In the electronic structure theory, the SO splitting of the states is found when the total spin is nonzero. In the case of group theory, however, whether J is an integer or a half-integer is important (cf., eqs 4 and 5). Even when the SOC contributes, if J is an integer (that is, an even number of electrons in the system), the representation of the system is the same as the point group of the NR case. The double group is applied only when the number of electrons is odd.