## 2 Methodology
### Time-Dependent Density Functional Theory  
The properties like excitonic energy transfer and spectral density are time-dependent phenomena and their properties in light harvesting complexes can be computed utilizing the time-dependent density functional theory (TDDFT). Several studies have been reported in the literature for computing the energy transfer and excitonic energy within the biological complexes associated with the photosynthesis process utilizing the TDDFT [113; 114; 115; 116; 117]. More details about the application of DFT in long-range interaction, along with its use in the large molecular systems, can be found in [118; 119; 120; 121]. The theory behind the applicability of TDDFT for large molecules is briefly highlighted below. In order to describe the application of the TDDFT to explain the properties of the biomolecules like BChls present in the FMO complex in their respective states, we will follow the procedure given in [118]. The time-dependent Kohn-Sham equation Equation (28) for an independent particle system can be derived assuming the existence of an effective potential \(v_{eff}\), whose orbitals \(\Psi(r,t)\) yield the same charge density as for the interacting system [118; 122; 123]  
\[\bigg{[}-\frac{1}{2}\
abla^{2}+v_{eff}(r,t)\bigg{]}\Psi(r,t)=-i\frac{\partial}{ \partial t}\Psi(r,t). \tag{28}\]  
The effective potential can be expressed as the sum of applied field (perturbation) (\(v(t)\)) and self-consistent field (\(v_{SCF}(r,t)\)):  
\[v_{eff}(r,t)=v(t)+v_{SCF}(r,t). \tag{29}\]  
Furthermore, the self-consistent field (\(v_{SCF}(r,t)\)) is expressed as:  
\[v_{SCF}(r,t)=\int\frac{\rho(r,t)}{|r-r^{\prime}|}+v_{xc}(r,t), \tag{30}\]  
where the quantity \(v_{xc}(r,t)\) is the exchange-correlation potential that can be expressed as the functional derivative of the exchange-correlation action \(A_{xc}\) as:  
\[v_{xc}[\rho](r,t)=\frac{\delta A_{xc}[\rho]}{\delta\rho(r,t)}\approx\frac{ \delta E_{xc}[\rho_{t}]}{\delta\rho_{t}(r)}=v_{xc}[\rho_{t}](r), \tag{31}\]  
where the unknown functional (\(A_{xc}\)) of \(\rho\) over both time and space is approximated by the exchange-correlation functional (\(E_{xc}\)) of the time-independent Kohn-Sham theory which is a function of \(\rho_{t}\) of space at fixed \(t\). In most of the time-dependent phenomena involving excitations such as the photosynthesis process, the energy transfer is supposed to be perturbed by the incident light. Therefore, in the presence of small perturbations of the system, \(\delta v(t)\) will give the following effective potential [118]:  
\[\delta v_{eff}(r,t)=\delta v(t)+\delta v_{SCF}(r,t), \tag{32}\]  
where \(\delta v_{SCF}(r,t)\) is the linear response of the self-consistent field arising from the change in the charge density as given by (in frequency representation):  
\[\delta\rho(r,\omega)=\sum_{ai}\delta P_{ai}(\omega)\Psi_{a}(r)\Psi_{i}^{*}(r) +\sum_{ia}\delta P_{ia}(\omega)\Psi_{i}(r)\Psi_{a}^{*}(r), \tag{33}\]  
where \(\delta P_{st}(\omega)\) is the linear response of the Kohn-Sham density matrix in the basis of the unperturbed molecular orbitals (\(i\), \(j\) for occupied; \(a\), \(b\) for virtual; \(s\), \(t\), \(u\), \(v\) for general orbitals). Furthermore, the linear response of the Kohn-Sham density matrix to the applied field utilizing the elementary results from time-dependent perturbation theory can be expressed as:  
\[\delta P_{st}(\omega)=\frac{\Delta n_{st}}{(\epsilon_{s}-\epsilon_{t})- \omega}\delta v_{st}^{eff}(\omega), \tag{34}\]  
where \(\Delta n_{st}\) is the difference in occupation numbers, \(\Delta n_{st}=1\) for \(st=ai\) and \(\Delta n_{st}=-1\) for \(st=ia\).