Charge-displacement analysis via natural orbitals for chemical valence in the four-component relativistic framework

We have recently introduced a simple yet powerful tool for analyzing quantitatively the coordination bond in terms of the donation and back-donation constituents of the Dewar-Chatt-Duncanson model. Our approach is based on the decomposition, via natural orbitals for chemical valence (NOCV), of the so-called charge-displacement (CD) function into additive chemically meaningful components (Bistoni et al. J. Chem. Phys. 2015, 142, 084112). The method, referred to as NOCV/CD, provides clear-cut measures of donation and back-donation charge flows following bond formation, and its robustness has been demonstrated by a tight correlation of the related charge-transfer estimates with experimental observables. In this paper we extend the NOCV/CD analysis scheme to the four-component relativistic framework, which includes spin-orbit coupling variationally. This formalism is incorporated into a recently developed, highly efficient parallel version of the relativistic Dirac-Kohn-Sham (DKS) program BERTHA (Rampino et al. J. Chem. Theory Comput. 2014, 10, 3766-3776). We test the accuracy and numerical stability of this new implementation through the analysis of the convergence properties of the basis sets employed to expand the DKS spinor solution and those used to linearize the electronic density in the density-fitting algorithm speeding up the evaluation of the DKS matrix. An illustration of NOCV/CD analysis in the relativistic framework is also given through a study of the metal- carbonyl coordination bond in a series of [M-CO]+ (M = Cu, Ag, Au) complexes of group 11 metals, where relativistic effects, including spin-orbit coupling, are found to play an important role.