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Crystals are materials that consist of the same pattern of atoms repeated throughout 3D space, on a grid of boxes called unit cells. To describe a specific crystal structure, you just need to define the size and shape of the unit cell, plus the type and positions of atoms within the cell. In practice, most crystals have additional symmetry, for example rotation or reflection. This symmetry is described using a branch of mathematics called group theory, and it turns out there are 230 distinct space groups that a crystal structure might fall into.

Scientists are aware that real crystals tend to have high symmetry. So in order to predict the structure of new materials, it is reasonable to guess possible space groups first, then insert atoms in a way that preserves the symmetry of those space groups. The basic idea is to generate a large number of random crystal structures in this way, then optimize them using force-field methods or DFT to calculate their energies. The structure with the lowest total energy is the one that is most likely to exist in the real world. Using computer simulations to do this is often much faster and cheaper than running physical experiments.

To ensure that symmetry is preserved when adding atoms, we can look at special positions within the unit cell called Wyckoff positions. Wyckoff positions define what symmetry is needed for objects located at different locations in the cell. To visualize this, imagine cutting out holes on a paper snowflake before unfolding it. If you cut a hole in the middle corner where two edges meet, you will end up with a single hole after unfolding. Mathematically, this is because the corner is a high-symmetry Wyckoff position. If you instead cut a hole in the middle of the paper, away from the edges, then you will end up with multiple holes after unfolding. We can apply the same idea to construct a symmetrical crystal. When we want to insert an atom into a Wyckoff position, we can make sure that the result is still symmetrical by making copies of that atom in the right places (e.g. by "unfolding the snowflake").

Doing this for molecular crystals is trickier, because you have to worry about the orientations of the molecules in addition to their positions. For example, imagine that instead of cutting a circular hole in the corner of the snowflake (similar to an atom), you instead cut out a letter of the alphabet (similar to a molecule). Now if you were to unfold the paper, you would end up with a shape that didn't match the original. Analogously, putting an asymmetrical molecule in a symmetrical Wyckoff position would result in extra atoms that didn't match the desired molecular structure. PyXtal solves this problem by automatically detecting molecular symmetry and determining which Wyckoff positions are compatible with which molecular orientations.

Simulating the wave equation works by splitting space up into a grid of tiny cells. You define the value of the field (the quantity you care about, like height, pressure, etc.) at each cell, as well as how quickly the value is changing there (the "velocity"). Then you start moving forward in time by small steps. For every time step, at each cell, you update the field value using the velocity, and you update the velocity using the acceleration.

This is a basic example of converting a differential equation into a cellular automaton. This technique can also be applied to more complicated equations, and is widely used in computational physics.

For the wave equation specifically, the acceleration is proportional to the second spatial derivative (the "Laplacian") of the field value. On a 2D grid, this can easily be calculated by comparing the value at each cell to the sum of its neighbors.

One nice property of the wave equation is that it is easy to extend to higher dimensions. The method for simulating a 3D system is basically the same as for 1D or 2D. You can even simulate a lower-dimensional object waving around in a higher-dimensional space. The simplicity of the calculation also makes it easy to code and suitable for running on a GPU.

One big challenge is ensuring numerical stability. If your cells or your time steps are too large, the simulation tends to "blow up" into chaotic noise. But if you go too small, you'll be doing more computation than you need to. To find the right balance, the CFL conditions can be used to determine how big your step sizes can be without losing stability.