A novel method is introduced in order to treat the dissipative dynamics of quantum systems interacting with a bath of classical degrees of freedom. The method is based upon an extension of the Nos\'e-Hoover chain (constant temperature) dynamics to quantum-classical systems. Both adiabatic and nonadiabatic numerical calculations on the relaxation dynamics of the spin-boson model show that the quantum-classical Nos\'e-Hoover chain dynamics represents the thermal noise of the bath in an accurate

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... d simple way. Numerical comparisons, both with the constant energy calculation and with the quantum-classical Brownian motion treatment of the bath, show that the quantum-classical Nos\'e-Hoover Chain dynamics can be used to introduce dissipation in the evolution of a quantum subsystem even with just one degree of freedom for the bath. The algorithm can be computationally advantageous in modeling, within computer simulation, the dynamics of a quantum subsystem interacting with complex molecular environments.