Application of the relativistic electron localization function to study the electronic structure of superheavy elements

A formula for calculating the relativistic electron localization function (RELF) within the framework of the Dirac-Fock method is obtained. An approach similar to that used earlier in [A.D. Becke and K.E. Edgecombe, The Journal of Chemical Physics $\mathbf{92}$, 5397 (1990)] in deriving an expression for the nonrelativistic electron localization function (ELF) is applied. It is demonstrated that the expression for RELF differs from the expression for ELF with replacement of the nonrelativistic electron density by its relativistic counterpart. Relativistic calculations of ELF and RELF for a number of superheavy elements are performed and the results are compared. By several examples it is shown that the ELF value equal to 0.5 does not necessarily correspond to the distribution density of homogeneous electron gas.