Intermediate coupling scheme

Luyken in intermediate coupling [21] The perturbation (PM) calculations by Statz and co-workers [22], who for the 4s-4p transition complex followed both the LS and an intermediate coupling schemes The analytical model potential (MP) employed by Ganas [23], and the theoretical analysis of their observations, performed in the framework of the Relativistic Hartree-Fock (HFR) approximation, by Quinet et al. [16]. The different experimental data sources are quoted at the foot of the tables. For the few transitions for which they were available, we have also included the data of the critical compilations (CC) by Vujnovic and Wiese [28] and by Wiese etal. [29]. 

If the atom or ion is situated in an environment of different atoms or ions, as, for instance, ions in a soUd, the surrounding hgands exert on it a further interaction, which is called the crystal field interaction Hcp and enters Eq. (9). One has to compare Hcf with Hi and H2, in order to decide whether it is or not a small perturbative term. In actinide solids, it is usually found that Hcf is of the same order of magnitude as Hi and H2, so that intermediate coupling schemes are necessary which include Hcf as well. (For a more exhaustive treatment of couplings in actinides, see Chap. D.)... 

In the presence of Coulomb correlation only, the wave function is characterized by the total spin S = SSj and the total angular momentum L = 2,1 of the 5 f electrons, and the total momentum J is given by Hund s rule (J = L S). Important spin orbit coupling will mix LS multiplets and only J remains a good quantum number. The Russell-Saunders coupling scheme is no longer valid and an intermediate coupling scheme is more appropriate. 

Due to the localization of the 5 f electrons in the oxides a final state multiplet structure is expected. Therefore, these spectra have been compared with final state intensities calculated in an intermediate coupling scheme which accoimts for the strong spin-orbit... 

We shall not present here complete sets of selection rules for all transitions in question. For these transitions they become much more complex, e.g. already for transition (25.24), the pentagon conditions can occur. Interested readers may find them in [173]. Selection rules differ considerably for the various coupling schemes. They change if we account for the relativistic corrections to the non-relativistic E/c-transition operators. However, all selection rules involving intermediate momenta are more or less approximate. Due to the presence of interaction between coupled momenta a certain intermediate coupling scheme takes place, and these selection rules are violated. 

As Lefebvre-Brion and Field [61] point out, the only coupling cases for which the electronic and nuclear motions can be separated are cases (a) and (c) consequently only in these cases can potential curves be defined unambiguously and accurately. However, as we have already pointed out, Hund s coupling cases are idealised descriptions and for most molecules the actual coupling corresponds to an intermediate situation. Moreover, the best description of the vector coupling often changes as the molecular rotation increases. In this section we consider the nature of the intermediate coupling schemes in more detail some of these will appear elsewhere in this book in connection with the observed spectra of specific molecules. 

The spectra of lanthanide ions are not as sensitive to the environment as the transition metal ions. The small changes in absorption spectra of Eu(NC>3)3 and EuCb were attributed to higher symmetry of the environment in the nitrate compound [128,129]. Later systematic investigations on the band intensities of the rare earth ions were made [130— 135]. The procedure involving an intermediate coupling scheme was used in the calculation of energy levels [123,125]. 

Theoretical calculations on the rare earth oxide systems using an intermediate coupling scheme show good agreement with experimentally observed spectra [18,19]. 

J1 t/w II / ) is the matrix element of the transition between the ground and excited state of the sensitizer and activator respectively (the calculation of these matrix elements in the intermediate-coupling scheme is now a well known procedure and may be found in references (9) and (79)) S is the overlap integral / is the interionic distance is a numerical factor that depends on the orientation of the coordinate axis and (Af rA 4/>2 is the expectation value of the radial integral of the 4/orbital. 

It turns out that what is required to make the calculated energy levels conform to the experimental levels is to use some sort of "intermediate" coupling scheme, that is- a method of coupling spin and orbit states that... 

The Pb has been treated with an intermediate coupling scheme Using an intermediate coupling scheme to describe the Pb atom, this value is reduced to 402 kJ/mol... 

It is usual to call the AR = 0 situation the weak-coupling scheme, AR > 0 the intermediate-coupling scheme, and AR > 0 the strong-coupling scheme. The word coupling relates to the coupling between the electrons and the vibrations of the center considered. The value of AR measures the strength of this interaction. 

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