We thank the reviewer for their time and effort in reviewing our work. Below, we address the raised questions.

Note: please refer to the last rebuttal comment for references

Weakness 1 Overstated Claims of State-of-the-Art Performance: Although the paper claims to achieve state-of-the-art (SOTA) performance, it falls significantly behind previously published models such as JODO, EQGAT-Diff, and the more recent Semla-Flow—all of which were published before ICLR 2025. The performance gap is quite substantial. For larger and more realistic molecules in the GEOM Drugs dataset, only 16% have all valid atoms according to the paper's model, whereas other models are approaching perfect scores on these metrics. Question1 Your model underperforms compared to JODO, EQGAT-Diff, and Semla-Flow but claims SOTA performance. Can you add a comparison to one of these models?

Stability/neutrality difference: We respectfully disagree - the "stability" metric reported by Jodo [1], EQGAT-DIFF [2], and SemlaFlow[3] does not correspond to the "neutrality" metric we report. Those works consider charge $\ne 0$ atoms as stable, whereas our "neutrality" definition only considers 0-charge to be "neutral". Please refer to "Weakness 3" section for details on the "neutrality" metric.

Differences in GEOM benchmark: We also note the GEOM benchmark discrepancy: [1] uses only the lowest energy conformation, [5] uses the lowest 5 energy conformations and we use the lowest 30, similar to [4]. Removing higher energy conformations should improve our performance, but it is difficult to speculate by how much, and therefore hard to compare on GEOM-drugs.

Neutrality metric for aromatic methods compared to Kekule (ours): Finally, please also note how we compute neutrality in explicit aromatic settings (remark ($\ddagger$)), where C has both valencies 4 and 4.5 considered correct (Appendix I).

Below we report JODO [1], our method, and MiDi [5] results (for context). Our method still achieves the best or highly competitive performance across metrics, with a disadvantage on TV\textsubscript{a}, which likely stems from the lack of hard-constrained atom count generation (ablation on $\beta$, Table 3 of main text). We now add JODO to our benchmark tables.

QM9:

GEOM:

Weakness 2 Equivariance Requirement Not Essential for High Performance: Some models that operate directly in E(3) space are eliminating the need for equivariance. For instance, Simplified Scalable Conformer Generation (MCF) demonstrates significant performance improvements in conformer generation without the requirement for equivariance. While MCF tackles a slightly different task, it illustrates that we can develop equivariance-unaware models for molecular modeling that achieve genuine SOTA performance.

We agree that eliminating architecture invariance constraints in favor of model scaling is a promising future direction and thank the Reviewer for the reference supporting our work. However, E(3) modeling is well-motivated as it encodes symmetry constraints of 3D objects and therefore guarantees to obtain similar representations for similar molecules, and has therefore been employed in many recent works [4,5,6,7,9]. We, therefore, deem our theoretical contributions highly valuable, in the context of molecules, tackling: i) the lack of chirality-awareness in E(3) modeling; ii) intractable chiral-aware extension of E(3) methods (Section 3, Chirality and isometry groups).

We thank the reviewer for their time and effort in reviewing our work. Below, we address the raised questions.

Weakness 1.1 The first weakness is the method's computational complexity scales with the volume of the 3D grid (H×W×D), making it potentially intractable for larger molecules. As the author mentioned chirality in amino acids, will be a bottleneck for implementing the model in bio- and pharma-tasks. The current implementation requires high-resolution grids to maintain accuracy, leading to significant memory requirements.

We agree with the reviewer that this is an important aspect to consider. We renamed Section 4.5 to "Architecture choice, rotation invariance, and scaling" to highlight the complexity analysis of our method. Appendix J now also contains an analysis of the memory overhead, where we compare favorably on large batch sizes (please see "General rebuttal", "Memory scaling" and Appendix J, Figure 20). We also note the "Question 3" response, "Resolution" parameter, and the new ablation we include in Appendix C - these show that utilizing lower resolution may be possible, which would allow larger molecules to be fit by the model. Still, whole proteins are unlikely to work as fields, but conditioning on "pockets" could be feasible.

Question 1 Could adaptive grid resolutions or sparse representations help reduce memory requirements while maintaining accuracy?

Interesting ideas! We believe both approaches would likely require different architecture parameterizations for viability. We briefly experimented with DiT [1] early on in the method development, and compared it against UNet, but used the same atom $\pmb{u}\_a$ and bond $\pmb{u}\_b$ tensor fields for both, and therefore had to restrict to a shallow Transformer because of scaling constraints ($|\mathcal{G}|^2$ scaling, with $\mathcal{G}$ being the evaluated points). Utilizing adaptive resolutions/sparse representations may, however, bring considerable benefits and is likely a promising future avenue.

[1] Peebles et al., "Scalable Diffusion Models with Transformers". IEEE/CVF 2023.

Question 2 How sensitive is the model's performance to the choice of field parameters ($\sigma$, $\gamma$)? What's the theoretical basis for choosing these values?/ Weakness 1.2 Another limitation is the analysis of field parameter sensitivity ($\sigma$, $\gamma$) is not very clear./ Question 3 Why did the authors choose 0.33Å resolution for the grid, and how would different resolutions affect the quality-efficiency trade-off?

Thank you for noticing this. Field parameter analysis should prove useful for future development: we now add ablations of field parameters in Appendix C (Figures 8 and 9). The following were added in Appendix C:

The field hyperparameters were selected during early experimentation. We focused on the effect of field hyperparameters on fitting $u\_a(x), u\_b(x)$ to the generated fields $\pmb{u}\_{a},\pmb{u}\_{b}$ (Equations 13 and 14 and Appendix E). We compute the "extraction accuracy", which measures how many molecules have the same molecular graph extracted from the fields they create. We additionally report the average "distance from original", computed as the average Euclidean distance between the correct atom positions and the ones extracted. For both, we determine how robust they are for various levels of noise applied to the fields. Ablation of field hyperparameters for the resulting diffusion model's performance was not performed, due to heavy computational resources required.

Field component standard deviation $\sigma$: Intuitively, larger standard deviations $\sigma$ would facilitate smaller receptive field CNN kernels to capture longer distance neighboring RBF components. At the same time, we require that molecular graphs and conformations are correctly extracted. Figure 8 shows how increasing $\sigma$ and $r$ worsens the noise robustness. However, the more coarse resolution of $0.44$ would allow for larger architectures. Since it is sensible to assume larger architectures will produce less noise, this option is very sensible for future development.

Field component weight $\gamma$: The component weights may be thought of as input an input normalization parameter, and we therefore did not consider other values that $0.176$, which should normalize input to (roughly), values between $[0,1]$. We believe this parameter is unlikely to change the model's performance.

Resolution $r$: In general, it is desired to have as low resolution as possible, while ensuring that conformer and graphs can still be extracted with sufficient accuracy. Figure 9 shows how lowering the resolution from 0.33 to 0.44Å worsens "extraction accuracy" and "distance from original" across noise levels. We note, however, that $r=0.44$ may still be a valid choice, as it would make larger NN parametrizations feasible, and therefore the field noise levels are likely going to be smaller.

We thank the reviewer for their time and effort in reviewing our work. Below, we address the raised questions.

Weakness 1 Although alignment methods are employed, the proposed method cannot guarantee the rotation invariance.

Irrespective of an initial molecule's orientation, the algorithm always deterministically rotates it s.t., the largest variation axes are aligned (in this order) on the height, width, and depth. Compared to previous methods that use E(3) invariant NN and can generate molecules with any rotation, our method (on optimality) will only generate molecules according to this frame of reference.

The specific orientation choice is related to the NN's generalization. Ideally, similar molecules will have similar orientations, resulting in their field densities being aligned well. Intuitively, aligning molecules on the highest axes variance would place various conformer categories (e.g., long and planar molecules, respectively), in roughly the same orientations, allowing our $\theta$ NN's activations to be similar for similar molecules. The degree to which this is true will determine how efficiently our NN's parameters are used, and in turn, the performance, (e.g, Table 3, using rotation augmentation FMG$\_{aug}$'s 82.6 Mol neutrality increases to FMG$\_{\beta=2}$'s 91.5, when the reference rotations in Section 3.5 are used).

Weakness 2 As shown in table 2, the molecule graph quality metric is not good on the QM9 dataset.

We argue that relative to other methods, our method maintains competitive MST and bond type distribution TV$\_b$. Our TV$\_a$ performance is sub-optimal, but we explain how this is likely a result of our model trading this property in favor of neutral molecule generation, which is not possible for point-cloud methods, as they are hard-constrained to generate a pre-specified number of atoms. Our ablation study provides further indication for this hypothesis, since, in Table 3, we obtain significant improvement on TV$\_a$, as a result of adding atom number conditioning $N$ (from no conditioning, FMG$\_{\beta=0}$ to FMG$\_{\beta=2}$).

Question 1 What's the trade-off between losing rotation invariance and gaining the ability of learning chirality?

Chirality is one 3D property that influences the physical, chemical, and biological properties of molecules, similar to all other isomeric molecular properties. Therefore, modeling molecules with E(3) invariance will always be limited. We argue that chirality awareness is especially important in organic chemistry, as enzymes/proteins are chiral, and often skew the distribution of optimal and safe drugs. This motivated us to explore the limitations of invariance groups wrt. chirality, proving the lack of chiral awareness in E(3), and further exploring computational constraints of adapting E(3) to SE(3), which is prohibitive ($\mathcal{O}(N^4)$).

In novel compound generation (joint generation of molecule and conformer graphs $p(C,G)$), rotational invariance is stated as a strict requirement for good generalization. However, we argue that non-invariant methods may still be employed successfully (e.g, Table 3, using rotation augmentation FMG$\_{aug}$'s 82.6 Mol neutrality increases to FMG$\_{\beta=2}$'s 91.5, when the reference rotations in Section 3.5 are used). Further evidence of this is shown in a parallel line of work that considers the generation of $p(C|G)$, where results suggest that good performance can be achieved without restricting the NN parametrizations [1,2].

[1] Yuyang Wang et al., "Swallowing the Bitter Pill: Simplified Scalable Conformer Generation". ICML 2024.

[2] Josh Abramson et al., "Accurate structure prediction of biomolecular interactions with AlphaFold 3". Nature 2024.

Thank you for your thoughtful feedback. We hope that your concerns have been satisfactorily addressed, and we would be greatful if the same will be reflected in stronger support from you for this work. Otherwise, we hope to further engage and address further concerns and suggestions.

We thank the reviewer for their time and effort in reviewing our work. Below, we address the raised questions.

Weakness 1 The impact of this approach on memory overhead requires further investigation. How does the memory overhead compare to point cloud representations? Could you provide insights on how memory usage scales with increasing molecule size and greater diversity of atom and bond types?

We agree with the reviewer that this is an important aspect to consider. We renamed Section 4.5 to "Architecture choice, rotation invariance, and scaling" to highlight our complexity analysis, which now includes the memory overhead. Please refer to the "General Rebuttal", section "Scalability" for a short explanation and Appendix J. Appendix J for an in-depth analysis.

Thank you for your suggested analysis and helping us improve our manuscript. We hope our new experiment has addressed the memory overhead aspects, and hope that you can provide further suggestions if not.