In general, the statement that core electrons occupy orbitals with high symmetry and valence electrons occupy orbitals with low symmetry is not accurate. The distribution of electrons in an atom's orbitals depends on the specific electronic configuration and the arrangement of the atoms in a molecule.
The electronic configuration of an atom determines the distribution of electrons among different energy levels and orbitals. Core electrons refer to the electrons in the inner shells or energy levels, closer to the nucleus. These electrons generally have lower energy and are tightly bound to the nucleus. They tend to occupy spherically symmetric orbitals, such as the 1s, 2s, and 2p orbitals in the case of the first three periods of the periodic table.
On the other hand, valence electrons are the outermost electrons in an atom that participate in chemical bonding. These electrons occupy the highest energy levels or valence shells. The valence electrons may be distributed among various orbitals, including s, p, d, and f orbitals, depending on the atom and its position in the periodic table. The symmetry of the valence orbitals depends on the atom's electronic configuration and molecular geometry.
The symmetry of orbitals in molecules is determined by the overall symmetry of the molecule. The molecular geometry, which is influenced by the arrangement of atoms and the type of chemical bonds formed, affects the symmetry of molecular orbitals. Valence electrons in a molecule can occupy a variety of molecular orbitals, and their symmetry will depend on the molecular geometry and the type of molecular orbitals involved (e.g., sigma bonds, pi bonds, etc.).
So, to summarize, both core and valence electrons can occupy orbitals with various symmetries, and the specific symmetry will depend on the electronic configuration of the atom or the molecular geometry in the case of molecules.