Extension of ligand-field theory to encompass bridged structures. Emphasis on the angular overlap model

Kohn–Sham DFT and ligand-field theory — Is there a synergy?

In forming electronic states of the partially filled shell of transition-metal atomic and molecular systems, real, symmetry-based, fixed, Kohn–Sham eigenorbitals can be used to bridge KS-states with strong-field, ligand-field states. Thereby, DFT computations, restrained by the use of these frozen orbitals of the so-called average-of-configuration type, allow a central-field modeling of the partially filled shell whose Hamiltonian matrix consists of mutually orthogonal diagonal and non-diagonal parts, of which only the former can be computed. Mutually orthogonal operators of ligand-field theory are particularly suited to parameterize the energy “data” obtained from the bridges between molecular Kohn–Sham DFT states and ligand-field states. With the d2 configuration as the simplest example encompassing both ligand-field and interelectronic repulsion, each one-electron parameter, though defined by energy differences of perturbed d orbitals, is associated with a 45 × 45, diagonal, theoretical, strong-field-type coefficient matrix of the ligand field repulsion model (LFR), which is mapped in a one-to-one fashion onto a likewise diagonal KS-DFT computational energy matrix. For sets of mutually orthogonal operators, the mapping determines the value of any such ligand-field parameter as a scalar product between the DFT matrix and the coefficient matrix of the associated ligand-field operator. Each and every two-electron parameter of LFR is in the same strong-field function basis associated with a 45 × 45 coefficient matrix that includes a non-diagonal part. This matrix, nevertheless, by the formation of a scalar product with the appropriate diagonal, computational DFT matrix, provides the value of the two-electron parameter. In spite of the lacking non-diagonal DFT information, its non-diagonal elements of the two-electron interelectronic repulsion matrices are indirectly accessible through the parameterization based upon the computed diagonal DFT matrices combined with the mapping of the DFT energy results onto the parametric LFR. In this way, LFR delivers back to DFT a quantification of the deviation of the systems’ eigenbasis from the DFT-computed states, which are defined by having unit occupation numbers. This work focuses firstly on using the LFR model for forming a full DFT energy matrix and dissecting it into mutually orthogonal one- and two-electron parts and secondly on the use of the two-electron parts to obtain a complete ligand-field image of a nephelauxetic, molecular atom, intrinsic of the chemical system.

Bridging Kohn - Sham DFT and the Angular Overlap Model. Ligand-Field Parameters and Bond Covalencies in Tetrahedral Complexes

Kohn–Sham density functional theory (DFT), constrained by the average-of-configuration computations, allows the valence shell of regular tetrahedral chlorido complexes of a complete series of 3d transition metal ions to be orbitally compared. The concept of classificational parentage provides a handle on the discussion of the energetic ordering of all the valence orbitals and illuminates an almost identical ordering for all the systems. Only the participation of the metal 4s orbital in bonding causes a few minor fluctuations. The partially filled ‘3d’ molecular orbitals sit in an energy window framed by completely filled ‘ligand orbitals’ on the low-energy side and an empty metal ‘4s’ orbital on the high-energy side. Regular tetrahedral symmetry requires the halides to be linearly ligating and this property is stable within the ‘experimental’ uncertainty for small distortions. By lowering the symmetry towards the planar configuration, keeping the equivalence of the ligands stable, the information content of the computations was doubled and the angular overlap energy parameters referring to the individual ligands obtained. The orbital energies of the partially filled shell depend linearly on the Angular Overlap Model (AOM) parameters eλ, the slope being the sum of the squares of the single-ligand λ angular overlaps (λ = σ and π). Mulliken population analysis shows the contents of the appropriate ligand orbitals in the ‘d’ orbitals to vary in parallel with the molecular orbital AOM energies and to increase pronouncedly with the oxidation number z. Results for tetraoxidoferrate(vi) show a remarkable resemblance with the chloride complexes of even the divalent metal ions. However, although the bonding orbitals are more π-bonding, the totally symmetrical bonding orbitals use M_4s less in the oxido complex. The sensitivity of covalency and spectroscopic energy parameters towards radial distortions are examined and show Werner-type complexes and the high-valent FeO42– to behave somewhat differently.

Inclusion of the ligand field contribution in a polarizable molecular mechanics: SIBFA-LF

To account for the distortion of the coordination sphere that takes place in complexes containing open-shell metal cations such as Cu(II), we implemented, in sum of interactions between fragments ab initio computed (SIBFA) molecular mechanics, an additional contribution to take into account the ligand field splitting of the metal d orbitals. This term, based on the angular overlap model, has been parameterized for Cu(II) coordinated to oxygen and nitrogen ligands. The comparison of the results obtained from density functional theory computations on the one hand and SIBFA or SIBFA-LF on the other shows that SIBFA-LF gives geometric arrangements similar to those obtained from quantum mechanical computations. Moreover, the geometric improvement takes place without downgrading the energetic agreement obtained from SIBFA. The systems considered are Cu(II) interacting with six water molecules, four ammonia or four imidazoles, and four water plus two formate anions.