The transfer of thermal energy at the nanoscale is critical for understanding and designing advanced technologies. Ultra-thin membranes separated by nanometre-scale gaps, subjected to temperature gradients, present a unique platform to explore novel energy transfer mechanisms. In this work we examine the interaction energy between two reconstructed silicon nanomembranes using a density-functional-based tight-binding (DFTB) approach, focusing on its impact on thermal conductance across a nanogap. By coupling the DFTB method with a harmonic atomistic model, we calculate the vibrational modes (phonons) and the equilibration times, which are directly related to the thermal conductance. Our findings show that surface reconstruction and the relative alignment of facing dimer structures significantly influence the phononic contribution to thermal conductance. Although the harmonic model simplifies the interactions of the real system,our results agree well with previous studies, demonstrating that this model captures key aspects of phonon-mediated heat transfer. Overall, our approach provides a computationally efficient method for understanding phononic heat transfer across nanogaps, with implications for designing nanoscale thermal management systems.