I know that some molecules have symmetry elements, and that we can associate operators to them. These operaors commute with the molecular electronic hamiltonian $H$, so the wavefunction can be chosen as a simultaneous eigenfunction of all these operators and $H$. When we use a basis set to expand the MOs, using, for exemple, gaussian-type functions, we can take linear combinations of these functions to construct symmetry adapted linear combinations (SALCs), that are eigenfunctions of the symmetry operators. My question is: when we do a HF calculation of the molecular orbitals of a molecule using programs like Gaussian, are the MOs SALCs MOs ( independent of the basis )? I am asking this because I was studyng the SFC proceddure using Szabo and I don't remember the book to comment about these questions about symmetry.
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Gaussian provides an option to respect symmetry during a calculation. In this case Gaussian will use symmetry adapted orbitals based on the detected point group and will show you in the ouptput to which irreducible representation an orbital belongs.
But you don't have to use symmetry in your calculation and in most cases it is done without it simply due to the fact that small deviations in geometry "break" your symmetry. Especially during geometry optimization on larger molecules with flexible alkyl-residues. The Hartree-Fock method does not really depend on symmetry. Symmetry can speed up calculations but it is not a requirement to carry it out.
