Crystallography: How to sample *unique* structures from configuration space? (binary alloy supercell)

I'm working with a binary alloy system (call it 'AB'), and I'd like to generate a few unique configurations for DFT purposes.

My primitive cubic unit cell consists of 4 atoms; i.e. 4 positions. Thus, if I want 25% A, 75% B, I'll assign 1 atom of type A and 3 atoms of type B to my 4 positions. For instance, in fractional coordinates, this looks like:

• A: 0 0 0
• B: 0.5 0.5 0
• B: 0.5 0 0.5
• B: 0 0.5 0.5

In this example, there is only 1 unique configuration. For instance, say I swap the positions of the first two atoms. This resulting coordinate list looks different, but it would still describe the exact same configuration because it is equivalent via symmetry operations:

• A: 0.5 0.5 0
• B: 0 0 0
• B: 0.5 0 0.5
• B: 0 0.5 0.5

This isn't really an issue for the 4-atom system, because the number of possible combinations is so small (only 4) that it's easy to check them. However, the problem scales rapidly. Assuming I stick with 25% A and 75% B percentage, if I generate a:

• 8-atom supercell: I have (8 choose 2) = 28 possible combinations.
• 16-atom supercell: I have (16 choose 4) = 1820 possible combinations.
• 32-atom supercell: I have (32 choose 8) = 10518300 possible combinations.
• 64 atom supercell: I have (64 choose 16) = 4.885e+14 possible combinations.

...and so forth. Considering that the system I would like to work with has a size of 128 atoms, proper sampling needs to be done intelligently.

My question is: How can I determine the number of UNIQUE combinations? I would like to sample from the set of unique combinations, instead of from the set of possible combinations. Given the large configuration space, this needs to be done in a computationally efficient manner.

This problem seems so fundamental that I would have expected someone to have found some sort of solution to it, but I haven't been able to find one yet.

(... I suspect the answer has something to do with utilizing space groups and symmetry operations. However, I'm relatively new to group theory / crystallographic concepts, so advice from someone more well-versed in these topics would be greatly appreciated.)

• +1 Welcome to our new community and thank you for contributing your question here!! We hope to see much more of you in the future! How did you find us? Jul 27 at 8:39
• Are you only interested in the cases where the fractional coordinates are rational numbers with small denominators? And besides, if your sample size is much smaller than the total number of combinations, then you don't need to worry about the equivalence of certain combinations, since exploiting the equivalence only increases your sampling efficiency marginally. Jul 27 at 12:51
• Thanks for the welcome! I've seen how helpful Stack Exchange is in the computer science community; I was wistfully hoping that there'd be a hub for the materials modelers, so I went looking for one. Happy to have found this! Jul 28 at 3:30
• @wzkchem5 I haven't yet seen an instance where a fractional coordinate is irrational, so yes. Thank you for your input in regards to the sampling. Would you be able to point me to any references as well? Jul 28 at 3:38