The main goal of the SCOML project is to develop a robust computational framework capable of performing excited-state dynamics in transition-metal compounds using the Tully Surface Hopping (TSH) technique. Since a fully ab initio treatment of this problem is computationally unfeasible, we developed a machine-learning-based approach.
Two systems have been investigated. The first is a toy model consisting of an octahedrally coordinated Fe(II)(CO)6 complex. This molecule, with its relatively small number of atoms, was chosen to build and test the computational workflow. The second system is the experimentally characterized Fe(II)(ptz)6 compound, for which experimental measurements are available, allowing direct comparison with theory.
In terms of ab initio electronic structure methods, different possibilities were explored to balance accuracy with computational efficiency. After careful testing, we opted to construct our database using N-Electron Valence State Perturbation Theory (NEVPT2) on top of a Complete Active Space Self-Consistent Field (CASSCF) wavefunction. For the toy system, we employed an active space of six d-electrons in five orbitals (6e,5o). For Fe(II)(ptz)6, we included two additional σ-bonding doubly occupied orbitals, giving a total active space of ten electrons in seven orbitals (10e,7o). In the calculations, we considered four singlet states, six triplet states, and three to five quintet states.
The computational tool we developed automatically links three main components:
(i) ORCA for the ab initio calculations,
(ii) a PyTorch wrapper for training a neural-network model of the electronic properties (modified SPaiNN), and
(iii) SHARC to perform TSH simulations using machine-learning predictions.
This entire workflow is managed by a Python code we developed, named pyVC, which is freely available on GitHub.
Particular attention was devoted to the training of Spin–Orbit Coupling (SOC) elements, which are essential for describing the spin-crossover transition. Because a large number of states are coupled by SOC in each calculation, obtaining accurate training is challenging. To address this, we developed a strategy to train Spin–Orbit Coupling Reduced Matrix Elements (RMEs) instead. RMEs are directly related to SOC elements and can be recovered from them using the Wigner–Eckart theorem. The advantages of this approach are twofold:
1) RMEs are defined at the spin-free level of the wavefunction, so the number of independent elements is significantly smaller than for the SOC matrix itself.
2)SOC elements are complex numbers, and learning their relative phases between real and imaginary parts is non-trivial. In contrast, RMEs are purely imaginary, which reduces the learning problem to a single component.
Another source of inaccuracy arises from the possible swapping of electronic roots between database entries, which complicates the training of coupling elements. To mitigate this issue, we tested a strategy based on molecular orbital rotations and alignment of CI vectors, ensuring consistent wavefunction character across geometries.
Beyond SOCs, pyVC can also train on energies, forces, and non-adiabatic coupling (NAC) vectors. In this project, however, we bypassed NAC training entirely and instead employed a curvature-based approximation to estimate NACs at intersystem crossing points.
Finally, the full code was applied to the Fe(II)(ptz)6 complex. Our preliminary results demonstrate the capability of the approach to simulate spin-crossover transitions, with good qualitative agreement with experimental data.
The newly developed code pyVC is flexible and not restricted to transition-metal-based systems. It has the potential to assist the study of excited-state dynamics whenever multi-reference methods are required to capture strong electronic correlation. The results of this project pave the way for systematic computational studies of light-induced spin transitions, helping to close the gap between experimental observations and theoretical modeling.
Throughout the process of testing and developing the code, a large number of calculations were performed. The resulting databases, containing all the data, are openly accessible for reuse.
