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I see that Gaussian has feature to set periodic boundary conditions specifying the parameter Tv in the input file. Does it do it via the plane wave basis set? Also, the question is whether it is efficient to use Gaussian to compute the relaxed lattice constant, band structure and DOS for solids or it is better to use programs like ABINIT.

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Even if it is possible to calculate periodic structures in Gaussian, I would not suggest it. Better option is to use codes devoted to calculation of solids, among others the ABINIT you mentioned, or e.g. Quantum Espresso.

Gaussian does not use plane wave basis set, will be slower and in general less powerful than codes devoted to solids.

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