SlowGuess/NeurIPS_2023_OCR huggingface dataset:
- All models performed better on alkanes or the alkyl part of the molecules, e.g., the linear nonane, the branched  $t$ -butyl group, and the cyclic cyclohexane. One exception is cubane, which adopts an unusually sharp  $90^{\circ}$  bonding angle that would be highly strained as compared to the  $109.45^{\circ}$  angle of a tetrahedral carbon. In this example, InfGCN was still able to make a relatively accurate prediction.
- The predictions were worse for atoms with high electronegativity (F, O, N), even if the errors are normalized by the number of electrons. From a chemical aspect, a higher density will lead to a better "polarizing" ability to distort the electron cloud of the covalent atom, leading to more complicated higher-order interactions between atomic orbitals  ${}^{4}$  . For example,the carbon atom in  ${\mathrm{{CF}}}_{4}$  has a significant positive partial charge,but DeepDFT overestimated its density (in pink). Noticeably, InfGCN can estimate the oxygen density with great accuracy, e.g., in glycerol, di-  $t$  -butyl ether, and isoamyl acetate.
- The predictions were even worse for a conjugated and aromatic system where electrons are highly delocalized<sup>5</sup>. Delocalization of electrons allows for long-distance interactions which are harder to capture. The presence of electron-donating groups (EDGs) and electron-withdrawing groups (EWGs) contributes greatly to the conjugated system. For example,
4The electron cloud actually won't undergo any polarizing or hybridization procedure, but it is still beneficial to think of it in this way, as it provides an intuitive interpretation of the linear combination of atomic orbitals (LCAO). In practice, such ideas are still widely used in chemistry.
The term delocalized is also not accurate for describing electron density. However, as mentioned above, it is a convenient way of building molecular orbitals from atomic orbitals, and so are the concepts of EDG and EWG mentioned in the following text.
Figure 7: Resonance forms of aniline and nitrobenzene with formal charges.
the amino group  $-\mathrm{NH}_{2}$  is an EDG when bonding to a conjugated system like benzene, facilitating ortho- and para-electrophilic reactions. In contrast, the nitro group  $-\mathrm{NO}_{2}$  is an EWG that facilitates ortho- and para-nucleophilic reactions (See Fig.7). It can be seen from the visualization that the ortho- and para-positions of aniline are underestimated (in green) and those of nitrobenzene are overestimated (in pink) with DeepDFT and other models. For cytosine and fluorouracil, the amide (lactam) tautomeric form predominates at pH 7, further making the electron structures more complicated to achieve accurate predictions. More examples of the conjugated systems include the nitro group itself where the density of oxygen is overestimated and the amide group in asparagine where the density of the amide oxygen is underestimated and that of nitrogen is overestimated.
Table 5: More results on QM9.
Name	File ID	Structure	Ground Truth	InfGCN	CNN	DeepDFT	DeepDFT2	EGNN	DimeNet	DimeNet++	GNO	FNO	LNO
ammonia	2	NH3	3.16	22.79	5.34	2.02	12.27	11.45	15.22	57.87	157.19	36.00
urea	20	HNNH2	0.99	3.56	3.33	1.24	9.63	9.83	9.94	44.72	29.73	22.31
acetone oxime	49	HO=CH2	0.92	2.72	2.90	1.02	12.54	13.04	12.60	31.79	30.03	21.52
furan	52	1.40	3.81	2.86	1.68	12.05	12.51	12.38	30.68	39.98	25.38
tetrafluoro-methane	184	CF4	1.76	4.83	1.95	1.00	9.79	6.48	7.75	34.19	36.31	27.36
glycerol	397	HO=CH2OH	0.79	2.52	2.77	0.88	11.73	11.36	11.03	33.05	29.21	23.78
cyclohexane	658	0.54	1.95	1.75	0.78	13.21	12.65	12.60	30.48	30.80	19.33
Table 5: More results on QM9. (Continued)
aniline	940	NH2	1.22	4.67	2.85	1.30	12.62	12.80	12.16	35.29	31.35	24.60
cytosine	4318	HN NH2	1.25	5.62	2.96	1.37	10.35	10.18	10.26	40.31	24.58	20.80
cubane	19116	0.74	8.22	2.76	1.27	11.82	13.03	12.31	28.61	32.88	23.46
purine	24537	N NH2	1.51	16.34	2.90	1.44	11.41	11.25	10.98	25.98	22.07	20.45
di-t-butyl ether	57520	0.78	2.28	2.40	1.04	12.37	12.79	11.88	55.77	27.71	29.73
isoamyl acetate	60424	CH2=CH	0.67	2.29	2.67	0.87	12.41	12.13	11.65	12.51	27.16	30.92
asparagine	61439	NH2CH2=CH	1.02	3.71	3.12	1.11	10.25	10.05	10.10	44.62	25.34	20.53
nonane	114514	0.65	2.54	2.38	0.85	12.86	12.53	12.41	37.23	26.02	20.92
nitrobenzene	131915	1.44	22.93	3.41	1.45	11.00	10.87	11.16	47.66	31.38	22.41
fluorouracil	132701	0.98	4.59	2.61	0.97	10.06	9.97	9.90	51.36	23.77	16.10