Relativistic definitions of atoms in molecules with the modified Dirac equation

In quantum chemistry, the region associated with atoms in molecules (AIMs) is determined using the basin definition through the action integral of the total Lagrangian density. This can be associated with different Hamiltonians, such as Schrödinger, Dirac, or the modified Dirac Hamiltonian. The latter two differ only in the associated metric matrix: while the Dirac Hamiltonian is the 4 × 4 identity matrix, the modified Dirac Hamiltonian is a diagonal matrix composed by the 2 × 2 identity matrix and a 2 × 2 diagonal matrix with elements T̂/2mc2. It was shown by Cioslowski and Karwowski that when the Dirac Hamiltonian is considered, the total Lagrangian density is zero at every point within the molecular volume, making it impossible to partition the molecular electronic structure into basins. Moreover, the nonrelativistic total Lagrangian density derived from the Dirac Hamiltonian is also zero at every point, and a heuristic term must be added to obtain the basin definition in the Quantum Theory of Atoms in Molecules (QTAIM) developed by Bader. In contrast, the total Lagrangian density associated with the modified Dirac Hamiltonian is nonzero at every point within the molecular volume, and the basin can be defined in a relativistic framework. Taking the nonrelativistic limit of this Lagrangian density, the standard nonrelativistic basin definition within the QTAIM approach is recovered.