Spin-dynamics simulations can demand more memory than a practical computer can provide. RadicalPy tackles that bottleneck with a hybrid method that combines chemical kinetics, semiclassical modeling, and quantum techniques—and reproduces experimental magnetic-field effects.

Radical pairs—electron-hole pairs and polaron pairs—are transient reaction intermediates used across physics, chemistry, and biology, including organic semiconductors, spintronics, quantum computing, solar cells, chemical reactions, and quantum biology. Yet quantitative analysis of radical pair phenomena has historically been successful by only a few select groups.

RadicalPy addresses that access problem with an intuitive open-source framework in Python. The toolbox implements classical, semiclassical, and quantum simulation methodologies, along with a radical pair kinetic rate equation solver, Monte Carlo-based spin dephasing rate estimations, and molecule database functionalities.

Its new kine-quantum method combines classical rate equations, semiclassical techniques, and quantum techniques, while offering time- and wavelength-resolved magnetic field effect simulations. Existing toolboxes such as EasySpin and Spinach focus on magnetic resonance techniques, including NMR and EPR, and offer little or no designated functionality for radical pair-specific problems.

MolSpin is an open-source alternative implemented in C plus plus with a custom scripting language, an approach described as necessary for larger simulations. But that approach has a significantly higher barrier of entry, making it inadequate for small- and medium-sized simulations and rapid prototyping.

The radical pair mechanism has gained popularity with nonspecialists through its proposed involvement in biological magnetoreception and quantum biology. At the same time, the lack of reproducibility in magnetic-field effects on radical pairs in biological reactions calls for a standardized method to simulate experimental results.

RadicalPy aims to become a standard for the community to use and develop. RadicalPy includes classical, semiclassical, and quantum simulation methods in one toolbox. Its isotope and molecule database includes spin multiplicities, magnetogyric ratios, and hyperfine coupling constants.

Those resources allow the Liouville-von Neumann equation to be solved in both Hilbert and Liouville space with relative ease. The motivation for kine-quantum is the current limitation in performing accurate simulations of experimental results, especially the memory requirements of quantum simulations.

Fifteen nuclei have been simulated with a large computer cluster, but those simulations do not include all nuclei of the radical pair—twenty-seven nuclei for the FAD-Trp radical pair—and are currently not feasible. The memory requirements grow as two to the power of N squared in Hilbert space and as two to the power of N to the fourth power in Liouville space, where N represents the number of spins.

To alleviate this problem, kine-quantum uses the Schulten-Wolynes semiclassical methodology, which has a constant memory requirement by restricting the Hamiltonian to two to the power of eight elements in Liouville space. Figure One presents the kine-quantum workflow: kinetic rate equations and exchange or dipolar relaxation feed into an initial radical-pair density matrix, which is then combined with reference spectra.

The pipeline produces wavelength- and time-dependent magnetic-field-effect spectra, illustrated by the lambda-t-MARY plots on the right, while the lower plot compares simulated time traces with data. This matters because RadicalPy integrates photochemical kinetics with semiclassical and quantum spin dynamics in an open-source Python framework, with simulations taking seconds to minutes.

Kine-quantum incorporates chemical rate equations into quantum simulations, allowing spin-spin interactions and relaxation superoperators to be used together. It also includes reference spectra to produce time- and wavelength-resolved magnetic field effects, which the method calls kine-quantum.

For flavin adenine dinucleotide photochemistry at acidic pH, the new methodology reproduces the experimental magnetic field effect data. The same simulation produces the magnetic field effect shown in Figure 3 and the transient absorption and fluorescence kine-quantum spectra shown in Figure 4.

The rate equations naturally produce a kinetics superoperator: each row and column corresponds to a state of the system, and the connecting rate constants appear as matrix entries. That matrix is also the adjacency-matrix representation of the rate equations as a directed graph.

The kinetics superoperator is combined with the Liouvillian to form the kine-quantum superoperator. Figure three benchmarks MARY simulations in terms of runtime and accuracy. Panel A compares Hilbert- and Liouville-space calculations as the number of spins increases, while marking the new approach in Liouville space using only two spins; simulations continued until the laptop’s memory capacity was exceeded.

Panel B compares the simulated magnetic-field-effect curves with experimental data: pure quantum simulations do not reproduce the experiment successfully, whereas the kine-quantum approach models the observed result. The comparison tests quantum simulation methods against kine-quantum for an experimental MARY spectrum for FAD at acidic pH, using a commercial laptop.

Hilbert-space simulations do not capture the shape of the experimental data and overestimate the low-field effect, while Liouville-space simulations including relaxation also fail to reproduce the experimental data. The six-spin Liouville-space quantum simulation is closest to the experimental data, but kine-quantum reproduces the data exceedingly well.

The six-spin model required about one hour to complete, compared with twenty-one seconds for kine-quantum, and the benchmarks show that kine-quantum outperforms traditional quantum-based MARY simulations in accuracy and performance. Figure 4 brings together the FAD molecular structure with kine-quantum simulations of transient absorption and fluorescence.

The three-dimensional plots map magnetic-field, wavelength, and time dependencies, while panel C examines how the normalized B one-half time evolution changes with the decay-rate parameter. This matters because FAD’s radical pair is coupled to a triplet excited state, so the method can describe magnetic-field effects alongside time- and wavelength-resolved spectra in a single simulation.

The new method faithfully reproduces the time-resolved spectra for FAD at acidic pH. The accurate transient absorption and fluorescence spectra result from including random fields relaxation and singlet-triplet dephasing. That example shows the versatility of the approach and its potential use for experimentalists in the field.

Figure five connects molecular motion to magnetic-field sensitivity for an FAD and tryptophan radical pair inside an AOT reverse micelle. The molecular-dynamics traces show changing radical separation and exchange interaction, while the autocorrelation analysis gives an exchange-interaction correlation time of one point six five nanoseconds and an estimated singlet–triplet dephasing rate of one point one times ten to the power of seven per second.

The resulting time-resolved MARY simulation and Lorentzian fits show how the magnetic response evolves over time. The exchange interaction has a correlation time of one point six five nanoseconds, and that value is used to calculate the spin dephasing rate with the function estimations dot k underscore S T D.

The resulting spin dephasing rate is one point one times ten to the seventh per second. That rate is consistent with experimental results in radical pairs in reverse micelles and is the main cause of spin relaxation in encapsulated radical pairs.

Figure six connects the proposed photochemical pathway to the protein structure of European robin cryptochrome four a, showing FAD alongside a tetrad of tryptophan residues. Panel B visualizes singlet-yield anisotropies for FAD radical anion pairs with either Z radical or W D H radical, comparing conditions without and with dipolar coupling.

The figure matters because it shows how a protein-constrained radical pair and anisotropic spin interactions can shape magnetic-field sensitivity. The two radical pair systems show the importance of including more than one hyperfine coupling constant when modeling biochemical reactions at geomagnetic field strengths of fifty microtesla.

A one-proton radical pair overexaggerates the magnitude of the low-field effect. This mechanism is believed to grant migratory animals their magnetosensing ability, and it is different from the mechanism involved in higher magnetic fields.

The final example moves from radical pairs to singlet fission, where one singlet exciton is split into two triplet excitons. The first direct experimental evidence for singlet fission came from magnetic field effects on crystalline tetracene in the late nineteen sixties.

Singlet fission has since gained attention as a way to increase solar-cell efficiency, while magnetic field effects help spin chemistry and spintronics researchers understand singlet fission in materials and photovoltaic devices. Figure seven links anthracene singlet fission to magnetic-field-dependent delayed fluorescence.

Panel A shows photoexcitation, formation of a strongly coupled triplet pair, and pathways involving free triplet excitons and triplet–triplet annihilation. Panel B separates the singlet, triplet, and quintet energy manifolds by exchange interaction; the calculated fluorescence effect appears at the marked singlet–quintet level anticrossings, showing why only specific magnetic fields produce a response.

RadicalPy offers classical and semiclassical simulation methods as additions to the spin chemistry, spintronics, and quantum biology communities. The toolbox can simulate a selection of experiments with relatively little input from the user.

Its backend handles tedious computational aspects of simulations, alleviating that burden for the user. Resonance techniques, oscillating magnetic field effects, electrically detected magnetic resonance, and optically detected magnetic resonance are currently under development or planned for extension.

Further resonant field-based approaches, including gamma-COMPUTE and local optimization theory, will also be supported. RadicalPy brings classical, semiclassical, and quantum simulations into an intuitive Python framework. Its kine-quantum method improves both feasibility and accuracy, while the examples show how the toolbox can connect simulations to real chemical, biological, and materials experiments.