L-S coupling

In L-S coupling, we need to determine the following sums in order to deduce the spectroscopic state of an atom ... 

The energy levels of the free Cr3+-ion (d ) are quite well known (4). A quantitative description with the Slater-Condon-Shortley method (5) using the Racah-parameters B and C ) and the L—S-coupling parameter A does not lead to a satisfactory agreement between theory and experiment this can be achieved only by assuming different C/R-ratios for different doublet-levels (Table 1). The depression of the P-level is especially remarkable. The deviations from the three-parameter theory are less pronounced for Ni + or Co + (2), for example. 

L—S-coupling is not included, because the energetic effects are negligible... 

Table 1. Energy-level diagram for the free C -3+ ion. (Calculated values include L—S coupling to first order) A was chosen to be 90 cm J...

Table 2. Energy-level diagram for C -3+ in an octahedral crystal field (L—S coupling not included) for the special choices and 30 (lowest doublet-levels only observed...

Within L-S coupling the ground state multiplet is leading to an effective moment Heff = 3.38 Xb (see Table 3) which is slightly higher than the experimental value Peff = 3.2 Pb thus J mixing is small and the ground state is essentially H4 (Chan and Lam ... 

The interactions of type (1) are known as L-S coupling or Russell-Saunders coupling. From the vector addition rule and the constraint that the values must differ by one in the unit of h/2n, the possible values of L and S are ... 

In L-S coupling, the total electronic angular momentum (spin and orbital) is defined by the quantum number J whose allowed values are... 

Spectroscopic terms arising from equivalent electronic configurations in L-S coupling... 

We are now in position to derive the electronic states arising from a given electronic configuration. These states have many names spectroscopic terms (or states), term symbols, and Russell-Saunders terms, in honor of spectroscopists H. N. Russell and F. A. Saunders. Hence, the scheme we use to derive these states is called Russell-Saunders coupling. It is also simply referred to as L-S coupling. 

In the L-S coupling scheme we have just discussed, it is assumed that the electronic repulsion is much larger than the spin-orbit interaction. This assumption certainly holds for lighter elements, as we have seen in the case of carbon atom (Fig. 2.3.2). However, this assumption becomes less and less valid as we go down the Periodic Table. The breakdown of this coupling scheme is clearly seen from the spectral data given in Table 2.3.3. 

Note that in Table 2.3.4 labels 1 and 2 are for convenience only. It is important to bear in mind that electrons are indistinguishable. The five electronic states derived using this scheme may be correlated to those derived with the L-S coupling scheme in Fig. 2.3.2. This correlation diagram is shown in Fig. 2.3.4. 

L-S coupling electronic interaction > spin-orbit coupling... 

The effect of filling the lanthanide f-shell on gas ion size is simply due to internal 4/ electron-electron repulsion which is reflected in the L/S coupling, i.e. the magnetic... 

The choice of a single function from either set (36) or (37) does not permit such a useful physical interpretation, and may indeed lead to difficulties as the internuclear distance is varied. Thus if one chooses just the perfectly paired function from the set (36), as -R-> 00 one finds each N atom is described by a curious non-stationary state - the so-called valence state of the atom, about which there has been so much discussion in the literature.18 The choice of the set of functions (36) in which orbitals participating in a bond are directly coupled to each other is just the VB theory as proposed by Slater and Pauling,19 whereas the set (37) formed from atoms in specific L-S coupled states corresponds to the spin-valence theory employed by Heitler.20... 

Fig. 3. Addition of magnetic moments in an atom with large L-S coupling. Vector magnitudes ...

For L — S coupling (so-called Russell-Saunders coupling), which is to combine the s of every electron to get S, and combine the I of every electron to obtain L initially and finally to combine S and L to get J ... 

The order of the energy levels arising from any configuration can be predicted, in the case of L-S coupling, by rules due to Hund (1925) ... 

Ions in which the separation of states with different / values is small compared with kT (weak L-S couplings). Here the orbital contribution is not quenched but is uncoupled from the spin so that a resultant J does not occur. The resultant moment is... 

Finally, it must be stressed that Hund s rules (and indeed the whole L-S coupling picture) do not generally apply well to heavy atoms. However, magnetic measurements confirm their approximate validity in the transition series Sc to Fe. The occupation of the 3d and 4s orbitals is indicated in the following scheme ... 

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