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.