Space Group Constrained Crystal Generation

Thanks for your suggestions! We provide the following responses to your concerns:

W1: The model details are not provided, some details about the model are missing. This makes me a little confused about what kind of model they are using, e.g., invariant ones or Equivariant ones.

Thank you for raising this concern. Our denoising model is invariant to O(3) transformations on lattices, as we denoise the invariant representation $\mathbf{k}$ rather than $\mathbf{L}$. Additionally, the model is translation invariant with respect to fractional coordinates, since it takes the relative coordinates as input, which remain unchanged under translation.

We have further improved the clarity of our model description in Section 4.4 and provided a more detailed illustration of the model architecture in Figure 4, Appendix B.1. Specifically, our denoising model consists of four main modules:

• Embedding Module combines the atom embedding and the sinusoidal time embedding into an MLP to acquire the initial node feature.
• Message-Passing Module in Eq. (12-13) iteratively updates the node feature $\mathbf{h}_i$ via the invariant representation $\mathbf{k}$ of the lattice, the node feature $\mathbf{h}_j$ and the relative fractional difference $\mathbf{f}_j - \mathbf{f}_i$ between all other nodes.
• Output Module in Eq. (14-15) yields the denoising terms without considering the space group constraints.
• Projection Module finally enforces the constraints by projecting the lattice onto a specific crystal family in Eq. (16) and aligning the atoms belonging to the same Wyckoff position in Eq. (17-18).

W2: How they construct the crystal structures from reduced number of atoms N' is not clear.

Sorry for the unclearity. Given a crystal structure with $N'$ Wyckoff positions, we denote that each Wyckoff position contains $n_s$ atoms satisfying $1\leq s\leq N'$. Here, $N=\sum_{s=1}^{N'}n_s$ represents the total number of atoms in a unit cell. We construct the whole crystal structures from reduced number of atoms $N'$ in this way:

1. For each $s$-th Wyckoff position, we use the notations $\mathbf{a}'_s$ and $\mathbf{f}'_s$ to respectively represent the atom type and basic fractional coordinate of Wyckoff position.
2. For each atom $s_i$ in the $s$-th Wyckoff position, we denote the atom type and fractional coordinate as $\mathbf{a}\_{s\_i}, \mathbf{f}\_{s\_i}$ (note that $1\leq s_i \leq n_s$).
3. First, we ensure that all atoms in one Wyckoff position have the same type, meaning $\mathbf{a}\_{s\_i}=\mathbf{a}_{s}$.
4. Next, for each atom $s_i$, we determine its fractional coordinate using the basic fractional coordinate and its corresponding transformation pair $(\mathbf{R}\_{s_i},\mathbf{t}\_{s_i})$. Specifically, we calculate $\mathbf{f}\_{s\_i}$ by $\mathbf{f}\_{s_i}=\mathbf{R}\_{s_i}\mathbf{f}'\_s+\mathbf{t}\_{s_i}$.
5. After iterating over all atoms and all Wyckoff positions, we finally obtain the the whole crystal structures.

The above explanations have been added into Appendix B.3 to address the reviewer's concern. Once again, we appreciate the reviewer for the positive recognition of our work and hope the reviewer's concerns regarding clarity are well-addressed.

Dear Reviewer 34Zd,

We greatly appreciate your thoughtful insights and valuable comments on our work. As the reviewer-author discussion period is ending, we would like to kindly remind you that we are looking forward to your feedback on our response.

We have provided more details about the architecture of the denoising model, and the relationship between the Wyckoff positions and individual atoms. If you have any further questions on our work, please do not hesitate to let us know.

Best regards,

Authors

Thanks for your constructive comments! We provide more explanations to address your concerns as follows.

W1: The relationship of the method with DiffCSP, on which it builds, should be better clarified, so as to more clearly highlight the contribution of the space group constraint.

Thanks for your suggestions, and we have further highlight the difference from DiffCSP in the revised version. Here, we summarize the difference in two main points:

1. DiffCSP directly generates the lattice matrix $\mathbf{L}$, while this paper instead generates the coefficients $k$ of the basis in Eq. (12) and (14) to better fulfil the space group constraint. Moreover, according to Proposition 1, the correspondence between the space of the coefficients $k$ and that of the inner product $\mathbf{L}^\top\mathbf{L}$ is bijective. This means our model enjoys the same expressivity with DiffCSP. This is supported by the ablation study in Table 4 where it showcased the comparable performance between DiffCSP and DiffCSP-k.
2. In Eq. (17), we first project the denoising term in the subspace of Wyckoff position via the pseudo-inverse of the matrix $\mathbf{R}$ of each atom, and then compute the mean of the projected denoising terms of the atoms in the same Wyckoff position. This constraint is not considered in the original DiffCSP but is crucial for adapting to the diffusion framework described in Section 4.3.

W2: The presentation of Table 2.

Thanks for your comment. We agree that the results of DiffCSP++ with GT space groups, which are typically unavailable in real-world settings and should be reported as an upper bound on achievable performance. We have reflected this point in Table 2 by renaming DiffCSP++ as DiffCSP++(w/ GT) to avoide the confusion in the revised paper.

W3 & Q1: Does DiffCSP++ in Table 3 use GT space group? is this to be expected?

No, DiffCSP++ in Table 3 does NOT use GT space group. Sorry for the confusion. Table 3 is for the ab initio generation task, where the atom types, coordinates and the entire lattice matrix are jointly generated. For the generation of each crystal, we randomly select a sample from the training set as the template for the space group. Note that we did no apply GT space group in this task, as the generation is not based on the templates from the testing set. Therefore, the results in Table 3 are consistent with the problem domain, and our method can be fairly compared with other approaches and applicable in the real-world setting.

Dear Reviewer GFzi,

Thanks again for your insightful comments, and this is a kind reminder that as the reviewer-author discussion period is ending soon, we are looking forward to your feedback on our response.

In our response and the latest revision, we have elaborated on the distinctions between our proposed method and the original DiffCSP, as well as clarified the application of GT. Thank you once again for your time and expertise. Please let us know if you have any further questions.

Best regards,

Authors

Thanks for your valuable comments! We have revised our paper according to your kind suggestions.

W1: Proposition 1 and 2 - are simple extensions of two known results from linear algebra - why is this presented without citations - to linear algebra books from Strang, Roman, etc.?

Thanks for your suggestion. For Proposition 1, the polar decomposition is well-known and we have added the citation about the polar decomposition in the revised paper. For Proposition 2, the 6 orthogonal bases of $3\times 3$ symmetric matrices are specifically designed by us. Although there could be other forms of orthongoal bases, our paper choose the one that straightfowardly distinguishes between different crystal families, as presented in Table 1. We provided a proof of Proposition 2 in Appendix A.2, and have included a citation in the revised paper to support that the space of $3\times 3$ symmetric matrices has a dimension of 6.

W2: Novelty is definitely present - but not something completely unexpected and draws and builds upon existing literature.

Thanks for recognizing our novelty. Here, we would like to further highlight the challengs by injecting space group into the diffusion process. In crystallology, it is well known that space group is closely related important properties of crystals, hence it will be beneficial if the crystal generation model can spontaneously maintain a given space group constraint. However, this is a non-trivial task, which presents two main challenges:

1. As listed in Table 1, the lattice must conform to a specific shape according to the space group.
2. Symmetric atoms (namely the atoms in the same orbit) must be updated synchronously, as illustrated by the dashed lines in Figure 2.

To address these challenges, we divide the constraint into two manageable parts: the basis constraint of the lattice matrix and the Wyckoff position constraint of the fractional coordinates, each of which is exactly retained during the generation process. Overall, while it is a natural enhancement of existing literature by further considering space group constraints, the way we implement the constraints is novel and non-straightforward.

Hi Kaifan,

Thanks for your interest in our work. We have included a discussion of PCVAE in Section 2 of the revised manuscript. While these two works are related in the part of the lattice constraint, our DiffCSP++ greatly differs from PCVAE in several key aspects:

1. Different Tasks. Given a space group, PCVAE focses only on predicting the lattice, whereas our DiffCSP++ aims at generating a whole crystal structure including both the lattice and atom coordinates/types. Since our task is more challenging, we require to elaborate a generation model that is capable of capturing the distribution of the crystal structure under the space group constraint. In our paper, we employ the diffusion model owing to its powerful generation ability, other than the VAE model used in PCVAE.
2. Different Implementation of Lattice Constraint. PCVAE represents the lattice using six lattice parameters, i.e., $a, b, c, \alpha, \beta, \gamma$, and predicts the parameter if its corresponding coefficient $p=1$ (see Table 3 in the PCVAE paper). However, since the edge parameters should be possitive and the angle parameters should fall into the range of $[0, 2\pi]$, it is nontrivial to ensure a valid prediction if we particularly apply a diffusion model whose output range is $[-\infty, \infty]$. Therefore, in our paper, we employs the logarithmic representation $\mathbf{k}$ of the lattice in Table 1. In this way, different lattice constraints correspond to different dimensions of $\mathbf{k}$, and more importantly, the range of $\mathbf{k}$ is $\mathbb{R}^6$ which is can be naturally generated by a diffusion model. We further discuss the relationship between $\mathbf{k}$ and the lattice parameters for each crystal family in Appendix A.3.
3. Atom Coordinate Constraint. Besides the constraint on lattices, our approach additionaly constrains the atom fractional coordinates with the help of Wyckoff position, providing a more comprehensive treatment of space group constraints.