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We derive a $c-$number Generalised Langevin Equation (GLE) describing theevolution of the expectation values $\left\langle x_{i}\right\rangle_{t}$ ofthe atomic position operators $x_{i}$ of an open system. The latter is coupledlinearly to a harmonic bath kept at a fixed temperature. The equations ofmotion contain a non-Markovian friction term with the classical kernel {[}L.Kantorovich - PRB 78, 094304 (2008){]} and a zero mean \emph{non-Gaussian}random force with correlation functions that depend on the initial preparationof the open system. We used a density operator formalism without assuming thatinitially the combined system was decoupled. The only approximation made inderiving quantum GLE consists in assuming that the Hamiltonian of the opensystem at time $t$ can be expanded up to the second order with respect tooperators of atomic displacements $u_{i}=x_{i}-\left\langlex_{i}\right\rangle_{t}$ in the open system around their exact atomic positions$\left\langle x_{i}\right\rangle_{t}$ (the "harmonisation" approximation). Thenoise is introduced to ensure that sampling many quantum GLE trajectoriesyields exactly the average one. An explicit expression for the pair correlationfunction of the noise, consistent with the classical limit, is also proposed.Unlike the usually considered quantum operator GLE, the proposed $c-$numberquantum GLE can be used in direct molecular dynamic simulations of open systemsunder general equilibrium or non-equilibrium conditions.

c-number Quantum Generalised Langevin Equation for an open system