Russell-Saunders coupling scheme

Consider a free atom or molecule in which several electrons occupy but do not fill a set of orbitals, which need not all be degenerate. There will usually be many ways in which the electrons can be distributed, some of which will be of lower energy than others. The differences in energy will be determined by electron repulsion and the more stable arrangements will be those with the least electron repulsion destabilization. So, electron-electron repulsion will be relatively small if the electrons occupy orbitals which are spatially well separated it will be reduced yet more if the electrons have a high spin multiplicity, i.e. have parallel spins, because two electrons with parallel spins can never be in the same orbital. 

It is to be emphasized that this table is a book-keeping exercise only it cannot be assumed that the actual functions are as indicated in it (the actual function, for instance, is a linear combination of all functions listed with resultant spin and orbital angular momenta of 0). 

A final word about spin-orbit coupling. Its effect is to cause small splittings between functions which we have so far regarded as degenerate. These splittings may be observed spectroscopically and values of spin-orbit coupling constants thus obtained (Section 9.3). 

Further reading An excellent treatment which goes well beyond that given above but which is both easy and, often, fun to read is Orbitals, Terms and States by M. Gerloch, Wiley, Chichester and New York, 1986. 

In this appendix are presented three different methods of obtaining the linear combinations of Table 6.2, repeated as Table A6.1, the ligand a orbital symmetry-adapted linear combinations for an octahedral complex. There is much to be learnt from a careful comparison of all three methods. 

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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... 

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. 

The method used in the calculations follows that explained in detail in Ref. [43]. The input quantum defects pa defined with respect to the Russell Saunders coupling scheme, which is the appropriate short-range basis, are given in Ref. 43. At energies corresponding to v = 100 the total number of open and closed channels in the final KF matrix is 414. 

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. 

This is the Russell-Saunders coupling scheme free-ion terms, set up as in the previous section, are perturbed by this Hamiltonian. 

If the approximation is made that HL > //ER, then we have the reverse of the Russell-Saunders coupling scheme. The individual electron total angular momenta, specified by /, couple together to give the total angular momentum for the set of electrons. The equivalent of equation (33) becomes... 

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... 

Hund mles apply if the Russell-Saunders coupling scheme is valid. Sometimes the first rule is apphed to molecules. 

Relativistic effects 27, 50 Russell Saunder coupling scheme 134... 

The electronic configurations of the lanthanides are described by using the Russell-Saunders coupling scheme. Values of the quantum numbers S and L corresponding to the lowest energy are derived in the conventional manner. These are then expressed for each ion in the form of a ground term with the symbolism that S, P, D, F,. .. correspond to L = 0, 1,... 

In this description, the Russell-Saunders coupling scheme is assumed, whereby spin-orbit coupling acts as a perturbation to crystal field effects. It is necessary to use a new notation to describe the splitting of crystal field levels by spin-orbit coupling. This involves double group notation. Some points to note in connection with this are as follows 659). 

The ground state for the (3d) configuration is S. Since L = 0 there can be no spin-orbit coupling within this term. However, there may be spin-orbit interaction with excited states. In the Russell-Saunders coupling scheme the selection rules on the non-vanishing matrix elements are... 

For almost all purposes, the Russell-Saunders coupling scheme is adequate for the specification of the energy levels of an isolated many-electron atom. In general, it is not necessary to work directly with the vectors S, L and /. Instead, many electron quantum numbers (not vectors), S, L and J, are used to label the energy levels in a simple way. The method of derivation is set out in Section SI.3.1. The value of S is not used directly but is replaced by the spin multiplicity, 25-1-1. Similarly, the total angular momentum quantum number, L, is replaced... 

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