Paper: Space Group Constrained Crystal Generation (ICLR-2024)

Reviewer comment (Reproducibility):
How they construct the crystal structures from reduced number of atoms N' is not clear.

Author response:
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 $1\leq s_i \leq n_s$).
5) First, we ensure that all atoms in one Wyckoff position have the same type, meaning $\mathbf{a}\_{s\_i}=\mathbf{a}_{s}$.
6) 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 $\mathbf{f}\_{s_i}=\mathbf{R}\_{s_i}\mathbf{f}'\_s+\mathbf{t}\_{s_i}$.
7) 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.

Reviewer comment (Novelty):
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.?

Author response:
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.

Reviewer comment (Novelty):
Novelty is definitely present - but not something completely unexpected and draws and builds upon existing literature.

Author response:
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.