Coupling of angular momenta

The strength of coupling between the spin and orbital motions of the electrons, referred to a spin-orbit coupling, depends on the atom concerned. 

The spin of one electron can interact with (a) the spins of the other electrons, (b) its own orbital motion and (c) the orbital motions of the other electrons. This last is called spin-other-orbit interaction and is normally too small to be taken into account. Interactions (a) and (b) are more important and the methods of treating them involve two types of approximation representing two extremes of coupling. 

In quantum mechanics, the total angular momentum operator / is simply the sum of the two individual angular-momentum vector operators 

Since the uncoupled states )i Wy(i)) and y2W (2)) h form a complete basis, the (direct) product basis built from them is a complete basis for the Hilbert space representation of the total eigenstate JMj) in accord with Eq. (4.13). However, in this product basis is not diagonal. We may expand the coupled state Mj) into the product basis of the uncoupled states 

It is important to stress that the above equation holds for J — ji + j2 only, which implies that / and / )2- However, a general, but significantly more complicated expression can be derived [75]. 

Related to the Clebsch-Gordan vector coupling coefficients are Wigner s 3j-symbols 

The possible values for the quantum number wii are -jy..jy and by counting those possibilities, we find there are + 1 of them. This is the multiplicity of states from the ji angular momentum. The total number of states for the whole system is a product of multiplicities, (2/i + 1x2/2 +1). 

Maximum resultant z-component = (maximum value of mi + maximum value of m2)h 

This must also equal the maximum z-component quantum number of the resultant vector, designated M, times h. That is. 

Since any z-component quantum number can take on only certain values because of the length of the associated vector (i.e., M = -/. /), knowing the maximum value of M means knowing /. Thus, we conclude that one possible value for the resultant / quantum number is/i+/2. 

When the process is continued to the next step down in the z-component of the resultant vector, it turns out that there has to be another possible / value, this one being ji + ]2 - 2. And this pattern continues until one reaches the minimum value, ji - /21- Thus, the rule for adding angular momenta is that the possible values for the resultant or total / quantum number span a range given by the quantum numbers associated with the source vectors  

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Section treats the spatial, angular momentum, and spin symmetries of the many-electron wavefunctions that are formed as antisymmetrized products of atomic or molecular orbitals. Proper coupling of angular momenta (orbital and spin) is covered here, and atomic and molecular term symbols are treated. The need to include Configuration Interaction to achieve qualitatively correct descriptions of certain species electronic structures is treated here. The role of the resultant Configuration Correlation Diagrams in the Woodward-Hoffmann theory of chemical reactivity is also developed. 

G. Gaigalas, 0. Scharf, S. Fritzsche, Maple procedures for the coupling of angular momenta. VIII. Spin-angular coefficients for single-shell configurations, Comput. Phys. Commun., 166, 141-169 (2005). 

Because of the large number of angular momentum observables involved in the state vectors of systems of more than one electron, it helps greatly in keeping track of the quantum numbers to express the coupling of angular momenta in terms of the symbols of Wigner, which have useful symmetries. 

Clebsch-Gordon coefficients - A set of coefficients used to describe the vector coupling of angular momenta in atomic and nuclear physics. 

The principal obstacle in the study of molecular potential functions more complicatinteraction constants. The isotope effect, discussed in Sec. 8-5, is particularly useful since it provides additional frequencies. Another property which has sometimes been used is the coupling of angular momenta of rotation and vibration, which affects the rotational fine structure- (see Appendix XVI). Centrifugal disl oi tion as observed in the pure rotational spectrum in the microwave region can also provide additional data." ... 

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