Vector coupling rule

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 ... 

The L and S values are those from which the / value was formed via the vector coupling rule. These formulae strictly apply only for the magnetism of free-ion levels. They provide a good aproximation for the magnetism of lanthanide complexes, as we shall note in Chapter 10, but provide no useful account of the magnetic properties of d block compounds. 

At first instance, the terms types which arise from this configuration of spin-orbital are written, such as ( s+dl by using the vector coupling rule, for which the terms L which arise from the combination of the orbital moments /j (of the electron no. 1) and (of the electron no. 2) are laying between and I and analogously possess a total moment of spin S with values between ( j+ j) I successively decreasing by one unit. 

The interactions of type (1) are known as L-S coupling or Russell-Saunders coupling. From the vector addition rule and the constraint that the values must differ by one in the unit of h/2n, the possible values of L and S are ... 

The tensorial structure of the spin-orbit operators can be exploited to reduce the number of matrix elements that have to be evaluated explicitly. According to the Wigner-Eckart theorem, it is sufficient to determine a single (nonzero) matrix element for each pair of multiplet wave functions the matrix element for any other pair of multiplet components can then be obtained by multiplying the reduced matrix element with a constant. These vector coupling coefficients, products of 3j symbols and a phase factor, depend solely on the symmetry of the problem, not on the particular molecule. Furthermore, selection rules can be derived from the tensorial structure for example, within an LS coupling scheme, electronic states may interact via spin-orbit coupling only if their spin quantum numbers S and S are equal or differ by 1, i.e., S = S or S = S 1. 

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... 

In order that the products C and D lie on the same potential energy surface as the reactants A and B, the total spin has to be conserved during the reaction. The spins and of the reactants may be coupled according to vector addition rules in such a way that the total spin of the transition state can have the following values ... 

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. 

Antisymmetric matrix, non-adiabatic coupling, vector potential, Yang-Mills field, 94-95 Aromaticity, phase-change rule, chemical reaction, 446-453 pericyclic reactions, 447-450 pi-bond reactions, 452-453 sigma bond reactions, 452 Aromatic transition state (ATS), phase-change rule, permutational mechanism, 451-453... 

As a rule, the density of states for molecular lattice vibrations is negligible as compared to that for crystal phonons. Therefore, the K-mode of a molecular lattice is coupled with the crystal phonons specified by the same wave vector K. Besides, the low-frequency collective mode m of adsorbed molecules can be considered as a... 

The last transition is forbidden because the demands from the angular momentum coupling and the parity requirement are mutually exclusive the coupling of the orbital angular momenta requires the vector addition L + = 0 with L = 1 and hence also = 1 on the other hand, the parity selection rule requires = even, and both conditions cannot be fulfilled simultaneously. Therefore, only five transitions are expected for the K-LL Auger spectrum in neon, and these can be identified in Fig. 3.3. 

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