Coupling schemes, electron interaction

The tensorial structure of the spin-orbit operators can be exploited to reduce the number of matrix elements that have to be evaluated explicitly. According to the Wigner-Eckart theorem, it is sufficient to determine a single (nonzero) matrix element for each pair of multiplet wave functions the matrix element for any other pair of multiplet components can then be obtained by multiplying the reduced matrix element with a constant. These vector coupling coefficients, products of 3j symbols and a phase factor, depend solely on the symmetry of the problem, not on the particular molecule. Furthermore, selection rules can be derived from the tensorial structure for example, within an LS coupling scheme, electronic states may interact via spin-orbit coupling only if their spin quantum numbers S and S are equal or differ by 1, i.e., S = S or S = S 1. 

A partly filled shel > exhibits a number of states of different energies which arise as a result of the interactions or couplings of the electrons in the shell. These states can be determined using the Russell-Saunders coupling scheme (Hund s rules) (Figgis, 1966). A characteristic property of a state is the spin multiplicity which is related to the number of unpaired electrons in a shell. A singlet state has a spin multiplicity of one (two electrons of opposite spin), a doublet state has a multiplicity of two and... 

Ethynylated dihydrofullerenes serve as precursors for buckydumbbells (Scheme 3.3). Coupling of the desilylated compound 10 with CuCl leads to the dimer 11 [24]. Reaction of Cjq with the acetylide Li-C=C-Li leads to the dimer with a bridge consisting of one acetylene unit only. Electronic interaction between the fullerene-units in these two buckydumbbells is negligible. Further examples of CgQ-acetylene-hybrids synthesized by using cyclopropanation reactions are shown in Chapter 4. 

For all-electron calculations, we used the atomic HFDB code [35, 36] which allows one to account for the Breit interactions both in the framework of the first-order perturbation theory (PT-1) and by the self-consistent way as well as to account for different models of nuclear charge distribution. For test calculations with (G)RECPs, the atomic Hartree-Fock code in the -coupling scheme (hfj) [17] was used (that was quite sufficient for... 

To a first approximation each of several electrons in such a partly filled shell may be assigned its own private set of one-electron quantum numbers, n, /, m, and s. However, there are always fairly strong interactions among these electrons, which make this approximation unrealistic. In general the nature of these interactions is not easy to describe, but the behavior of real atoms often approximates closely to a limiting situation called the L-S or Russell-Saunders coupling scheme. 

The existence of two coupling schemes for a shell of equivalent electrons is conditioned by the relative values of intra-atomic interactions. If the non-spherical part of electronic Coulomb interactions prevails over the spin-orbit, then LS coupling takes place, otherwise the jj coupling is valid. As we shall see later on, for the overwhelming majority of atoms and ions, including fairly highly ionized ones, LS coupling is valid in a shell of equivalent electrons, that is why we shall pay the main attention to it. 

As we have already seen in Chapters 11 and 12, the realization of one or another coupling scheme in the many-electron atom is determined by the relation between the spin-orbit and non-spherical parts of electrostatic interactions. As the ionization degree of an atom increases, the coupling scheme changes gradually from LS to jj coupling. The latter, for highly ionized atoms, occurs even within the shell of equivalent electrons (see Chapter 31). 

A total set of selection rules consists of the sum of all selection rules both exact and approximate . Transition is forbidden if at least one selection rule is violated. The transitions may be forbidden to a different extent -by one, two, three, etc. violated conditions. If an electronic transition is between complex configurations, then, as we shall see in the next section, there may be a large number of selection rules. However, the majority of them, especially with regard to the quantum numbers of intermediate momenta, are rather approximate, even when a specific pure coupling scheme is valid. This is explained by the presence of interaction between the momenta. 

As we have seen in Chapter 11, the energy levels of atoms and ions, depending on the relative role of various intra-atomic interactions, are classified with the quantum numbers of different coupling schemes (11.2)— (11.5) or their combinations. Therefore, when calculating electron transition quantities, the accuracy of the coupling scheme must be accounted for. The latter in some cases may be different for initial and final configurations. Then the selection rules for electronic transitions are also different. That is why in Part 6 we presented expressions for matrix elements of electric multipole (Ek) transitions for various coupling schemes. 

The LSJ-coupling scheme introduced above is called the Russell-Saunders coupling scheme [RSa25], It is based on the validity of equ. (1.9). The other extreme coupling case follows if the spin-orbit interaction dominates the Coulomb interaction between the electrons. This is called the jjJ-coupling scheme and requires that... 

Using the basis functions which follow from the approximate Hamiltonian H° of equ. (1.3), it is the residual interaction H — H° which causes the Auger transitions. This operator, however, reduces to the Coulomb interaction if more than one electron changes its orbital.) Within the LS-coupling scheme this transition operator requires the following selection rules... 

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