Definitions of atoms in molecules with the modified Dirac equations

In this work, we present calculations of the basins and electronic population at the relativistic level, with expressions derived from a total Lagrangian density associated with the modified Dirac Hamiltonian [Zapata-Escobar and Maldonado, J. Chem. Phys. 163, 024310 (2025)]. Calculations were carried out using a local implementation for the numerical construction of the basins and the corresponding integrations of the electronic population. The basins' evaluation and electronic population calculations were obtained on both relativistic and nonrelativistic expressions, reproducing 99.9% of the total charge in the HX (X = F, Cl, Br, I, At) and SnH4 molecules. We found that the relativistic effects cause a contraction of the electron density around the nuclei of the heavy atoms, while for the hydrogen atom, the electronic population decreases, leading to a decrease in the dipole moment. In addition, we present the continuity equation from the modified Dirac Hamiltonian in order to interpret the scalar field used to obtain the basin with this scheme of calculation. Finally, we propose a new Lagrangian density associated with the Dirac Hamiltonian, from which it is possible to obtain an expression for the zero-flux condition, thereby recovering the nonrelativistic basin expression within the quantum theory of atoms in molecules approach directly from the relativistic formulation without adding a heuristic term.