L-S coupling scheme

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. 

For light element atoms an alternative L-S coupling scheme is applied (this is also termed the Russell-Saunders scheme) which is based on a reverse assumption n < V. Non-relativistic (or quasi-relativistic) spatial orbitals are used as a basis set l,mt) in creating the electron configurations electron term state vectors L, ML, S, Ms). Now the addition of the angular momenta proceeds, being based on the assumption... 

In the second step, we introduce the Spin-Orbit (SO) interaction, V q, in the ith-atom within the L-S coupling scheme, i.e., Vg = —where, is the spin-orbit coupling constant for the ith-atom,L its orbital angular momentum along the Z-axis, and its total... 

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. 

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 ... 

We have based the discussion on a scheme in which we first added the individual electronic orbital angular momenta to form a total-orbital-angular-momentum vector and did the same for the spins L = S,- L, and S = 2i S,. We then combined L and S to get J. This scheme is called Russell-Saunders couplit (or L-S coupling) and is appropriate where the spin-orbit interaction energy is small compared with the interelec-tronic repulsion energy. The operators L and S commute with + W,ep, but when is included in the Hamiltonian, L and no longer commute with H. (J does commute with + //rep + Q ) If the spin-orbit interaction is small, then L and S almost commute with (t, and L-S coupling is valid. 

In order to describe the hyperfine levels of HD+ we must couple together the angular momenta due to complete rotation of the molecule (N), electron spin (S=i), nuclear spin of the proton (iH i) and nuclear spin of the deuteron (Iiy=l). The coupling scheme suggested by the relative magnitudes of the interactions is ... 

The simple L, S model falls victim to the Z" increase of the L S coupling between electron spin (S) and the angular momentum (L) of the orbitals in many-electron atoms at about Z = 30. Heavier elements need to be described by the /, J scheme where J = L + S. 

The j-dependent REP may be averaged over the quantum number ] = l 1/2, whereby spin-orbit contributions are suppressed [11, pp. 359, 374] and the more familiar A-S coupling scheme may be used [2, 14]. Such an averaged potential (AREP) was applied in a CAS-MC-SCF calculation (for CAS see [15]), followed by a first-order Cl treatment (FOCI) [9]. Spin-orbit effects were later introduced by a relativistic Cl method (RCI, first applied in [14, 16])... 

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. 

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. 

---