In the context of atomic orbitals, an open shell is a valence shell which is not completely filled with electrons or that has not given all of its valence electrons through chemical bonds with other atoms or molecules during a chemical reaction. Atoms generally reach a noble gas configuration in a molecule. The noble gases (He, Ne, Ar, Kr, Xe, Rn) are less reactive and have configurations 1s2 (He), 1s22s22p6 (Ne), 1s22s22p63s23p6 (Ar), etc.
For molecules it signifies that there are unpaired electrons. In molecular orbital theory, this leads to molecular orbitals that are singly occupied. In computational chemistry implementations of molecular orbital theory, open shell molecules have to be handled by either the restricted Open-shell Hartree-Fock method or the unrestricted Hartree-Fock method.
Likewise a closed shell or closed shell configuration is obtained with a completely filled valence shell. This configuration is very stable. In another meaning a closed shell configuration corresponds to state with all molecular orbitals doubly occupied or empty (a singlet state). Open shell molecules are more difficult to study computationally
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