Nuclear spin coordinates

In the last section, we showed rot to have the eigenvalue (—1/ for inversion of the nuclei. Since inversion amounts to interchanging the nuclear spatial coordinates, and since the nuclear spin coordinates do not occur in proV this is also the eigenvalue of rot for the operator which interchanges the space and spin coordinates of the two nuclei. 

The selection rule (4.138) differs from previously discussed selection rules in that it holds well for nonradiative transitions, as well as for radiative transitions. In deriving (4.138), we made no reference to the operator d, beyond the statement that it did not involve the nuclear spin coordinates. For any time-dependent perturbation that does not involve nuclear spin, the selection rule (4.138) will hold. Thus molecular collisions will not cause nonradiative transitions between symmetric and antisymmetric rotational levels of a homonuclear diatomic molecule. If we somehow start with all the molecules in symmetric levels, the collisions will not populate the antisymmetric levels. 

As for diatomic molecules (Section 4.8), the wave function of a poly- " atomic molecule should include the nuclear spin coordinates, as well as... 

The function PeI is independent of nuclear spin, so that the summation over nuclear-spin coordinates in (7.13) is... 

To derive the Hamiltonian operator corresponding to (8.104), we replace pe and pN by the corresponding operators. The resulting Hamiltonian contains the electron s spatial coordinates r, as well as electronic and nuclear spin coordinates. In this chapter, we are dealing only with spin... 

The inversion operation i which leads to the g/u classification of the electronic states is not a true symmetry operation because it does not commute with the Fermi contact hyperfine Hamiltonian. The operator i acts within the molecule-fixed axis system on electron orbital and vibrational coordinates only. It does not affect electron or nuclear spin coordinates and therefore cannot be used to classify the total wave function of the molecule. Since g and u are not exact labels, it was realised by Bunker and Moss [265] that electric dipole pure rotational transitions were possible in ll], the g/u symmetry breaking (and simultaneous ortho-para mixing) being relatively large for levels very close to the dissociation asymptote. The electric dipole transition moment for the 19,1 19,0 rotational transition in the ground electronic state was calculated... 

The quantum numbers tliat are appropriate to describe tire vibrational levels of a quasilinear complex such as Ar-HCl are tluis tire monomer vibrational quantum number v, an intennolecular stretching quantum number n and two quantum numbers j and K to describe tire hindered rotational motion. For more rigid complexes, it becomes appropriate to replace j and K witli nonnal-mode vibrational quantum numbers, tliough tliere is an awkw ard intennediate regime in which neitlier description is satisfactory see [3] for a discussion of tire transition between tire two cases. In addition, tliere is always a quantum number J for tire total angular momentum (excluding nuclear spin). The total parity (symmetry under space-fixed inversion of all coordinates) is also a conserved quantity tliat is spectroscopically important. 

Let us discuss further the pemrutational symmetry properties of the nuclei subsystem. Since the elechonic spatial wave function t / (r,s Ro) depends parameti ically on the nuclear coordinates, and the electronic spacial and spin coordinates are defined in the BF, it follows that one must take into account the effects of the nuclei under the permutations of the identical nuclei. Of course. 

If deuterio acids are used then ij -HD complexes are formed these are particularly useful in establishing the retention of substantive H-H bonding in the coordinated ligand by observation of a 1 1 1 triplet in the proton nmr spectrum (the proton signal being split by coupling to deuterium with nuclear spin 7 = 1). 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

A few thioether-ligated copper(II) complexes have been reported, however (cf. Section 6.6.3.1.2) (417) (essentially square planar), (418) (two crystalline forms one TBP and other SP),361 (419) (SP),362 (420) (SP),362 (421) (TBP),362 (422) (SP),363 (423) (SP),363 (424) (two independent complexes SP and octahedral),364 (425) (TBP).364 In the complexes (420) and (421), EPR spectra revealed that the interaction between the unpaired electron and the nuclear spin of the halogen atom is dependent on the character of the ligand present. For (424) and (425), spectral and redox properties were also investigated. Rorabacher et al.365 nicely demonstrated the influence of coordination geometry upon CV/Cu1 redox potentials, and reported structures of complexes (426) and (427). Both the Cu1 (Section 6.6.4.5.1) and Cu11 complexes have virtual C3v symmetry. 

Figure 1. Spherical coordinates used to calculate the dipolar interaction between two nuclear spins in a strong magnetic field H .

Chemical bonds can have covalent character, and EPR spectroscopy is an excellent tool to study covalency An unpaired electron can be delocalized over several atoms of a molecular structure, and so its spin S can interact with the nuclear spins /, of all these atoms. These interactions are independent and thus afford additive hyperfine patterns. An unpaired electron on a Cu2+ ion (S = 1/2) experiences an / = 3/2 from the copper nucleus resulting in a fourfold split of the EPR resonance. If the Cu is coordinated by a... 

By = xiytZi) and spin coordinates of individual electrons, n represents the number of electrons in the molecule. For many cases it is appropriate to include nuclear charges as well and to define a molecular charge distribution Q (R) ... 

J = 1,3,5 — are antisymmetric with respect to the nuclear coordinates. It follows that homonuclear diatomic molecules with anti-symmetric nuclear spin wave functions (nuclei with half-integer I = 1/2, 3/2...) can combine only with symmetric rotational functions (even J = 0,2,4...), while those with symmetric nuclear spin wave functions (even I) can combine only with antisymmetric rotational functions... 

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