Evaluating First-Principles Electron-Phonon Couplings: Consistency Across Methods and Implementations

Electron-phonon coupling (EPC) is fundamental for understanding the behavior of molecules and crystals, influencing phenomena such as charge transport, energy transfer, phase transitions, and polaron formation. Accurate computational methods to calculate EPCs from first principles are essential, but their complexity has resulted in a variety of computational strategies, raising concerns about their mutual consistency. In this study, we provide a systematic benchmark of methods for EPC calculation by comparing two fundamentally different {\it ab initio} methodologies. We investigate Gaussian-type orbital methods based on the \textsc{CP2K} code and plane-wave-based projector-augmented-wave (PAW) methods combined with maximally localized Wannier functions, as implemented in \textsc{VASP} and \textsc{wannier90}. In addition, we further distinguish between the derivative--of--Hamiltonian ($dH$) and derivative--of--states ($d\psi$) approaches for obtaining EPC parameters. The comparison is conducted on a representative set of organic molecules, including pyrazine, pyridine, bithiophene, and quarterthiophene, varying significantly in size and flexibility. We find excellent agreement across implementations and basis sets when employing the same computational approach ($dH$ or $d\psi$), demonstrating robust consistency between the numerical schemes. However, noticeable deviations occur when comparing the $dH$ and $d\psi$ approaches within each code and for specific cases discussed in detail. Our findings emphasize the reliability of EPC computations using the $dH$ method and caution against potential pitfalls associated with the $d\psi$ approach, providing guidance for future EPC calculations and model parameterizations.