Lanthanide standard model

In conclusion, it is clear that the standard model makes an excellent first approximation to the magnetic properties of lanthanide materials, but to understand lanthanide materials on a detailed individual basis, a more sophisticated approach is required. 

In principle, all possible excitations contribute to the photoelectron spectrum and the proper quantum mechanical amplitude must be calculated. For the lanthanides, the atomic limit corresponds to the assumption that the photoelectron spectrum is dominated by those processes, where the photon hits a particular ion and causes an excitation on that ion without disturbing the remainder of the crystal. In the standard model, the lanthanide ion would initially be in its bivalent / configuration with the Hund s rule ground state multiplet (Table 1 in Section 2.2), and would be transferred into some multiplet within configuration... 

The determination of the electronic structure of lanthanide-doped materials and the prediction of the optical properties are not trivial tasks. The standard ligand field models lack predictive power and undergoes parametric uncertainty at low symmetry, while customary computation methods, such as DFT, cannot be used in a routine manner for ligand field on lanthanide accounts. The ligand field density functional theory (LFDFT) algorithm23-30 consists of a customized conduct of nonempirical DFT calculations, extracting reliable parameters that can be used in further numeric experiments, relevant for the prediction in luminescent materials science.31 These series of parameters, which have to be determined in order to analyze the problem of two-open-shell 4f and 5d electrons in lanthanide materials, are as follows. 

An electrostatic hydration model has been applied to the trivalent lanthanide and actinide ions in order to predict the standard free energy (AG°) and enthalpy (AHt) of hydration for these series. Assuming crystallographic and gas-phase radii for Bk(III) to be 0.096 and 0.1534 nm, respectively, and using 6.1 as the primary hydration number, AG298 was calculated to be -3357 kJ/mol, and A/Z298 was calculated to be -3503 kJ/mol (187). 

Retention of Rohrschneider-McReynolds standards of selected chiral alcohols and ketones was measured to determine the thermodynamic selectivity parameters of stationary phases containing (- -)-61 (M = Pr, Eu, Dy, Er, Yb, n = 3, R = Mef) dissolved in poly(dimethylsiloxane) . Separation of selected racemic alcohols and ketones was achieved and the determined values of thermodynamic enantioselectivity were correlated with the molecular structure of the solutes studied. The decrease of the ionic radius of lanthanides induces greater increase of complexation efficiency for the alcohols than for the ketone coordination complexes. The selectivity of the studied stationary phases follows a common trend which is rationalized in terms of opposing electronic and steric effects of the Lewis acid-base interactions between the selected alcohols, ketones and lanthanide chelates. The retention of over fifty solutes on five stationary phases containing 61 (M = Pr, Eu, Dy, Er, Yb, n = 3, R = Mef) dissolved in polydimethylsiloxane were later measured ". The initial motivation for this work was to explore the utility of a solvation parameter model proposed and developed by Abraham and coworkers for complexing stationary phases containing metal coordination centers. Linear solvation... 

The idea of building the crystal field of transition-metal and lanthanide compounds as a superposition of single ligand contributions was first expressed by Griffith (1964), but it was effectively introduced by Bradbury and Newman (1967) and developed in subsequent work. It is utilized to standardize the analysis of crystal field data. A large amount of information on and beyond the subject appears in a review paper by Newman and Ng (1989) which stresses the relationship of the superposition model with the angular overlap model often preferred for d electrons. 

There are two kinds of problems when an attempt is made to interpret the f-electron spectra using the J-0 theory. One of them is conceptual. The standard realization of the Judd-Ofelt model is based on the assumption that only the central ion is perturbed by the environment, while in fact there is a mutual interaction between two subsystems, the central ion and the surrounding ligands. As a consequence the dynamic model has been introduced [29-32]. In this model the lanthanide ion plays a static role. The dipoles on the ligands, which result from the presence of the lanthanide ion, induce in turn the multi-poles on the central ion. This interaction between the multi-poles on the central ion and the dipoles on the ligands is the origin of additional contributions to the transition amplitude, and consequently to the intensity parameters Q, ... 

It is interesting to note that the dynamic part of the transition amplitude is independent of any excited configuration and its evaluation is rather straightforward. Since the static and dynamic models contribute to the intensity parameters simultaneously it is possible to verify their relative magnitude, or at least their signs. The results of such an analysis [36] demonstrated that the radial terms of the second-order static (standard Judd-Ofelt) effective operators are negative for all the lanthanide ions, and the dynamic radial parts are positive the angular parts are the same for all members of the lanthanide family. 

Here another source of a conceptual problem of the seeond order-approach appears. The standard formulation of the J-O theory, even if extended by the dynamic coupling model, is based on the single configuration approximation. This means that in such a description all the eleetron correlation effects are neglected and it is well known that the transition amplitude strongly depends on them. At this point also the spin-orbit interactions should be taken into consideration as possibly important in the description of the spectroscopic patterns of the lanthanides. In the case of all of these possibly important physical mechanisms there is a demand for an extension of the standard Judd-Ofelt formulation. The transition amplitude in equation (10.17) has to be modified by the third-order contributions that originate from various perturbing operators introduced in addition to the crystal field potential that plays a... 

The two-particle nature of Coulomb interaction in equation (10.27) is the reason that among the third-order contributions to the transition amplitude, in addition to one particle effective operators (as in the standard J-O approach), two particle objects are also present. However, the numerical analysis based on ab initio calculations performed for all lanthanide ions, applying the radial integrals evaluated for complete radial basis sets (due to perturbed function approach), demonstrated that the contributions due to two-particle effective operators are relatively negligible [11,44-58]. This is why here they are not presented in an explicit tensorial form (see for example Chapter 17 in [13]). At the same time it should be pointed out that two-particle effective operators, as the only non-vanishing terms, play an important role in determining the amplitude of transitions that are forbidden by the selection rules of second- and the third-order approaches. This is the only possibility, at least within the non-relativistic model, to describe the so-called special transitions like, 0 <—> 0 in Eu +, for example, as discussed above. 

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