We present a computational methodology for the screening of a chemical space of 1025 substituted norbornadiene molecules for promising kinetically stable molecular solar thermal (MOST) energy storage systems with high energy densities that absorb in the visible part of the solar spectrum. We use semiempirical tight-binding methods to construct a dataset of nearly 34 000 molecules and train graph convolutional networks to predict energy densities, kinetic stability, and absorption spectra and then use the models together with a genetic algorithm to search the chemical space for promising MOST energy storage systems. We identify 15 kinetically stable molecules, five of which have energy densities greater than 0.45 MJ/kg, and the main conclusion of this study is that the largest energy density that can be obtained for a single norbornadiene moiety with the substituents considered here, while maintaining a long half-life and absorption in the visible spectrum, is around 0.55 MJ/kg.