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We propose an energy-efficient power allocation algorithm for the multi-user millimeter-wave (mmWave) rate-splitting multiple access (RSMA) downlink with hybrid precoding and quality of service (QoS) constraints. The proposed scheme is applicable to the physical layer design of future wireless networks, such as the 6G cellular downlink, in which a transmitter equipped with multiple antennas must communicate unicast messages to multiple receivers simultaneously. First, we use a low-complexity design to define the analog and digital precoders in closed form. Second, we define an energy efficiency (EE) maximization problem to jointly optimize the power allocation among streams and the common stream rate allocation among users. We then solve the problem using a combination of Dinkelbach’s algorithm and difference of convex functions (DC) programming methods. Simulation results show that the proposed RSMA scheme offers EE improvements over a comparable space division multiple access (SDMA) power allocation scheme in scenarios with perfect and imperfect channel state information at the transmitter. Lastly, we present extensive numerical experiments that suggest that the computational complexity of the proposed RSMA energy-efficient power allocation algorithm can be reduced using the interior-point method such that the computational efficiency of RSMA is comparable to that of SDMA.