# Multiplicity (chemistry)

In spectroscopy and quantum chemistry, the multiplicity of an energy level is defined as 2S+1, where S is the total spin angular momentum.[1][2][3] States with multiplicity 1, 2, 3, 4, 5 are respectively called singlets, doublets, triplets, quartets and quintets.[2] The multiplicity is also equal to the number of unpaired electrons plus one.[4]

## Atoms

The multiplicity is often equal to the number of possible orientations of the total spin relative to the total orbital angular momentum L, and therefore to the number of near–degenerate levels that differ only in their spin–orbit interaction energy.

For example, the ground state of the carbon atom is a 3P state. The superscript three (read as triplet) indicates that the multiplicity 2S+1 = 3, so that the total spin S = 1. This spin is due to two unpaired electrons, as a result of Hund's rule which favors the single filling of degenerate orbitals. The triplet consists of three states with spin components +1, 0 and –1 along the direction of the total orbital angular momentum, which is also 1 as indicated by the letter P. The total angular momentum quantum number J can vary from L+S = 2 to L–S = 0 in integer steps, so that J = 2, 1 or 0.[1][2]

However the multiplicity equals the number of spin orientations only if S ≤ L. When S > L there are only 2L+1 orientations of total angular momentum possible, ranging from S+L to S-L.[2][3] The ground state of the nitrogen atom is a 4S state, for which 2S + 1 = 4 in a quartet state, S = 3/2 due to three unpaired electrons. For an S state, L = 0 so that J can only be 3/2 and there is only one level even though the multiplicity is 4.

## Molecules

Most stable organic molecules have complete electron shells with no unpaired electrons and therefore have singlet ground states. This is true also for inorganic molecules containing only main-group elements. Important exceptions are dioxygen (O2) as well as methylene (CH2) and other carbenes.

However, higher spin ground states are very common in coordination complexes of transition metals. A simple explanation of the spin states of such complexes is provided by crystal field theory.

### Dioxygen

The highest occupied orbital energy level of dioxygen is a pair of degenerate π* orbitals. In the ground state of dioxygen, this energy level is occupied by two electrons of the same spin, as shown in the molecular orbital diagram. The molecule therefore has two unpaired electrons and is in a triplet state.

In contrast, the first excited state of dioxygen has two electrons of opposite spin in the π* level so that there are no unpaired electrons. In consequence it is a singlet state and is known as singlet oxygen.

### Carbenes

In organic chemistry, carbenes are molecules which have carbon atoms with only have six valence electrons and therefore disobey the octet rule.[5] Carbenes generally split into singlet carbenes and triplet carbenes, named for their spin multiplicities. Both have two non-bonding electrons; in singlet carbenes these exist as a lone pair and have opposite spins so that there is no net spin, while in triplet carbenes these electrons have parallel spins.[6]

## References

1. ^ a b Atkins & de Paula 2006, p. 353.
2. ^ a b c d Levine 1991, p. 308.
3. ^ a b "multiplicity (spin multiplicity)". IUPAC (International Union of Pure and Applied Chemistry). 2006. doi:10.1351/goldbook.M04062. Retrieved 30 March 2018.
4. ^ Miessler & Tarr 1999, p. 33.
5. ^ Clayden et al. 2001, p. 1055.
6. ^ Clayden et al. 2001, p. 1061.