Here is the way I usually think about the problem: we do not start with “spins” as isolated arrows. We start with electrons in a molecule. Electronic-structure theory tells us where those electrons are and how they interact. From that we build a spin Hamiltonian. Only then do we ask how the spin state evolves and what an experiment can actually observe.
Electronic structure
Start with the electrons
If I put a molecular geometry on the board, the first question is not yet “how does the spin precess?” It is: what electronic state does this geometry support? Within the Born–Oppenheimer approximation we freeze the nuclei for the moment and solve the electronic problem. In atomic units, a useful schematic Hamiltonian is
The first two electronic terms are familiar: kinetic energy and attraction to the nuclei. The hard part is the electron–electron repulsion. Every practical electronic-structure method is, in one way or another, a strategy for dealing with this many-electron problem without explicitly solving the exact wavefunction for every electron coordinate.
So what is the difference between HF, DFT and “correlated” methods?
A single Slater determinant. Exchange is treated exactly within that determinant, but dynamical electron correlation is missing.
Works through the electron density and a Kohn–Sham reference system. In principle exact; in practice the exchange–correlation functional is the approximation.
Methods such as MP2 or coupled cluster add correlation beyond a single determinant, usually at substantially higher computational cost.
Needed when several electronic configurations are genuinely important—for example near bond breaking, degeneracies or strongly correlated states.
For photochemistry we also need excited states. TD-DFT is often the practical workhorse, while wavefunction and multireference approaches become important when charge transfer, double-excitation character or near-degeneracy makes a single-reference description unreliable.
State mixing and an avoided crossing
Here is the smallest model that already shows something important. Two localized states have an energy offset (Delta) and interact through a coupling (t):
Away from the crossing, the lower state is mostly localized on one diabatic state.
The model is deliberately tiny, but the lesson is general: interaction changes both energies and state character. That same logic appears in charge transfer, excited-state mixing and many effective Hamiltonians.
Effective spin description
Now compress the electronic problem into spin interactions
Once the electronic state is known, we usually do not want to carry the full electronic wavefunction through a spin-dynamics simulation. Instead, we project the relevant physics onto a much smaller spin space. For a pair of radicals a useful schematic Hamiltonian is
This is where electronic structure and spin dynamics meet. The symbols in this Hamiltonian are not arbitrary fitting decorations: they encode the underlying electron density, spin–orbit coupling and geometry.
What else can appear in the Hamiltonian?
Nuclear Zeeman interactions, nuclear quadrupole tensors for nuclei with (I>1/2), zero-field splitting for higher-spin states, microwave or radiofrequency driving fields, and additional exchange or anisotropic terms depending on the experiment.
Time evolution
What does the spin Hamiltonian actually do?
For a closed pure state, the answer is the time-dependent Schrödinger equation. In spin chemistry and magnetic resonance we very often deal with ensembles, incomplete information and environmental coupling, so the density matrix is usually the more useful language:
The commutator gives coherent evolution. The relaxation superoperator (mathcal R) represents the fact that the spin system is not isolated from molecular motion, solvent, vibrations and the rest of its environment. Once ( ho(t)) is known, an observable follows from (langle O angle=mathrm{Tr}[ hohat O]).
Why not just propagate a wavefunction?
You can, if the problem is a pure closed state or if you use a stochastic unraveling of an open-system equation. But a density matrix naturally represents statistical mixtures, decoherence and ensemble averages. This is why Liouville-space formulations are so common in EPR, NMR and radical-pair theory.
(T_1), (T_2) and dephasing are not the same thing
(T_1) describes longitudinal population relaxation toward equilibrium. (T_2) describes decay of transverse coherence. A useful decomposition is
where (T_phi) is the pure-dephasing time. So (T_2) is not simply defined by (T_1). In the absence of pure dephasing one obtains the upper limit (T_2=2T_1); additional dephasing makes (T_2) shorter.
Larmor precession of an electron spin
Before adding hyperfine coupling, relaxation or a second electron, it is worth understanding the simplest motion. An approximately isotropic electron spin in a static field precesses at
At 1 mT an electron with (gapprox2) precesses at roughly 28 MHz.
The numerical frequency and period are physical. The animation itself is slowed down enormously so that the motion is visible.
Spin chemistry
How can spin motion change chemistry?
This is the key step in radical-pair chemistry. Photoexcitation or thermal electron transfer can create two radicals whose electron spins are correlated. The pair may start, for example, in a singlet state. Different magnetic interactions on the two radicals then change the spin character in time.
What do singlet and triplet actually mean?
For two electron spins (1/2), the singlet is (lvert S angle=(lvertalphaeta angle-lvertetaalpha angle)/sqrt2). The triplet manifold contains (lvert T_+ angle=lvertalphaalpha angle), (lvert T_0 angle=(lvertalphaeta angle+lvertetaalpha angle)/sqrt2), and (lvert T_- angle=lvertetaeta angle). The labels describe the coupled two-electron spin state, not two separate classical arrows.
If singlet and triplet radical pairs have different reaction channels, the chemical product yield depends on the spin dynamics. In a simple first-order picture, a singlet product yield can be written schematically as
provided the reaction kinetics are included consistently in the evolution of ( ho(t)). This equation is the bridge from an evolving quantum state to a chemical observable.
A minimal singlet–triplet mixing model
A real radical pair can contain many nuclear spins and four electronic spin states. But a two-level model is enough to see what coupling and detuning do:
Coupling is currently strong enough to overcome most of the detuning, so large-amplitude S–T oscillations remain possible.
This model is intentionally pedagogical. A realistic radical pair additionally contains (T_+), (T_0), (T_-), nuclear spins, anisotropy, orientation dependence, relaxation, molecular motion and spin-selective reaction kinetics.
Molecular motion & open systems
The Hamiltonian is often moving too
In a protein, solvent or flexible donor–acceptor system, the geometry changes continuously. That means the magnetic interactions can become time-dependent:
This compact equation is easy to underestimate. A side-chain rotation can change a hyperfine tensor. A donor–acceptor distance can change exchange coupling by orders of magnitude. Protein motion can reorient anisotropic (g)- and dipolar tensors. So molecular dynamics is not merely “structural decoration” around the spin calculation—it can determine the spin dynamics itself.
Interactions can be motionally averaged.
Fluctuations can drive efficient relaxation and strongly modify coherent dynamics.
The system behaves more like an ensemble of quasi-static conformations.
This multiscale connection is a recurring theme of my current work and of the development of MolSpin.
What do we actually measure?
Translate the dynamics into an experiment
A simulation becomes useful only when it predicts an observable. Different experiments interrogate different parts of the same Hamiltonian and dynamics.
Selected reading
Where these ideas appear in my work
Modeling spin relaxation in complex radical systems using MolSpin
Open-system dynamics and relaxation in molecular spin systems.
J. Comput. Chem. (2023) →Spin Dynamics of Radical Pairs Using the Stochastic Schrödinger Equation in MolSpin
Stochastic state-vector propagation as an alternative route to radical-pair dynamics.
J. Chem. Theory Comput. (2024) →Importance of Polarizable Embedding for Absorption Spectrum Calculations of Arabidopsis thaliana Cryptochrome 1
How the molecular environment changes electronic excitation energies in a flavoprotein chromophore.
J. Phys. Chem. B (2024) →Revealing the Impact of g-Tensor Anisotropy on the Charge Recombination in Donor–Acceptor Dyads Under High Magnetic Fields
A direct example of an electronic-structure-derived magnetic interaction controlling spin-dependent kinetics.
JACS (2025) →Magnetosensitivity of Model Flavin–Tryptophan Radical Pairs in a Dynamic Protein Environment
How molecular motion and fluctuating interactions influence magnetosensitivity.
J. Phys. Chem. B (2025) →Weak Radiofrequency Field Effects on Biological Systems Mediated through the Radical Pair Mechanism
A broader view of radical-pair physics, weak RF fields and biological magnetic-field effects.
Chemical Reviews (2025) →Reaction-yield detected magnetic resonance spectroscopy of radical pairs in cryptochrome-4a
Connecting radical-pair spin dynamics to a reaction-yield-detected resonance experiment.
Free Radic. Biol. Med. (2026) →Multiscale modeling approaches in biomolecular physics
How atomistic simulation, electronic structure and quantum observables can be connected across scales.
Advances in Physics: X (2026) →