  The end-to-end energy - energy correlations of random transverse-field
quantum Ising spin chains are computed using a generalization of an
asymptotically exact real-space renormalization group introduced previously.
Away from the critical point, the average energy - energy correlations decay
exponentially with a correlation length that is the same as that of the spin -
spin correlations. The typical correlations, however, decay exponentially with
a characteristic length proportional to the square root of the primary
correlation length. At the quantum critical point, the average correlations
decay sub-exponentially as $\bar{C_{L}}\sim e^{-const\cdot L^{1/3}}$, whereas
the typical correlations decay faster, as $\sim e^{-K\sqrt{L}}$, with $K$ a
random variable with a universal distribution. The critical energy-energy
correlations behave very similarly to the smallest gap, computed previously;
this is explained in terms of the RG flow and the excitation structure of the
chain. In order to obtain the energy correlations, an extension of the
previously used methods was needed; here this was carried out via RG
transformations that involve a sequence of unitary transformations.
