The spin Hamiltonian is the bridge
Electronic-structure theory gives us molecular magnetic parameters. Spin dynamics needs those parameters arranged into an effective Hamiltonian. This module is about understanding what each term means physically, what assumptions hide inside it, and which conventions you must state before a number becomes meaningful.
Read a spin Hamiltonian term by term and state the associated units and sign convention.
Distinguish Zeeman, hyperfine, exchange, dipolar, quadrupole and zero-field-splitting physics.
Recognize when tensor orientation and the molecular-to-laboratory frame are essential.
Overview
Write only the physics you actually need
A useful generic spin Hamiltonian is
You rarely need every term at once. A radical pair of two organic \(S=\tfrac12\) radicals may need electron Zeeman, hyperfine, exchange and dipolar interactions. A transition-metal complex with \(S>1/2\) can instead make zero-field splitting central. A nucleus with \(I>1/2\) can add quadrupole structure.
Zeeman interaction
The external field defines the basic energy scale
Why the \(g\)-factor is more than a fitting number
What it is: Electron spin carries a magnetic moment, so an external field lifts the degeneracy of spin projections.
What it changes: The energy separation grows approximately linearly with field: this is the Zeeman splitting that sets the Larmor frequency.
What you observe: The field/frequency position of magnetic-resonance transitions.
What it is: The proportionality between magnetic field and electron-spin magnetic energy. Molecular orbital character and SOC make it molecule-specific and often anisotropic.
What it changes: Different principal \(g\)-values give different precession frequencies for different molecular orientations.
What you observe: Orientation-dependent EPR resonance fields and \(g\)-strain when conformations have slightly different tensors.
For an electron spin, the Zeeman term is usually written
If \(\mathbf g=g\mathbf 1\), the interaction is isotropic. In a molecule, spin–orbit coupling and the local electronic structure generally make \(\mathbf g\) a tensor. The resonance therefore depends on how the molecule is oriented relative to the magnetic field.
The nuclear Zeeman interaction is much smaller because the nuclear magneton is much smaller than the Bohr magneton:
Hyperfine coupling
Nuclei tell you where the unpaired electron lives
Hyperfine coupling turns electronic spin density into a nuclear fingerprint
What it is: An isotropic interaction proportional, in the simplest picture, to the unpaired spin density at the nucleus.
What it changes: It shifts electron-spin energies according to the nuclear-spin projection but does not depend on molecular orientation.
What you observe: Isotropic hyperfine splittings in solution EPR and NMR-related spin-polarization effects.
What it is: The anisotropic magnetic interaction between the distributed electron spin density and the nuclear magnetic moment.
What it changes: It makes the coupling depend on the orientation of the electron–nucleus geometry relative to the field.
What you observe: Anisotropic EPR/ENDOR patterns and orientation-dependent radical-pair dynamics.
The hyperfine interaction between an electron spin and a nucleus is
It is useful to split the tensor into an isotropic and a traceless anisotropic part,
The isotropic Fermi-contact contribution is closely related to the spin density at the nucleus. The anisotropic contribution reflects the spatial distribution of the unpaired spin and behaves like an electron–nuclear dipolar interaction.
Why can a proton far from the formal radical centre still have a hyperfine coupling?
Because spin density can be transferred through bonds or delocalized through a conjugated system. Hyperfine couplings are therefore often a much more sensitive probe of the actual electronic structure than a simple Lewis structure suggests.
Two electron spins
Exchange and dipolar coupling are physically different
Two mechanisms couple electron spins—and they scale very differently with geometry
What it is: A quantum-mechanical interaction arising from antisymmetry of the many-electron wavefunction together with orbital overlap and electron correlation.
What it changes: It changes the singlet–triplet energy separation and can suppress or enhance singlet–triplet mixing.
What you observe: Singlet–triplet gaps, magnetic coupling constants and strong geometry sensitivity, often approaching exponential distance dependence.
What it is: The direct magnetic interaction between two spatially separated electron magnetic moments.
What it changes: It depends on distance as roughly \(r^{-3}\) and on the orientation of the inter-spin vector.
What you observe: Orientation-dependent splittings, distance information in EPR and radical-pair anisotropy.
For this lecture I will use the exchange convention
Exchange originates from the antisymmetry of the electronic wavefunction and orbital overlap. It can change extremely rapidly with geometry. Other communities use \(-2J\,\mathbf S_1\cdot\mathbf S_2\), so the sign and factor of two are not universal.
The through-space magnetic dipolar interaction has a completely different origin. Using dimensionless spin operators and the point-dipole approximation,
The key signatures are the \(r^{-3}\) distance dependence and the strong orientation dependence.
Electron–electron dipolar geometry
At \(90^\circ\), the secular orientation factor is positive and equal to one.
The plotted angular factor is the familiar high-field secular orientation factor. The full dipolar Hamiltonian is tensorial; do not use this one number as a substitute for the full interaction when non-secular terms matter.
Higher spins & nuclei
Quadrupole and zero-field splitting add new structure
Higher spin quantum numbers introduce interactions that do not exist for spin-1/2
What it is: Nuclei with \(I>\tfrac12\) possess a non-spherical electric quadrupole moment that interacts with the local electric-field gradient.
What it changes: It splits nuclear-spin sublevels even without changing the electron-spin state and can mix nuclear projections.
What you observe: Additional EPR/ENDOR/ESEEM structure and nuclear-frequency shifts.
What it is: A compact description of anisotropic splitting inside an electron-spin multiplet with \(S>\tfrac12\).
What it changes: \(D\) sets the dominant axial splitting and \(E\) measures rhombicity in the principal-axis convention.
What you observe: Zero-field and low-field level separations and characteristic triplet/high-spin EPR patterns.
Nuclei with \(I>1/2\) have an electric quadrupole moment that can interact with the electric-field gradient:
For an electron spin \(S>1/2\), spin–spin and spin–orbit effects can split the spin sublevels even at zero external field. In a common principal-axis convention,
\(D\) measures the axial part and \(E\) the rhombic part in this convention. The tensor form is more general, and—as always—the sign convention and units need to be stated explicitly.
Orientation & motion
A tensor is only meaningful together with its molecular frame
Anisotropy means that the interaction depends on direction
What it is: A direction-dependent generalization of a scalar coupling. Its principal values describe the interaction along three mutually orthogonal principal axes.
What it changes: Rotating the molecule changes the component of the interaction projected onto the laboratory magnetic-field direction.
What you observe: Orientation-dependent resonance fields, splittings and relaxation rates in crystals, powders and partially ordered samples.
What it is: The molecular directions in which a symmetric interaction tensor is diagonal and can be described by its principal values.
What it changes: They determine how electronic structure is geometrically tied to the measured anisotropy.
What you observe: Angular patterns in single-crystal EPR and characteristic turning points in powder spectra.
What it is: Time-dependent rotation of the molecular frame relative to the laboratory field.
What it changes: Fast tumbling averages anisotropic interactions; intermediate motion modulates them and contributes to relaxation.
What you observe: Motional narrowing in solution and temperature/viscosity-dependent linewidths.
An anisotropic \(g\)-tensor or hyperfine tensor is not just three numbers. It has principal values and principal axes. Rotating the molecule relative to the field rotates the tensor into the laboratory frame and changes the observed interaction.
In a rigid crystal, \(\mathbf R\) is fixed. In a tumbling molecule or protein, it becomes time-dependent. That is the point where the spin Hamiltonian naturally connects to molecular dynamics and relaxation theory.
Selected reading
Examples from my work
Revealing the Impact of g-Tensor Anisotropy on the Charge Recombination in Donor–Acceptor Dyads Under High Magnetic Fields
A direct example of an anisotropic spin-Hamiltonian term changing recombination kinetics.
JACS (2025) →Peculiar Differences between Two Copper Complexes Containing Similar Redox-Active Ligands
DFT and multiconfigurational electronic-structure analysis for transition-metal systems.
Inorg. Chem. (2024) →Multiscale modeling approaches in biomolecular physics
How molecular structure, electronic interactions and quantum observables are connected across scales.
Advances in Physics: X (2026) →Key external literature
Where to read next
These are deliberately selected from outside my own work: foundational papers or reviews that are especially useful for this topic.
Quantum chemical calculations of spectroscopic properties of metalloproteins and model compounds: EPR and Mössbauer properties
F. Neese · Current Opinion in Chemical Biology (2003). A concise bridge between electronic structure and experimentally fitted spin-Hamiltonian parameters.
Open DOI →Prediction of molecular properties and molecular spectroscopy with density functional theory: From fundamental theory to exchange-coupling
F. Neese · Coordination Chemistry Reviews (2009). A broader review of magnetic response, spectroscopy and exchange coupling from DFT.
Open DOI →