Angular Momentum Rules

In Chapters 4, 5, and 7, we deal with the angular momentum of electrons due to their orbital motions and spins and also with angular momentum due to molecular rotation. The conclusions we arrive at are based partly on physical arguments related to experience with macroscopic bodies. While this is the most natural way to introduce such concepts, it is not the most rigorous. In this appendix we shall demonstrate how use of the postulates discussed in Chapter 6 leads to all of the relationships we have introduced earlier. The approach is based entirely on the mathematical properties of the relevant operators, making no appeal to physical models. 

According to postulate II (Section 6-3), we must first find the classical physical expressions for the quantity of interest in terms of x, y, z, px, py, pz, and t. We know that the angular momentum for an object of mass m moving with velocity u in a circular orbit 

Here i, j, k are unit vectors pointing, respectively, along the x, y, z Cartesian axes. A cross product is taken by expanding the following determinant  

The coefficient for i is the magnitude of the x-component of angular momentum, etc. That is, 

This gives the classical expressions we need for the x, y, and z components of angular momentum and for the square of its magnitude. 

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Here the combination of the reactants to form the intermediate violates both the spin rule and the orbital angular momentum rule. This reaction appears to be slow at low ion energy (23). Consider Reaction 7 ... 

The spin rule is satisfied, but the orbital angular momentum rule is not. The reaction is apparently fast at low ion energies (4) hence, if there is an important selection rule in the combination of reactants, it is seemingly the spin rule. Conservation of spin in combining reactants is probably more likely than conservation of orbital angular momentum, since the latter will be more strongly coupled to collision angular momentum. 

The calculation of the matrix elements (38) and (39) is for small elementary cells the most time-consuming part of the (R)FPLO approach. For the overlap matrix S, one- and two-center integrals have to be provided while the Hamiltonian matrix requires the calculation of one-, two- and three-center integrals. As both the orbital and potential functions involved are well localized, only a limited number of multi-center integrals have to be calculated. The one- and two-center-integrals are further simplified by the application of angular momentum rules to one- and two-dimensional integrations, respectively. There are however two points which make the calculation of these matrix elements (in principle) much more involved for the relativistic approach. At first, the... 

The angular momentum rule found its most accurate and striking application in Bohr s interpretation of the hydrogen spectrum, where an analogous postulate was made about the permissible states of an electron circulating around a nucleus, and it may be regarded apart from a trivial correction as established. 

The terms deduced from the chart for the (3<2) (4 7 configuration are =P, D, P, S, G, ip, D, P, and S. Following the spin-multiplicity and angular-momentum rules, the G state should be most stable. 

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

Any operator J, which satisfies the commutation rule Eq. (7-18), represents quantum mechanical angular momentum. Orbital angular momentum, L, with components explicitly given by Eq. (7-1), is a special example5 of J. 

The proof of the theorem affirming that J8 is a proper quantum mechanical angular momentum involves only an expansion of (Ji + J2) x (Ji + J2) with subsequent use of the commutation rules for Jj and J2, and the fact that Jj and J2 commute because they act in... 

The s, therefore, satisfy angular momentum commutation rules. Since each of these matrices has eigenvalues 1 and 0, they form a representation of the angular momentum operators for spin 1. 

Experimental data on nitrogen obtained from spin-lattice relaxation time (Ti) in [71] also show that tj is monotonically reduced with condensation. Furthermore, when a gas turns into a liquid or when a liquid changes to the solid state, no breaks occur (Fig. 1.17). The change in density within the temperature interval under analysis is also shown in Fig. 1.17 for comparison. It cannot be ruled out that condensation of the medium results in increase in rotational relaxation rate primarily due to decrease in free volume. In the rigid sphere model used in [72] for nitrogen, this phenomenon is taken into account by introducing the factor g(ri) into the angular momentum relaxation rate... 

A similar defect is also inherent to the operator f(1), which rules the angular momentum relaxation according to... 

There does not seem to be any selection rule such as conservation of spin or orbital angular momentum which this reaction does not satisfy. It is also not clear that overall spin conservation, for example, is necessary in efficient reactions (5, 16, 17, 20). Further, recent results (21) seem to show a greatly enhanced (20 times) reaction rate when the N2 is in an excited vibrational state (vibrational temperature 4000 °K. or about 0.3 e.v.). This suggests the presence of an activation energy or barrier. A barrier of 0.3 e.v. is consistent with the low energy variation of the measured cross-section in Figure 1. 

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... 

Note that, throughout this discussion, we have used lower-case letters when refering to orbitals and upper-case when we mean many-electron wavefunctions. There arises the question of, what are the relationships between I and L, or between s and S . They are determined by the vector coupling rule. This states that the angular momentum for a coupled (i.e. interacting) pair of electrons may take values ranging from their sum to their difference (Eq. 3.11). 

The vector coupling rule applies to all forms of angular momentum ... 

All this is summarized in Fig. 3-12. The energy ordering of the free-ion terms is not determined by consideration of angular momentum properties alone and in general yields only to detailed numerical computation. The ground term - and only the ground term - may be deduced, however, from some simple rules due to Hund. 

Consider now spin-allowed transitions. The parity and angular momentum selection rules forbid pure d d transitions. Once again the rule is absolute. It is our description of the wavefunctions that is at fault. Suppose we enquire about a d-d transition in a tetrahedral complex. It might be supposed that the parity rule is inoperative here, since the tetrahedron has no centre of inversion to which the d orbitals and the light operator can be symmetry classified. But, this is not at all true for two reasons, one being empirical (which is more of an observation than a reason) and one theoretical. The empirical reason is that if the parity rule were irrelevant, the intensities of d-d bands in tetrahedral molecules could be fully allowed and as strong as those we observe in dyes, for example. In fact, the d-d bands in tetrahedral species are perhaps two or three orders of magnitude weaker than many fully allowed transitions. 

Finally we observe from Fig. 1 the magnitude of Gw qz) decreases for increasing at fixed w. Thus, the only way to fulfill the Bethe sum rule at arbitrarily large values of q will be to include basis functions of arbitrarily high angular momentum. This confirms a previously reached conclusion [12],... 

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