# Diatomic molecule

A space-filling model of the diatomic molecule dinitrogen, N2

Diatomic molecules are molecules composed of only two atoms, of either the same or different chemical elements. The prefix di is of Greek origin, meaning "two". If a diatomic molecule consists of two atoms of the same element, such as hydrogen (H2) or oxygen (O2), then it is said to be homonuclear. Otherwise, if a diatomic molecule consists of two different atoms, such as carbon monoxide (CO) or boron monoxide (BO), the molecule is said to heteronuclear.

## Homonuclear molecules

A periodic table showing the elements that exist as homonuclear diatomic molecules under typical laboratory conditions.

The only chemical elements that are stable two atom homonuclear molecules at standard temperature and pressure (STP), are hydrogen (H2), nitrogen (N2) and oxygen (O2), plus the halogens fluorine (F2) and chlorine (Cl2). Those diatomic elements that are gaseous at STP, when grouped together with the monatomic noble gases, such as argon, are called "elemental gases" or "molecular gases" to distinguish them from molecules that are also chemical compounds. The noble gases do not form diatomic molecules: This can be explained using molecular orbital theory.

The halogens bromine (Br2) and iodine (I2) can also form diatomic gas at slightly elevated temperatures.[1]
Other elements that can form two atom molecules are phosphorus (P2) and sulfur (S2) although neither of these molecules are stable in atmospheric conditions.

## Heteronuclear molecules

All other diatomic molecules are chemical compounds of two different elements, for example, nitric oxide (NO). Many different elements combine to form heteronuclear diatomic molecules, and this phenomenon, in general, depends on temperature and pressure. Many chemical compounds form diatomic molecules when evaporated.

## Occurrence

Hundreds of diatomic molecules have been characterized[2] in the terrestrial environment, laboratory, and interstellar medium. About 99% of the Earth's atmosphere is composed of two diatomic molecules: oxygen (21%) and nitrogen (78%). The natural abundance of hydrogen (H2) in the Earth's atmosphere is only on the order of parts per million, but H2 is, in fact, the most abundant diatomic molecule in nature. The interstellar medium is, indeed, dominated by hydrogen atoms.

The bond in a homonuclear diatomic molecule is non-polar. In most diatomic molecules, the elements are nonidentical. Prominent examples include carbon monoxide, nitric oxide, and hydrogen chloride, but other important examples include gaseous MgO, SiO, and many other species not normally considered diatomic because they polymerize near room temperature. All halogens are diatomic, excepted astatine.

Elements that consist of diatomic molecules, under typical laboratory conditions of 1 bar and 25 °C, include hydrogen (H2), nitrogen (N2), oxygen (O2), and the halogens (although it is not yet known whether astatine forms diatomic astatine molecules[3]).[4] Other elements form homonuclear diatomics when evaporated, but these diatomic species repolymerize at lower temperatures. For example, heating ("cracking") elemental phosphorus gives diphosphorus, P2.

## Molecular geometry

Diatomic molecules cannot have any geometry but linear, as any two points always lie in a line. This is the simplest spatial arrangement of atoms after the sphericity of single atoms.[5]

## Historical significance

Diatomic elements played an important role in the elucidation of the concepts of element, atom, and molecule in the 19th century, because some of the most common elements, such as hydrogen, oxygen, and nitrogen, occur as diatomic molecules. John Dalton's original atomic hypothesis assumed that all elements were monatomic and that the atoms in compounds would normally have the simplest atomic ratios with respect to one another. For example, Dalton assumed water's formula to be HO, giving the atomic weight of oxygen as eight times that of hydrogen, instead of the modern value of about 16. As a consequence, confusion existed regarding atomic weights and molecular formulas for about half a century.

As early as 1805, Gay-Lussac and von Humboldt showed that water is formed of two volumes of hydrogen and one volume of oxygen, and by 1811 Amedeo Avogadro had arrived at the correct interpretation of water's composition, based on what is now called Avogadro's law and the assumption of diatomic elemental molecules. However, these results were mostly ignored until 1860. Part of this rejection was due to the belief that atoms of one element would have no chemical affinity toward atoms of the same element, and part was due to apparent exceptions to Avogadro's law that were not explained until later in terms of dissociating molecules.

At the 1860 Karlsruhe Congress on atomic weights, Cannizzaro resurrected Avogadro's ideas and used them to produce a consistent table of atomic weights, which mostly agree with modern values. These weights were an important pre-requisite for the discovery of the periodic law by Dmitri Mendeleev and Lothar Meyer.[6]

## Excited electronic states

Diatomic molecules are normally in the lowest or ground state, which is also known as the $X$ state. When a gas of diatomic molecules is bombarded by energetic electrons, the molecules are excited to higher electronic states, such as occurs, for example, in the natural aurora, high-altitude nuclear explosions, and rocket-born electron gun experiments.[7] The excitation can also occur when the gas absorbs light or other electromagnetic radiation. The excited states are unstable and naturally relax back to the ground state. Over various short time scales after the excitation (typically a fraction of a second, or sometimes longer than a second if the excited state is metastable), transitions occur from the higher to lower electronic states and ultimately to the ground state, and each transition results in the emission of a photon. This emission is known as fluorescence. Successively higher electronic states are traditionally named $A$, $B$, $C$, etc. (but this convention is not always followed, and sometimes lower case letters and alphabetically out-of-sequence letters are used, as can be seen in the example given below). The excitation energy must be greater than or equal to the energy of the electronic state in order for the excitation to occur.

In quantum theory, an electronic state of a diatomic molecule is represented by

$^{2S+1} \Lambda (v)$

where $S$ is the total electronic spin quantum number, $\Lambda$ is the total electronic angular momentum quantum number along the internuclear axis, and $v$ is the vibrational quantum number. $\Lambda$ takes on values 0, 1, 2, …, which traditionally are represented by the electronic state symbols $\Sigma$, $\Pi$, $\Delta$,…. For example, the following table lists the common electronic states (without vibrational quantum numbers), along with the energy of the lowest vibrational level ($v=0$) of diatomic nitrogen (N2), the most abundant gas in the Earth's atmosphere.[8] In the table, the subscripts and superscripts after $\Lambda$ give additional quantum mechanical details about the electronic state.

State Energy ($T_0$, cm$^{-1}$)
$X ^1\Sigma_g^+$ 0.0
$A ^3\Sigma_u^+$ 49754.8
$B ^3\Pi_g$ 59306.8
$W ^3\Delta_u$ 59380.2
$B' ^3\Sigma_u^-$ 65851.3
$a' ^1\Sigma_u^-$ 67739.3
$a ^1\Pi_g$ 68951.2
$w ^1\Delta_u$ 71698.4

## Energy levels

The molecular term symbol is a shorthand expression of the angular momenta that characterize the electronic quantum state of a diatomic molecule, which is an eigenstate of the electronic molecular Hamiltonian. It is also convenient, and common, to represent a diatomic molecule as two-point masses connected by a massless spring. The energies involved in the various motions of the molecule can then be broken down into three categories: the translational, rotational, and vibrational energies.

### Translational energies

The translational energy of the molecule is simply given by the kinetic energy expression:

$E_{trans}=\frac{1}{2}mv^2$

where m is the mass of the molecule and v is its velocity.

### Rotational energies

Classically, the kinetic energy of rotation is

$E_{rot} = \frac{L^2}{2 I} \,$
where
$L \,$ is the angular momentum
$I \,$ is the moment of inertia of the molecule

For microscopic, atomic-level systems like a molecule, angular momentum can only have specific discrete values given by

$L^2 = l(l+1) \hbar^2 \,$
where l is a non-negative integer and $\hbar$ is the reduced Planck constant.

Also, for a diatomic molecule the moment of inertia is

$I = \mu r_{0}^2 \,$
where
$\mu \,$ is the reduced mass of the molecule and
$r_{0} \,$ is the average distance between the centers of the two atoms in the molecule.

So, substituting the angular momentum and moment of inertia into Erot, the rotational energy levels of a diatomic molecule are given by:

$E_{rot} = \frac{l(l+1) \hbar^2}{2 \mu r_{0}^2} \ \ \ \ \ l=0,1,2,... \,$

### Vibrational energies

Another way a diatomic molecule can move is to have each atom oscillate—or vibrate—along a line (the bond) connecting the two atoms. The vibrational energy is approximately that of a quantum harmonic oscillator:

$E_{vib} = \left(n+\frac{1}{2} \right)\hbar \omega \ \ \ \ \ n=0,1,2,.... \,$
where
n is an integer
$\hbar$ is the reduced Planck constant and
$\omega$ is the angular frequency of the vibration.

### Comparison between rotational and vibrational energy spacings

The spacing, and the energy of a typical spectroscopic transition, between vibrational energy levels is about 100 times greater than that of a typical transition between rotational energy levels.

## Hund's cases

The good quantum numbers for a diatomic molecule, as well as good approximations of rotational energy levels, can be obtained by modeling the molecule using Hund's cases.

• Huber, K. P. and Herzberg, G. (1979). Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules. New York: Van Nostrand: Reinhold.
• Tipler, Paul (1998). Physics For Scientists and Engineers : Vol. 1 (4th ed.). W. H. Freeman. ISBN 1-57259-491-8.