## 2 Theory of the Optical Response with Inhomogeneous Broadening  
A detailed description of the quantum model of PSIIRC without taking into account of the inhomogeneous broadening has been given in our previous published paper [40]. Considering the multimode Brownian oscillator model to calculate the spectral density and the lineshape function for cofactors of PSIIRC, and the exciton theory to simulate the optical response of a system of coupling pigments [9], the final expressions in terms of the modified Redfield theory [43; 44] for absorption \(\sigma_{abs}^{ext}(\omega)\), linear \(\sigma_{LD}^{ext}(\omega)\), circular \(\sigma_{CD}^{ext}(\omega)\) dichroism and steady-state fluorescence \(\sigma_{fl}^{ext}(\omega)\) of PSIIRC were written as follows:  
\[\sigma_{abs}^{ext}(\omega)\approx\frac{\omega}{\pi}\sum_{\alpha}^{N}\mathbf{d }_{\alpha}^{2}Re\int_{0}^{\infty}dte^{i(\omega-\epsilon_{\alpha})t}e^{-g_{ AAA}(t)}e^{-0.5K_{aa}t} \tag{1}\]  
\[\sigma_{CD}^{ext}(\omega)\approx\frac{\omega}{\pi}\sum_{\alpha}^{N}\mathbf{R }_{\alpha}Re\int_{0}^{\infty}dte^{i(\omega-\epsilon_{\alpha})t}e^{-g_{ AAA}(t)}e^{-0.5K_{aa}t} \tag{2}\]  
\[\sigma_{LD}^{ext}(\omega)\approx\frac{\omega}{\pi}\sum_{\alpha}^{N}\left[d_{ \alpha}^{2}-\frac{1}{2}\left(d_{\alpha}^{2}+d_{\alpha}^{2}\right)\right]Re \int_{0}^{\infty}dte^{i(\omega-\epsilon_{\alpha})t}e^{-g_{ AAA}(t)}e^{-0.5K_{aa}t} \tag{3}\]  
\[\sigma_{fl}^{ext}(\omega)\approx\frac{\omega^{3}}{\pi}\sum_{\alpha}^{N}\frac{ (\mathbf{nd}_{\alpha})^{2}e^{\epsilon_{\alpha}\beta}}{\sum_{\alpha}e^{ \epsilon_{\alpha}\beta}}Re\int_{0}^{\infty}dte^{i(\omega-\epsilon_{\alpha}+2 \lambda_{AAA})t}e^{-g_{AAA}*(t)}e^{-0.5K_{aa}t} \tag{4}\]  
where \(\epsilon_{\alpha}\) and \(\mathbf{d}_{\alpha}^{2}=d_{\alpha}^{2}+d_{\alpha}^{y^{2}}+d_{\alpha}^{z^{2}}\) are the eigenvalues of the Frenkel Hamiltonian of PSIIRC and the corresponding squares of the Q\({}_{y}\) transition moments of Chhs and their components, \(g_{AAA}(t)\) is the lineshape function in the exciton representation, \(K_{aa}\) is the relaxation rates of the exciton states, \(\lambda_{aaaa}=-\underset{\tau\to\infty}{\lim}Im\left[\frac{d_{GaAs}(\tau)}{d \tau}\right]\) is the reorganization energy, which represents the influence of vibronic modes on the exciton state, \(\alpha=1,2,\ldots,N\) enumerates the exciton states, N is the number of pigments in PSIIRC [40], \(\beta=1/kT\), \(\mathbf{R}_{\alpha}=\sum_{nm}c_{n}^{\alpha}c_{m}^{\alpha}r_{nm}(\mathbf{d}_{n }\times\mathbf{d}_{m})\) is the rotation strength matrix of PSIIRC needed to estimate the effect of circular dichroism, \(c_{n}^{\alpha}\) are the eigenvectors and \(r_{nm}\) are distances between Chhs. It is worth adding that PSIIRC consists of two types of pigments optically active at 600-700 nm: Chhs and pheopbytins. In fact, they have to be characterized by two different spectral densities; however, we used the same spectral density for all pigments in our simulations. Generally, this simplification will not affect the results of the test of MCM and DE working together, but to analyze the energy transfer and charge separation in PSIIRC in detail, two spectral densities have to be considered. To incorporate the inhomogeneous broadening employing MCM, the diagonal elements of the Frenkel Hamiltonian PSIIRC must be modified by a fluctuation parameter:  
\[H_{ext}^{j}= \begin{pmatrix}\Omega_{1}+\delta_{1}^{j}&J_{12}&&J_{1N}\\ &\cdots&\\ J_{21}&&\Omega_{2}+\delta_{2}^{j}&&J_{2N}\\ &\vdots&&\ddots&\vdots\\ J_{N1}&&J_{N2}&&\cdots&\Omega_{N}+\delta_{N}^{j}\end{pmatrix}  \tag{5}\]where \(\delta_{n}^{j}=\left\{\delta_{1}^{j},\,\delta_{2}^{j},\,\ldots,\delta_{N}^{j}\right\}\) is a set of fluctuations generated by a Gaussian random number generator. The output of the generator is controlled by the full width at half maximum \(\Delta=FWHM_{\Omega}/2\sqrt{2\cdot ln2}\), \(j=1,2,\ldots\), \(MC\) is the index of MCM realizations and MC is the maximum number of MCM realizations.