Total spin angular momentum quantum

Now the total spin-angular momentum quantum number S is given by the number, n, of unpaired electrons times the spin angular momentum quantum number s for the electron, that is, S = nil. Substitution of this relationship into Eq. (5.11) yields an alternative form of the spin-only formula. 

For many-electron light atoms, the Russell-Saunders coupling rules prevail One combines the orbital angular momenta lt of each electron, treated as a vector, to form the total orbital angular momentum quantum number (and vector) L = h one similarly couples the spin angular momentum quantum numbers s, into a total spin angular momentum quantum number S = s > then one adds L and S to get the total angular momentum vector... 

L = total orbital angular momentum quantum number S = total spin angular momentum quantum number J = total angular momentum quantum number... 

When there are two electrons in a molecule, their total spin angular momentum quantum number S is given by Equation 1.9. 

The product mn = 2SA + 1X25 + 1) is the number of possible total spin angular momentum quantum numbers S of the encounter complex that are allowed by combination of the quantum numbers SA and SB of the reaction partners as defined by Equation 2.30. 

S = total spin angular momentum quantum number... 

So far in this chapter we have not included the effect of electron spin. As mentioned in Chapter XI, molecules may possess singlet, doublet, triplet states, etc., according as the total spin angular momentum quantum number S is equal to 0, J, 1, . For molecules, just as for atoms, the selection rule is AS = 0 this rule must be considered in applying the results given in Table 14 3. Tables 14 1 and 14 2 are correct only for singlet states, which are by far the most important. For a detailed account of the way the states are modified when S 5 0, the reader is referred to a textbook on molecidar spectroscopy. ... 

Suppose a quantum system consists of a pair of spin- particles, and that the system is prepared in such a way that (1) the total spin angular momentum is zero (i.e. the... 

Here L, S, and J are the quantum numbers corresponding to the total orbital angular momentum of the electrons, the total spin angular momentum, and the resultant of these two. Hund predicted values of L, S, and J for the normal states of the rare-earth ions from spectroscopic rules, and calculated -values for them which are in generally excellent agreement with the experimental data for both aqueous solutions and solid salts.39 In case that the interaction between L and S is small, so that the multiplet separation corresponding to various values of J is small compared with kT, Van Vleck s formula38... 

Fio. 2-11.—The interaction of the spin angular momentum vectors of two electrons to form a resultant total spin angular momentum vector, corresponding to the value of the total spin quantum number S = 0 or S = 1. 

We now consider many-electron atoms. We will assume Russell-Saunders coupling, so that an atomic state can be characterized by total electronic orbital and spin angular-momentum quantum numbers L and S, and total electronic angular-momentum quantum numbers J and Mj. (See Section 1.17.) The electric-dipole selection rules for L, J, and Mj can be shown to be (Bethe and Jackiw, p. 224)... 

The value of J depends on the total orbital angular momentum quantum number, L, and the total spin angulur momentum quantum number, S (Appendix C). Some calculated and experimental magnetic moments for lanthanide complexes are shown in Table 11.25... 

L is the quantum number specifying the total orbital angular momentum for the term, 5 the total spin angular momentum. Each of these momenta has components in any chosen direction, z say, which take on the integral values Lz, from L to -L, or S. from S to -S, respectively. There are 1L + 1 values of L, and 2S + 1 values of Sz, each with appropriate wave functions. Consequently, a term specified by L and S is (2L + + l)-fold degenerate. 

In the extreme case where the spin-orbit interaction is much larger than electronic repulsion, total orbital angular momentum L and total spin angular momentum S are no longer good quantum numbers. Instead, states are defined by total angular momentum J, which is the vector sum of all the total angular momenta y values of the individual electrons ... 

Multiplicity Spin Multiplicity) The number of possible orientations, calculated as 2S -L 1, of the spin angular momentum corresponding to a given total spin quantum number (S), for the same spatial electronic wavefunction. A state of singlet multiplicity has S = 0 and 2S -i- 1 = 1. A doublet state has S = 1/2, 2S -i- 1 = 2, etc. Note that when S > L (the total orbital angular momentum quantum number) there are only 21 -t 1 orientations of total angular momentum possible. 

As in the case of the orbital angular momentum, the upper case 5 here signifies the total spin angular momentum of aU the unpaired electrons in the atom outside of closed shells. The filled, inner subshells contribute nothing to 5. It follows that if the configuration has only a single electron that is not in a filled subsheU, the 5 quantum number is simply the quantum number for the single electron. 

It is a fundamental fact of quantum mechanics, that a spin-independent Hamiltonian will have pure spin eigenstates. For approximate wave functions that do not fulfill this criterion, e.g. those obtained with various unrestricted methods, the expectation value of the square of the total spin angular momentum operator, (5 ), has been used as a measure of the degree of spin contamination. is obviously a two-electron operator and the evaluation of its expectation value thus requires knowledge of the two-electron density matrix. 

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