Relativistic definitions of atoms in molecules with the modified Dirac equation

Abstract

In quantum chemistry, the region associated with atoms in molecules (AIMs) is determined using the basin definitionthrough the action integral of the total Lagrangian density. This can be associated with different Hamiltonians, suchas Schrödinger, Dirac, or the modified Dirac Hamiltonian. The latter two differ only in the associated metric matrix;while for the Dirac Hamiltonian is the 4x4 identity matrix, for the modified Dirac Hamiltonian is a diagonal matrixcomposed by the 2x2 identity matrix and a 2x2 diagonal matrix with elements T̂ /2mc2 . It was shown by Cioslowski andKarwowski that when the Dirac Hamiltonian is considered, the total Lagrangian density is zero at every point withinthe molecular volume, making impossible to partition the molecular electronic structure into basins. Moreover, thenonrelativistic total Lagrangian density derived from the Dirac Hamiltonian is also zero at every point, and a heuristicterm must be added to obtain the basin definition in the Quantum Theory of Atoms in Molecules (QTAIM) developedby Bader. In contrast, the total Lagrangian density associated with the modified Dirac Hamiltonian is nonzero at everypoint within the molecular volume, and the basin can be defined in a relativistic framework. Taking the nonrelativisticlimit of this Lagrangian density, the standard nonrelativistic basin definition within the QTAIM approach is recovered.