When we do Hartree-Fock computations by solving the Roothaan equations $FC = SC \varepsilon$ is it a Galerkin method?
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$\begingroup$ Also, I think this question is a good fit here, but if you have any questions about electronic structure theory that don't pick up steam here, I would say you should try to ask them on Matter Modeling SE $\endgroup$– TyberiusDec 12, 2021 at 22:02
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Yes, that equation comes about from making the approximation that we can write the wavefunction in terms of a finite basis.
You won't typically see the phrase Galerkin used in the chemical literature, but the linear combination of atomic orbitals (LCAO) approach is a Galerkin method. Here is at least one paper, written by a mathematician about electronic structure methods, describing it as a Galerkin method.
