Angular momentum coupling

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Coupling in molecules
Rovibrational coupling
Vibronic coupling
Rovibronic coupling
Spin-spin coupling
Spin-orbit coupling

In quantum mechanics, the orbital and spin angular momentum of bodies can interact in angular momentum coupling. These interactions in atoms are used in spectroscopy.

In astronomy, spin-orbit coupling is the ratio between the frequency with which a planet or other celestial body spins about its own axis to that with which it orbits another body. This is more commonly known as orbital resonance.

General theory and detailed origin

In general, angular momentum coupling describes an interaction between the rotation of two objects. In quantum mechanics, coupling also exists between angular momenta belonging to different Hilbert spaces of a single object, e.g. its spin and its orbital angular momentum.

In quantum mechanics, one usually expands the quantum states of composed systems (i.e. made of subunits like two hydrogen atoms or two electrons) in basis sets which are made of direct products of quantum states which in turn describe the subsystems individually. If the subsystems are invariant with respect to rotations, according to Noether's theorem, they can be chosen as eigenstates of the angular momentum operator (and of its component along any arbitrary z axis). The subsystems are therefore correctly described by a set of l, m quantum numbers (see angular momentum for details). In that basis set, the elements of the total Hamiltonian that are not vanishing between basis functions characterized by different l, m quantum numbers couple the angular momenta of the subsystems. This happens in the case the total Hamiltonian does not commute with the angular operators acting on the sub systems only. The angular momentum coupling terms are the terms of the total Hamiltonian which do not commute with the subsystem individual angular momenta (or are not invariant with respect to the rotations of the subsystems).

Spin-orbit coupling

The behavior of atoms and smaller particles is well described by the theory of quantum mechanics, in which each particle has an intrinsic angular momentum called spin and specific configurations (of e.g. electrons in an atom) are described by a set of quantum numbers. Collections of particles also have angular momenta and corresponding quantum numbers, and under different circumstances the angular momenta of the parts add in different ways to form the angular momentum of the whole. Angular momentum coupling is a category including some of the ways that subatomic particles can interact with each other.

In atomic physics, spin-orbit coupling describes a weak magnetic interaction, or coupling, of the particle spin and the orbital motion of this particle, e.g. the electron spin and its motion around an atomic nucleus. One of its effects is to separate the energy of internal states of the atom, e.g. spin-aligned and spin-antialigned that would otherwise be identical in energy. This interaction is responsible for many of the details of atomic structure.

In Nuclear Magnetic Resonance spin-spin coupling is a kind of weak interaction between the spin of a nucleus and the spins of nearby nuclei. This interaction frequently causes an NMR resonance to split into multiple closely separated resonances. The splitting of NMR lines can be used to extract detailed information about the structure and conformation of molecules. The spin-spin coupling that exists between nuclear spin and electronic spin is responsible for the atomic hyperfine structure.

LS coupling

In light atoms electron spins s_i interact among themselves so they combine to form a total spin angular momentum S. The same happens with orbital angular momenta l_i, forming a single orbital angular momentum L. The interaction between the quantum numbers L and S is called Russell-Saunders coupling or LS coupling. Then S and L add together and form a total angular momentum J:

[\mathbf J = \mathbf L + \mathbf S] where [\mathbf L = \sum_i \mathbf_i] and [\mathbf S = \sum_i \mathbf_i]

This is an approximation which is good as long as any external magnetic fields are weak. In larger magnetic fields, these two momenta decouple, giving rise to a different splitting pattern in the energy levels (the Paschen-Back effect.), and the size of LS coupling term becomes small.

For an extensive example on how LS-coupling is practically applied, see the article on Term symbols.

jj coupling

In heavier atoms the situation is different. In atoms with bigger nuclear charges, spin-orbit interactions are frequently as large or larger than spin-spin interactions or orbit-orbit interactions. In this situation, each orbital angular momentum l_i tends to combine with each individual spin angular momentum s_i, originating individual total angular momenta j_i. These then add up to form the total angular momentum J

[\mathbf J = \sum_i \mathbf j_i = \sum_i (\mathbf_i + \mathbf_i)]

This description, facilitating calculation of this kind of interaction, is known as jj coupling.

Spin-spin coupling

Spin-spin coupling or spin-spin splitting is the coupling of the spin angular momentum states of different particles. H-NMR depends on the spin of the ¹H hydrogen nucleus (a proton). Multiple absorptions on the NMR spectrum arising from a single proton are due to the interaction among the hydrogen nuclei under examination and other nearby NMR-active nuclei.

The number of peaks in the spin-spin coupling are denoted as singlet, doublet, triplet, quartet, quintet etc. collectively called multiplets. The number of peaks in a simple multiplet for a proton is proportional to the number of equivalent ¹H atoms connected by two or three bonds to the proton being observed. That is, a proton bonded to a carbon is split by the protons bonded to any other carbons bonded to that carbon.

Protons that have x equivalent neighbouring protons have x + 1 peaks: this is called the x+1 rule. For example, if you have the molecule CH₃-CH₂-CH₃, there would be two multiplet peaks on the ¹H-NMR. The methyl (CH₃ group) protons would appear as a triplet (3), because the adjacent carbon has two protons on it (x = 2, number of peaks = 2 + 1=3). The methylene (CH₂ group) protons would appear as a septet (7), because there are two adjacent groups of three protons (x = 6, number of peaks = 6 + 1 = 7).

Also each multiplet has a ratio of intensities for its peaks, as follows (also see Pascal's triangle):

Singlet 1

Doublet 1:1

Triplet 1:2:1

Quartet 1:3:3:1

Quintet 1:4:6:4:1

Septet 1:6:15:20:15:6:1

This means that the peak in the middle of the multiplet will be the tallest peak, and its ratio depends on the number of peaks.

There are also more complex spin splitting patterns. These signals do not follow the peak intensity ratios as shown above, because of overlapping signals from adjacent protons. These complex splitting signals mostly occur with cyclic and aromatic compounds.

Term symbols

Term symbols are used to represent the states and spectral transitions of atoms, they are found from coupling of angular momenta mentioned above. When the state of an atom has been specified with a term symbol, the allowed transitions can be found by applying selection rules found by considering which transitions that would conserve angular momentum. A photon has spin 1, and when there is a transition with emission or absorption of a photon the atom will need to change state to conserve angular momentum. The term symbol selection rules are. ΔS=0, ΔL=0,±1, Δl=±1, ΔJ=0,±1

Relativistic effects

In very heavy atoms, relativistic shifting of the energies of the electron energy levels accentuates the spin-orbit coupling effect, normally very weak and neglected to first order when atomic physics is first taught to chemistry students. Thus, for example, uranium molecular orbital diagrams must directly incorporate relativistic symbols when considering interactions with other atoms.

Nuclear coupling

In atomic nuclei, the spin-orbit interaction is much stronger than for atomic electrons, and is incorporated directly into the nuclear shell model. In addition, unlike atomic-electron term symbols, the lowest energy state is not L - S, but rather, l + s. All nuclear levels whose l value (orbital angular momentum) is greater than zero are thus split in the shell model to create states designated by l + s and l - s. Due to the nature of the shell model, which assumes an average potential rather than a central Coulombic potential, the nucleons that go into the l + s and l - s nuclear states are considered degenerate within each orbital (e.g. The 2p3/2 contains four nucleons, all of the same energy. Higher in energy is the 2p1/2 which contains two equal-energy nucleons).

External links

• [LS and jj coupling]
• [Term symbol]

 

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