Atomic multiplets

The spin-orbit interaction in free atoms alters the atomic terms l i) = L, ML, S, Ms) into atomic multiplets, denoted as L,S,J,Mj) = JM). The energies of the multiplets are given as follows 

With the help of the replacement theorem (Section 1.4) these matrix elements can be replaced by those of the total angular momentum operator 

The reduced matrix element of a vector operator (tensor operator of the first rank) is expressed in terms of the 67-symbol as follows 

Without loss of generality we take the field parallel to the z-axis 

It should be mentioned that the orbital and spin angular momentum vectors define the electronic magnetic momentum vector 

---

In lanthanide complexes, the ligand field created by the surrounding ligands will split the atomic /-multiplets into several components. The latter are doubly degenerate (Kramers doublets (KDs)) for systems with odd number of electrons and non-degenerate (in the absence of symmetry) for systems with even number of electrons. 

A complete decomposition of the ab initio computed CF matrix in irreducible tensor operators (ITOs) and in extended Stevens operators. The parameters of the multiplet-specific CF acting on the ground atomic multiplet of lanthanides, and the decomposition of the CASSCF/RASSI wave functions into functions with definite projections of the total angular momentum on the quantization axis are provided. 

Atomic multiplet Spin free states J multiplet Kramers Doublets... 

As depicted in Fig. 3, one can follow several paths for obtaining the CFMs. An alternative to the above process was developed by Konig and Krem-mer [61,62], They followed the path (a) - (c) - (b) starting with the basis set of atomic multiplets their weak-field theory required that the symmetry-adapted coefficients = (JM JFya) enter the transformation... 

There is no need to pass from the basis set of the ATs L,Mi,S,Ms) to the basis set of the atomic multiplets (LS),J,Mj) since such a unitary transformation does not lead to a gain in the computational effort. In the basis set of the L,Mi, S, Ms) functions, the operator Vax is diagonal but the operator Hso has off-diagonal matrix elements. In contrast, in the basis set of the (LS),J,Mj) kets, the operator Hso is diagonal, but the operator Vax has off-diagonal matrix elements. Therefore, none of these basis sets is appropriate for considering the Zeeman term as a small perturbation. 

Figure 3a shows photoemission data from a Gd C82 [7] film measured with Al Ka x-rays compared to the empty fullerene C82. Even at a glance it is easy to see that the x-ray photoemission spectrum of Fig. 3a differs strongly from that shown in Fig. 2 now the C 2s/2p-derived crand 7tMO s are much less distinct and the spectrum is dominated by a large structure centred at a binding energy of about 11 eV. The C82 spectrum can be taken as a good approximation for the C 2s/2p contribution to the photoemission spectrum, and thus can be used to subtract out the non-Gd related emission, resulting in the spectrum shown in Fig. 3b. This pure Gd emission can then be simulated using a simple atomic multiplet...

Fig. 11 Lines+symbols experimental x-ray absorption data at the L2j3 edge of Sc in Sc2 C84 recorded at the temperatures indicated. The energy resolution was 100 meV. The grey solid lines represent broadened atomic multiplet calculations for a purely ionic trivalent Sc (d° initial state—>2pd final state) and divalent Sc (d1 initial state— 2pd2 final state) ion...

In these charge-transfer atomic multiplet calculations, the effective formal valency of the encaged Sc ions is given by the ratio of the d° and d1 initial-state contributions, which represent trivalent and divalent Sc, respectively. Figure 12 shows how the spectrum would evolve from the pure d° initial state... 

Fig. 13 Direct comparison of the experimental Sc-L23 x-ray absorption spectra of Sc2 C84 (line+symbols) with a simulation (grey solid line) based upon an atomic multiplet model with an initial state composed of 61% d° and 39% d L configurations. For details,see Ref. [34]...

With this example of data from the dimetallofullerene Sc2 C84, the efficacy of combining x-ray absorption and charge-transfer atomic multiplet calculations in investigating the valency, charge transfer and cage-metal interaction of more complex, hybridised systems such as Sc2 C84 has been demonstrated. Based on the valence sensitivity of the measurement, we are confident in overruling... 

Simulation of the data within the framework of charge-transfer atomic multiplet calculations shows that this hybridisation yields an effective Sc 3d electron count of about 0.4 electrons, or a valency of 2.6. 

In this paper we modify and extend this approach in several ways. In particular, we consider the magnetic fine structure effects in the presence of a uniform electric field F for ls2p Pj- excited states of helium. We introduce two separate differential polarizabihties to describe the quadratic part of the electric field splitting and three differential hyperpolarizabilities to describe the terms the order of in the fine-structure splitting of the atomic multiplet s2p Pj. We have developed a calculational approach that allows correct estimation of potential contributions due to continuum spectra to the dipole susceptibilities j3 and 7. In the next section we briefly outline our method. The details of the calculations of the angular and radial matrix elements have been described elsewhere [8,9] and are omitted here for brevity. Atomic units are used throughout. 

The Stark effect on the magnetic fine structure occurs as a result of disturbance of atomic levels under the influence of the relativistic and correlation effects as well as the interaction with external electric field F. If the fields are weak enough the centre of multiplet is shifted and there occurs the splitting of sublevels of atomic multiplet n, L, J. The dipole moment induced in an atom by a uniform electric field F is for most purposes expressed as a linear function of F, but higher... 

Here 7q (nLJ) corresponds to the scalar part of hyperpolarizability, 72 (nLJ) corresponds to the rank 2 tensor part of hyperpolarizabihty, 74 (nLJ) determines the tensor part of the rank 4. As the held strength F increases, the sphtting of the level may reach values comparable to the distance between levels of the same parity (the components of the hne structure of an atomic multiplet) and, therefore, the level shift AEnLJM can be found by solving the secular equation... 

In the perturbation theory for degenerate states the resonant hyperpolarizability is determined by the tensor part of polarizability [9] and may be extracted out of the fourth-order terms self-consistently in the case of nondegenerate perturbation theory the resonant part appears for separate sublevels of an atomic multiplet. The numerical results are listed in Table 2. 

Unfortunately, the model does not predict the magnitudes of these parameters, on which the transition depends. It is even conceivable in the presence of atomic multiplet structure, or of term-dependent localisation effects internal to the atom, etc., that the semiempirical parametrisation might absorb effects not really intended in the model, such as atomic multiplet splittings or term dependence of the orbitals. 

The implicit assumption in the previous model is that atomic structure has little influence on the degree of / electron localisation in the solid. There are difficulties in reconciling this view with any persistence of atomic multiplet structure, and with the fact that changes of localise tion are clearly associated with certain regions of the Periodic Table. For this reason, various attempts have been made to explore what properties of atoms might survive in solids and, perhaps, provide a quasiatomic mechanism to drive changes of valence. 

In this sense, atomic multiplet theory provides complementary information about the valence state. One can take the view that this information should be used, and then blended in some way with the conceptual framework of the Anderson single-impurity model, so that the matrix elements coupling the / electrons to the conduction band can continue to play the decisive role in determining the extent of / electron localisation. 

In addition to the main multiplet lines, one additional weak feature appears systematically several eV above each M/y and My multiplet. Its separation from the main lines (3-5 eV) is too large as compared to the observed separation (1.7 eV) between tetravalent and trivalent 3d94/n+1 final atomic multiplets for any explanation in terms of multiplet splittings. It has been suggested that the weak feature originates from excitation to... 

However Allen [641] and others have argued against an atomic interpretation. It is probably fair to say that atomic multiplet theory and the condensed matter models have not yet been completely reconciled. This question is important high temperature superconductors of the form RBa2Cu306, where R can be any member of the rare-earth series of elements except Ce and Pr, have attracted much attention. A fluctuation in valence involving 4/ electrons is likely to be involved in the emergence of high temperature superconductivity for these materials. 

The most interesting case is photoemission of 4/ electrons in the rare earths as noted in the previous section, because of the collapsed nature of the 4/ orbitals, the photoemission spectrum can be interpreted completely even in the solid by atomic multiplet theory, and this applies also to magnetic circular dichroism. Thole and van der Laan [642] have derived sum rules for magnetic dichroism in rare-earth 4/ photoemission. They have shown that the integrated intensity is simply the sum over each sublevel of its occupation number times the total transition probability from that sublevel to the continuum shell. Polarisation effects in the 4/ photoemission spectra of rare earths are very large, and this tool based on quasiatomic analysis is of considerable significance it provides a new... 

---