orbital, which is the mathematical function ψ(r,θ,φ).

However, the electron is much more likely to be found in particular regions around the atom than in others. Given this, a boundary surface can be drawn so that the electron has a high probability to be found somewhere within the surface, and all regions outside the surface have low values. Even though the precise placement of the surface is arbitrary, any reasonably compact determination must follow a pattern specified by the behavior of |ψ(r,θ,φ)|2, the square of the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

(also called magnitude or modulus) of the complex-valued wave function. This boundary surface is sometimes what is meant when the "shape" of an orbital is referred to. Sometimes the ψ function will be graphed to show its phases, rather than the |ψ(r,θ,φ)|2 which shows probability density but has no phases (which have been lost in the process of taking the absolute value, since ψ(r,θ,φ) is a complex number). |ψ(r,θ,φ)|2 orbital graphs tend to have less spherical, thinner lobes than ψ(r,θ,φ) graphs, but have the same number of lobes in the same places, and otherwise are recognizable. This article, in order to show wave function phases, shows mostly ψ(r,θ,φ) graphs.

The lobes can be viewed as interference patterns between the two counter rotating "m" and "-m" modes, with the projection of the orbital onto the xy plane having a resonant "m" wavelengths around the circumference. For each m there are two of these +<-m> and -<-m>. For the case where m=0 the orbital is vertical, counter rotating information is unknown, and the orbital is z-axis symmetric. For the case where l=0 there are no counter rotating modes. There are only radial modes and the shape is spherically symmetric. For any given n, the smaller l is, the more radial nodes there are. Loosely speaking n is energy, l is analogous to eccentricity
Orbital eccentricity
The orbital eccentricity of an astronomical body is the amount by which its orbit deviates from a perfect circle, where 0 is perfectly circular, and 1.0 is a parabola, and no longer a closed orbit...

, and m is orientation.

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

Also in general terms, determines an orbital's shape, and its orientation. However, since some orbitals are described by equations in complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s, the shape sometimes depends on also.

The single s-orbitals () are shaped like spheres. For n = 1 the sphere is "solid" (it is most dense at the center and fades exponentially outwardly), but for n = 2 or more, each single s-orbital is composed of spherically symmetric surfaces which are nested shells (i.e., the "wave-structure" is radial, following a sinusoidal radial component as well). See illustration of a cross-section of these nested shells, at right. The s-orbitals for all n numbers are the only orbitals with an anti-node (a region of high wave function density) at the center of the nucleus. All other orbitals (p, d, f, etc.) have angular momentum, and thus avoid the nucleus (having a wave node at the nucleus).

The three p-orbitals for n = 2 have the form of two ellipsoids with a point of tangency at the nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

(the two-lobed shape is sometimes referred to as a "dumbbell
Dumbbell
The dumbbell, a type of free weight, is a piece of equipment used in weight training. It can be used individually or in pairs .-History:...

"). The three p-orbitals in each shell
Electron shell
An electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" , followed by the "2 shell" , then the "3 shell" , and so on further and further from the nucleus. The shell letters K,L,M,.....

are oriented at right angles to each other, as determined by their respective linear combination of values of .
Four of the five d-orbitals for n = 3 look similar, each with four pear-shaped lobes, each lobe tangent to two others, and the centers of all four lying in one plane, between a pair of axes. Three of these planes are the xy-, xz-, and yz-planes, and the fourth has the centres on the x and y axes. The fifth and final d-orbital consists of three regions of high probability density: a torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

with two pear-shaped regions placed symmetrically on its z axis.

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

For each s, p, d, f and g set of orbitals, the set of orbitals which composes it forms a spherically symmetrical set of shapes. For non-s orbitals, which have lobes, the lobes point in directions so as to fill space as symmetrically as possible for number of lobes which exist for a set of orientations. For example, the three p orbitals have six lobes which are oriented to each of the six primary directions of 3-D space; for the 5 d orbitals, there are a total of 18 lobes, in which again six point in primary directions, and the 12 additional lobes fill the 12 gaps which exist between each pairs of these 6 primary axes.

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
Spherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...

. These shapes are not unique, and any linear combination is valid, like a transformation to cubic harmonic
Cubic harmonic
In fields like computational chemistry and solid-state and condensed matter physics the so called atomic orbitals, or spin-orbitals, as they appear in textbooks on quantum physics, are often partially replaced by cubic harmonics for a number of reasons....

s, in fact it is possible to generate sets where all the d's are the same shape, just like the px, py, and pz are the same shape.

### Orbitals table

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
Radium is a chemical element with atomic number 88, represented by the symbol Ra. Radium is an almost pure-white alkaline earth metal, but it readily oxidizes on exposure to air, becoming black in color. All isotopes of radium are highly radioactive, with the most stable isotope being radium-226,...

. ψ graphs are shown with - and + wave function phases shown in two different colors (arbitrarily red and blue). The pz orbital is the same as the p0 orbital, but the px and py are formed by taking linear
combinations of the p+1 and p-1 orbitals (which is why they are listed under the m=±1 label). Also, the p+1 and p-1 are not
the same shape as the p0, since they are pure spherical harmonics
Spherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...

.
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

### Understanding why atomic orbitals take these shapes

The shapes of atomic orbitals can be understood qualitatively by considering the analogous case of standing waves on a circular drum
Vibrations of a circular drum
The vibrations of an idealized circular drum, essentially an elastic membrane of uniform thickness attached to a rigid circular frame, are solutions of the wave equation with zero boundary conditions....

. 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 illustration), 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 the path of an atomic-sized black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

, or some other imaginary particle which is able to fall with increasing velocity from space directly through the Earth, without stopping or being affected by any force but gravity, and in this way falls through the core and out the other side in a straight line, and off again into space, while slowing from the backwards gravitational tug. If such a particle were gravitationally bound to the Earth it would not escape, but would pursue a series of passes in which it always slowed at some maximal distance into space, but had its maximal velocity at the Earth's center (this "orbit" would have an orbital eccentricity
Orbital eccentricity
The orbital eccentricity of an astronomical body is the amount by which its orbit deviates from a perfect circle, where 0 is perfectly circular, and 1.0 is a parabola, and no longer a closed orbit...

of 1.0). If such a particle also had a wave nature, it would have the highest probability of being located where its velocity and momentum were highest, which would be at the Earth's core. In addition, rather than be confined to an infinitely narrow "orbit" which is a straight line, it would pass through the Earth from all directions, and not have a preferred one. Thus, a "long exposure" photograph of its motion over a very long period of time, would show a sphere.

In order to be stopped, such a particle would need to interact with the Earth in some way other than gravity. In a similar way, all s electrons have a finite probability of being found inside the nucleus, and this allows s electrons to occasionally participate in strictly nuclear-electron interaction processes, such as electron capture
Electron capture
Electron capture is a process in which a proton-rich nuclide absorbs an inner atomic electron and simultaneously emits a neutrino...

and internal conversion
Internal conversion
Internal conversion is a radioactive decay process where an excited nucleus interacts with an electron in one of the lower atomic orbitals, causing the electron to be emitted from the atom. Thus, in an internal conversion process, a high-energy electron is emitted from the radioactive atom, but...

.

Below, a number of drum membrane vibration modes are shown. The analogous wave functions of the hydrogen atom are indicated. 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 modes

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 an antinode 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-symmmetry 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 modes

d-type modes

## Orbital energy

In atoms with a single electron (hydrogen-like atom
Hydrogen-like atom
A hydrogen-like ion is any atomic nucleus with one electron and thus is isoelectronic with hydrogen. Except for the hydrogen atom itself , these ions carry the positive charge e, where Z is the atomic number of the atom. Examples of hydrogen-like ions are He+, Li2+, Be3+ and B4+...

s), the energy of an orbital (and, consequently, of any electrons in the orbital) is determined exclusively by . The orbital has the lowest possible energy in the atom. Each successively higher value of has a higher level of energy, but the difference decreases as increases. For high , the level of energy becomes so high that the electron can easily escape from the atom. In single electron atoms, all levels with different within a given are (to a good approximation) degenerate, and have the same energy. [This approximation is broken to a slight extent by the effect of the magnetic field of the nucleus, and by quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

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 the intrinsic properties of its orbital, but also on its interactions with the other electrons. These interactions depend on the detail of its spatial probability distribution, and so the energy level
Energy level
A quantum mechanical system or particle that is bound -- that is, confined spatially—can only take on certain discrete values of energy. This contrasts with classical particles, which can have any energy. These discrete values are called energy levels...

s of orbitals depend not only on but also on . Higher values of are associated with higher values of energy; for instance, the 2p state is higher than the 2s state. When = 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 = 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 of higher atomic number, the of electrons becomes more and more of a determining factor in their energy, and the principal quantum numbers of electrons becomes less and less important in their energy placement.

The energy sequence of the first 24 subshells (e.g., 1s, 2p, 3d, etc.) is given in the following table. Each cell represents a subshell with and 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.
1 1
2 2 3
3 4 5 7
4 6 8 10 13
5 9 11 14 17 21
6 12 15 18 22
7 16 19 23
8 20 24

Note: empty cells indicate non-existent sublevels, while numbers in italics indicate sublevels that could 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
Electron configuration
In atomic physics and quantum chemistry, electron configuration is the arrangement of electrons of an atom, a molecule, or other physical structure...

). 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
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

). These quantum numbers include the three that define orbitals, as well as s
Spin quantum number
In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given particle...

, or spin quantum number
Spin quantum number
In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given particle...

. Thus, two electrons may occupy a single orbital, so long as they have different values of . However, only two electrons, because of their spin, can be associated with each orbital.

Additionally, an electron always tends to fall to the lowest possible energy state. It is possible for it to occupy any orbital so long as it does not violate the Pauli exclusion principle, but if lower-energy orbitals are available, this condition is unstable. The electron will eventually lose energy (by releasing a photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

) 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
Periodic table
The periodic table of the chemical elements is a tabular display of the 118 known chemical elements organized by selected properties of their atomic structures. Elements are presented by increasing atomic number, the number of protons in an atom's atomic nucleus...

. 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 , it consists of elements whose outermost electrons fall in the th shell. Niels Bohr
Niels Bohr
Niels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...

was the first to propose (1923) that the periodicity
Periodic table
The periodic table of the chemical elements is a tabular display of the 118 known chemical elements organized by selected properties of their atomic structures. Elements are presented by increasing atomic number, the number of protons in an atom's atomic nucleus...

in the properties of the elements might be explained by the periodic filling of the electron energy levels, resulting in the electronic structure of the atom.

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

The following is the order for filling the "subshell" orbitals, which also gives the order of the "blocks" in the periodic table:
1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p

The "periodic" nature of the filling of orbitals, as well as emergence of the s, p, d and f "blocks" is more obvious, if this order of filling is given in matrix form, with increasing principal quantum numbers starting the new rows ("periods") in the matrix. Then, each subshell (composed of the first two quantum numbers) is repeated as many times as required for each pair of electrons it may contain. The result is a compressed periodic table, with each entry representing two successive elements:
  1s 2s 2p 2p 2p 3s 3p 3p 3p 4s 3d 3d 3d 3d 3d 4p 4p 4p 5s 4d 4d 4d 4d 4d 5p 5p 5p 6s (4f) 5d 5d 5d 5d 5d 6p 6p 6p 7s (5f) 6d 6d 6d 6d 6d 7p 7p 7p 

The number of electrons in an electrically neutral atom increases with the atomic number
Atomic number
In chemistry and physics, the atomic number is the number of protons found in the nucleus of an atom and therefore identical to the charge number of the nucleus. It is conventionally represented by the symbol Z. The atomic number uniquely identifies a chemical element...

. The electrons in the outermost shell, or valence electron
Valence electron
In chemistry, valence electrons are the electrons of an atom that can participate in the formation of chemical bonds with other atoms. Valence electrons are the "own" electrons, present in the free neutral atom, that combine with valence electrons of other atoms to form chemical bonds. In a single...

s
, tend to be responsible for an element's chemical behavior. Elements that contain the same number of valence electrons can be grouped together and display similar chemical properties.

### Relativistic effects

For elements with high atomic number 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
Mercury (element)
Mercury is a chemical element with the symbol Hg and atomic number 80. It is also known as quicksilver or hydrargyrum...

(which results from 6s electrons not being available for metal bonding) and the golden color of gold
Gold
Gold is a chemical element with the symbol Au and an atomic number of 79. Gold is a dense, soft, shiny, malleable and ductile metal. Pure gold has a bright yellow color and luster traditionally considered attractive, which it maintains without oxidizing in air or water. Chemically, gold is a...

and caesium
Caesium
Caesium or cesium is the chemical element with the symbol Cs and atomic number 55. It is a soft, silvery-gold alkali metal with a melting point of 28 °C , which makes it one of only five elemental metals that are liquid at room temperature...

(which results from narrowing of 6s to 5d transition energy to the point that visible light begins to be absorbed).

In the Bohr Model
Bohr model
In atomic physics, the Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing attraction,...

, an electron has a velocity given by , where Z is the atomic number, is the fine-structure constant
Fine-structure constant
In physics, the fine-structure constant is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. Being a dimensionless quantity, it has constant numerical value in all systems of units...

, 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
Dirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...

, which accounts for relativistic effects, the wavefunction of the electron for atoms with Z > 137 is oscillatory and unbounded. The significance of element 137, also known as untriseptium
Untriseptium
Untriseptium , also known as eka-dubnium or element 137, is a hypothetical chemical element which has not been observed to occur naturally, nor has it yet been synthesised. Due to drip instabilities, it is not known if this element is physically possible...

, was first pointed out by the physicist Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

. 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. See Extension of the periodic table beyond the seventh period.

There are no nodes in relativistic orbital densities, although individual components of the wavefunction will have nodes.

## Transitions between orbitals

Under quantum mechanics, each quantum state has a well-defined energy. When applied to atomic orbitals, this means that each state has a specific energy, and that if an electron is to move between states, the energy difference is also very fixed.

Consider two states of the Hydrogen atom:

State 1) n=1, l=0, ml=0 and s=+

State 2) n=2, l=0, ml=0 and s=+

By quantum theory, state 1 has a fixed energy of E1, and state 2 has a fixed energy of E2. 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 E2 - E1. 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 E2 - E1 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 only jump to one of the orbitals, it cannot jump to a state between orbitals. The result is that only photons of a specific frequency will be absorbed by the atom. This creates a line in the spectrum, known as an absorption line, which corresponds to the energy difference between states 1 and 2.

The atomic orbital model thus predicts line spectra, which are observed experimentally. This is one of the main validations of the atomic orbital model.

The atomic orbital model is nevertheless an approximation to the full quantum theory, which only recognizes many electron states. The predictions of line spectra are qualitatively useful but are not quantitatively accurate for atoms and ions other than those containing only one electron.

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