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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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... Hermann et al. [11] reviewed the application of neural networks to model trial wave functions, highlighting how machine learning can boost both the flexibility and precision of variational methods in predicting the ground state energies of quantum systems. The effectiveness of the variational approach in determining ground state energies across numerous complex systems has been well-documented [12][13][14][15][16][17][18][19]. Additionally, recent research has focused on developing more advanced trial wave functions to further improve the accuracy of the variational method. ...

... Inserting the integrals solutions in Eq. (16)(17)(18)(19)(20)(21) with the normalization constant obtained in (14) into (15) ...

... Using the change of variable 2 1 2 y b r the following integrals (42)(43)(44)(45)(46)(47)(48)(49)(50) can be evaluated with the help of the standard solutions given in (16)(17)(18)(19)(20)(21) ...

In this article, we demonstrated the effectiveness of the variational approach in solving the bound state solutions of the Schrödinger equation for solvable potential functions. The method has been extended to determine the exact analytical excited states energy spectra for the Coulomb and Pseudo-harmonic functions. By minimizing the total energy, the variational parameters were precisely derived. The results for these potential functions align perfectly with those obtained through other methods in the existing literature. Additionally, the approach can be easily applied to systems with trial wave functions involving Laguerre polynomials and other orthogonal functions.

... 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. ...

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: ...

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.

The electronic structure of metal–organic frameworks (MOFs) containing transition metal (TM) ions represents a significant and largely unresolved computational challenge due to limited solutions to the quantitative description of low-energy excitations in open d-shells. These excitations underpin the magnetic and sensing properties of TM MOFs, including the observed remarkable spin-crossover phenomenon. We introduce the effective Hamiltonian of crystal field approach to study the d–d spectrum of MOFs containing TM ions; this is a hybrid QM/QM method based on the separation of crystal structure into d- and s,p-subsystems treated at different levels of theory. We test the method on model frameworks, carbodiimides, and hydrocyanamides and a series of M-MOF-74 (M = Fe, Co, Ni) and compare the computational predictions to experimental data on magnetic properties and Mössbauer spectra.

Here we presented a pretreatment method for measurement of Fe-55 and Ni-63 in leaching solution of hardened cement. This work was divided into two parts, the first part describes the development of pretreatment method, the second part describes the pretreatment method validation. The results indicated that the chemical recovery of Fe(III) and Ni(II) were 92.25% ± 3.21% and 83.40% ± 4.52%, respectively. Impurity ion Co-60 and Sr-90 removal rate was more than 99%. It was proved that this ion exchange and extraction process could be used as a pretreatment method for leaching solution of cement solidification.

Modeling of structure and properties of molecules and materials (crystals/solids) on the basis of their electronic structure is one of the most important consumers of computer resources (processor time, memory and storage). The known attempts to improve its efficiency reduce to massive parallelization. This approach ignores enormous diversity of types of structures and behaviors of molecules and materials. Moreover, this diversity is by no means reflected in the paradigm currently dominating the field of molecular/material modeling.

A model electronic Hamiltonian to describe ligand exchange reactions of palladium(II) complexes with pyridine (Py) and tridentate (L) ligands was developed. It was shown that the model Hamiltonian can adequately reproduce the structures and potential energies of the reactant/product, intermediate, and transition state of the ligand exchange reaction of [PdPy4]²⁺ with free Py. The model Hamiltonian was extended to describe reactions of multi-metal complexes and was adequately applied to describe various clusters, [PdaLbPyc]2a+, in the self-assembly of an octahedron-shaped coordination capsule, [Pd6L8]¹²⁺. The heterogeneity in the energetics of intermediate species [PdaLbPyc]2a+ was strongly suggested by the calculations, and the underlying microscopic interactions were clarified with the geometrical motif. The present framework provides a way to examine the reaction mechanisms of complex metal ligand self-assembly, which can be complementary microscopic information to the recently investigated novel experimental results for the real time evolutions.

In the last chapter of this book, we employ the general methods developed in the previous chapters in the context of a specific class of molecular systems known as coordination compounds. The simplest statement about this class of systems is negative: it is poorly describable by transitional (classical) MM methods. The reasons are twofold and manifest themselves differently for different subclasses of coordination compounds. The first is the non-directional character of coordination bonds and their unsaturability. This feature is common to all types of coordination centers that manifest a wide variety of coordination numbers and coordination polyhedra. The second important source of problems relates to the coordination compounds of transition metal ions with open d-shells. In this case, the situation is that each of the multiple electronic states available for the open d-shell produces corresponding PES which all lie in a narrow energy interval and may intersect due to their sophisticated dependence on molecular geometry. This produces a picture in which a unique PES usually assumed in the classical MM models does not exist, but a bunch of them is present, all of which must be uniformly treated. As a result of these two sources of problems, the material of this chapter is divided into two parts. In the first we consider the factors responsible for the “unspecific” behavior – a group of electrons occupying three-dimensionally delocalized orbitals of the central atom (ion) of the complex and its close vicinity. This produces a mechanistic picture of PES of coordination compounds capable of reproducing the effects of ligand mutual influence. In the second we address the transition metal complexes with open d-shells. We apply the general hybrid methodology to develop a true QM/MM scheme of including sufficiently quantum subsystem (the d-shell) in the general classical (MM treated) environment.

In this chapter we provide a hybrid perspective of the methods of molecular modeling present in the literature. The widespread viewpoint is that hybrid modeling is a rather specific, restricted field in the otherwise universal modeling realm of quantum chemistry. From this perspective, classical models of molecular potential known as Molecular Mechanics seem a completely foreign subject that has to be artificially attached to the quantum description. This presupposes the problems of the process of developing quantum-classical junctions. We take a different view of this area, based on the general scheme of variable separation as presented in the previous chapter. On that basis we analyze, the entire realm of molecular modeling and arrive at a conclusion that basically all modeling methods employ – although largely implicitly – the electron variable separation. This forms the hybrid perspective of molecular modeling mentioned above. Based ou it, we present a short review of the methods of quantum chemistry, including a description of the unsolved problems of semi-empirical quantum chemistry and suggest solutions to these problems, on the basis of the patterns of variable separations which are alternative to those accepted in traditional semi-empirical quantum chemistry. Finally we use the general scheme of variable separation to classify the existing methods of hybrid molecular modeling (in the narrow sense) and clarify the origins of the problems these methods face. Some ways of solving or avoiding these problems are suggested.

A circular scale of time is proposed and applied to the calculation of the Rayleigh-Schrödinger perturbation energy for a non-degenerate state

The basic starting point for calculating ab initiowave functions of molecules is generally the Hartree—Fock (HF) wave function, which in the simplest case involves two electrons (one with each spin) in each orbital Φi with the total wave function antisymmetrized in order to satisfy the Pauli principle $$ a[({\phi _{1}}\alpha )({\phi _{1}}\beta )({\phi _{2}}\alpha )({\phi _{2}}\beta )...({\phi _{n}}\alpha )({\phi _{n}}\beta )] = a[({\phi _{1}}{\phi _{1}}{\phi _{2}}{\phi _{2}}...{\phi _{n}}{\phi _{n}}\alpha \beta \alpha \beta ...\alpha \beta )] $$ (1) Here a is the antisymmetrizer or determinant operator* and α and β are the usual spin functions. In Eq. (1) as elsewhere, we arrange products of spatial functions and spin functions in order of increasing electron numbers.

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Molecular-orbital calculations within the INDO framework are reported for the hexafluoro, hexachloro, and hexaaquo metallate (II) ions of the first transition series. The parameterization scheme and choice of orbital exponents have been detailed. The results indicate that the method in its present form gives fairly satisfactory predictions of molecular properties.

When the atomic-scale geometry of an interface or surface is known, its properties can be estimated and understood in terms of elementary theory of its electronic structure, or can be accurately calculated using full local-density theory. This is illustrated in terms of semiconductor heterojunctions and ideal semiconductor surfaces. The necessity for self-consistency is discussed in this connection. It is less certain that the atomic-scale geometries can be reliably predicted. This is discussed in terms of reconstruction of semiconductor surfaces, and the silicon surface in particular.

Some of the challenges facing scientists in the area of the electronic structure and properties of solids are reviewed. At a time when computational advances have made possible high-quality calculations, not even conceivable 10 years ago, the article stresses the importance of going beyond the raw output of the calculation and understanding where the result originates. The areas selected for study include the Fermi surface and charge density waves, the stability of solids and the structures of solids under pressure, metal-insulator transitions, the method of moments, superconductivity, and the use of a relatively new technique for the study of chemical bonding, the electron localization function (ELF). 77 refs., 16 figs., 1 tab.