Bonding analysis using localized relativistic orbitals: Water, the ultrarelativistic case and the heavy homologues $H2X$ (⁠$X=Te$⁠, Po, eka-Po)

We report the implementation of Pipek-Mezey [J. Chem. Phys. 90, 4916 (1989)] localization of molecular orbitals in the framework of a four-component relativistic molecular electronic structure theory. We have used an exponential parametrization of orbital rotations which allows the use of unconstrained optimization techniques. We demonstrate the strong basis set dependence of the Pipek-Mezey localization criterion and how it can be eliminated. We have employed localization in conjunction with projection analysis to study the bonding in the water molecule and its heavy homologues. We demonstrate that in localized orbitals the repulsion between hydrogens in the water molecule is dominated by electrostatic rather than exchange interactions and that freezing the oxygen $2s$ orbital blocks polarization of this orbital rather than hybridization. We also point out that the bond angle of the water molecule cannot be rationalized from the potential energy alone due to the force term of the molecular virial theorem that comes into play at nonequilibrium geometries and which turns out to be crucial in order to correctly reproduce the minimum of the total energy surface. In order to rapidly assess the possible relativistic effects we have carried out the geometry optimizations of the water molecule at various reduced speed of light with and without spin-orbit interaction. At intermediate speeds, the bond angle is reduced to around 90°, as is known experimentally for $H2S$ and heavier homologues, although our model of ultrarelativistic water by construction does not allow any contribution from $d$ orbitals to bonding. At low speeds of light the water molecule becomes linear which is in apparent agreement with the valence shell electron pair repulsion (VSEPR) model since the oxygen $2s1∕2$ and $2p1∕2$ orbitals both become chemically inert. However, we show that linearity is brought about by the relativistic stabilization of the $(n+1)s$ orbital, the same mechanism that leads to an electron affinity for eka-radon. Actual calculations on the series $H2X$ (⁠$X=Te$⁠, Po, eka-Po) show the spin-orbit effects for the heavier species that can be rationalized by the interplay between SO-induced bond lengthening and charge transfer. Finally, we demonstrate that although both the VSEPR and the more recent ligand close packing model are presented as orbital-free models, they are sensitive to orbital input. For the series $H2X$ (⁠$X=O$⁠, S, Se, Te) the ligand radius of the hydrogen can be obtained from the covalent radius of the central atom by the simple relation $rlig(H)=0.67rcov(X)+27$ (in picometers).