Coupled spin systems single spins

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. 

In a coupled spin system, the number of observed lines in a spectrum does not match the number of independent z magnetizations and, fiirthennore, the spectra depend on the flip angle of the pulse used to observe them. Because of the complicated spectroscopy of homonuclear coupled spins, it is only recently that selective inversions in simple coupled spin systems [23] have been studied. This means that slow chemical exchange can be studied using proton spectra without the requirement of single characteristic peaks, such as methyl groups. 

Equation (32) gives the phase of the vector in question for a single nucleus at the time of exchange. In the case of weakly coupled spin systems, this relationship remains valid and just has to be amended with index j (the value of tojP is either oofP or o>j depending on the parity of r). 

In the TPPI method a single data set with 512 increments is collected. In each successive increment the phase of the 90° pulse at the end of the period is incremented by 90° with respect to the phase of the corresponding pulse in the previous increment. (An equivalent experiment can be performed in which the phases of the pulses before the ti period are shifted by 90°). This is equivalent to changing the reference frame in so that the transmitter in the dimension appears to be shifted to one edge of the spectrum. After performing a real Fourier transformation, all peaks will appear to be shifted to one side of the transmitter in /. The main disadvantage of this technique is that phase distortions can appear for resonances in strongly coupled spin systems. 

In any strongly coupled spin system, each line in the spectrum is a mixture of transitions of various nuclei, which depends on the chemical shifts and coupling constants. When a chemical exchange happens, all the spectral parameters change with it. Therefore, a magnetization (or a coherence) that was associated with a single spectral line in one site may be spread among several lines in the other site. In order to deal with these complexities, we must use the density matrix. 

ENDOR transitions can be easily understood in temis of a simple system consisting of a single unpaired electron spin (S=2) coupled to a single nuclear spin (1=2). The interactions responsible for the various... 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

In contrast, in a two-spin system the two nuclei coupled with each other by the coupling constant, J, will have four energy levels available for transitions (Fig. 5.56). Such a system not only has single-quantum coher-... 

In the carbon-13 experiments so far discussed, only a single radio-frequency pulse has been used to irradiate the spin system. This gave us information on the chemical shifts of the carbon nuclei in the molecule. The coupled spectrum obtained using gated decoupling (1.2.2) told us how many protons are bound to any one carbon atom however, this experiment requires a lot of time. There are however other experiments which give us this information... 

On the basic of relaxation theory the concept of TROSY is described. We consider a system of two scalar coupled spins A, I and S, with a scalar coupling constant JIS, which is located in a protein molecule. Usually, I represents H and S represents 15N in a 15N-1H moiety. Transverse relaxation of this spin system is dominated by the DD coupling between I and S and by CSA of each individual spin. An additional relaxation mechanism is the DD coupling with a small number of remote protons, / <. The relaxation rates of the individual multiplet components in a single quantum spectrum may then be widely different (Fig. 10.3) [2, 9]. They can be described using the single-transition basis opera-... 

In order to study the decoherence effect, we examined the time evolution of a single spin coupled by exchange interaction to an environment of interacting spin bath modeled by the XY-Hamiltonian. The Hamiltonian for such a system is given by [104]... 

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