Exe.py: Ab initio fine structure parameters for trigonal defect qubits within the E$\otimes$e Jahn-Teller case

Trigonal solid-state defects are often subjects of spontaneous symmetry breaking driven by the $E\otimes e$ Jahn-Teller effect, reflecting strong electron-phonon coupling. These systems, particularly paramagnetic defect qubits in solids are central for quantum technology applications, where accurate knowledge of their fine-structure parameters $-$ shaped by the complex interplay of spin-orbit and electron-phonon interactions $-$ is essential. We introduce the Exe.py code part of the jahn-teller-dynamics package, a Python code that implements the first-principles approach of [Phys. Rev. X 8, 021063 (2018)] to accurately compute the spin-orbit-phonon entanglement in trigonal defects utilizing the output from density functional theory calculations (DFT). By employing $\Delta$SCF calculations, the method extends naturally to excited states and predicts fine-structure parameters of zero-phonon lines (ZPLs), including Zeeman shifts under external magnetic fields. The approach is applicable not only to solid-state defects but also to Jahn-Teller active trigonal molecules such as the $X$CH$_3$ family. We demonstrate the capabilities of Exe.py through applications to negatively charged Group-IV$-$vacancy (G4V) defects in diamond: SiV$^-$, GeV$^-$, SnV$^-$, PbV$^-$ and the neutral N$_3$V$^0$ defect in diamond, and the CH$_3$O methoxy radical.