  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 and 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.
