## 2 Methodology
### Normal Mode Analysis  
Here, we will describe the mathematical formulation of the normal mode analysis following the procedure described in [60; 111]. In order to perform the normal mode analysis associated with the spectral density of FMO light harvesting complex, first, we define a standard time-dependent Hamiltonian for the pigment-protein complex responsible for the light harvesting phenomena as [60]:  
\[H=\sum_{mn}H_{mn}(t)|m\rangle\langle n|+\sum_{\xi}\frac{\hbar\omega_{\xi}}{4}(P _{\xi}^{2}+Q_{\xi}^{2}), \tag{20}\]  
where \(|m\rangle\) represents a state where only pigment \(m\) is excited and all other pigments are in their electronic ground state, the diagonal element \(H_{mm}(t)\) is the electronic transition energy between the local excited state \(|m\rangle\) and the ground state, i.e., the site energy of the pigment \(m\), while the off-diagonal element \(H_{mn}(t)\) with \(\mathrm{m}\
eq\mathrm{n}\) is the excitonic coupling between states \(|m\rangle\) and \(\langle n|\). Furthermore, the protein environment is described by a set of harmonic oscillators obtained from a normal mode analysis where the \(Q_{\xi}\) and \(P_{\xi}\) are the dimensionless coordinates and momenta of the systems, respectively, which are expressed in terms of the creation and annihilation operators of vibrational quanta, \(C_{\xi}^{\dagger}\) and \(C_{\xi}\), respectively, of normal mode \(\xi\) as [60]:  
\[Q_{\xi}=(C_{\xi}^{\dagger}+C_{\xi})\quad\text{and}\quad Q_{\xi}=i(C_{\xi}^{ \dagger}-C_{\xi}). \tag{21}\]  
Figure 5: (Color online) Low temperature absorption spectra for the wild-type (WT) FMO complexes (black lines) and the mutants (red lines). (**A**–**H**) show the absorption spectra of FMO complexes carrying mutations in amino acid residues near BChls 1–8, respectively. The blue dashed lines represent the mutant minus wild-type absorption difference spectra. (This figure is reproduced with permission from [44]). Alternatively, these dimensionless coordinates and momenta can also be expressed in terms of the mass-weighted normal coordinates \(q_{\xi}\) and momenta \(p_{\xi}\) as [60]:  
\[Q_{\xi}=\sqrt{\frac{2\omega_{\xi}}{\hbar}}q_{\xi}\quad\text{and}\quad P_{\xi}= \sqrt{\frac{2}{\omega_{\xi}\hbar}}p_{\xi}. \tag{22}\]  
The time dependence of \(H_{mn}(t)\) for the light harvesting complex can be introduced from the nuclear dynamics caused by the coupling between the pigment and protein, which, following [60], can be expressed as the Taylor series expansion of the small displacements \(R_{j}(t)\) of atoms from their equilibrium positions \(R_{j}^{(0)}\) as:  
\[H_{mn}(t)\approx H_{mn}^{(0)}+\sum_{j}(\
abla_{j}H_{mn}|_{0})\cdot(R_{j}(t)-R _{j}^{(0)}), \tag{23}\]  
where \(H_{mn}^{(0)}\) and \(\
abla_{j}H_{mn}|_{0}\) are the values of the Hamiltonian and its gradient taken w.r.t the three Cartesian coordinates of the \(j\)th atom, respectively, at the equilibrium position of nuclei in the electronic ground state. In addition, the displacement of the \(j\)th atom in terms of its weighted mass \(M_{j}\) can be expressed as [60; 111]  
\[R_{j}(t)-R_{j}^{(0)}=M_{j}^{-\frac{1}{2}}\sum_{\xi}A_{j}^{(\xi)}q_{\xi}(t), \tag{24}\]  
where \(A_{j}^{(\xi)}\) contains the contributions of the \(j\)th atom to the eigenvector of normal mode \(\xi\). From the above equations, the total time-dependent Hamiltonian can be expressed as (see details, e.g., in [60; 111])  
\[H_{mn}(t)\approx H_{mn}^{(0)}+\sum_{\xi}\hbar\omega_{\xi}g_{\xi}(m,n)Q_{\xi}(t), \tag{25}\]  
where \(g_{\xi}(m,n)\) describes the coupling between the two sites and is given by  
\[g_{\xi}(m,n)=\omega_{\xi}^{-\frac{3}{2}}(2\hbar)^{-\frac{1}{2}}\sum_{j}M_{j}^ {-\frac{1}{2}}(\
abla_{j}H_{mn}|_{0})A_{j}^{(\xi)}. \tag{26}\]  
Furthermore, \(g_{\xi}(m,n)\) is computed by using the Poisson-TrEsp method and details about the same can be found in [108; 111; 112].