Chemical Bonding of a Molecular Transition-Metal Ion in a Crystalline Environment

A theoretical description of the chemical bonding of transition metals and other complex metal atoms in molecules and solids has been formulated in terms of a spin-unrestricted self-consistent-field cluster model. This model permits the accurate calculation, from first principles, of electronic energy levels and wave functions for the cluster, but requires relatively little computer time. The cluster, which may be a free polyatomic molecule, part of a macromolecule, or a polyatomic complex in an ordered or disordered solid, is geometrically partitioned into contiguous atomic, interatomic, and extramolecular regions. The one-electron Schr\"odinger equation is numerically integrated within each region in the partial-wave representation for spherically averaged and volume-averaged potentials which include Slater's $X\ensuremath{\alpha}$ statistical approximation to exchange correlation. The wave functions and their first derivatives are joined continuously throughout the cluster via multiple-scattered-wave theory similar to that developed originally by Korringa. The effects of a particular environment on the cluster are described by boundary conditions, e.g., the matching of the solutions of Schr\"odinger's equation in the extramolecular region to those within the atomic regions at an artificial spherical boundary surrounding the entire cluster. This numerical procedure is repeated, using the wave functions obtained at each iteration to generate a charge density and new potential, until self-consistency is attained. The model is illustrated for the tetrahedrally coordinated permanganate ion (Mn${\mathrm{O}}_{4}^{\ensuremath{-}}$) in the stabilizing field of a typical crystalline environment KMn${\mathrm{O}}_{4}$. The spin-unrestricted cluster calculation leads to a ground-state electronic structure of Mn${\mathrm{O}}_{4}^{\ensuremath{-}}$ which is consistent with the observed Van Vleck paramagnetism of permanganate crystals. Computer-generated contour maps of the cluster wave functions and charge densities are presented, showing the formation of directed chemical bonds. Both $\ensuremath{\pi}"$ and "$\ensuremath{\sigma}$ bonding" between oxygen $2p$ electrons and manganese $3d$ electrons are shown to be important. In conjunction with Slater's transition-state theory, which accurately describes the effects of orbital relaxation in optical transitions, the cluster model is used to interpret the measured optical properties of permanganate crystals. The characteristic purple color of KMn${\mathrm{O}}_{4}$ is shown to originate from the induced transfer of electrons between the oxygen ligands and central manganese atom in the Mn${\mathrm{O}}_{4}^{\ensuremath{-}}$ cluster. Finally, with the use of larger clusters, the theoretical model is extended to the chemical bonding of transition-metal impurities in semiconductors and transition metals which are the biologically active centers of certain enzymes and proteins.