## 2 Methodology
### Hierarchical Equations of Motion (HEOM)  
The hierarchical equations of motion (HEOM) plays an important role in studying the photosynthetic electronic energy transfer by quantifying the site-dependent reorganization dynamics of protein environments [53; 98; 99]. In particular, the HEOM theory has been extensively used to: (a) account for various linear and nonlinear spectra in molecular and solid state materials, (b) evaluate charge and exciton transfer rates in biological systems, (c) simulate resonant tunneling and quantum ratchet processes in nanodevices, and (d) explore quantum entanglement states in quantum information theories [100]. Notably, the electronic exciton dynamics in the light harvesting complexes are also coupled to the vibrations and other mechanical modes that could induce a possibility of the memory effects in this process. Prolonged coherent dynamics is predicted in the light harvesting complexes due to the slow dissipation of reorganization energy to the vibrational environment [53]. The HEOM and its solution provide a better way for understanding the excitation and energy transfer phenomena in the light harvesting complexes by also accounting for the nonlocal time effects [53; 98; 99; 101; 102]. Importantly, the excitation energies of each and every pigment or the sites of the FMO complex have to be determined for studying the excited state phenomenon based on the absorption of the light at different sites of the FMO protein. The study of excitonic dynamics of the different sites of the pigments in the FMO complex has been performed in [103]. The detailed theoretical analysis of electronic excitation energy transfer through the FMO pigment-protein complex has been provided in [104]. In the earlier studies, there were only 7 BChls reported for the FMO complex and, accordingly, the excitonic Hamiltonian was described by 7\(\times\)7 matrix [50; 53]. Moreover, there has also been experimental evidence for the existence of the eighth pigment in the FMO protein [17; 18], thus the Hamiltonian for the system comprising of 8\(\times\)8 matrix can be expressed as:  
\[H_{\text{FMO}}=\begin{bmatrix}12,405&-87.7&5.5&-5.9&6.7&-13.7&-9.9&21\\ -87.7&12,530&30.8&8.2&0.7&11.8&4.3&42\\ 5.5&30.8&12,210&-53.5&-2.2&-9.6&6&0.6\\ -5.9&8.2&-53.5&12,320&-70.7&-17.0&-63.3&-1.3\\ 6.7&0.7&-2.2&-70.7&12,480&81.1&-1.3&33\\ -13.7&11.8&-9.6&-17.0&81.1&12,630&39.7&-7.9\\ -9.9&4.3&6.0&-63.3&-1.3&39.7&12,440&-9.3\\ 21&42&0.6&-1.3&3.3&-7.9&-9.3&12,430\end{bmatrix} \tag{11}\]  
Furthermore, the values of the site energies and their coupling during the photosynthetic excitation are calculated by using Equation (12) (see Section 2.4) by fitting the optical spectra from the excitonic coupling between different pigments of the FMO complex. The diagonal components of Equation (11) denote the site energies and the off-diagonal elements denote the coupling between different sites. Notably, the elements of the Hamiltonian matrix presented in Equation (11) have been determined by fitting the optical spectra obtained by utilizing experimentally determined shapes for the spectral densities. More details about the process of testing the Hamiltonian via calculation of the excitation spectra using experimental and computational techniques can be found in [50; 105; 106; 107; 103; 108].