Energy efficiency of three common hybrid automatic repeat request (HARQ) schemes including Type I HARQ, HARQ with chase combining (HARQ-CC) and HARQ with incremental redundancy (HARQ-IR), is analyzed and joint power allocation and rate selection to maximize the energy efficiency is investigated in this paper. Unlike prior literature, time-correlated fading channels is considered and two widely concerned quality of service (QoS) constraints, i.e., outage and goodput constraints, are also considered in the optimization, which further differentiates this work from prior ones. Using a unified expression of asymptotic outage probabilities, optimal transmission powers and optimal rate are derived in closed-forms to maximize the energy efficiency while satisfying the QoS constraints. These closed-form solutions then enable a thorough analysis of the maximal energy efficiencies of various HARQ schemes. It is revealed that with low outage constraint, the maximal energy efficiency achieved by Type I HARQ is <formula><tex>$\frac{1}{4\ln2}$</tex></formula> bits/J, while HARQ-CC and HARQ-IR can achieve the same maximal energy efficiency as <formula><tex>$\frac{\kappa_\infty}{4\ln2}$</tex></formula> bits/J where <formula><tex>$\kappa_\infty = 1.6617$</tex></formula>. Moreover, time correlation in the fading channels has a negative impact on the energy efficiency, while large maximal allowable number of transmissions is favorable for the improvement of energy efficiency. The effectiveness of the energy-efficient optimization is verified by extensive simulations and the results also show that HARQ-CC can achieve the best tradeoff between energy efficiency and spectral efficiency among the three HARQ schemes.