Turn spin dynamics into chemistry
A radical pair is one of the cleanest places where quantum spin dynamics becomes chemically observable. The spins evolve coherently, but the singlet and triplet parts of the state can react differently. Change the spin evolution and you can change the reaction yield.
Construct the singlet/triplet basis for two electron spins and explain why it is not a classical arrow picture.
Identify the magnetic interactions that generate singlet–triplet interconversion.
Connect spin evolution to spin-selective reaction yields and magnetic-field effects.
Formation
How do we get a spin-correlated radical pair?
A radical pair is both a chemical intermediate and a coupled two-spin quantum system
What it is: A molecular species with at least one unpaired electron, giving it an electron spin and a magnetic moment.
What it changes: The unpaired electron makes the species paramagnetic and sensitive to Zeeman, hyperfine and spin–spin interactions.
What you observe: EPR signals, characteristic reactivity and spin-dependent transient spectroscopy.
What it is: Two radicals created or brought together within one reaction sequence, often by photoinduced electron transfer.
What it changes: Their two electron spins can retain correlation from the precursor state and evolve coherently before the radicals separate or recombine.
What you observe: Magnetic-field-dependent reaction yields, transient EPR and spin-selective products.
What it is: A non-classical relation between the two electron spins inherited from how the pair was formed, commonly singlet or triplet character.
What it changes: It determines which spin-selective reaction channels are initially allowed and provides the starting condition for singlet–triplet dynamics.
What you observe: Initial spin polarization and different recombination behaviour for singlet-born versus triplet-born pairs.
Photoexcitation followed by electron transfer is a common route. If a singlet precursor undergoes spin-conserving electron transfer, the newly formed radical pair starts with strong singlet character. A triplet precursor can instead populate a triplet-born radical pair.
Two electron spins
Singlet and triplet are coupled two-spin states
The singlet–triplet basis describes correlation between two spins
What it is: An antisymmetric two-electron spin state with total spin \(S=0\). The individual electrons do not have independent fixed up/down labels.
What it changes: It can react through singlet-selective chemical channels and can coherently mix with triplet character when the two radicals experience different magnetic interactions.
What you observe: Singlet-product yield and singlet-selective recombination probability.
What it is: Three symmetric two-electron spin states with total spin \(S=1\).
What it changes: Triplet character opens different reaction pathways and responds differently to Zeeman, exchange and dipolar interactions.
What you observe: Triplet products, triplet EPR signatures and field-dependent reaction yields.
For two spin-\(\tfrac12\) electrons there are four coupled states:
Hamiltonian
What drives singlet–triplet interconversion?
Singlet–triplet mixing requires the two radicals to become magnetically distinguishable
What it is: Different nuclei and spin-density patterns create different local magnetic fields on the two electron spins.
What it changes: The electrons accumulate different phases, converting singlet character into triplet character and back.
What you observe: Nuclear-spin-dependent oscillations and magnetic-field effects in reaction yield.
What it is: If the two radicals have different effective \(g\)-values, their Zeeman precession frequencies differ in an external field.
What it changes: The relative electron-spin phase grows at a field-dependent rate, especially important at higher fields.
What you observe: Field-strength-dependent singlet–triplet mixing and high-field magnetic effects.
What it is: A short-range electronic interaction that directly changes the singlet–triplet energy gap.
What it changes: Large \(|J|\) can energetically isolate singlet and triplet states and suppress weak hyperfine-driven mixing.
What you observe: Distance/conformation-sensitive reaction kinetics and shifted spin-transition conditions.
What it is: A through-space magnetic interaction between the two electron spins.
What it changes: It introduces orientation-dependent splittings and can mix or separate triplet sublevels depending on geometry.
What you observe: Directional magnetic response and geometry-sensitive radical-pair dynamics.
A useful schematic radical-pair Hamiltonian is
The crucial ingredient is usually a difference between the magnetic environments of the two radicals. If both electron spins experienced exactly the same Hamiltonian, there would be much less opportunity to change the total singlet/triplet character. Hyperfine asymmetry, \(g\)-tensor differences and anisotropic interactions provide the required inequivalence.
A compact mathematical test is to ask whether the Hamiltonian commutes with the singlet projector \(\hat P_S=|S\rangle\langle S|\). Under closed dynamics,
If \([\hat H,\hat P_S]=0\), that Hamiltonian term cannot by itself change the singlet population. Terms that make the two radicals magnetically inequivalent generate non-zero matrix elements between singlet and triplet sectors and therefore drive S–T interconversion.
For the simplest isotropic \(\Delta g\) mechanism, the relative electron precession frequency is
This shows why \(\Delta g\)-driven mixing strengthens with magnetic field, whereas hyperfine-driven mixing can already be efficient at low field.
Interactive
A minimal S–T mixing model
This model deliberately throws away most of the real radical-pair complexity. It keeps only one effective singlet state, one effective triplet state, an ordinary-frequency coupling \(v\) and detuning \(\delta\), both expressed in Hz or MHz. That is enough to see resonance and off-resonance behaviour.
Coupling versus detuning
Coupling is currently strong enough to overcome most of the detuning.
Interactive
See how exchange competes with field-driven \(\Delta g\) mixing
The previous widget treats coupling and detuning abstractly. A more physical reduced \(\{|S\rangle,|T_0\rangle\}\) model separates two roles: exchange produces an S–T energy gap, while magnetic inequivalence produces an off-diagonal coupling. For the simplest isotropic \(\Delta g\) contribution,
\(v_0\) represents a field-independent effective mixing channel, for example a reduced hyperfine-asymmetry contribution. The \(\Delta g\) term grows linearly with field because the two electrons acquire different Zeeman frequencies. The eigenvalue gap, also in ordinary-frequency units, is
Exchange separates S and T; magnetic inequivalence mixes them
Exchange still defines a substantial S–T energy gap, but field-dependent \(\Delta g\) mixing is no longer negligible.
The differential Zeeman contribution is represented in the \(S/T_0\) basis as an off-diagonal coupling. Real radical pairs additionally contain \(T_\pm\), nuclear-spin manifolds, anisotropic tensors, electron–electron dipolar coupling and often time-dependent \(J\) and \(D\).
Spin-selective reaction
The observable is usually not the spin state itself
Chemical kinetics acts as the detector of the quantum spin state
What it is: A chemical reaction whose rate depends on whether the radical pair has singlet or triplet spin character because orbital symmetry and spin conservation favour different product channels.
What it changes: It continuously converts spin populations into chemical loss, so reaction kinetics and spin dynamics compete on the same timescale.
What you observe: Different singlet/triplet product yields and field-dependent recombination kinetics.
What it is: The probability per unit time for a particular chemical channel to remove or transform the radical pair.
What it changes: A very fast rate can terminate the pair before substantial spin mixing; a very slow rate allows more coherent evolution but also more time for relaxation.
What you observe: Radical-pair lifetime, transient decay and integrated reaction yield.
What it is: The time-integrated amount of product formed through a chosen spin-selective channel.
What it changes: It compresses the entire history of spin evolution and reaction into an experimentally accessible scalar observable.
What you observe: Magnetic-field effects reported as changes in fluorescence, absorption, product concentration or related chemical signals.
If singlet and triplet radical pairs react through different channels, the time-dependent spin character controls product formation. The key modelling point is that chemistry must act during the spin propagation rather than being attached only after a closed-system trajectory has finished.
This is the important conceptual bridge: a quantum spin state evolves on nanosecond or microsecond timescales, while the experiment may report only a final chemical yield. The next section makes that simultaneous spin–reaction dynamics explicit.
From spin state to product yield
Reaction kinetics continuously measures the evolving singlet and triplet character
A useful way to connect the density matrix to chemistry is through singlet and triplet projectors, \(\hat P_S\) and \(\hat P_T\). In the standard Haberkorn description of first-order spin-selective loss, the reaction operator enters the equation of motion itself:
The corresponding integrated singlet yield is then
The reaction therefore does not wait until spin evolution is finished. Increasing \(k_S\) can increase the instantaneous probability of singlet reaction while simultaneously shortening the time available for further singlet–triplet mixing. This competition is why a reaction rate cannot be interpreted independently of the Hamiltonian and relaxation timescales.
Magnetic fields
Why can weak fields matter at all?
A magnetic field does not need to supply the reaction energy. It only needs to change the relative spin evolution before the radicals react or separate. That can alter the fraction of time spent in reactive singlet or triplet character.
Static fields change Zeeman splittings and level structure. Oscillating RF or microwave fields can drive transitions when they are resonant with spin-energy differences. Whether an effect survives depends on the competition between coherent dynamics, relaxation, molecular motion and reaction kinetics.
Dynamic environments
Proteins make the Hamiltonian time-dependent
In a protein, \(J\), \(\mathbf D\), hyperfine tensors and even \(g\)-tensors fluctuate because the molecular geometry fluctuates. A compact way to write this is
This is where molecular dynamics, electronic structure and spin dynamics have to meet. A single optimized structure can be informative, but it may miss the distribution and time correlation of the interactions that actually control the spin evolution.
Selected reading
Examples from my work
Magnetosensitivity of Model Flavin–Tryptophan Radical Pairs in a Dynamic Protein Environment
How fluctuating protein environments influence magnetic-field sensitivity.
J. Phys. Chem. B (2025) →Weak Radiofrequency Field Effects on Biological Systems Mediated through the Radical Pair Mechanism
A broad theoretical and experimental perspective on weak-field radical-pair effects.
Chemical Reviews (2025) →Revealing the Impact of g-Tensor Anisotropy on the Charge Recombination in Donor–Acceptor Dyads Under High Magnetic Fields
How \(g\)-tensor anisotropy modifies spin-dependent recombination.
JACS (2025) →Reaction-yield detected magnetic resonance spectroscopy of radical pairs in cryptochrome-4a
Detecting spin resonance through a chemical reaction yield.
Free Radic. Biol. Med. (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.
Magnetic field effects in chemical kinetics and related phenomena
U. E. Steiner and T. Ulrich · Chemical Reviews (1989). A foundational review of spin chemistry and magnetic-field effects in radical reactions.
Open DOI →The Radical-Pair Mechanism of Magnetoreception
P. J. Hore and H. Mouritsen · Annual Review of Biophysics (2016). A tutorial review connecting radical-pair spin chemistry to biological magnetoreception.
Open DOI →Density matrix description of spin-selective radical pair reactions
R. Haberkorn · Molecular Physics 32, 1491–1493 (1976). A foundational density-matrix formulation of spin-selective radical-pair reaction kinetics.
Open DOI →