spectroscopy Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...

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 A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The te ...

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 bir ...

, quartets and quintets. In the

ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than 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 Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party i ...

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–or ...

energy. For example, the ground state of a

carbon Carbon () is a chemical element with the symbol C and atomic number 6. It is nonmetallic and tetravalent In chemistry, the valence (US spelling) or valency (British spelling) of an element is the measure of its combining capacity with o ...

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 In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin). If s is the particle's s ...

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 Nitrogen is the chemical element with the symbol N and atomic number 7. Nitrogen is a nonmetal and the lightest member of group 15 of the periodic table, often called the pnictogens. It is a common element in the universe, estimated at se ...

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 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 chemist ...

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: *A ...

(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" ma ...

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 A molecular orbital diagram, or MO diagram, is a qualitative descriptive tool explaining chemical bonding in molecules in terms of molecular orbital theory in general and the linear combination of atomic orbitals (LCAO) method in particular. A f ...

. 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 Singlet oxygen, systematically named dioxygen(singlet) and dioxidene, is a gaseous inorganic chemical with the formula O=O (also written as or ), which is in a quantum state where all electrons are spin paired. It is kinetically unstable at ambie ...

. 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

organic chemistry Organic chemistry is a subdiscipline within chemistry involving the scientific study of the structure, properties, and reactions of organic compounds and organic materials, i.e., matter in its various forms that contain carbon atoms.Clayden, J.; ...

s are molecules which have carbon atoms with only six electrons in their valence shells and therefore disobey the

octet rule The octet rule is a chemical rule of thumb that reflects the theory that main-group elements tend to bond in such a way that each atom has eight electrons in its valence shell, giving it the same electronic configuration as a noble gas. The rule i ...

. 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