A symmetry orbital is a linear combination of atomic orbitals transforming as a component of an irreducible representation of the molecular point group. A molecular orbital is a one-electron wavefunction for the molecule, obtained as an eigenfunction of the chosen effective one-electron Hamiltonian operator. Symmetry organizes the eigenproblem, but a symmetry label alone does not determine its coefficients.

A symmetry orbital is a molecular orbital when it is also an eigenfunction of the effective one-electron Hamiltonian operator. If an irreducible representation occurs once in a complete invariant model space, a symmetry-invariant Hamiltonian acts as a scalar on that copy by Schur's lemma. Its basis functions are then degenerate eigenfunctions. When the same symmetry occurs in several copies, those copies can mix; one must diagonalize the corresponding block. Merely having a one-dimensional symmetry label is insufficient if other orbitals have that same label.