Nuclear magnetic resonance effective” spin Hamiltonians

The development of the effective Hamiltonian has been due to many authors. In condensed phase electron spin magnetic resonance the so-called spin Hamiltonian [20,21] is an example of an effective Hamiltonian, as is the nuclear spin Hamiltonian [22] used in liquid phase nuclear magnetic resonance. In gas phase studies, the first investigation of a free radical by microwave spectroscopy [23] introduced the ideas of the effective Hamiltonian, as also did the first microwave magnetic resonance study [24], Miller [25] was one of the first to develop the more formal aspects of the subject, particularly so far as gas phase studies are concerned, and Carrington, Levy and Miller [26] have reviewed the theory of microwave magnetic resonance, and the use of the effective Hamiltonian. 

There are a variety of techniques for the determination of the various parameters of the spin-Hamiltonian. Often applied are Electron Paramagnetic or Spin Resonance (EPR, ESR), Electron Nuclear Double Resonance (ENDOR), Electron Electron Double Resonance (ELDOR), Nuclear Magnetic Resonance (NMR), occassionally utilizing effects of Chemically Induced Dynamic Nuclear Polarization (CIDNP), Optical Detection of Magnetic Resonance (ODMR), Atomic Beam Spectroscopy and Optical Spectroscopy. The extraction of the magnetic parameters from the spectra obtained by application of these and related techniques follows procedures which may in detail depend on the technique, the state of the sample (gaseous, liquid, unordered solid, ordered solid) and on spectral resolution. For particulars, the reader is referred to the general references (D). 

We turn now to the corresponding studies of the isotopic species D2 and HD. The deuterium nucleus has spin Id equal to 1, so that the two equivalent deuterium nuclei in D2 have their spins coupled to give total nuclear spin / equal to 2, 1 or 0. The states with / equal to 2 or 0 correspond to ortho-D2, whilst that with / equal to 1 is known as para-T>2. The molecular beam magnetic resonance studies have been performed on para-D2, in the. 1 = 1 rotational level. Formally, therefore, the effective Hamiltonian is the same as that described above for experimental studies of ortho-H2, also in the J = 1 rotational level. There is one extremely important difference, however, in that the... 

We calculate the effects of the Hamiltonian (8.105) on these zeroth-order states using perturbation theory. This is exactly the same procedure as that which we used to construct the effective Hamiltonian in chapter 7. Our objective here is to formulate the terms in the effective Hamiltonian which describe the nuclear spin-rotation interaction and the susceptibility and chemical shift terms in the Zeeman Hamiltonian. We deal with them in much more detail at this point so that we can interpret the measurements on closed shell molecules by molecular beam magnetic resonance. The first-order corrections of the perturbation Hamiltonian are readily calculated to be... 

The rotational and Zeeman perturbation Hamiltonian (X) to the electronic eigenstates was given in equation (8.105). It did not, however, contain terms which describe the interaction effects arising from nuclear spin. These are of primary importance in molecular beam magnetic resonance studies, so we must now extend our treatment and, in particular, demonstrate the origin of the terms in the effective Hamiltonian already employed to analyse the spectra. Again the treatment will apply to any molecule, but we shall subsequently restrict attention to diatomic systems. 

The spectrum follows rules similar to those described in chapter 8 for the molecular beam magnetic resonance study of 7Li79Br, which also has a + ground state and two nuclear spins of 3/2. In that case the transitions studied were nuclear spin transitions within the J = 1 level, but the effective Hamiltonian is similar in the two cases. The Hamiltonian used by Low, Varberg, Connelly, Auty, Howard and Brown [89] contained rotational, quadrupolar and nuclear spin-rotation terms,... 

The evolution of the density matrix is governed by Eq. (2.10) in which the Hamiltonian for the spin system must be specified. It is noted here that the relaxation effects arising from dissipative interactions between the spin system and the lattice have not been included in the equation. The nuclear spin Hamiltonian contains only nuclear spin operators and a few phenomenological parameters that originate from averaging the full Hamiltonian for a molecular system over the lattice coordinates. These magnetic resonance parameters can, at least in principle, be deduced by quantum chemical calculations [2.3]. The terms that will be needed for discussion in this monograph will be summarized here. 

Electron Spin Echo Envelope Modulation (ESEEM) and pulse Electron Nuclear Double Resonance (ENDOR) experiments are considered to be two cornerstones of pulse EPR spectroscopy. These techniques are typically used to obtain the static spin Hamiltonian parameters of powders, frozen solutions, and single crystals. The development of new methods based on these two effects is mainly driven by the need for higher resolution, and therefore, a more accurate estimation of the magnetic parameters. In this chapter, we describe the inner workings of ESEEM and pulse ENDOR experiments as well as the latest developments aimed at resolution and sensitivity enhancement. The advantages and limitations of these techniques are demonstrated through examples found in the literature, with an emphasis on systems of biological relevance. 

The theory of the magnetic hyperfine interactions in NCI is essentially the same as that already described for the PF radical in the previous section, except that the nuclear spins / are 1 for 14N and 3/2 for 35C1. The form of the effective Hamiltonian for the quadrupole interaction and its matrix elements for two different quadrupolar nuclei was described in some detail in chapter 8 when we discussed the electric resonance spectra of CsF and LiBr. We now use the same case (b) hyperfine-coupled basis set as was used for PF. The quadrupole Hamiltonian for the two nuclei can be written as the sum of two independent terms as follows ... 

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