Quantum Dynamics Approaches
In PyFREC, quantum dynamics simulations are implemented with the quantum master equation formalism.3, 18, 27 The Hamiltonian consists of a pure electronic part (\(H_{\text{el}}\)) that describes coupled electronic excited states, a vibrational part (\(H_{\text{vib}}\)) that includes molecular vibrations of fragments, and electron-vibrational couplings (\(H_{\text{el}-\text{vib}}\)):
\(H=H_{\text{el}}+H_{\text{vib}}+H_{\text{el}-\text{vib}}\) (11)
Huang-Rhys factors are used to describe electron-vibrational coupling:
\(H_{\text{el}-\text{vib}}=\sum_{i=1}^{N}{\sum_{k=1}^{\left.\ n(i\right)}{\hslash\omega_{\text{ik}}\sqrt{S_{\text{ik}}}\ \left(a_{\text{ik}}^{\dagger}+a_{\text{ik}}\right)}}\left.\ \left|i\right.\ \right\rangle\left\langle\left.\ i\right|\right.\ \)(12)
where \(\omega_{\text{ik}}\) is the vibrational mode, the Huang-Rhys factors \(S_{\text{ik}},\ a_{\text{ik}}^{\dagger}\) and\(a_{\text{ik}}\) are standard raising and lowering operators,N is the number of fragments, \(\left.\ n(i\right)\) is the number of vibrational modes coupled to the electronic state i . A Lindblad-type quantum master equation is written as:
\(\frac{\text{dρ}}{\text{dt}}=-\frac{i}{\hslash}\left[H,\rho\right]+\mathcal{L}_{\text{deph}}\left(\rho\right)+\mathcal{L}_{\text{vib}}\left(\rho\right)\)(13)
where \(\rho\) is the density matrix, \(\mathcal{L}_{\text{deph}}\) is the electronic dephasing operator (to describe interactions with the environment) and \(\mathcal{L}_{\text{vib}}\) is the vibrational damping operator. Sample quantum dynamics of the fully coherent regime determined by the\(\frac{i}{\hslash}\left[H,\rho\right]\) term in Eq. 13 is shown in Figure 5. The dynamics that include all terms from Eq. 13 is presented in Figure 6.
The parameters such as electronic decoherence rate and vibrational damping rates3 in the Lindblad equations are entered as parameters. The user is free to choose desirable parameters based on other calculations or empirical (spectroscopic) findings as a part of the user input. The user may either choose quantum master equation or Förster theory calculations.