Symmetry crystal field parameters

Table 2.1 Non-forbidden crystal field parameters that a group of point charges may contribute collectively, as a function of their point-group symmetry.

The problem of estimating crystal field parameters can be solved by considering the CFT/LFT as a special case of the effective Hamiltonian theory for one group of electrons of the whole A -electronic system in the presence of other groups of electrons. The standard CFT ignores all electrons outside the d-shell and takes into account only the symmetry of the external field and the electron-electron interaction inside the d-shell. The sequential deduction of the effective Hamiltonian for the d-shell, carried out in the work [133] is based on representation of the wave function of TMC as an antisymmetrized product of group functions of d-electrons and other (valence) electrons of a complex. This allows to express the CFT s (LFT s or AOM s) parameters through characteristics of electronic structure of the environment of the metal ion. Further we shall characterize the effective Hamiltonian of crystal field (EHCF) method and its numerical results. 

The crystal field does, however, have a dramatic effect on the magnetic anisotropy of lanthanide complexes. For complexes of less than cubic symmetry the three principal values of the susceptibility tensor are unequal. For uniaxial symmetry, Xx — Xy Xz and for biaxial symmetry Xx Xy Xz- Very extensive studies632-640 have been carried out on the single crystal susceptibilities of the D3d lanthanide hexakis(antipyrene) triiodides over the temperature range 80-300 K, and crystal field parameters were obtained. This crystal field-induced anisotropy is responsible for the effectiveness of lanthanide complexes as NMR shift reagents, and single crystal anisotropies of lanthanide complexes have been determined in this connection also.563... 

General relationships between AOM and crystal field parameters are shown in Table 23. Using the AOM one can easily compute the electronic energy levels, inclusive of spin-orbit coupling, without any symmetry assumption or perturbation procedure, and it is also easy to account for the different chemical natures of the ligands and for differences in bond distances. It is also possible to handle anisotropic n interactions, which can be expected to occur with pyridine or pyridine iV-oxide ligands.366,367 General review articles on the AOM and its applications have already appeared.364,368-371... 

In fact, even approximate cubic symmetry seems to be rare for lanthanoid or actinoid element complexes.50 In low symmetry the number of crystal field parameters necessary to account for the system can be quite large. On the other hand, the spectra of lanthanoid complexes contain many... 

The unperturbed eigenfunctions have complete spherical symmetry and Vc can be expanded in terms of the crystal field parameters, A , and the usual spherical harmonics Y (0, 97) to give... 

The derivation of a general crystal-field independent method for polymetallic lanthanide complexes related to eq. (51) is precluded by the consideration of variable numbers of different crystal-field parameters (maximum n) depending on the exact symmetry of the axial complex (see sect. 4.1.1, Rigault et al. (2000a)). 

Axial symmetry in trimetallic lanthanide complexes requires the location of the metal ions along the molecular threefold or fourfold axes. Since the terminal coordination sites are different from the central coordination site for symmetry reasons, two different crystal-field parameters 2terminal ancj /)2ccntral must be considered. Equation (61) holds for the general case of three (n = 3) magnetically non-coupled lanthanide metal ions packed along the symmetry axis and eqs. (62)-(64) can be used for homotrimetallic axial complexes. To the best of our knowledge, only one partial study of the NMR data for a >3-symmetrical axial trimetallic complex has been reported (Bocquet et al., 2002 Floquet et al., 2003 see sect. 5.1.2). The Dih-symmetrical complexes [R3(L16-3H)2(OH2)6]3+ do not fit the requirements for axial symmetry since the metal ions are located on mirror planes and not on the threefold axis, but... 

The C are tensor operators, whose matrix elements again can be calculated exactly, whereas the crystal-field parameters Bk are regarded as adjustable parameters. The number of parameters for this potential is greatly reduced by the parity and triangular selection rules and finally by the point symmetry for the f-element ion in the crystal. Detailed information about the crystal-field potential has been given for example by Gorller-Walrand and Binnemans (1996). 

Further interesting and intensively studied systems are provided by MYX compounds. In particular, the crystal-field parameters have been determined for BOF Eu3+ (R = La, Gd Y = Br, Cl), BOCl Pr3+ (R = La, Pr, Gd) and MFCl Sm2+ (M = Ba, Ca, Sr). For the corresponding references see table 1. As an example for these compounds, fig. 7 shows the crystal-field parameter shifts up to 16 GPa for BOCl Pr3+ (B = La, Pr, Gd) obtained by Bungenstock et al. (2000b). The Pr3+ ion in BOC1 is surrounded by four O2- and five Cl- ions. According to the site symmetry C v five crystal-field parameters must be taken into account ... 

The crystal-field parameters introduced in sect. 4.1 still contain all the structural information about the local environment. Therefore, a direct comparison of crystal-field parameters derived from different hosts, even with the same site symmetry, is not reasonable. In addition, the crystal-field parameters cannot be directly related to the distance and angle variations induced by the high-pressure application. Widely used models which extract the structural information from the crystal-field parameters are the angular-overlap (Jprgensen et al., 1963) and superposition model (Bradbury and Newman, 1967). In the case of f elements, the superposition model has been employed widely for the analysis of crystal-field parameters. 

Here Bk s stand for the crystal field parameters (CFP), and Ck(m) are one-electron spherical tensor operators acting on the angular coordinates of the mth electron. Here and in what follows the Wyboume notation (Newman and Ng, 2000) is used. Other possible definitions of CFP and operators (e.g. Stevens conventions) and relations between them are dealt with in a series of papers by Rudowicz (1985, 2000,2004 and references therein). Usually, the Bq s are treated as empirical parameters to be determined from fitting of the calculated energy levels to the experimental ones. The number of non-zero CFP depends on the symmetry of the RE3+ environment and increases with lowering the symmetry (up to 27 for the monoclinic symmetry), the determination of which is non-trivial (Cowan, 1981). As a result, in the literature there quite different sets of CFP for the same ion in the same host can be found (Rudowicz and Qin, 2004). 

Chapter 5 summarizes the crystal field spectra of transition metal ions in common rock-forming minerals and important structure-types that may occur in the Earth s interior. Peak positions and crystal field parameters for the cations in several mineral groups are tabulated. The spectra of ferromagnesian silicates are described in detail and correlated with the symmetries and distortions of the Fe2+ coordination environments in the crystal structures. Estimates are made of the CFSE s provided by each coordination site accommodating the Fe2+ ions. Crystal field splitting parameters and stabilization energies for each of the transition metal ions, which are derived from visible to near-infrared spectra of oxides and silicates, are also tabulated. The CFSE data are used in later chapters to explain the crystal chemistry, thermodynamic properties and geochemical distributions of the first-series transition elements. 

The actual site symmetry of the Pr3+ ion in the lattice is C3[,. However, the doubly degenerate levels of D3j, still retain their degeneracy in C3h and the additional crystal-field parameter that arises in C3[, is not significant. The selection rules for electric dipole radiation in C3j, symmetry, however, are different. 

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