# DECIDE: A Deterministic Mixed Quantum-Classical Dynamics Approach
## 3 DECIDE Mixed Quantum-Classical Dynamics  
In this section, we derive the DECIDE equations of motion [40]. Let us start by considering a fully quantum system with a time-independent Hamiltonian \(\hat{H}=\hat{H}_{\text{S}}(\hat{x})+\hat{H}_{B}(\hat{X})+\hat{H}_{C}(\hat{x}, \hat{X})\), where \(\hat{H}_{\text{S}}\) is the subsystem Hamiltonian, \(\hat{H}_{\text{B}}\) is the bath Hamiltonian, \(\hat{H}_{\text{C}}\) is the subsystem-bath coupling Hamiltonian, \(\hat{x}\) denotes a set of generalized coordinates that provides a complete description of the state of the subsystem, and \(\hat{X}=(\hat{R},\hat{P})\) denotes the set of position and momentum operators of the bath. In the Heisenberg picture, the dynamics of the subsystem and bath coordinates are given by the quantum Liouville equation, namely  
\[\frac{d}{dt}\hat{x}(t) = \frac{i}{\hbar}\big{[}\hat{H},\hat{x}(t)\big{]}=\frac{i}{\hbar}e ^{i\hat{\ell}t}\big{[}\hat{H},\hat{x}\big{]}\equiv\frac{i}{\hbar}\big{(}[\hat {H},\hat{x}]\big{)}(t)\] \[\frac{d}{dt}\hat{X}(t) = \frac{i}{\hbar}\big{[}\hat{H},\hat{X}(t)\big{]}=\frac{i}{\hbar}e ^{i\hat{\ell}t}\big{[}\hat{H},\hat{X}\big{]}\equiv\frac{i}{\hbar}\big{(}[\hat {H},\hat{X}]\big{)}(t), \tag{12}\]  
where \(e^{i\hat{\ell}t}\hat{A}=e^{i\hat{H}t/\hbar}\hat{A}e^{-i\hat{H}t/\hbar}\). In the above equations, the time arguments outside the brackets indicate that one should first evaluate the commutator, then apply the time dependence. Taking a partial Wigner transform over the bath DOF (defined in terms of the bath DOF at \(t=0\)) of the above equations and retaining only zero-order terms in \(\hbar\) in the resulting Moyal product expansion yields,  
\[\frac{d}{dt}(\hat{x}(t))_{W} = \frac{i}{\hbar}\big{(}e^{i\hat{\ell}t}\big{)}_{W}e^{\hbar\Lambda/ 2i}\big{(}[\hat{H},\hat{x}]\big{)}_{W}\approx\frac{i}{\hbar}e^{i\hat{\ell}_{W} t}\big{(}[\hat{H},\hat{x}]\big{)}_{W}\] \[\frac{d}{dt}(\hat{X}(t))_{W} \approx \frac{i}{\hbar}e^{i\hat{\ell}_{W}t}\big{(}[\hat{H},\hat{X}]\big{)} _{W}\, \tag{13}\]  
Assuming that the subsystem and bath coordinates are initially decoupled, evaluating the partial Wigner transform of the commutators in the above equations, and retaining only first-order terms in \(\hbar\) in the resulting Moyal product expansion leads to the DECIDE equations of motion,  
\[\frac{d}{dt}(\hat{x}(t))_{W} \approx \frac{i}{\hbar}e^{i\hat{\mathcal{L}}_{W}t}\big{(}\big{[}\hat{H}_{W},\hat{x}\big{]}\big{)}=\frac{i}{\hbar}\big{(}\big{[}\hat{H}_{W},\hat{x}\big{]} \big{)}(t)\] \[\frac{d}{dt}(\hat{X}(t))_{W} \approx -e^{i\hat{\mathcal{L}}_{W}t}\big{(}\big{\{}\hat{H}_{W},X\big{\}}_{ a}\big{)}=-\big{(}\big{\{}\hat{H}_{W},X\big{\}}_{a}\big{)}(t), \tag{14}\]  
where the antisymmetrized Poisson bracket is given by \(\{\hat{H}_{W},\hat{A}_{W}\}_{a}=\frac{1}{2}\{\hat{H}_{W},\hat{A}_{W}\}-\frac{ 1}{2}\{\hat{A}_{W},\hat{H}_{W}\}\). (N.B. : The assumption that the subsystem and bath are initially decoupled leads to \(([\hat{H},\hat{x}])_{W}=[\hat{H}_{W},\hat{x}]\) and \(([\hat{H},\hat{X}])_{W}=-\frac{\hbar}{\hat{t}}\{\hat{H}_{W},\hat{X}\}_{a}\).) It should be noted that the partially Wigner-transformed Hamiltonian retains an operator character because the transform was not performed on the subsystem DOF. In addition, to account for noncommutativity in the above equations, one has to replace any coupling terms in \(H_{C}\) with their Weyl-ordered symmetric forms, e.g., \(\hat{x}X\) would be replaced by \(\frac{1}{2}(\hat{x}X+X\hat{x})\). The errors caused by the approximations above become significant when the subsystem dynamics is highly non-Markovian, namely when the dependence on the initial bath coordinates is important [40]. This can be the case for very low bath temperatures, very strong subsystem-bath coupling, and very slow baths, i.e., when memory effects are very pronounced [22; 40].