Molecular qubits based on transition metal complexes are attractive because of their chemical tunability: by changing ligands and geometry, one can control spin dynamics. The key challenge is extending spin lifetime, which is limited by spin–phonon relaxation. For spin 1/2 in copper(II) porphyrins, existing calculations based on normal modes in the harmonic approximation systematically overestimated the contribution of low-frequency "rippling" distortions. Experiments using resonance Raman scattering and inelastic neutron scattering pointed to the dominance of metal–nitrogen stretching vibrations in the 100–300 cm⁻¹ range, but no theoretical tool could automatically extract such modes at finite temperature. This work joins attempts to account for anharmonicity via molecular dynamics, but neighboring approaches either stayed within harmonic analysis or did not yield physically interpretable coupling coefficients.
For three copper porphyrins, CuOEP, CuTPP, and CuTiPP, first-principles molecular dynamics was run at 300 K for about 3 ps with a 0.5 fs time step. Every fifth femtosecond frame, the g-tensor was computed, and its fluctuations along five channels (xx, yy, zz, perpendicular, and isotropic) were decomposed into contributions from normal mode amplitudes and their pairwise products. The coefficients of this expansion were determined by LASSO regression with cross-validation, which automatically selected significant modes and mode pairs while discarding noise. Normal modes were obtained by diagonalizing the Hessian at the optimized geometry, and atomic displacements were projected onto them after removing center-of-mass motion and rotation.
The regression showed that in all three systems the dominant spin–vibration coupling lies in the 100–300 cm⁻¹ range, not below 100 cm⁻¹ as earlier spectral density analysis suggested. For CuOEP, the main contribution to the gzz channel comes from a mode near 289 cm⁻¹; for CuTPP, a cluster around 265–268 cm⁻¹; for CuTiPP, modes above 300 cm⁻¹ with a secondary peak near 112–123 cm⁻¹. The low-frequency rippling modes that dominate the spectral density turned out to be weakly coupled to the spin. Bilinear coefficients, responsible for two-phonon (Raman) processes, are densest for CuTiPP and sparsest for CuOEP. Plugging the extracted coefficients into the formulas for direct and Raman channels reproduced the experimental order of relaxation times: CuTPP relaxes fastest, while CuOEP and CuTiPP are slower and close to each other. The temperature dependence of the relaxation rate shows nonmonotonic curvature, reflecting the crossover from the direct to the Raman process.
The work offers an alternative to analytical approaches based on the Redfield equation: instead of computing coupling derivatives at a single equilibrium geometry, coefficients are extracted from finite-temperature dynamics using regularized regression. This allows automatic separation of one- and two-phonon channels and accounts for anharmonic effects. Unlike neighboring studies that either limited themselves to the harmonic approximation or lacked mode resolution, this method yields physically interpretable coupling coefficients for each g-tensor channel. The approach does not claim analytical rigor, but shows that temporal information from molecular dynamics is necessary for a correct description of relaxation.
Describing processes on nanosecond and microsecond timescales will require trajectories many orders of magnitude longer, which is beyond the reach of ab initio molecular dynamics. Acceleration methods will be needed, such as machine-learned potentials or multiscale schemes. Extending to higher-spin systems, including spin–orbit coupling, Orbach processes, and spin–spin interactions, are natural next steps.
The method applies to any molecular qubit where the g-tensor is sensitive to vibrations. It could accelerate materials design for quantum computing by pointing out which vibrational modes should be suppressed through chemical modification.
Test the method's transferability to other classes of molecular qubits, including complexes with higher spin and strong spin–orbit coupling. Develop hybrid schemes that combine short first-principles dynamics with machine-learned potentials for long trajectories.
For the chemistry of molecular qubits, an open question remains: how to predict coherence and relaxation times in the condensed phase, where intermolecular interactions and lattice phonons can change the picture. This method works for an isolated molecule and does not account for environmental effects, limiting direct comparison with experiments in crystals or on surfaces.
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What the work does
The paper proposes a method to extract spin-phonon coupling coefficients from short molecular dynamics trajectories using LASSO regression, separating one-phonon and two-phonon channels. Applied to three copper porphyrins, it reproduced the experimental order of relaxation times and the shape of the temperature dependence, showing the dominance of vibrations in the 100–300 cm⁻¹ range.
Where its strength is
Automatically identifying significant modes and mode pairs without a harmonic approximation for the dynamics is a concrete tool that can be transferred to other systems and generalized.
Where it could go next
Worth trying
Why this may matter
If the method proves transferable, it could become a standard tool for rapid assessment of spin-phonon coupling in molecular qubits, bridging the gap between computation and experiment.
Nearby works in our archive
The archive is sparse around this work: the closest review is only 0.53, against a usual 0.64. It looks like few have drilled nearby.
The three porphyrins studied differ only in the substituents around the macrocycle periphery, yet their g-tensors fluctuate differently: the least distorted CuOEP has fluctuation amplitudes comparable to the most distorted CuTiPP, while the intermediate CuTPP shows noticeably smaller ones. Static geometry does not predict dynamics.