# DECIDE: A Deterministic Mixed Quantum-Classical Dynamics Approach
## 4 Hydrogen Bond Model
### Dynamics Simulation Details  
Using Equation (16) and the Hamiltonian in Equation (19), one may obtain the DECIDE equations of motion for the proton transfer model in the quantum harmonic oscillator basis, viz.,  
\[\frac{d}{dt}x^{\alpha a^{\prime}}(t) = \frac{i}{\hbar}[\hat{H}_{W},\hat{x}]^{\alpha a^{\prime}}=\frac{ p^{\alpha a^{\prime}}}{m}\] \[\frac{d}{dt}p^{\alpha a^{\prime}}(t) = \frac{i}{\hbar}[\hat{H}_{W},\hat{p}]^{\alpha a^{\prime}}=-A^{0}x^ {\alpha a^{\prime}}-k_{1}X^{\alpha a^{\prime}}\] \[\frac{d}{dt}X^{\alpha a^{\prime}}(t) = -\{\hat{H}_{W},X\}^{\alpha a^{\prime}}=\frac{p^{\alpha a^{\prime }}}{m}\] \[\frac{d}{dt}p^{\alpha a^{\prime}}(t) = -\{\hat{H}_{W},P\}^{\alpha a^{\prime}}=-k_{1}x^{\alpha a^{\prime }}, \tag{22}\]  
where, for example, \(x^{\alpha a^{\prime}}=\langle\alpha|\hat{x}|a^{\prime}\rangle\) and \(|\alpha\rangle\) is the \(a^{th}\) harmonic oscillator state. Given that the basis set is composed of 12 functions, the total number of coupled differential equations is \(12^{2}\times 4\).  
We assume the initial state of the composite system to be factorized, i.e., \(\hat{p}(0)=\hat{p}_{S}(0)\rho_{B,W}(0)\), where \(\hat{p}_{S}(0)\) is the initial density operator of the subsystem (viz., the proton) and \(\rho_{B,W}(0)\) is the initial Wigner distribution of the bath (viz., the solvent coordinate). In our simulations, the system is initialized in the adiabatic ground state \(|\psi_{0}\rangle\) of the Hamiltonian in Equation (19), such that \(\hat{p}_{S}(t)=|\psi_{0}\rangle\langle\psi_{0}|\). In the harmonic oscillator basis, the initial values of the matrix elements of the subsystem coordinates are given by the following analytical expressions:  
\[x^{\alpha a^{\prime}}(0) = \frac{1}{2}\Big{(}\hbar^{2}/2 mA^{0}\Big{)}^{1/4}\Big{(}\sqrt{ \alpha+1}\delta_{\alpha+1,a^{\prime}}+\sqrt{\alpha^{\prime}+1}\delta_{\alpha, a^{\prime}+1}\Big{)},\] \[p^{\alpha a^{\prime}}(0) = i\Big{(}2\hbar^{2}mA^{0}\Big{)}^{1/4}\Big{(}\sqrt{\alpha+1} \delta_{\alpha+1,a^{\prime}}-\sqrt{\alpha^{\prime}+1}\delta_{\alpha,a^{\prime }+1}\Big{)}. \tag{23}\]  
Figure 3: Ground- and first-excited state adiabatic potential energy surfaces of the 1D quadratic proton transfer model. \(V\) denotes a variational result, and \(A\) denotes an analytical result. The following parameter values were used to obtain these surfaces: \(A^{0}=3367.6\) cm\({}^{-1}\) Å\({}^{-2}\), \(k_{1}=1.7\times 10^{4}\) cm\({}^{-1}\) Å\({}^{-2}\), and \(m=1\) amu.  
The initial values of the matrix elements of the bath coordinates are given by  
\[X^{\text{n}\alpha^{\prime}}(0)=X(0)\delta_{\alpha,\alpha^{\prime}},\ P^{\text{n} \alpha^{\prime}}(0)=P(0)\delta_{\alpha,\alpha^{\prime}}, \tag{24}\]  
where \(X(0)/P(0)\) is the initial value of the bath position/momentum. In practice, \(X(0)\) and \(P(0)\) may be sampled from Wigner distributions corresponding to thermal equilibrium states. However, in this study, we choose \(X(0)=-0.2\) A and \(P(0)=50\) a.u., since we only generate a single trajectory for demonstration purposes. Starting from these initial conditions, the DECIDE equations of motion in Equation (22) are integrated using the [PERSON] fourth-order method [49] with a time step of 1 a.u. to simulate the dynamics of the system. In general, a particular integrator is chosen to achieve an optimal balance between efficiency and accuracy for a given system.  
The expectation value of a property \(\hat{A}(t)\) may be calculated according to  
\[\langle\hat{A}(t)\rangle=\sum_{\beta\beta^{\prime}}\int\,\text{d}X(0)\text{d}P (0)\ A^{\beta^{\prime}}(t)\rho_{S}^{\beta^{\prime}\beta}(0)\rho_{B,W}(0). \tag{25}\]  
However, since we only consider single trajectories for this demonstration, we take \(\rho_{B,W}(0)=1\). In particular, we are interested in the conservation of the total energy along a trajectory, which is calculated according to  
\[\langle\hat{H}(t)\rangle=\sum_{\alpha\alpha^{\prime}}H^{\text{n}\alpha^{ \prime}}(t)\rho_{S}^{\alpha^{\prime}\alpha}(0). \tag{26}\]