Coupled representation

Equation (2.3) describes line positions correctly for spectra with small hyperfine coupling to two or more nuclei provided that the nuclei are not magnetically equivalent. When two or more nuclei are completely equivalent, i.e., both instantaneously equivalent and equivalent over a time average, then the nuclear spins should be described in terms of the total nuclear spin quantum numbers I and mT rather than the individual /, and mn. In this coupled representation , the degeneracies of some multiplet lines are lifted when second-order shifts are included. This can lead to extra lines and/or asymmetric line shapes. The effect was first observed in the spectrum of the methyl radical, CH3, produced by... 

The spin-orbit coupling term in the Hamiltonian induces the coupling of the orbital and spin angular momenta to give a total angular momentum J = L + S. This results in a splitting of the Russell-Saunders multiplets into their components, each of which is labeled by the appropriate value of the total angular momentum quantum number J. The character of the matrix representative (MR) of the operator R(0 n) in the coupled representation is... 

This matrix may be reduced by the same prescription as was used earlier in eq. (11), namely by forming the coupled representation... 

The purpose of this review is to discuss the main conclusions for the electronic structure of benzenoid aromatic molecules of an approach which is much more general than either MO theory or classical VB theory. In particular, we describe some of the clear theoretical evidence which shows that the n electrons in such molecules are described well in terms of localized, non-orthogonal, singly-occupied orbitals. The characteristic properties of molecules such as benzene arise from a profoundly quantum mechanical phenomenon, namely the mode of coupling of the spins of the n electrons. This simple picture is furnished by spin-coupled theory, which incorporates from the start the most significant effects of electron correlation, but which retains a simple, clear-cut visuality. The spin-coupled representation of these systems is, to all intents and purposes, unaltered by the inclusion of additional electron correlation into the wavefunction. 

To derive this result, we express a typical matrix element in terms of the coupled representation on the one hand and in terms of the decoupled on the other ... 

The diamagnetic terms were not calculated by Ramsey [7] for the coupled representation, which was used only for analysis of the low field spectra. 

The structure of the 9 x 9 energy matrix in the coupled representation is shown in table 8.2, with the explicit matrix elements given below. 

Table 8.2. Energy matrix forE[2 in the J = 1 level in the coupled representation... 

Additional terms involving the scalar and tensor interactions between the two nuclear spins were found to be too small to be significant in the first study, but we will meet them later. We encountered the quadrupole term in (8.171) in our earlier discussion of the D2 molecule, and obtained the following results for the matrix elements in the coupled representation ... 

The matrix elements of the quadrupole interaction are calculated in various places, for different coupling cases, in the main text. Here we shall carry out the calculation in a case (a) coupled representation, which will enable us to define the nuclear quadrupole moment, the electric field gradient, and the quadrupole coupling constant. 

The presence of two nuclear spins means that there is considerable choice in the selection of basis functions the reader who wishes to practice virtuosity in irreducible tensor algebra is invited to calculate the matrix elements in the different coupled representations that are possible In fact the sensible choice, particularly when a strong magnetic field is to be applied, is the nuclear spin-decoupled basis set t], A N, S, J, Mj /N, MN /H, MH). Again note the possible source of confusion here MN is the space-fixed component of the nitrogen nuclear spin /N, not the space-fixed component of N. This nuclear spin-decoupled basis set was the one chosen by Wayne and Radford in their analysis of the NH spectrum. 

We have chosen to use the hyperfine-coupled representation, where for 12CH, F is equal to J 1 /2. An appropriate basis set is therefore t], A N, A S, J, /, F), with MF also important when discussing Zeeman effects. As usual the effective zero-field Hamiltonian will be, at the least, a sum of terms representing the spin-orbit coupling, rigid body rotation, electron spin-rotation coupling and nuclear hyperfine interactions, i.e. 

The 77 coupling representation is generally useful in the reduction of the Lippmann—Schwinger equations since it applies to situations where spin—orbit coupling is not negligible. The quantum numbers used in the representation are defined in table 7.1. Primed and double-primed quantities are used to distinguish different angular-momentum states. 

This form is used in the LS-coupling representation of the potential matrix element. For jj coupling we use... 

This approximation has been shown to have at least semiquantitative validity over the whole energy range. It is unacceptable only for excitations involving a change of target spin s in the LS-coupling representation. Here exchange amplitudes make the only contribution and the factorisation approximation is too severe. 

The spectroscopic factors are critical quantities in determining the accuracy of a configuration-interaction calculation of the structure of the ion. The ion state /) is written in the weak-coupling representation as... 

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