# DECIDE: A Deterministic Mixed Quantum-Classical Dynamics Approach
## 1 Introduction  
Most systems of interest in chemistry and biology contain very large numbers of degrees of freedom (DOF). An exact simulation of the dynamics of such a system requires a fully quantum mechanical treatment of the entire system, which is computationally prohibitive due to the exponential scaling with the number of DOF. One approach to overcoming this issue is to treat a few DOF of interest quantum mechanically (i.e., subsystem) and the remainder classically (i.e., environment or bath) [1, 2, 3, 4]. Within this approach, the quantum subsystem is described in terms of a Hilbert space and the classical environment in terms of a phase space of positions and momenta. Situations in which the quantum subsystem is described by a non-Hermitian Hamiltonian have also been considered [5]. Previously, these _mixed quantum-classical_ techniques have been applied to a wide range of phenomena, e.g., proton transfer reactions [6, 7], electron transfer reactions [8, 9], proton-coupled electron transfer reactions [10, 11, 12, 13, 14, 15, 16], exciton transport in photosynthetic complexes [17, 18, 19], heat transport in molecular junctions [20, 21, 22, 23, 24], metamolecules [25], and magnetic molecules [26, 27]. The quantum-classical Liouville equation (QCLE) has served as a starting point for simulating the dynamics of mixed quantum-classical systems [2, 28, 29]. Over the years, a hostof techniques have been developed based on approximate solutions of the QCLE [3; 30]. The most accurate of these techniques are the stochastic surface-hopping solutions [6; 31; 32; 33; 34; 35; 36], but they require extremely large ensembles of trajectories to obtain converged results and, as a result, will be computationally prohibitive for many systems. On the other hand, the Poisson Bracket Mapping Equation (PBME) approach provides a highly computationally efficient approximate solution of the QCLE, but its applicability is mainly restricted to systems with weak subsystem-bath couplings [8; 37; 38; 39]. Recently, the \"Deterministic evolution of coordinates with initial decoupled equations\" (DECIDE) method was developed, which offers a favourable balance between computational economy and accuracy [40]. DECIDE has two main advantages compared to the stochastic QCLE-based methods: (i) It is a deterministic method that requires the integration of \(L^{2}(L^{2}-2+2N)\) coupled differential equations (where \(L\) is the number of basis functions used to represent the subsystem and \(N\) is the number of environmental DOF). Typically, only a few thousand trajectories are required to obtain converged results, in contrast to several orders of magnitude more trajectories in the case of the stochastic QCLE-based methods; (ii) There is no need to diagonalize the Hamiltonian matrix on-the-fly. DECIDE has demonstrated great promise in solving the QCLE with high accuracy and low computational cost for a number of model systems, including the spin-boson model [40; 41], Fenna-Matthews-Olson model [40; 41], a three-state photo-induced electron transfer model [40], nonequilibrium spin-boson model [21; 22; 42], and a quantum battery model [43]. To date, the DECIDE equations have been represented in _complete_ spin, subsystem, and adiabatic energy bases and successfully applied to simulate the dynamics of Hamiltonians expressed in these bases. However, the feasibility of solving the DECIDE equations in an _incomplete_ position basis has yet to be explored. In this review, we explore this by first considering a one-dimensional, quadratic model of a hydrogen bond coupled to a solvent coordinate. The review is organized as follows. In Section 2, we first discuss the Wigner-Weyl transform, which has been widely used in mixed quantum-classical dynamics, as it provides an invertible mapping between Hilbert space operators and phase space functions. In Section 3, we derive the DECIDE equations of motion and briefly discuss their underlying approximations and energy conservation.