In spectroscopy and

quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...

, the multiplicity of an energy level is defined as ''2S+1'', where ''S'' is the total spin angular momentum. States with multiplicity 1, 2, 3, 4, 5 are respectively called singlets, doublets,

triplets A multiple birth is the culmination of one multiple pregnancy, wherein the mother gives birth to two or more babies. A term most applicable to vertebrate species, multiple births occur in most kinds of mammals, with varying frequencies. Such ...

, quartets and quintets. In the ground state of an atom or molecule, the unpaired electrons usually all have parallel spin. In this case the multiplicity is also equal to the number of unpaired electrons plus one.

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 In quantum physics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orb ...

energy. For example, the ground state of a carbon atom is ³P. 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 electron In chemistry, an unpaired electron is an electron that occupies an orbital of an atom singly, rather than as part of an electron pair. Each atomic orbital of an atom (specified by the three quantum numbers n, l and m) has a capacity to contain ...

s, as a result of

Hund's rule Hund's rule of maximum multiplicity is a rule based on observation of atomic spectra, which is used to predict the ground state of an atom or molecule with one or more open electronic shells. The rule states that for a given electron configuration ...

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. 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. 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 Organic may refer to: * Organic, of or relating to an organism, a living entity * Organic, of or relating to an anatomical organ Chemistry * Organic matter, matter that has come from a once-living organism, is capable of decay or is the product ...

molecules have complete electron shells with no unpaired electrons and therefore have singlet ground states. This is true also for

inorganic In chemistry, an inorganic compound is typically a chemical compound that lacks carbon–hydrogen bonds, that is, a compound that is not an organic compound. The study of inorganic compounds is a subfield of chemistry known as ''inorganic chemistr ...

molecules containing only

main-group element In chemistry and atomic physics, the main group is the group of elements (sometimes called the representative elements) whose lightest members are represented by helium, lithium, beryllium, boron, carbon, nitrogen, oxygen, and fluorine as arrange ...

s. Important exceptions are

dioxygen There are several known allotropes of oxygen. The most familiar is molecular oxygen (O2), present at significant levels in Earth's atmosphere and also known as dioxygen or triplet oxygen. Another is the highly reactive ozone (O3). Others are: * ...

(O₂) as well as methylene (CH₂) and other

carbene In organic chemistry, a carbene is a molecule containing a neutral carbon atom with a valence of two and two unshared valence electrons. The general formula is or where the R represents substituents or hydrogen atoms. The term "carbene" m ...

s. However, higher spin ground states are very common in

coordination complex A coordination complex consists of a central atom or ion, which is usually metallic and is called the ''coordination centre'', and a surrounding array of bound molecules or ions, that are in turn known as ''ligands'' or complexing agents. Many ...

es of

transition metal In chemistry, a transition metal (or transition element) is a chemical element in the d-block of the periodic table (groups 3 to 12), though the elements of group 12 (and less often group 3) are sometimes excluded. They are the elements that can ...

s. A simple explanation of the spin states of such complexes is provided by

crystal field theory Crystal field theory (CFT) describes the breaking of degeneracies of electron orbital states, usually ''d'' or ''f'' orbitals, due to a static electric field produced by a surrounding charge distribution (anion neighbors). This theory has been used ...

Dioxygen

The highest occupied orbital energy level of dioxygen is a pair of antibonding 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 and second excited states of dioxygen are both states of singlet oxygen. Each has two electrons of opposite spin in the π* level so that S = 0 and the multiplicity is 2S + 1 = 1 in consequence. In the first excited state, the two π* electrons are paired in the same orbital, so that there are no unpaired electrons. In the second excited state, however, the two π* electrons occupy different orbitals with opposite spin. Each is therefore an unpaired electron, but the total spin is zero and the multiplicity is 2S + 1 = 1 despite the two unpaired electrons. The multiplicity of the second excited state is therefore not equal to the number of its unpaired electrons plus one, and the rule which is usually true for ground states is invalid for this excited state. Molecular orbital scheme for the three forms of oxygen

Molecular orbital scheme for the three forms of oxygen

Carbenes

In organic chemistry,

s are molecules which have carbon atoms with only six electrons in their valence shells and therefore disobey the octet rule. 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 In chemistry, a lone pair refers to a pair of valence electrons that are not shared with another atom in a covalent bondIUPAC ''Gold Book'' definition''lone (electron) pair''/ref> and is sometimes called an unshared pair or non-bonding pair. Lone ...

and have opposite spins so that there is no net spin, while in triplet carbenes these electrons have parallel spins.

References

Bibliography