# Atomic orbital

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An atomic orbital is the description of the behavior of an electron in an atom according to quantum mechanics. It is a particular type of electron wavefunction, and as such it describes the probability of the electron being in any location, and its energy (see Electron orbital for more background details).

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## General information

The simplest atomic orbitals are those that occur in an atom 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.

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.

Analytically, the orbitals are considered as separate entities. However, in general, an electron's behavior is not fully described by a single orbital. Electron states are best represented by "mixtures" (linear combinations) of multiple orbitals.

The quantum number n first appeared in the Bohr model, now outmoded. It determined, 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 dispels this theory, but confirms that these orbitals are closely related. 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".

## Mathematical characterization

### Derivation

As all electron orbitals, atomic orbitals are solutions to the Schrdinger equation. In this case, the potential term is the potential given by Coulomb's law:

[itex]V = -\frac{1}{4 \pi \epsilon_0} \frac{Ze^2}{r}[itex]

where

• The first term is a constant, usually abbreviated by the letter k,
• Z is the atomic number,
• e is the elementary charge,
• r is the magnitude of the distance from the nucleus.

The wavefunction is a function of three spatial variables, so that after removing the time-dependence, the Schrdinger equation is a partial differential equation in three variables. However, since the potential is spherically symmetric, it is profitable to write the equation in spherical coordinates. In this form, any individual eigenfunction ψ can be written as a product of three single-variable functions, often denoted as follows:

[itex]\psi(r, \theta, \phi) = R(r)f(\theta)g(\phi)[itex]

(It is customary in atomic physics that θ represents the polar angle (colatitude) and φ the azimuthal angle, contrary to the convention in some other disciplines.) This modification is computationally valuable. Written in terms of these three factors, the Schrdinger equation looks very formidable, but through separation of variables, it can be reduced to three separate equations, each in one variable.

Two separations are required, resulting in two separation constants. A third arbitrary constant results from the application of boundary conditions to R. The equations given below use a form of the separation constants that seems arbitrary, but it simplifies matters later on.

[itex]\frac{1}{R(r)} \frac{d}{dr} \left ( r^2 \frac{dR}{dr} \right ) + \frac{2 \mu r^2}{\hbar^2}(E-V(r)) = l(l+1)[itex]
[itex]\frac{1}{g(\phi)} \frac{d^2 g(\phi)}{d\phi^2} = -m^2[itex]
[itex]l(l+1)\sin ^2(\theta) + \frac{\sin(\theta)}{f(\theta)} \frac{d}{d\theta} \left [ \sin(\theta) \frac{df}{d\theta} \right ] = m^2[itex]

where:

### Results

In addition to [itex]\ell[itex] and m, a third arbitrary integer, called n, emerges from the boundary conditions placed on R. The functions R, f and g that solve the equations above depend not only for their values but for their very form on the values of these integers, called quantum numbers. As a result, it is customary to subscript the functions with the values of the quantum numbers they depend on. The forms of the functions are:

[itex]\psi = C_{nlm}\, R_{nl}(r)\, f_{lm}(\theta)\, g_m(\phi)[itex]
[itex]R_{nl}(r) = \frac{a_0}{r} e^{-\frac{r}{a_0n}} \mathcal{L}_{nl} \left( \frac{r}{a_0} \right)[itex]
[itex]f_{lm}(\theta) = \frac{(\sin\theta)^{|m|}}{2^l l!} \left [ \frac{d}{d(\cos\theta)} \right ]^{l+|m|}(\cos ^2(\theta)-1)^l \qquad[itex] (the associated Legendre functions)
[itex]g_m(\phi) = e^{im\phi}[itex]

where:

[itex]a_0 = {{4\pi\varepsilon_0\hbar^2}\over{\mu Ze^2}}[itex]

The functions f and g are sometimes consolidated into the function [itex]Y(\theta, \phi) = f(\theta)g(\phi)[itex]. This function is a spherical harmonic.

The values of the integer quantum numbers are subjected to restrictions (see "Limitations on the quantum numbers" below).

### Angular momentum

Each atomic orbital is associated with an angular momentum L. It is a vector (spatial), and its magnitude is given by:

[itex]\left| \mathbf L \right| = \hbar \sqrt{l(l+1)}[itex]

The projection of this vector onto any arbitrary direction is quantized. (This does not mean that the angular momentum of an actual electron is quantized in the same way, since the electron's wavefunction is a linear combination of orbitals.) If the arbitrary direction is called z, the quantization is given by:

[itex]L_z = m_l\hbar[itex]

where [itex]m_l[itex] is restricted as described below. This value is always less than the total angular momentum. Thus, if the [itex]\mathbf L[itex]-vector is measured in some direction, it will not lie entirely in that direction; part of it will lie in perpendicular directions. This allows the uncertainty principle to stand. It mandates that no two components of [itex]\mathbf L[itex] may be known at once. If one component were known to be equal to the total [itex]\mathbf L[itex], the other two would necessarily be zero.

These two relations do not give the total angular momentum of the electron. For that, electron spin must be included.

This quantization of angular momentum closely parallels that proposed by Niels Bohr (see Bohr model) in 1913, with no knowledge of wavefunctions.

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

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

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

[itex]l=0[itex] 1 2 3 4 ...
[itex]n=1[itex] [itex]m_l=0[itex]
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 [itex]n[itex]- and [itex]\ell[itex]-values. [itex]n[itex] is represented by its numerical value, but [itex]\ell[itex] 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 [itex]n=2[itex] and [itex]\ell=0[itex] 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 [itex]\psi^2[itex], the square of the wavefunction. This boundary surface is what is meant when the "shape" of an orbital is mentioned.

Generally speaking, the number [itex]n[itex] determines the size and energy of the orbital: as [itex]n[itex] increases, the size of the orbital increases.

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

Four of the five [itex]d[itex]-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 [itex]xy[itex]-, [itex]xz[itex]-, and [itex]yz[itex]-planes, and the fourth has the centres on the [itex]x[itex] and [itex]y[itex] axes. The fifth and final [itex]d[itex]-orbital consists of three regions of high probability density: a torus with two pear-shaped regions placed symmetrically on its [itex]z[itex] axis.

## Orbital energy

In atoms with a single electron (essentially hydrogen), the energy of an orbital (and, consequently, of any electrons in the orbital) is determined exclusively by [itex]n[itex]. The [itex]n=1[itex] orbital has the lowest possible energy in the atom. Each successively higher value of [itex]n[itex] has a higher level of energy, but the difference decreases as [itex]n[itex] increases. For high [itex]n[itex], 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 [itex]n[itex] but also on [itex]\ell[itex]. Higher values of [itex]\ell[itex] are associated with higher values of energy; for instance, the 2p state is higher than the 2s state. When [itex]\ell[itex] = 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 [itex]\ell[itex] = 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 [itex]n[itex] and [itex]\ell[itex] 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.

[itex]s[itex] [itex]p[itex] [itex]d[itex] [itex]f[itex] [itex]g[itex]
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 ([itex]n[itex], [itex]\ell[itex], and [itex]m_\ell[itex]), as well as (the hitherto unmentioned) s. Thus, two electrons may occupy a single orbital, so long as they have different values of [itex]s[itex].

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 [itex]i[itex], it consists of elements whose outermost electrons fall in the [itex]i[itex]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 [itex]\ell[itex]-state (but the [itex]n[itex] associated with that [itex]\ell[itex]-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.

## Related topics

• The Orbitron (http://www.shef.ac.uk/chemistry/orbitron/), a visualization of all common and uncommon atomic orbitals, from 1s to 7g
• David Manthey's Orbital Viewer (http://www.orbitals.com/orb/index.html) renders orbitals with n ≤ 30
• Java orbital viewer applet (http://www.falstad.com/qmatom/)

## References

• Tipler, Paul & Ralph Llewellyn (2003). Modern Physics (4th ed.). New York: W. H. Freeman and Company. ISBN 0-7167-4345-0

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