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Angular momentum coupling

Angular momentum coupling is a fundamental procedure in quantum mechanics for constructing the total angular momentum of a system from the individual angular momenta of its constituents, such as electrons in an atom, by combining orbital and spin components according to specific quantum rules.^[1] This process is essential for understanding the structure of atomic and molecular energy levels, as it determines how the magnitudes and orientations of these angular momenta interact to produce quantized total states.^[2] In multi-electron atoms, two primary coupling schemes are employed: LS coupling (also known as Russell-Saunders coupling), where the total orbital angular momentum L and total spin angular momentum S are first found, and then coupled to give the total angular momentum J; this scheme predominates in lighter atoms due to weaker spin-orbit interactions.^[1] Conversely, jj coupling pairs the orbital and spin angular momenta of each individual electron before combining these pairs, which becomes more relevant in heavier atoms where relativistic effects strengthen spin-orbit coupling. The resulting states are denoted by term symbols of the form ^{2S+1}L_J, where the letter denotes the value of the total orbital angular momentum quantum number L (e.g., S for L=0, P for L=1), $2S+1 is the spin multiplicity with S the total spin quantum number, and J the total angular momentum quantum number.^[1] The mathematical framework for this coupling relies on Clebsch-Gordan coefficients, which provide the expansion coefficients for expressing coupled total angular momentum eigenstates in terms of uncoupled basis states, ensuring conservation of the total z-component and satisfying triangle inequalities for the quantum numbers.^[3] These coefficients are crucial for calculating transition probabilities and selection rules in spectroscopy, explaining phenomena like the fine structure splitting in atomic spectral lines and the anomalous Zeeman effect, where magnetic fields further split energy levels based on J.^[2] Applications extend to nuclear physics, quantum computing, and molecular dynamics, where precise angular momentum descriptions underpin reactivity and optical properties.^[4]

Fundamentals of Angular Momentum

Definition and Quantum Operators

Angular momentum coupling refers to the quantum mechanical process of combining multiple angular momentum vectors to determine the total angular momentum of a system, such as the total angular momentum \mathbf{J} resulting from the vector sum of orbital angular momentum \mathbf{L} and spin angular momentum \mathbf{S}.^[5] This coupling arises in systems like atoms or particles where individual angular momenta interact, leading to quantized total states that describe the system's rotational properties. In quantum mechanics, the orbital angular momentum operator is defined as \mathbf{L} = \mathbf{r} \times \mathbf{p}, where \mathbf{r} is the position operator and \mathbf{p} is the momentum operator.^[6] The spin angular momentum \mathbf{S} is an intrinsic operator associated with particles, such as electrons, and does not derive from spatial motion.^[7] These operators satisfy the fundamental commutation relations of angular momentum algebra, given by

[L_x, L_y] = i \hbar L_z, \quad [L_y, L_z] = i \hbar L_x, \quad [L_z, L_x] = i \hbar L_y,

with cyclic permutations for the spin operators as well.^[8] These relations ensure that the components of angular momentum do not commute, reflecting the non-classical nature of quantum rotation.^[9] The eigenvalues of the angular momentum operators characterize the possible measurement outcomes. For orbital angular momentum, the magnitude squared L^2 has eigenvalues \hbar^2 l(l+1) where l = 0, 1, 2, \dots, and the z-component L_z has eigenvalues m_l \hbar with m_l = -l, -l+1, \dots, +l.^[10] For electron spin, the spin quantum number is s = 1/2, yielding S^2 eigenvalue \hbar^2 s(s+1) = (3/4) \hbar^2 and S_z eigenvalues \pm (1/2) \hbar.^[11] The theoretical framework for angular momentum operators and their coupling emerged in the 1920s through the development of quantum mechanics, with key contributions from Werner Heisenberg's matrix mechanics in 1925 and Erwin Schrödinger's wave mechanics in 1926, which incorporated quantized angular momentum to explain atomic spectra.^[12]

Conservation and Commutation Relations

In quantum mechanics, the conservation of angular momentum arises from the rotational invariance of the physical laws governing a system, as encapsulated by Noether's theorem. This theorem establishes a correspondence between continuous symmetries of the action and conserved quantities; specifically, invariance under infinitesimal rotations leads to the conservation of total angular momentum.^[13] For a Hamiltonian H that is rotationally invariant—meaning it commutes with the generators of rotations—the total angular momentum \mathbf{J} is a conserved observable.^[14] The total angular momentum operator \mathbf{J} is the vector sum of contributions from orbital angular momentum \mathbf{L}, spin \mathbf{S}, and any other relevant angular momenta, such as nuclear spin. Conservation holds if the commutator [H, \mathbf{J}] = 0, ensuring that the time evolution does not alter the expectation value of \mathbf{J}.^[15] This condition implies that \mathbf{J} shares eigenstates with H, making the eigenvalues of \mathbf{J}^2 and J_z good quantum numbers for energy eigenstates in such systems.^[16] The algebraic structure underlying angular momentum is the Lie algebra of the special unitary group SU(2), which governs the commutation relations of the components: [J_x, J_y] = i \hbar J_z and cyclic permutations.^[17] To facilitate the representation theory, ladder operators are defined as J_\pm = J_x \pm i J_y, which satisfy [J_z, J_\pm] = \pm \hbar J_\pm and raise or lower the magnetic quantum number m_j by \hbar while preserving the total angular momentum quantum number j. These operators generate the irreducible representations of SU(2), with dimension $2j + 1 for each j.^[18] In isolated atoms or molecules, where external torques are absent and the Hamiltonian is spherically symmetric, the total angular momentum \mathbf{J} is conserved, serving as a good quantum number that labels the states. For instance, in the absence of external fields, the eigenstates of an atom can be classified by the total j and m_j, enabling the systematic coupling of individual angular momenta within the system.^[16]

Addition of Angular Momenta

General Formalism

Angular momentum coupling in quantum mechanics provides the mathematical framework for combining the angular momenta of multiple particles or subsystems to form a total angular momentum. In the vector model, the total angular momentum operator \mathbf{J} is defined as the vector sum \mathbf{J} = \mathbf{j_1} + \mathbf{j_2}, where \mathbf{j_1} and \mathbf{j_2} are the individual angular momentum operators for two subsystems, assuming they commute [\mathbf{j_1}, \mathbf{j_2}] = 0.^[19] The possible magnitudes j of the total angular momentum satisfy the inequality |j_1 - j_2| \leq j \leq j_1 + j_2, with j, j_1, and j_2 differing by integers, reflecting the classical vector addition analogy quantized in quantum theory.^[20] The states in the coupled representation, denoted |j_1 j_2 j m\rangle (or simply |j m\rangle), are simultaneous eigenstates of the operators \mathbf{J}^2, \mathbf{J_z}, \mathbf{j_1}^2, and \mathbf{j_2}^2, with eigenvalues j(j+1)\hbar^2, m\hbar, j_1(j_1+1)\hbar^2, and j_2(j_2+1)\hbar^2, respectively (setting \hbar = 1 hereafter). In contrast, the uncoupled representation uses states |j_1 m_1, j_2 m_2\rangle, which are eigenstates of \mathbf{j_1}^2, \mathbf{j_2}^2, \mathbf{j_{1z}}, and \mathbf{j_{2z}}, but not necessarily of \mathbf{J}^2. The transformation between these bases is given by

|j m\rangle = \sum_{m_1, m_2} \langle j_1 m_1 j_2 m_2 | j m \rangle |j_1 m_1, j_2 m_2\rangle,

where the coefficients \langle j_1 m_1 j_2 m_2 | j m \rangle ensure orthogonality and completeness, with the sum restricted to m_1 + m_2 = m. This change of basis preserves the total projection m and diagonalizes the total angular momentum.^[19]^[20] The squared total angular momentum operator expands as \begin{equation*} \mathbf{J}^2 = \mathbf{j_1}^2 + \mathbf{j_2}^2 + 2 \mathbf{j_1} \cdot \mathbf{j_2}, \end{equation*} which highlights the interaction term $2 \mathbf{j_1} \cdot \mathbf{j_2} that couples the subsystems, leading to the degeneracy and splitting patterns in energy levels when the Hamiltonian commutes with \mathbf{J}^2. In the coupled basis, \mathbf{J}^2 |j m\rangle = j(j+1) |j m\rangle, simplifying calculations for systems invariant under rotations.^[19] For combining more than two angular momenta, recoupling schemes extend the formalism using transformation coefficients such as the Wigner 6j symbols for three angular momenta, which relate different pairing schemes like ( \mathbf{j_1} + \mathbf{j_2} ) + \mathbf{j_3} to \mathbf{j_1} + ( \mathbf{j_2} + \mathbf{j_3} ), and the 9j symbols for four, connecting multiple coupling paths in multi-particle systems. These symbols ensure the invariance of the total angular momentum under rebracketing and are essential for irreducible tensor methods in quantum mechanics.^[20]

Clebsch-Gordan Coefficients

Clebsch-Gordan coefficients, denoted \langle j_1 m_1 j_2 m_2 | J M \rangle, are the numerical factors that express the expansion of coupled angular momentum states in terms of uncoupled product states, or vice versa, in the context of adding two angular momenta \mathbf{J}_1 and \mathbf{J}_2 to form a total \mathbf{J} = \mathbf{J}_1 + \mathbf{J}_2.^[21] Specifically, they represent the overlap integrals between the uncoupled basis states |j_1 m_1\rangle |j_2 m_2\rangle and the coupled basis states | (j_1 j_2) J M \rangle, where M = m_1 + m_2 and J ranges from |j_1 - j_2| to j_1 + j_2 in integer steps.^[22] These coefficients ensure the coupled states transform irreducibly under rotations, preserving the total angular momentum quantum numbers.^[21] The coefficients are computed recursively by starting from the highest-weight state, where M = J = j_1 + j_2, for which \langle j_1 j_1 j_2 j_2 | J J \rangle = 1, and applying the lowering operator \mathbf{J}_- = \mathbf{J}_{1-} + \mathbf{J}_{2-} successively to generate lower M states within each J multiplet.^[23] This method exploits the commutation relations of angular momentum operators to derive relations that connect coefficients for adjacent M values, ensuring phase consistency via the Condon-Shortley convention where the coefficient for the highest state is positive real.^[21] Equivalent recursion formulas exist in terms of related 3j symbols. For example, the recursion in the projections m_2 and m_3 is

(j_1(j_1+1) - j_2(j_2+1) - j_3(j_3+1) - 2 m_2 m_3) \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix} = \sqrt{(j_2 - m_2)(j_2 + m_2 + 1)(j_3 - m_3 + 1)(j_3 + m_3)} \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 + 1 & m_3 - 1 \end{pmatrix} + \sqrt{(j_2 - m_2 + 1)(j_2 + m_2)(j_3 - m_3)(j_3 + m_3 + 1)} \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 - 1 & m_3 + 1 \end{pmatrix},

with Clebsch-Gordan coefficients related by \langle j_1 m_1 j_2 m_2 | J M \rangle = (-1)^{j_1 - j_2 + M} \sqrt{2J + 1} \begin{pmatrix} j_1 & j_2 & J \\ m_1 & m_2 & -M \end{pmatrix}.^[23] For common low-dimensional cases, explicit values are tabulated. Consider the coupling of two spin-1/2 angular momenta (j_1 = j_2 = 1/2), yielding total J = 1 (triplet, symmetric) or J = 0 (singlet, antisymmetric). The non-zero coefficients are: | j_1 | m_1 | j_2 | m_2 | J | M | \langle j_1 m_1 j_2 m_2 | J M \rangle | |---------|---------|---------|---------|-------|-------|---------------------------------------| | 1/2 | 1/2 | 1/2 | 1/2 | 1 | 1 | 1 | | 1/2 | 1/2 | 1/2 | -1/2 | 1 | 0 | \sqrt{1/2} | | 1/2 | -1/2 | 1/2 | 1/2 | 1 | 0 | \sqrt{1/2} | | 1/2 | -1/2 | 1/2 | -1/2 | 1 | -1 | 1 | | 1/2 | 1/2 | 1/2 | -1/2 | 0 | 0 | \sqrt{1/2} | | 1/2 | -1/2 | 1/2 | 1/2 | 0 | 0 | -\sqrt{1/2} | These values illustrate the symmetry properties and normalization.^[21] The Clebsch-Gordan coefficients satisfy orthogonality and completeness relations, such as

\sum_{m_1 m_2} \langle j_1 m_1 j_2 m_2 | J M \rangle \langle j_1 m_1 j_2 m_2 | J' M' \rangle = \delta_{J J'} \delta_{M M'},

which confirms the coupled states form an orthonormal basis, and the sum rule

\sum_{J} (2J + 1) |\langle j_1 m_1 j_2 m_2 | J M \rangle|^2 = (2j_1 + 1)(2j_2 + 1),

expressing the dimensionality of the tensor product space.^[23] These properties ensure unitarity of the transformation matrix between bases.^[21] In applications, Clebsch-Gordan coefficients facilitate basis changes in time-independent perturbation theory, where uncoupled representations of perturbations (e.g., in molecular or nuclear systems) are projected onto coupled states to compute energy corrections. They are also central to the Wigner-Eckart theorem, which factorizes matrix elements of tensor operators T_q^{(k)} as \langle \alpha J M | T_q^{(k)} | \alpha' J' M' \rangle = (-1)^{J - M} \begin{pmatrix} J & k & J' \\ -M & q & M' \end{pmatrix} \langle \alpha J || T^{(k)} || \alpha' J' \rangle, reducing the number of independent elements via the reduced matrix element and a single Clebsch-Gordan coefficient (or equivalent 3j symbol). This theorem simplifies calculations in atomic and particle physics spectra.^[21]

Atomic Coupling Schemes

LS Coupling

LS coupling, also known as Russell-Saunders coupling, is a scheme used to describe the angular momentum coupling in multi-electron atoms, particularly those with low atomic numbers where electrostatic interactions dominate over spin-orbit effects. In this approximation, the orbital angular momenta \mathbf{l}_i of the individual electrons are first vectorially added to yield the total orbital angular momentum \mathbf{L} = \sum_i \mathbf{l}_i, while their spin angular momenta \mathbf{s}_i are similarly coupled to form the total spin angular momentum \mathbf{S} = \sum_i \mathbf{s}_i. The resulting \mathbf{L} and \mathbf{S} are then coupled to produce the total angular momentum \mathbf{J} = \mathbf{L} + \mathbf{S}. This stepwise coupling is valid for light atoms because the spin-spin and orbit-orbit interactions are stronger than the spin-orbit interaction, allowing the latter to be treated as a perturbation after establishing the LS basis. The possible values of the quantum numbers follow from the addition rules: L ranges from the maximum possible value (\sum l_i) down to 0 (or 1 if required by symmetry) in integer steps, subject to selection rules and Pauli exclusion for equivalent electrons; S from \sum s_i down to 0 or 1/2 in steps of 1; and J from |L - S| to L + S in unit steps. Clebsch-Gordan coefficients determine the allowed combinations when adding \mathbf{L} and \mathbf{S}. The energy levels in the absence of spin-orbit coupling depend primarily on L and S, with degeneracy lifted by the perturbation, leading to multiplets split by J. This scheme provides a good description of atomic spectra for elements up to approximately zinc, where the fine structure is relatively small. Atomic energy levels are denoted by term symbols of the form ^{2S+1}L_J, where $2S+1 is the spin multiplicity, L is symbolized by letters (S for 0, P for 1, D for 2, F for 3, etc.), and J is the total angular momentum quantum number. The ground state term for a given configuration is predicted by Hund's rules: (1) the state with maximum S has the lowest energy due to minimized electron repulsion from parallel spins occupying different orbitals; (2) for a given S, the maximum L is lowest in energy from reduced Coulomb repulsion; (3) for less than half-filled shells, the minimum J (i.e., J = |L - S|) is the ground state, while for more than half-filled, the maximum J ( J = L + S ) is lowest. These rules arise from the variational principle applied to the electrostatic Hamiltonian and have been verified experimentally across many atomic spectra.^[24] A representative example is the carbon atom in its ground-state configuration $1s^2 2s^2 2p^2. The two equivalent p electrons (l=1, s=1/2) yield possible terms ^3P, ^1D, and ^1S, determined by Pauli exclusion and angular momentum addition, with nine microstates distributed as five for ^3P (S=1, L=1), three for ^1D (S=0, L=2), and one for ^1S (S=0, L=0). Applying Hund's rules, the ground term is ^3P (maximum S=1, then maximum L=1), further split into J=0,1,2 levels by spin-orbit interaction, with ^3P_0 as the lowest energy state. This matches observed spectral lines in the ultraviolet region.

jj Coupling

In the jj coupling scheme, the orbital angular momentum \mathbf{l}_i and spin angular momentum \mathbf{s}_i of each individual electron are coupled to form its total angular momentum \mathbf{j}_i = \mathbf{l}_i + \mathbf{s}_i, with possible values j_i = l_i \pm 1/2 for electrons with spin s_i = 1/2.^[25] The total angular momentum \mathbf{J} of the atom is then obtained by coupling all the individual \mathbf{j}_i: \mathbf{J} = \sum \mathbf{j}_i.^[26] This approach applies primarily to heavy atoms, where the spin-orbit interaction dominates over the electrostatic repulsion between electrons, making the individual spin-orbit couplings the primary interaction.^[26] In jj coupling, neither the total orbital angular momentum \mathbf{L} = \sum \mathbf{l}_i nor the total spin \mathbf{S} = \sum \mathbf{s}_i is conserved as a good quantum number; only the total J remains a good quantum number.^[25] The prevalence of jj coupling in heavy atoms stems from the rapid increase in spin-orbit interaction strength with atomic number Z, scaling approximately as Z^4 for hydrogen-like systems, which outpaces the slower Z-dependence of electron-electron repulsion.^[27] This makes the scheme particularly suitable for elements with Z > 40, such as lead (Z=82) or bismuth (Z=83), where pure jj coupling or near-jj limits describe the spectra more accurately than alternatives.^[26] For the mercury atom (Z=80) in its 6s6p excited configuration, the 6s electron contributes j=1/2, while the 6p electron contributes j=1/2 or j=3/2, yielding levels designated as ^3P_0, ^3P_1, and ^3P_2 (with ^1P_1 higher up).^[28] In atoms of intermediate Z, such as mercury, the energy levels often exhibit intermediate coupling, blending jj and LS characteristics to account for comparable spin-orbit and electron repulsion strengths.^[28]

Spin-Orbit Interaction

Physical Origin

The spin-orbit interaction arises fundamentally from the relativistic coupling between an electron's spin and its orbital motion in the electric field of the nucleus. In a classical electromagnetic picture, an electron orbiting the nucleus experiences an electric field \mathbf{E} produced by the nucleus. From the electron's rest frame, special relativity transforms this electric field into a magnetic field \mathbf{B} = -\frac{\mathbf{v} \times \mathbf{E}}{c^2}, where \mathbf{v} is the electron's velocity and c is the speed of light.^[29] This magnetic field exerts a torque on the electron's spin magnetic moment \boldsymbol{\mu} = -\frac{e g}{2 m c} \mathbf{S}, where e is the electron charge, g \approx 2 is the Landé g-factor, m is the electron mass, and \mathbf{S} is the spin angular momentum. The interaction energy is U = -\boldsymbol{\mu} \cdot \mathbf{B}, leading to an effective coupling between \mathbf{L} (orbital angular momentum) and \mathbf{S}. However, a naive calculation overestimates the precession rate by a factor of 2; Llewellyn Thomas resolved this in 1926 by accounting for the relativistic Thomas precession of the electron's spin frame, which contributes an equal but opposite effect, yielding a net factor of $1/2.^[29] In quantum mechanics, this manifests as the spin-orbit Hamiltonian H_{SO} = \xi(r) \mathbf{L} \cdot \mathbf{S}, where the radial function \xi(r) incorporates the Thomas factor: \xi(r) = \frac{1}{2 m^2 c^2} \frac{1}{r} \frac{dV}{dr} for a central potential V(r), with the $1/2 arising from the precession correction.^[29]^[30] A fully relativistic derivation emerges from the Dirac equation via the Foldy-Wouthuysen transformation, which expands the 4-component Dirac Hamiltonian H = c \boldsymbol{\alpha} \cdot \mathbf{p} + \beta m c^2 + V(r) into a non-relativistic 2-component form, decoupling positive and negative energy states order by order in v/c. At order (v/c)^2, even terms include the spin-orbit interaction; applying unitary transformations U = e^{iS} with S = -\frac{i}{2m c} \beta \boldsymbol{\alpha} \cdot \mathbf{p} eliminates odd operators, yielding the effective Hamiltonian with H_{SO} = \frac{1}{2 m^2 c^2} \frac{1}{r} \frac{dV}{dr} \mathbf{L} \cdot \mathbf{S} for a Coulomb potential.^[30] For hydrogen-like atoms with nuclear charge Z e, the strength scales as \xi \propto Z^4 / n^3, where n is the principal quantum number, due to the Z dependence in V(r) = -Z e^2 / r and the expectation value \langle 1/r^3 \rangle \propto Z^3 / n^3.^[31]

Effects on Atomic Spectra

Spin-orbit coupling introduces fine structure to atomic energy levels by splitting each term characterized by orbital angular momentum quantum number L and spin angular momentum quantum number S into $2S + 1 sublevels labeled by the total angular momentum quantum number J, ranging from J = |L - S| to J = L + S in integer steps. This splitting arises from the perturbative treatment of the spin-orbit Hamiltonian H_{SO}, which couples the spin and orbital motions of electrons. The energy shift for a given J is given by

\langle H_{SO} \rangle = \frac{1}{2} \xi [J(J+1) - L(L+1) - S(S+1)],

where \xi is the spin-orbit coupling constant, positive for less-than-half-filled shells and leading to an inverted ordering for more-than-half-filled shells.^[32] The Landé interval rule governs the spacings between these fine-structure levels, stating that the energy separation \Delta E between consecutive J values (specifically, between J and J+1) is proportional to J + 1, i.e., \Delta E \propto J + 1. This linear increase in intervals reflects the quadratic dependence in the energy formula, ensuring that higher-J levels are more widely separated. In atomic spectra, this fine structure manifests as closely spaced lines, resolvable only with high-resolution instruments, and the rule holds well in the regime of weak coupling where perturbation theory applies.^[32]^[33] A prominent example is the sodium D-line doublet in the emission spectrum of neutral sodium atoms, arising from transitions between the unsplit $3s ^2S_{1/2} ground state and the spin-orbit-split $3p ^2P excited state levels ^2P_{3/2} and ^2P_{1/2}. The wavelengths are 588.9950 nm for the ^2P_{3/2} \to ^2S_{1/2} (D_2) transition and 589.5924 nm for the ^2P_{1/2} \to ^2S_{1/2} (D_1) transition, corresponding to an energy splitting of approximately 17.2 cm^{-1} or 2.13 meV in the $3p state. This splitting, much smaller than the coarse structure separation to higher levels, exemplifies how spin-orbit effects refine the spectral lines observed in alkali metal vapors.^[34] The vector model of angular momentum further predicts the relative intensities (branching ratios) of these fine-structure components through the Clebsch-Gordan coefficients governing transition probabilities. For electric dipole transitions like the sodium D-lines, the intensity ratio follows the statistical weights $2J + 1; thus, the D_2 line (J = 3/2) is twice as intense as the D_1 line (J = 1/2), with an observed ratio near 2:1 under equilibrium conditions. This arises because the vector addition of \mathbf{L} and \mathbf{S} to form \mathbf{J} determines the overlap of wavefunctions, ensuring conservation of angular momentum in photon emission.^[35]

Spin-Spin Interaction

Electron-Electron Spin Coupling

The electron-electron spin coupling refers to the magnetic interactions between the spins of different electrons in an atom, distinct from spin-orbit effects. This coupling originates from the classical magnetic dipole-dipole interaction between the magnetic moments associated with electron spins, leading to an anisotropic contribution to the atomic Hamiltonian. In multi-electron atoms, this interaction is relatively weak compared to the Coulomb repulsion but plays a role in fine-structure corrections, particularly for states with multiple unpaired spins.^[36] The Hamiltonian describing the direct magnetic dipole-dipole spin-spin interaction is

H_{\mathrm{SS}} = -\frac{\mu_0 g_e^2 \mu_B^2}{4\pi} \sum_{i < j} \frac{\mathbf{S}_i \cdot \mathbf{S}_j - 3 (\mathbf{S}_i \cdot \hat{\mathbf{r}}_{ij}) (\mathbf{S}_j \cdot \hat{\mathbf{r}}_{ij}) }{r_{ij}^3},

where \mathbf{S}_i and \mathbf{S}_j are the spin angular momentum operators of electrons i and j, r_{ij} is the interelectronic distance, \hat{\mathbf{r}}_{ij} = \mathbf{r}_{ij}/r_{ij} is the unit vector along the line joining the electrons, g_e \approx 2 is the electron g-factor, and \mu_B is the Bohr magneton. This form arises from the relativistic Breit interaction in the Pauli approximation and is evaluated as a perturbation in non-relativistic atomic calculations. The anisotropic nature of this term—due to the angular dependence through \hat{\mathbf{r}}_{ij}—leads to orientation-dependent energy shifts that partially lift degeneracies within multiplet levels.^[36] In addition to the direct dipole interaction, an isotropic component of the electron-electron spin coupling emerges from the quantum exchange symmetry required by the Pauli exclusion principle. This exchange effect manifests as an effective Heisenberg Hamiltonian of the form J \mathbf{S}_1 \cdot \mathbf{S}_2, where J is the exchange parameter determined by the spatial overlap of electron wavefunctions and the Coulomb interaction; positive J favors antiparallel spins (singlet states) with lower energy. Unlike the dipole term, this isotropic exchange dominates the gross splitting between states of different total spin multiplicity and is incorporated perturbatively in the LS coupling scheme, where the total spin \mathbf{S} is the vector sum of individual electron spins. In multi-electron atoms, such exchange contributions, combined with the dipole term, cause fine splittings within spectroscopic terms of given L and S. A representative example is the helium atom in the $1s2s configuration, where the energy separation between the triplet (^3S) and singlet (^1S) states is approximately 0.78 eV, largely due to the isotropic exchange interaction expressed in Heisenberg form, with a smaller direct spin-spin dipole contribution of order $10^{-4} eV affecting the fine structure within the triplet. This exchange-driven splitting highlights how spin coupling enforces antisymmetric wavefunctions, lowering the energy of the spatially symmetric singlet relative to the antisymmetric triplet. In heavier multi-electron atoms, analogous spin-spin effects contribute to interval splittings in terms like ^3P, where the dipole term introduces deviations from the rigid Landé interval rule predicted by spin-orbit alone.

Impact on Energy Levels

The electron-electron spin-spin interaction introduces perturbative shifts to the energy levels of multi-electron atoms within a given LS term, where the anisotropic components cause small deviations from the Landé interval rule. The isotropic scalar part of the interaction, arising from quantum exchange effects in the Coulomb repulsion, dominates the splittings between different total spin multiplicities and can be expressed using the total spin \mathbf{S}, yielding a term of the form (A/2) [S(S+1) - \sum s_k(s_k+1)], with A < 0 typically leading to lower energy for larger total spin multiplicity. These exchange shifts are distinct from the magnetic contributions in the Breit-Pauli Hamiltonian, which are smaller and provide fine-structure corrections at order \alpha^2 Ry.^[37] In the context of Hund's rules, the spin-spin interaction promotes ferromagnetic alignment of electron spins in half-filled shells, favoring the maximum total spin S for the ground state to minimize overall energy through reduced Pauli repulsion in the antisymmetric spatial wavefunction. This alignment effect is particularly pronounced in configurations like p³ or d⁵, where parallel spins allow electrons to occupy degenerate orbitals without violating the Pauli exclusion principle, lowering the Coulomb repulsion energy.^[38] The spin-spin contribution to fine structure is typically smaller than that from spin-orbit coupling but becomes relatively more significant in light atoms (Z ≲ 20), where the spin-orbit term scales with Z⁴ and is weaker, allowing the spin-spin effects to influence level separations on the order of 0.1–1 cm⁻¹. In such systems, the interaction refines the ordering of multiplet components without dominating the overall term structure.^[37]

Term Symbols

Construction and Notation

The construction of term symbols begins with the electron configuration of an atom, where the total orbital angular momentum quantum number L and total spin angular momentum quantum number S are determined by coupling the individual electron angular momenta under the LS (Russell-Saunders) coupling scheme.^[39] The basic notation for a term symbol is ^{2S+1}L_J, where L is represented by a capital letter (S for L=0, P for L=1, D for L=2, etc.), $2S+1 is the spin multiplicity indicating the number of possible spin projections, and J is the total angular momentum quantum number ranging from |L - S| to L + S.^[39] This notation encapsulates the spectroscopic state of the atom, providing insight into its energy level and magnetic properties.^[39] The Pauli exclusion principle mandates that the total wavefunction of electrons must be antisymmetric under particle exchange, which imposes restrictions on the allowed combinations of orbital and spin states, particularly for equivalent electrons occupying the same subshell.^[39] For equivalent electrons, this antisymmetry requires that no two electrons share the same set of quantum numbers (n, l, m_l, m_s), leading to the exclusion of certain microstates—specific assignments of individual m_l and m_s values.^[40] Tables of allowed terms for common equivalent electron configurations, such as p^2 or d^3, are derived systematically to enumerate valid states while enforcing this principle, avoiding symmetric or antisymmetric violations in the spatial and spin parts of the wavefunction.^[39] To build the possible terms, one enumerates all allowed microstates and groups them by their total projections M_L = \sum m_l and M_S = \sum m_s, then subtracts the highest-weight states to identify the underlying L and S values.^[40] For the p^2 configuration (two equivalent p electrons with l=1), the possible M_L ranges from -2 to 2 and M_S from -1 to 1, yielding 15 microstates after Pauli exclusion; these decompose into the terms ^3P (S=1, L=1), ^1D (S=0, L=2), and ^1S (S=0, L=0), with no ^3D, ^1P, or other combinations permitted due to antisymmetry requirements.^[39]^[40] The full term symbol incorporates J to account for spin-orbit coupling, as in ^3P_0, ^3P_1, ^3P_2 for the p^2 terms, and may include a superscript for parity (even or odd, based on \sum l) and a prefix for the configuration, such as

2p^2 \, ^1D

.^[39] For complex atoms with many electrons or open shells, single-configuration approximations often fail, and modern computational methods like configuration interaction (CI) are employed to mix multiple configurations into accurate many-electron states, yielding refined term symbols that better match observed spectra. In CI approaches, Slater determinants from various configurations are linearly combined to satisfy the antisymmetry and diagonalize the Hamiltonian, enabling precise determination of terms in heavy or excited atoms.

Allowed Terms and Selection Rules

In the LS coupling scheme, the allowed terms for configurations involving equivalent electrons, such as d^n or f^n shells, are those that comply with the Pauli exclusion principle, ensuring the total wavefunction remains antisymmetric. These terms are identified by constructing all possible combinations of total orbital angular momentum L and total spin S, then applying projection operators to eliminate states that would violate the exclusion principle for indistinguishable electrons. For non-equivalent electrons, all vector couplings within the allowed ranges of L and S are permitted, but for equivalent electrons, many are forbidden; tables derived from systematic enumeration provide the complete lists for specific cases.^[41] Representative examples illustrate this restriction. For the p^2 configuration (l=1, N=2), the allowed terms are ^{3}P, ^{1}D, and ^{1}S, excluding higher-spin or duplicate L states that would require symmetric spatial parts for parallel spins. For the d^2 configuration (l=2, N=2), the permitted terms are ^{1}S, ^{3}P, ^{1}D, ^{3}F, and ^{1}G; the ground term is typically ^{3}F by Hund's rules. Similar tables exist for d^n up to d^{10} and f^n up to f^{14}, with recurring terms (e.g., multiple ^{2}D in d^3) distinguished by seniority numbers or labels like A/B for f^5 and f^9. These are compiled in standard references using group-theoretical methods or computational enumeration.^[41]^[42] For optical transitions between these terms, electric dipole (E1) radiation governs the allowed processes in LS coupling, subject to strict selection rules derived from the symmetry of the dipole operator. The rules are \Delta S = 0, \Delta L = 0, \pm 1 (excluding $0 \leftrightarrow 0), and \Delta J = 0, \pm 1 (excluding $0 \leftrightarrow 0), where the \Delta J restriction arises from conservation of angular momentum projected along the quantization axis. These ensure that only transitions preserving total spin multiplicity while changing the orbital configuration modestly are permitted, forbidding intercombination lines like singlet-to-triplet.^[43] A complementary requirement is the Laporte rule, which mandates a change in parity (\Delta \pi = -1, or g \leftrightarrow u) for the transition to be allowed, as the electric dipole operator is odd under spatial inversion. Atomic parity is determined by (-1)^{\sum l_i}, even (g) for even total l sum and odd (u) otherwise; thus, transitions within the same configuration (e.g., d \to d) are often forbidden unless vibronic or other perturbations mix parities. This rule, combined with the others, sharply limits observable spectral lines in light atoms.^[43] The relative intensities of allowed transitions, quantified by line strengths S, depend on the squared matrix element |\langle \psi_f | \mathbf{r} | \psi_i \rangle|^2, where \mathbf{r} is the dipole operator. In LS coupling, this factors into a radial integral (configuration-dependent) and an angular integral involving 3j or 6j symbols related to Clebsch-Gordan coefficients, which encode the \Delta L, \Delta S, and \Delta J selections while weighting the coupling probabilities. For instance, the line strength for a specific J \to J' transition within allowed terms is S = (2J+1) |\langle \alpha L S J || \mathbf{C}^{(1)} || \alpha' L' S' J' \rangle|^2, with the reduced matrix element incorporating CG coefficients; tables of these coefficients facilitate computation for d^n and f^n spectra.^[43]^[44]

Advanced Coupling Effects

Relativistic Corrections

In the Dirac theory of the hydrogen atom, relativistic corrections to angular momentum coupling arise naturally from the relativistic kinematics of the electron and its spin-orbit interaction, providing an exact treatment within the one-electron framework. The Dirac equation incorporates both the relativistic mass increase and the spin-magnetic moment coupling to the electromagnetic field, leading to energy levels that depend on the principal quantum number n and the total angular momentum quantum number j, but are degenerate in the orbital angular momentum l. This fine structure resolves the degeneracy of the non-relativistic Schrödinger levels, with the energy given by

E_{n j} = m c^2 \left[ 1 + \left( \frac{Z \alpha}{n - (j + 1/2) + \sqrt{(j + 1/2)^2 - (Z \alpha)^2}} \right)^2 \right]^{-1/2},

where m is the electron mass, c the speed of light, Z the atomic number, and \alpha the fine-structure constant; for low Z, this approximates to a relativistic correction of order \alpha^4 m c^2 / n^3. This formulation exactly accounts for the spin-orbit coupling in hydrogen-like atoms, predicting the observed splitting of spectral lines without additional ad hoc terms. Beyond the Dirac fine structure, quantum electrodynamics (QED) introduces further corrections due to vacuum fluctuations and radiative effects, most notably the Lamb shift, which lifts the $2S_{1/2}-$2P_{1/2} degeneracy not captured by the Dirac equation. The Lamb shift arises from the electron's interaction with virtual photons, effectively renormalizing the electron's self-energy and causing a small upward shift in s-states relative to p-states, on the order of $1057 MHz for the n=2 levels in hydrogen. Hans Bethe's seminal non-relativistic calculation treated the shift as the principal-value integral over vacuum excitations, yielding a value in close agreement with experiment and heralding the renormalization program in QED. Higher-order QED contributions, including vertex corrections and vacuum polarization, refine this to parts per million, confirming the theory's predictive power for angular momentum-coupled states. For multi-electron atoms, the Breit interaction provides relativistic corrections to the electron-electron coupling, accounting for retarded transverse photon exchange that modifies spin-spin, spin-other-orbit, and orbit-orbit interactions beyond the non-relativistic Darwin term. The Breit Hamiltonian includes terms like -\frac{\alpha}{2 r_{12}} (\boldsymbol{\alpha}_1 \cdot \boldsymbol{\alpha}_2 + \frac{(\boldsymbol{\alpha}_1 \cdot \mathbf{r}_{12})(\boldsymbol{\alpha}_2 \cdot \mathbf{r}_{12})}{r_{12}^2}), where \boldsymbol{\alpha} are Dirac matrices and r_{12} the inter-electron distance, introducing magnetic and retardation effects of order \alpha^4 / r. This interaction is crucial for fine-structure splittings in heavy atoms, where it competes with one-body Dirac corrections. Modern precision tests, such as the muon anomalous magnetic moment g-2, validate these QED-based relativistic corrections by probing the underlying theory to extraordinary accuracy, with implications for atomic angular momentum coupling through consistent determination of \alpha. The Fermilab Muon g-2 experiment's final 2025 measurement, achieving 127 parts per billion (0.127 ppm) precision, confirms previous results and aligns closely with updated Standard Model predictions from recent lattice QCD calculations, though some theoretical inconsistencies between data-driven and computational approaches persist, reducing but not fully resolving prior tensions.^[45] Deviations, if any, would signal new physics affecting fine-structure interpretations, but current concordance supports QED's role in coupling schemes.

Intermediate and Nuclear Coupling

In atomic physics, intermediate coupling arises in elements with moderate atomic numbers (mid-Z, such as transition metals), where neither the Russell-Saunders (LS) nor jj coupling schemes fully dominate due to comparable strengths of electrostatic, spin-orbit (H_SO), and spin-spin (H_SS) interactions.^[46] To accurately describe the energy levels, the full Hamiltonian H = H_elec + H_SO + H_SS is diagonalized in the basis of LS-coupled terms, yielding a mixing of states that deviates from pure coupling approximations.^[47] This approach accounts for small perturbations from ideal LS coupling, improving predictions for spectra in ions like those of neon or tungsten.^[48] Hyperfine structure emerges from the interaction between the total electronic angular momentum J and the nuclear spin I, described by the Hamiltonian H_hf = A I · J, where A is the hyperfine constant encoding magnetic dipole and electric quadrupole contributions.^[49] The Fermi contact term, arising from the interaction of the nuclear magnetic moment with s-electron spin density at the nucleus, dominates for light atoms and is proportional to the probability density |ψ(0)|^2.^[50] This term vanishes for non-s electrons unless spin polarization induces s-like density. A prominent example is the 21 cm line in neutral hydrogen, resulting from the hyperfine transition between the F=1 and F=0 states of the ground level, driven by the proton-electron spin interaction with energy splitting ΔE = (5/2) g_p μ_N μ_B B_hf / ħ ≈ 5.9 × 10^{-6} eV, corresponding to a frequency of 1420 MHz.^[51] Isotope shifts in hyperfine structure, such as in cadmium or dysprosium, arise from differences in nuclear spin I and mass across isotopes (e.g., ^113Cd vs. ^111Cd), altering the hyperfine constants and leading to measurable splittings on the order of MHz.^[52]^[53] In molecular extensions, hindered rotor coupling appears in diatomic spectra under weak torsional barriers, such as in van der Waals complexes or reaction intermediates, where rotational angular momentum J couples to vibrational modes, modifying energy levels beyond the rigid rotor model.^[54] For instance, in Li + CaH systems, low-anisotropy potentials lead to hindered rotation with oscillatory internal angular momentum states, influencing near-threshold bound levels.^[55]

References

[1]
8.4: Angular Momentum Coupling - Chemistry LibreTexts
Oct 30, 2023 · Any system that has more than one source of angular momentum will be subject to coupling between those forms of angular momentum.Term Symbols · The Microstate Method · Spin-Orbit Coupling · The Hole Rule
[2]
Quantized Angular Momentum - HyperPhysics
This kind of coupling gives an even number of angular momentum levels, which is consistent with the multiplets seen in anomalous Zeeman effects such as that of ...
[3]
Clebsch-Gordan Coefficient -- from Wolfram MathWorld
Clebsch-Gordan coefficients commonly arise in applications involving the addition of angular momentum in quantum mechanics. If products of more than three ...
[4]
Addition of angular momenta | Intro to Quantum Mechanics I Class ...
Angular momentum coupling occurs when two or more angular momenta in a quantum system interact · Clebsch-Gordan coefficients quantify the probability amplitudes ...
[5]
https://web.mit.edu/course/5/5.73/oldwww/Fall04/notes/VIII_Addition_of_Angular_Momenta.pdf
[6]
[PDF] 04 - Addition of Angular Momentum
An electron possesses orbital angular momentum, L, due to its motion in space. ▷ An electron also possesses an intrinsic spin angular momentum, S.
[7]
Angular Momentum Operators - Richard Fitzpatrick
The operator $L^2$, which represents the magnitude squared of the angular momentum vector, is defined \begin{displaymath} L^2 = L_x^{ 2} + L_y^{ 2} + L_z^{ 2}.
[8]
[PDF] Chapter 9 Angular Momentum Quantum Mechanical Angular ...
The other two relations, £ Ly, Lz ¤ = i¯h Lx and £Lz, Lx ¤ = i¯h Ly can be calculated using similar procedures. A Representation of Angular Momentum Operators.
[9]
Eigenvalues of Orbital Angular Momentum
The simultaneous eigenkets of $ L^2$ and $ L_z$ are completely specified by two quantum numbers, $ l$ and $ m$. These kets are denoted $ \vert l, m\rangle$.
[10]
[PDF] chapter 11 spin
It is customary to refer to the spin angular momentum quantum number as the spin. Electrons, protons, and neutrons all have spin 1/2. Photons have spin 1.
[11]
A history of Quantum Mechanics - MacTutor - University of St Andrews
Schrödinger in 1926 published a paper giving his equation for the hydrogen atom and heralded the birth of wave mechanics. Schrödinger introduced operators ...
[12]
17 Symmetry and Conservation Laws - Feynman Lectures - Caltech
Now we want to give an example of how we use the theorem of conservation of angular momentum in a specifically quantum physical problem. We look at break-up ...
[13]
[PDF] What is Noether's Theorem? | OSU Math
Abstract. Noether's theorem states that given a physical system, for every infinitesimal symmetry, there is a corresponding law of symme-.
[14]
Angular Momentum - Physics
The total angular momentum of an isolated physical system is a constant of motion. Classically and quantum mechanically, conservation of angular momentum is ...
[15]
[PDF] Chapter 13. Angular Momentum: General Theory
L that represents the orbital angular momentum in quantum mechanics, we replace, in Eq. 1, r and p with the appropriate operatorsr andp. This means that. L. =r ...
[16]
[PDF] Angular Momentum - JILA
It is thought that no fundamental particle can have spin higher than two. In nonrelativistic quantum mechanics, spin is introduced phenomenologically. A spin.
[17]
[PDF] angular - - ladder
Feb 13, 2020 · To summarize, if H is rotationally-invariant (spherically symmetric), then total angular momentum J is conserved. This is an example of ...
[18]
[PDF] Angular Momentum 1 Angular momentum in Quantum Mechanics
Classically, angular momentum is defined by. L = r × p . Since in QM we have. [xi,pj] = ihδij. it follows that [Li,Lj] 6= 0.
[19]
None
Below is a merged summary of the angular momentum coupling content from all provided segments, consolidating the information into a comprehensive response. To retain all details efficiently, I will use a table in CSV format for key concepts across the sources, followed by a narrative summary that integrates additional details and context. This approach ensures maximum density and clarity while preserving all information.
[20]
https://library.oapen.org/bitstream/handle/20.500.12657/50493/9789814415491.pdf
[21]
34.1 Special Notation
Clarification (effective with 1.0.14):: The reference for Clebsch–Gordan coefficients, Condon and Shortley (1935) , was replaced by Edmonds (1974) and Rotenberg ...
[22]
DLMF: §34.3 Basic Properties: 3⁢𝑗 Symbol ‣ Properties ‣ Chapter 34 3⁢𝑗,6⁢𝑗,9⁢𝑗 Symbols
### Summary of Recursion Relations for 3j Symbols
[23]
Hund's rules | Zeitschrift für Physik D Atoms,Molecules and Clusters
We review the present state of our undertanding of Hund's first and second rules, their domains of validity, and of generalizations in cases where the.Missing: paper | Show results with:paper
[24]
[PDF] LS-coupling and jj-coupling
Total angular momentum. • The interactions between electrons (angular ... ⁃ ⇒ jj-coupling will be relevant for heavy atoms. H2 = N. ∑ i=1 (-12V2 ri ...
[25]
The role of jj coupling on the energy levels of heavy atoms - SciELO
In this case, the jj terms are given by the term symbol (j1, j2, j3,...)J, where the values inside the parenthesis are the total angular momentum quantum numbers ...Missing: physics | Show results with:physics
[26]
Spin-Orbit Coupling - an overview | ScienceDirect Topics
... atomic nucleus or the band structure of a solid [24]. The spin–orbit coupling increases with the atomic number Z of the atom as Z4 in the case of a hydrogen ...
[27]
[PDF] Atomic Physics - Doyle Group | Harvard
atoms in the -LS-coupling scheme usually leads to an interval rule. Fìrrther examples of energy levels are given in the exercises at the end of this chapter.
[28]
[PDF] Origin of the Spin-Orbit Interaction - arXiv
Abstract. We consider a semi-classical model to describe the origin of the spin-orbit interaction in a simple system such as the hydrogen atom.
[29]
[PDF] The Foldy-Wouthuysen Transformation
In the pseudo-derivation of the spin-orbit term given in Notes 25 we glossed over the derivation of the extra factor of 1/2 due to Thomas precession ...
[30]
[PDF] Quantum Physics III Chapter 2: Hydrogen Fine Structure
Apr 2, 2022 · In the hydrogen atom the spin-orbit coupling arises because the electron is moving in the electric field of the proton. Since the electron ...Missing: strength | Show results with:strength
[31]
[PDF] Lecture 12 Atomic structure - TCM
Landé interval rule. Recall that separation of energies for states of different J arises from spin-orbit term contributionˆH2 (fine structure),. #|J,mJ,L,S|/i.
[32]
None
### Summary of Spin-Orbit Coupling Effects on Atomic Spectra
[33]
[PDF] Sodium D Line Data - Daniel A. Steck
May 27, 2000 · In this reference we present many of the physical and optical properties of sodium that are relevant to various quantum optics experiments.
[34]
[PDF] Atomic Spectra - Rutgers Physics
eliminated by observing the "D" lines ratio while varying the intensity from a hot sodium discharge with crossed polaroids. Further investigation showed.
[35]
Spin-Dependent Terms of the Breit–Pauli Hamiltonian Evaluated ...
The spin–spin interaction lifts the 3-fold degeneracy of each state, resulting the ΔE(2) = E 0 (2) – E ±1 (2) zero-field splitting, where the subscript of ...3. Methods And... · 4. Numerical Tests Of The... · 4.2. Molecular Applications
[36]
Fine Structure of the Level in Atomic : Theory and Experiment
Apr 7, 2005 · ... spin-spin (ss) terms of order α 2 Ry in the Breit-Pauli interaction ... light is of key importance for the accurate determination of the fine ...
[37]
Hund's Rules for Atomic Energy Levels - HyperPhysics
Hund's Rule #1: The term with the maximum multiplicity lies lowest in energy. The explanation of the rule lies in the effects of the spin-spin interaction.Missing: original paper
[38]
Atomic Spectroscopy - Different Coupling Schemes | NIST
Oct 3, 2016 · LS Coupling (Russell-Saunders Coupling). Some of the examples given below indicate notations bearing on the order of coupling of the electrons.Missing: carbon | Show results with:carbon<|control11|><|separator|>
[39]
https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas/atomic-spectroscopy-2
[40]
Atomic Spectroscopy - Allowed Terms | NIST
Oct 3, 2016 · The configuration l N has more than one allowed term of certain LS types if l > 1 and 2 < N < 4l (d 3 - d 7, f 3 - f 11, etc.). The ...
[41]
https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas/atomic-spectroscopy-1
[42]
Atomic Spectros. - Spectral Lines - NIST
Selection rules for discrete transitions. Electric dipole (E1) ("allowed"), Magnetic dipole (M1) ("forbidden"), Electric quadrupole (E2) ("forbidden") ...
[43]
[PDF] 1. Introduction
... coefficients resemble those produced in the corresponding line strengths from ... MsM L and M'SM' L must be chosen so that the Clebsch-Gordan coefficients in the ...
[44]
The Theory of Atomic Structure and Spectra - Robert D. Cowan
417 jj Coupling. 119. 418 Pair Coupling. 122. Si I 3pns. 125. 420 Other Coupling Schemes. 128. 422 Statistical Weights Complex Atoms. 130. 423 Quantitative ...
[45]
[PDF] arXiv:0809.4092v1 [physics.atom-ph] 24 Sep 2008
Sep 24, 2008 · The small deviations from the IS-coupling scheme are well accounted for by an exact diagonalization for the intermediate coupling case.
[46]
[1909.01991] Isotope shifts in $^{20,22}$Ne - arXiv
Sep 4, 2019 · Isotope shifts in ^{20,22}Ne -- Precision measurements and global analysis in the framework of intermediate coupling. We report new precision ...
[47]
[PDF] Hyperfine Structure in Atoms
Molecular hydrogen has a completely different hyperfine structure from atomic hydrogen, arising from the spin-spin interaction of the two protons in the ...
[48]
4.1: Physical origin of the hyperfine interaction - Chemistry LibreTexts
Mar 24, 2022 · The contribution to the hyperfine coupling from spin density in s orbitals on the nucleus under consideration is therefor called Fermi contact ...
[49]
21 cm line of atomic hydrogen | American Journal of Physics
Dec 1, 1994 · Discovered in 1951, the 21 cm hyperfine line of atomic hydrogen has become one of the best‐studied spectral features in radioastronomy.
[50]
The Isotope Shift in Hyperfine Structure | Phys. Rev.
The isotope shift due to a change of the nuclear radius by 1 percent is found to be larger than those observed.
[51]
Measurement of hyperfine structure and isotope shifts in the Dy
The isotope shifts and hyperfine coefficients were determined from each fit with uncertainties ranging from to 0.1 MHz to 10 MHz depending on the relative ...
[52]
Hindered asymmetric top states for chemical reactions *
A computational method for development of hindered asymmetric top rotational wavefunctions for atom-diatom chemical reactions is presented.
[53]
Statistics of near-threshold bound states for Li+CaH and Li+CaF
For low anisotropies, the diatomic molecule executes hindered rotation in the complex and the resulting internal rotation is only weakly coupled to the ...<|control11|><|separator|>