  Crystals are the foundation of numerous scientific and industrial
applications. While various learning-based approaches have been proposed for
crystal generation, existing methods seldom consider the space group constraint
which is crucial in describing the geometry of crystals and closely relevant to
many desirable properties. However, considering space group constraint is
challenging owing to its diverse and nontrivial forms. In this paper, we reduce
the space group constraint into an equivalent formulation that is more
tractable to be handcrafted into the generation process. In particular, we
translate the space group constraint into two parts: the basis constraint of
the invariant logarithmic space of the lattice matrix and the Wyckoff position
constraint of the fractional coordinates. Upon the derived constraints, we then
propose DiffCSP++, a novel diffusion model that has enhanced a previous work
DiffCSP by further taking space group constraint into account. Experiments on
several popular datasets verify the benefit of the involvement of the space
group constraint, and show that our DiffCSP++ achieves promising performance on
crystal structure prediction, ab initio crystal generation and controllable
generation with customized space groups.
