The generated ground truth atomic explanation dataset includes 3,619,970 atomwise and 1,555,262 bondwise explanation values. An explanation value was not generated for every atom, or bond, due to the option of no chemical operation (see Figures and) or if the paired molecule was not found in the dataset. The distribution of the number of explanation values in each molecule is shown in the left panel of Figure . On average, each molecule with explanation values contains 1.82 atomwise and 1.98 bondwise values. The fraction of molecules with more than five explanation values per molecule is small. The distribution of explanation values (right panel of Figure ) peaks at a value of 0 and does not show a dependency on the value of the chemical property, here the HOMO-LUMO gap. The distribution of explanation values depends strongly on the atom type, as shown in Figure . The distribution for most atom types is non-gaussian, asymmetric, and not centered at zero value due to the directional nature of the decision tree and chemical operations. For carbon atoms, the distribution of explanation values is substantially narrower than for other atom types and peaks at zero value, and the reason may be that only terminal carbons are eligible in this atomwise attribution definition. For boron, sulfur, and phosphorus, flatter and broader histograms are observed with negative average atomwise explanation values, indicating the diverse roles these elements play in impacting the HOMO-LUMO gap of different molecules and an overall tendency to decrease the gap. For germanium, arsenic, and selenium, the histogram is sparse due to the limited number of molecules in the PCQM4Mv2 dataset. A heavy-tailed distribution for silicon, phosphorus, and sulfur could be caused by the range of possible oxidation numbers for these elements. The bondwise attribution depends on whether the bond is part of a ring and on the size of the ring, as shown in Figure . The average explanation value of bonds outside of rings is more negative (with a larger amplitude) than the average explanation value of bonds in rings, indicating that unsaturated bonds outside of rings impact the HOMO-LUMO gap more than bonds in rings. The difference can be explained by the different chemical operations, 𝑜 , applied to find paired molecules, 𝑀 , as defined in Eq. ( ). For bonds outside of a ring, the valid chemical operation (changing to a single bond) only applies to unsaturated bonds. Hence, the statistics always reflect the impact on HOMO-LUMO gaps by unsaturated bonds, which correspond to degenerate molecular orbitals and smaller HOMO-LUMO gaps, leading to negative explanation values. In contrast, for bonds in rings, 𝑜 applies in two situations: saturating an unsaturated bond in a ring (a big impact on the HOMO-LUMO gap) and removing a saturated bond in a ring (less impact on the HOMO-LUMO gap). The average impact of these two situations of bonds in rings leads to a smaller amplitude of the explanation value. Another observation is that the average bondwise attribution becomes more negative as the ring size increases. A possible explanation is that very small rings (3-or 4-membered rings) are more likely to be saturated, so removing a bond in the ring is likely to have less impact on the HOMO-LUMO gap. Neighboring atoms (or bonds) in a molecule can have similar attributions. This assumption of smoothness for neighboring attributions can be validated with the ground truth dataset.