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Electronic Structure and Optical Spectra of Transition Metal Complexes via the Effective Hamiltonian Method

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A semiempirical effective Hamiltonian treatment is proposed for transition metal complexes, taking into accountd-electron correlations, weak covalency of the metal-ligand bonds and the electronic structure of the ligand sphere. The technique uses the variation wave function which differs from the usual Hartree-Fock antisymmetrized product of molecular orbitals extended over the whole complex. The scheme is implemented and parameters describing the metal-ligand interactions are adjusted to reproduced-d-excitation spectra of a number of octahedral MF 64− (M=Mn, Fe, Co, Ni) anions, Mn(FH) 62+ cation, CoCl 64− anion, and a tetrahedral CoCl 42− anion. The values of the parameters are reasonable, thus confirming the validity of the proposed scheme.
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... 7 On the other hand, an effective model Hamiltonian approach has been developed to incorporate both the M-L interactions and the d-electron repulsions explicitly, with having wavefunctions in a d-orbital space. 16,17 A more rigorous effective Hamiltonian approach named the effective Hamiltonian crystal field (EHCF) method 18,19 has been developed to describe d-d states of transition metal containing systems. In the next section, these quantum mechanical approaches for metal containing systems are focused with emphasis on their concepts. ...
... In the above model Hamiltonian approach, the effective Hamiltonian is derived in a simplified fashion for the use in modeling the Hamiltonian matrix elements. In the EHCF method, 18,19,21,22 an effective Hamiltonian for transition metal containing systems is derived in a more rigorous fashion. In the ab initio electronic structure calculations, the electron correlation effects can be described by taking account of all the electronic configurations including metal and ligands. ...
... [37][38][39] The EHCF method, which also describes the electronic d-d states by construction, has been applied to computing structures and d-d excitations of wider range of transition metal complexes. 18,19,21 For example, the EHCF approach combined with MM could successfully reproduce the structures and spin states of a wide range of Fe(II) and Co(II) complexes having mono-and poly-dentate nitrogen donor ligands. 19,21 The EHCF has been also extended to periodic solids by incorporating effects from band structures, which was tested for the periodic transition metal oxides such as MnO. ...
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This perspective highlights three theoretical and computational methods to capture the coordination self-assembly processes at the molecular level: quantum chemical modeling, molecular dynamics, and reaction network analysis. These methods cover the different scales from the metal-ligand bond to a more global aspect, and approaches that are best suited to understand the coordination self-assembly from different perspectives are introduced. Theoretical and numerical researches based on these methods are not merely ways of interpreting the experimental studies but complementary to them.
... In such hybrid approach, the required approximations can be derived systematically [17]. A specific realisation of this idea has led to the development of the effective Hamiltonian of crystal field (EHCF) method, dating back to 1991, which was originally developed and successfully applied to TM molecular complexes [18][19][20][21][22], including spin-crossover compounds of iron (II) [23,24]. Previously, EHCF has been applied to predict the ground state and optical spectra of insulating TM containing solids in the cluster approximation [25][26][27]. ...
... Derivation of the EHCF theory for periodic systems follows the same general steps as those for the method developed for molecules and finite clusters and described previously in Refs. [17][18][19][20]. In the framework of EHCF, the space of one-electron states, spanned by local atomic orbitals, is divided into two sub-spaces: (i) l-space spanned by s-and p-orbitals of TM atoms and all orbitals of other (light) elements; (ii) d-space spanned by dorbitals of TMs. ...
... Operator H RR is the key element as it renormalises the bare Hamiltonian on the account of coupling between the model many-electron states (1) and charge-transfer states which present in the theory only implicitly. This assumption is only acceptable when, within each d-shell, charge transfer states are significantly higher in energy than the d-d excited states, which is, ultimately, the boundary condition for the applicability of the present approach [19]. Averaging the effective Hamiltonian (2), obtained by the Löwdin partition, over the multiplier functions Ψ d and Ψ l yields two separate Hamiltonians for the d-and l-(sub)systems: ...
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Effective Hamiltonian of Crystal Field (EHCF) is a hybrid quantum chemical method originally developed for an accurate treatment of highly correlated d-shells in molecular complexes of transition metals. In the present work, we generalise the EHCF method to periodic systems containing transition metal atoms with isolated d-shells, either as a part of their crystal structure or as point defects. A general solution is achieved by expressing the effective resonance interactions of an isolated d-shell with the band structure of the crystal in terms of the Green's functions represented in the basis of local atomic orbitals. Such representation can be obtained for perfect crystals and for periodic systems containing atomic scale defects. Our test results for transition metal oxides (MnO, FeO, CoO, and NiO) and MgO periodic solid containing transition metal impurities demonstrate the ability of the EHCF method to accurately reproduce the spin multiplicity and spatial symmetry of the ground state. For the studied materials, these results are in a good agreement with experimentally observed d-d transitions in optical spectra. The proposed method is discussed in the context of modern solid state quantum chemistry and physics.
... or further notation ). It ultimately comes from the mixing of the states in the model configuration subspace—that with the fixed number of electrons in the d-shell—with those in the outer subspace— one spanned by the MLCT and LMCT states as depicted inFig- ure 1. This comprises original form of the effective Hamiltonian crystal field (EHCF) theory. [12] It inherits the form of the WF describing the ground and low-lying excited states of a TMC which the CFT uses implicitly. This move turned out to be very much successful numerically as we described previously many times. It was the first example of using explicitly the group product in quantum chemistry at least in the semi-empirical co ...
... This has been reached by taking into account the energies of the respective d-shells calculated by the EHCF method. Since the intrashell static correlations were of crucial importance here, the hybrid QM/MM-like incarnation of the EHCF contained its local version which can be briefly characterized as a method of sequential derivation and independent esti- mation [12] of parameters of the Angular Overlap Model (AOM) [5,17] —the successful empirical systematics of the spectrochemical data combined with the correlated calculation of the d-shell energy. It represents the crystal filed felt by the d-shells as a superposition of ligand-specific increments e l known as AOM parameters determined from experiment. ...
... where D 16 ð Þ LL j ð Þ are elements of the Green's functions in the local basis. More details can be found elsewhere. [12] There are Implications for extending this approach to the solid state. [18] Nephelauxetic effect ...
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