Multiplicity

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In quantum mechanics, multiplicity or degree of degeneracy is understood as the number of orthogonal states that have a certain eigenvalue in common for a certain observable . These states are therefore degenerate eigen-states of this observable.

One example is spin multiplicity , which relates to the observable total spin of an atomic shell . In the simplest example, the hydrogen atom , the ground state electron can adopt one of two orthogonal spin states. Without an external magnetic field, the two states have the same eigenvalue for energy and can therefore not be differentiated energetically, i.e. that is, they form a twofold degenerate energy level ; the multiplicity is 2 here, the level is a doublet. In a magnetic field, the level splits into two levels due to the Zeeman effect .

Correspondingly, with two electrons the state with total spin is called a singlet , because it does not split, and the state with total spin is called triplet , because it splits 3 times in the magnetic field. ${\ displaystyle S = 0}$ ${\ displaystyle S = 1}$

Generally, a total spin system has spin multiplicity . The independent states (and all their linear combinations ) have the same energy in many cases, but differ e.g. B. in the orientation of the spin with respect to a marked axis . This is expressed by the different eigenvalues of the z-component of the spin (see e.g. directional quantization in a magnetic field ): ${\ displaystyle S}$ ${\ displaystyle 2S + 1}$ ${\ displaystyle 2S + 1}$ ${\ displaystyle 2S + 1}$ ${\ displaystyle m_ {S}}$

{\ displaystyle m_ {S} = \ underbrace {-S, -S + 1, \ ldots, S-1, S} _ {2S + 1 \, {\ text {values: multiplicity}}}.}

An energy level with spin multiplicity can split up into a maximum of levels when additional interactions occur . In the line spectra of atoms, this leads to a fine structure . ${\ displaystyle 2S + 1}$ ${\ displaystyle 2S + 1}$

Spin multiplets
Spin quantum number ${\ displaystyle S}$	magn. QZ of the spin ${\ displaystyle m_ {S}}$	Multiplicity ${\ displaystyle 2S + 1}$	designation	Type
0	0	1	Singlet	Scalar boson
1/2	−1/2, +1/2	2	Doublet	Fermion
1	−1, 0, +1	3	Triplet	Vector boson
3/2	−3/2, −1/2, +1/2, +3/2	4th	quartet	Fermion
2	−2, −1, 0, +1, +2	5	quintet	Tensor boson

Multiplicity of atoms and molecules

In systems consisting of several electrons and / or atomic nuclei , a distinction is made between the spin multiplicity of the electrons and the spin multiplicity of the atomic nuclei.

Multiplicity of the electron spin

One-electron systems

The intrinsic angular momentum of an electron, as a quantum number of an elementary particle with the spin projected onto any spatial direction, has two possible settings: parallel or antiparallel. There is therefore an electronic doublet state. The multiplicity of the one-electron system is . ${\ displaystyle \ textstyle S = {\ frac {1} {2}}}$ ${\ displaystyle \ textstyle 2S + 1 = 2}$

Example: The electron of a single hydrogen atom H • (This could also be used as an example for a radical with zero paired electrons in the table below.)

Multi-electron systems

For atoms (or ions ) with several electrons and for molecules , the total spin quantum number of the entire electronic system must first be determined. For an atom with electrons is given by ${\ displaystyle S}$ ${\ displaystyle i}$ ${\ displaystyle S}$

{\ displaystyle S = \ left | \ sum _ {i} m_ {s_ {i}} \ right |,}

where is the spin quantum number of the i-th electron. Since the individual spins of paired electrons do not contribute to the overall spin due to their opposite orientation, it is sufficient to count the unpaired electrons. Your individual spin quantum numbers add up to the total spin quantum number ${\ displaystyle m_ {s_ {i}}}$ ${\ displaystyle s = + 1/2}$ ${\ displaystyle S = n _ {\ text {unpaired}} / 2.}$

As a simple example, the helium atom can serve as a 2-electron system, but the states as a singlet (parahelium) and as a triplet (orthohelium) are possible. ${\ displaystyle S = 0}$ ${\ displaystyle S = 1}$

system	example	Ground state electrons		Total spin quantum number ${\ displaystyle S}$	Multiplicity ${\ displaystyle 2S + 1}$	Basic state
system	example	paired	unpaired	Total spin quantum number ${\ displaystyle S}$	Multiplicity ${\ displaystyle 2S + 1}$	Basic state
most of the molecules	Hydrogen molecule HH	all (here 1x2)	0	0/2 = 0	2x0 + 1 = 1	Singlet
radical	Nitrogen monoxide • N = O or NO •	here 5x2	1	1/2	2x (1/2) +1 = 2	Doublet
Biradicals	Oxygen molecule • OO •	here 5x2	2	2/2 = 1	2x1 + 1 = 3	Triplet
Metal ions , especially the subgroup , and complexes		... x2	${\ displaystyle \ geq 2}$	${\ displaystyle \ geq 1}$	${\ displaystyle \ geq 3}$	Triplet, quartet, ...

The numerical value of the multiplicity is indicated in the term symbols in the left superscript, which are often used to characterize the quantum states of atoms and molecules.

Example: For hydrogen atoms (H) in the ground state, the term symbol is ² S _1/2 (multiplicity 2).

Meaning: selection rules, intercombination prohibition

The spin multiplicity plays an important role for the selection rules in spectroscopy in multi-electron systems. Electrical transitions take place particularly well when the coupling of the spins and thus the multiplicity is retained (permitted transition, e.g. fluorescence from the first excited singlet state to the singlet ground state).

In contrast, processes in which the multiplicity changes ( intercombination ) are considered forbidden according to the common usage in spectroscopy ( intercombination prohibition ). More precisely, this means that they usually take place only to a small extent or “slowly” (i.e. statistically seldom). B. in phosphorescence (transition from the lowest excited triplet state to the singlet ground state).

Nuclear spin multiplicity

The spin of the nucleons and their orbital angular momentum result in the total spin of the nucleus. This is usually referred to as nuclear spin, although the orbital angular momentum of the nucleons also contributes. The total angular momentum of the atom - also known as atomic spin - results from nuclear angular momentum and envelope angular momentum and can take on the values so that the multiplicity is for and for . ${\ displaystyle I}$ ${\ displaystyle I}$ ${\ displaystyle J}$ ${\ displaystyle I + J, I + J-1, \ ldots, | IJ |}$ ${\ displaystyle 2I + 1}$ ${\ displaystyle I \ geq J}$ ${\ displaystyle 2Y + 1}$ ${\ displaystyle J> I}$

literature

Walter Greiner: Theoretical Physics: Quantum Mechanics - Introduction . Harri Deutsch Verlag, 2005, ISBN 978-3-8171-1765-9 ( limited preview in Google book search).
Hermann hook, Hans Christoph Wolf: atomic and quantum physics . Springer, 2003, ISBN 978-3-642-18519-9 ( limited preview in Google book search).

Individual evidence

^ Theo Mayer-Kuckuk: Nuclear Physics: An Introduction . Springer-Verlag, 2013, ISBN 3-322-84876-0 , p. 55 ( limited preview in Google Book search - there is a typo with multiplicity: it has to be called instead of ). ${\ displaystyle J> I}$ ${\ displaystyle J <I}$

[1] Theo Mayer-Kuckuk: Nuclear Physics: An Introduction . Springer-Verlag, 2013, ISBN 3-322-84876-0 , p. 55 ( limited preview in Google Book search - there is a typo with multiplicity: it has to be called instead of ). ${\ displaystyle J> I}$ ${\ displaystyle J <I}$