We present a spectroscopic study of single quantum dot molecules (QDMs) formed by two closely stacked ${\text{In}}_{0.5}{\text{Ga}}_{0.5}\text{As}$ layers. The exciton fine structures as well as direct and indirect excitonic species associated with QDMs were identified by power dependent and polarization resolved microphotoluminescence measurements. As the temperature was increased, a directional energy transfer between the direct and indirect excitons in single QDMs was observed. A rate-equation model was developed to explain our data. We show that a phonon-assisted nonresonant tunneling of the hole between the two adjacent dots is responsible for such directional energy transfers in QDMs.