Linear regression (LR) and random forest (RF) methods were applied, being with VTS and X_LasY descriptor ensembles, to predict E H , E L , and ΔE HL . The mean absolute errors (MAE) of the testing sets are summarized in Tables and, and the corresponding training set results are provided in Table . The nonlinear RF models using X_Las05 descriptor ensembles, denoted as X_Las05_RF, appear to provide better MAEs in all three target properties than the corresponding linear models (X_Las05_LR). Current X_Las05_RF models also provide comparable predictivity in respect to the early RF results using extended connectivity fingerprints (ECP4) representation, however, demonstrated higher learning efficiency with using 88k training molecules (vs. 118k molecules for the ECP4_RF model). In Table , current linear models labeling as X_Las05_LR appear to be noticeably more accurate than the previous linear models using Elastic Net and Baysian Ridge methods coupled with ECP4 representations. ). 7 With lowering the penalty parameter to 0.1 during the process of descriptor extraction, one can generate substantially larger descriptor ensembles, being labeled as X_Las01, over X_Las05 as shown in Table . Consequently, all X_Las01_LR and X_Las01_RF models provide enhanced predictivity than the corresponding models using X_Las05 descriptor ensembles due to larger degrees of freedom in the descriptor space (see Table ). By examining the difference between R 2 and Q 2 of the training results in Table , all LR and RF models using VTS, X_Las01, and X_Las05 ensembles can be considered statistically meaningful due to the absence of overfitting (|R 2 -Q 2 | < 0.1). Figure . The flow charts of machine learning and deep learning approaches using QM9 dataset Despite all LR and RF models provide reasonable predictions for E H , E L , and ΔE HL of the organic molecules in QM9 dataset, the conventional chemical accuracy was still not reached using these linear and RF approaches. The current LR and RF models can achieve 3-6 times of chemical accuracy in terms of MAEs of E H , E L , and ΔE HL predictions. Faber et al. demonstrated that combining NN models with the graph representations, being derived from the three dimensional (3D) Cartesian coordinates, could significantly reduce MAEs of E H , E L , and ΔE HL predictions down to less than 0.1 eV. In Table , the results reported by Schütt et al. using a deep learning NN model (SchNET) provided the most accurate results for the E H , E L , and ΔE HL predictions where MAEs of E H and E L were less than 0.043 eV, except that of ΔE HL was at 0.063 eV. In this study, we reproduced the SchNET approach using fewer molecules for the training set (88k vs. the original size of 110k), and the corresponding models (annotated as Schnet-3d) can provide MAEs of E H , E L , and ΔE HL at 0.051, 0.041, 0.075 eV, respectively. Only minor deterioration was observed for the performance of these models in comparison with the original case. The training set used ~118k compounds of QM9 dataset in reference 7. GG and GC denote gated graph neural network and graph convolutional neural network models, respectively. Both models used training set of ~110k molecules of QM9 dataset in reference 41 (MPNN) and reference 38 (SchNET).