The Total Angular Quantum Number

A Systematic Correlation of the Properties of the f-Transition Metal Ions 

In Ref. (55), a cross-section of the properties of the lanthanides and the actinides were plotted to exemplify their linear variation with the L-values of the ions. Here we shall examine a wide variety of data to explore the validity of this linear relationship. 

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Spin Orbit Coupling of p2 With a p2 arrangement, resultant orbital quantum number values of L = 2,1 and 0, and also resultant quantum number values S = 1 and 0 are obtained. These may be coupled to give the total angular quantum number J. Each of these arrangement corresponds to an electronic arrangement called a spectroscopic state. 

Thus, the eigenstates of cannot be classified according to I, but to j, the total angular quantum number. These common eigenfunctions of and /I are then only simultaneous eigenfunctions of one additional component of and not of the other two because the components do not commute. 

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. 

In a free multielectron atom or ion, the spin and orbital angular moments of the electrons couple to give a total angular momentum represented in the Russell-Saunders scheme by the quantum number J. Since J arises from vectorial addition of L (the total orbital quantum number) and 5 (total spin quantum number), it may take integral (or half-integral... 

In this table, Ml is the total magnetic quantum number of the ion. Its maximum is the total orbital angular quantum number L. Ms is the total spin quantum number along the magnetic field direction. Its maximum is the total spin quantum number 5. / = L 5, is the total angular momentum quantum number of the ion and is the sum of the orbital and spin momentum. For the first seven ions (from La + to Eu +), J =L —S, for the last eight ions (from Gd + to Lu +), J = L + S. The spectral term consists of three quantum numbers, L, S, and J and may be expressed as. The value of L is indicated by S, P, D, F, G, H, and I for L = 0,... 

The atomic terms are characterized by the orbital angular momentum which is reflected in the value of the total orbital quantum number L. Just as the individual angular momenta are quantized, so are the resultants. For the example above, with = 1 for two p electrons, the maximum value of the resultant angular momentum is represented by the sum of the quantum numbers, and the minimum value is represented by the difference. Intermediate values are allowed too, provided they differ from the extremes by a whole number ... 

S is the total spin quantum number =. . ., L is the total angular mo-... 

Similarly, if we calculate the resultant value of the spin quantum number for the electrons outside the closed shells we obtain the total spin quantum number, S. The components of the total spin quantum numbers, S, S — 1,— (S — 1), —S, are the possible values of the quantum number E, which corresponds to Ms in the atom. There are 2S 1 values of E the multiplicity of the state is 25 -h 1. There is a total angular momentum quantum number, Q, which has the values... 

According to quantum mechanics, electron is the fermion with 1/2 spin quantum number, which is the eigenvalue of spin angular momentum. In a system comprised of two electrons, the total spin quantum number S is 0 or 1. The spin wave function of the eigenstates S = 0 and S = 1 can be demonstrated by ... 

Now let us move from the orbital quantum number to the spin quantum number. In Section 1.6, we stated that the spin quantum number, s, determines the magnitude of the spin angular momentum of an electron and has a value of j. For a 1-electron species, is the magnetic spin angular momentum and has a value of + or -j. We now need to define the quantum numbers S and Ms for multielectron species. The spin angular momentum for a multielectron species is given by equation 21.10, where S is the total spin quantum number. 

Note that in atomic systems with more than one valence electron the individual angular momentum vectors are summed up (L = Y jii, S = and J =L + S)to yield the total system quantum number L, S, J, Ml, Ms and Mj. 

We also need Pauli s exclusion principle which states that each box can contain only one electron. Hund s first rule requires that electrons be added to available boxes so as to maximize the total spin of the system, S = JliSi. Hund s second rule insists that, consistent with the first rule, the total magnetic quantum number Ml = itnii be maximized. It is also standard to define M/,(max)s L, the total orbital angular momentum quantum number. With these ideas in mind it is easy to write down the total spin and orbital quantum numbers for all possible 4/" configurations as shown in the following table. 

Compared to state 111 >, state 12> represents a reorientation of both the p orbital as well as the orbital angular momentum. This reorientation is accomplished by rotation of L and C in an opposite sense, so that the total projection quantum number (M=mj+m 2) remains 0. Physically the mixing of states 111 > and 12> implies that as soon as R becomes small enough that the I and H electrostatic potentials begin to differ significantly, the initially prepared orientation of the p-orbital will be scrambled. There is another implication in many semiclassical treatments of atomic and molecular collisions both the orientation and the magnitude of C are assumed to remain fixed during the collision [55,56]. The discussion in... 

Diamagnetic and paramagnetic substances develop magnetic moments that are respectively opposed to, and in the direction of an external magnetic field. The magnetic moments for the paramagnetic molecnles that we shall discuss in this book arise either primarily or almost entirely from the presence of one or more unpaired-electron spins, i.e. the total spin quantum number S is non-zero for such a molecule. For a molecule with spin quantum number S, the spin angular... 

Regardless of the nature of the intramolecular dynamics of the reactant A, there are two constants of the motion in a nnimolecular reaction, i.e. the energy E and the total angular momentum j. The latter ensures the rotational quantum number J is fixed during the nnimolecular reaction and the quantum RRKM rate constant is specified as k E, J). 

The simplest case is a transition in a linear molecule. In this case there is no orbital or spin angular momentum. The total angular momentum, represented by tire quantum number J, is entirely rotational angular momentum. The rotational energy levels of each state approximately fit a simple fomuila ... 

The presence of two angular momenta has as a consequence that only their sum, representing the total angular momentum in the case considered, necessary commutes with the Hamiltonian of the system. Thus only the quantum number K, associated with the sum, N, of and Lj,... 

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