Orbitals spin and

R designates a 6-dimensional two-electron/two-orbital spin- and spatial-symmetry nonadapted model space, while 4R is a 4-dimensional model space spanned by the four Ms = 0 determinants. 

In summary, the techniques described in this chapter allow us to derive expansions of the operators that correspond to physical quantities, in terms of irreducible tensors in the spaces of orbital, spin and quasispin momenta, and also to separate terms that can be expressed by operators whose eigenvalues have simple analytical forms. Since the operators of physical quantities also contain terms for which this separation is impossible, the following chapter will be devoted to the general technique of finding the matrix elements of quantities under consideration. 

As has been shown, second-quantized operators can be expanded in terms of triple tensors in the spaces of orbital, spin and quasispin angular momenta. The wave functions of a shell of equivalent electrons (15.46) are also classified using the quantum numbers L, S, Q, Ml, Ms, Mq of the three commuting angular momenta. Therefore, we can apply the Wigner-Eckart theorem (5.15) in all three spaces to the matrix elements of any irreducible triple tensorial operator T(JC K) defined relative to wave functions (15.46)... 

The operators of orbital, spin and quasispin angular momenta of the two-shell configuration are expressed in terms of sums of one-shell triple tensors (15.52) ... 

Using (17.61) we can additionally obtain the following relationships for the tensorial products that are complete scalars in orbital, spin and quasispin spaces ... 

These tensorial products have certain ranks with respect to the total orbital, spin and quasispin angular momenta of both shells. We shall now proceed to compute the matrix elements of such tensors relative to multi-configuration wave functions defined according to (17.56). Applying the Wigner-Eckart theorem in quasispin space to the submatrix element of tensor gives... 

The use of the tensorial properties of both the operators and wave functions in the three (orbital, spin and quasispin) spaces leads to a new very efficient version of the theory of the spectra of many-electron atoms and ions. It is also developed for the relativistic approach. 

L, S, J AAA L, S, J L, S, J A J j =jl +j2 orbital, spin, and total angular momenta quantum mechanical operators corresponding to L, S, and J quantum numbers that quantize L2, S2, and J2 operator that obeys the angular momentum commutation relations total (j) and individual (ji, j2, ) angular momenta, when angular momenta are coupled... 

This expression can be factorised into orbital, spin and rotational parts and then tidied up to produce an effective operator of the form... 

For paramagnetic diatomic molecules with net orbital magnetic moments, the g-factor depends on the complicated coupling between the orbital, spin, and rotational moments. 

The final three terms of the rotational operator in Eq. (3.1.13), which couple the orbital, spin, and total angular momenta, are responsible for perturbations between different electronic states ... 

States of individual atoms are usually described by quantum numbers L, S, and for the electronic orbital, spin, and total angular momenta, respectively. However, in scattCTing and bound-state problems involving pairs of atoms or molecules it is common to use lower-case letters for quantum numbers of individual collision partners and reserve capital letters for quantities that refer to the collision system (or complex) as a whole. Thus, in this subsection we will use I and s for the quantum numbers of a single helium atom and reserve L and S for the end-over-end angular momentum of the atomic pair and the total spin, respectively. 

First, let us define three new quantum numbers L, S, and J representing the overall quantum state of the subshell. Those quantum numbers are associated with the overall orbital, spin, and spin orbit angular momentum of the subshell, respectively. The idea behind those overall quantum numbers is that for an unfilled subshell, the sum of the orbital angular momentum t of each electron, as well as the sum of the spin angular momentum s of each electron may be different from zero. The sum of the total angular momentum j = f + s of each electron may thus also be different from zero. These new vectors are overall vectors because they enclose the contribution of all the electrons (it is a vector sum). The overall quantum numbers are easily calculated from the filling of the subsheU by summing the me values of each electron... 

The El selection rules may also be formulated in terms of the quantum numbers L, S, and J for the many-electron orbital, spin, and total angular momentum. These prove to be... 

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