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. States with multiplicity 1, 2, 3, 4, 5 are respectively called singlets, doublets, triplets, quartets and quintets.

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.

However the equality of multiplicity and number of spin orientations is only true when 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.

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