# Atomic orbital

A less formal description of the electrons in atoms can be found at Electron configuration.

In quantum mechanics, the state of an atom, i.e. the eigenstates of the atomic Hamiltonian, are expanded (see configuration interaction expansion and basis (linear algebra)) 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.)

In atomic physics, the atomic spectral lines correspond to transitions (quantum leaps) between quantum states of an atom. These states are labelled by a set of quantum numbers summarized in the term symbol and usually associated to particular electron configurations, i.e. by occupations schemes of atomic orbitals (e.g. $\displaystyle 1s^2 2s^2 2p^6$ for the ground state of Neon -- term symbol: $\displaystyle {}^1S_0$ ). 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 to a given transition. One can say for example 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 Pauli exclusion principle and cannot be distinguished from the other electrons in the atom. Moreover, it sometimes happens that the configuration interaction expansion converges very slowly and that one cannot speak about simple one-determinantal wave function at all. This is the case when electron correlation is large.

## Hydrogen-like atoms

Main article: Hydrogen-like atom

The simplest atomic orbitals are those that occur in an atom with a single electron, such as the hydrogen atom. In this case the atomic orbitals are the eigenstates of the hydrogen Hamiltonian. They can be obtained analytically (see 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.

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

A given (hydrogen-like) atomic orbital is identified by unique values of three quantum numbers: n, l, and ml. 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 orbital. 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. It determines, among other things, the distance of the electron from the nucleus; all electrons with the same value of n lay at the same distance. Modern quantum mechanics confirms that these orbitals are closely related. For this reason, orbitals with the same value of n are said to comprise an "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".

## Qualitative characterization

### Limitations on the quantum numbers

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 rules governing the possible values of the quantum numbers are as follows:

The principal quantum number n 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 a shell.

The orbital angular momentum quantum number $\displaystyle \ell$ is a non-negative integer. Within a shell where n is some integer n0, $\displaystyle \ell$ ranges across all (integer) values satisfying the relation $\displaystyle 0 \le \ell \le n_0-1$ . For instance, the n = 1 shell has only orbitals with $\displaystyle \ell=0$ , and the n = 2 shell has only orbitals with $\displaystyle \ell=0$ , and $\displaystyle \ell=1$ . The set of orbitals associated with a particular value of $\displaystyle \ell$ are sometimes collectively called a subshell.

The magnetic quantum number $\displaystyle m_\ell$ is also always an integer. Within a subshell where $\displaystyle \ell$ is some integer $\displaystyle \ell_0$ , $\displaystyle m_\ell$ ranges thus: $\displaystyle -\ell_0 \le m_\ell \le \ell_0$ .

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

$\displaystyle l=0$ 1 2 3 4 ...
$\displaystyle n=1$ $\displaystyle m_l=0$
2 0 -1, 0, 1
3 0 -1, 0, 1 -2, -1, 0, 1, 2
4 0 -1, 0, 1 -2, -1, 0, 1, 2 -3, -2, -1, 0, 1, 2, 3
5 0 -1, 0, 1 -2, -1, 0, 1, 2 -3, -2, -1, 0, 1, 2, 3 -4, -3, -2 -1, 0, 1, 2, 3, 4
... ... ... ... ... ... ...

Subshells are usually identified by their $\displaystyle n$ - and $\displaystyle \ell$ -values. $\displaystyle n$ is represented by its numerical value, but $\displaystyle \ell$ is represented by a letter as follows: 0 is represented by 's', 1 by 'p', 2 by 'd', 3 by 'f', and 4 by 'g'. For instance, one may speak of the subshell with $\displaystyle n=2$ and $\displaystyle \ell=0$ as a '2s subshell'.

### The shapes of orbitals

Any discussion of the shapes of electron orbitals is necessarily imprecise, because a given electron, regardless of which orbital it occupies, can at any moment be found at any distance from the nucleus and in any direction.

However, the electron is much more likely to be found in certain regions of the atom than in others. Given this, a boundary surface can be drawn so that the electron has a high probability to be found anywhere within the surface, and all regions outside the surface have low values. The precise placement of the surface is arbitrary, but any reasonably compact determination must follow a pattern specified by the behavior of $\displaystyle \psi^2$ , the square of the wavefunction. This boundary surface is what is meant when the "shape" of an orbital is mentioned.

Generally speaking, the number $\displaystyle n$ determines the size and energy of the orbital: as $\displaystyle n$ increases, the size of the orbital increases.

Also in general terms, $\displaystyle \ell$ determines an orbital's shape, and $\displaystyle m_\ell$ its orientation. However, since some orbitals are described by equations in complex numbers, the shape sometimes depends on $\displaystyle m_\ell$ also. $\displaystyle s$ -orbitals ($\displaystyle \ell=0$ ) are shaped like spheres. $\displaystyle p$ -orbitals have the form of two ellipsoids with a point of tangency at the nucleus. The three $\displaystyle p$ -orbitals in each shell are oriented at right angles to each other, as determined by their respective values of $\displaystyle m_\ell$ .

Four of the five $\displaystyle d$ -orbitals look similar, each with four pear-shaped balls, each ball 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 $\displaystyle xy$ -, $\displaystyle xz$ -, and $\displaystyle yz$ -planes, and the fourth has the centres on the $\displaystyle x$ and $\displaystyle y$ axes. The fifth and final $\displaystyle d$ -orbital consists of three regions of high probability density: a torus with two pear-shaped regions placed symmetrically on its $\displaystyle z$ axis.

## Orbital energy

In atoms with a single electron (essentially the hydrogen atom), the energy of an orbital (and, consequently, of any electrons in the orbital) is determined exclusively by $\displaystyle n$ . The $\displaystyle n=1$ orbital has the lowest possible energy in the atom. Each successively higher value of $\displaystyle n$ has a higher level of energy, but the difference decreases as $\displaystyle n$ increases. For high $\displaystyle n$ , the level of energy becomes so high that the electron can easily escape from the atom.

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 levels of orbitals depend not only on $\displaystyle n$ but also on $\displaystyle \ell$ . Higher values of $\displaystyle \ell$ are associated with higher values of energy; for instance, the 2p state is higher than the 2s state. When $\displaystyle \ell$ = 3, 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 $\displaystyle \ell$ = 4 the energy is pushed into the shell two steps higher.

The energy order of the first 24 subshells is given in the following table. Each cell represents a subshell with $\displaystyle n$ and $\displaystyle \ell$ given by its row and column indices, respectively. The number in the cell is the subshell's position in the sequence. Empty cells represent subshells that either do not exist or stand farther down in the sequence.

$\displaystyle s$ $\displaystyle p$ $\displaystyle d$ $\displaystyle f$ $\displaystyle g$
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

## 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 ($\displaystyle n$ , $\displaystyle \ell$ , and $\displaystyle m_\ell$ ), as well as (the hitherto unmentioned) s. Thus, two electrons may occupy a single orbital, so long as they have different values of $\displaystyle s$ .

Additionally, an electron always tries to occupy 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 speficied 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 $\displaystyle i$ , it consists of elements whose outermost electrons fall in the $\displaystyle i$ th shell.

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 $\displaystyle \ell$ -state (but the $\displaystyle n$ associated with that $\displaystyle \ell$ -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 number of electrons in a 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.