NMR of two coupled spins

Let us consider two /-coupled nuclear spins denoted as I and S. The simplest case to examine is that of heteronuclear interaction ( C and nuclei in the chloroform molecule, for example), so that the magnitude of the /-coupling is much smaller than the difference between the resonant frequencies of the two nuclei. This is named the AX system in the 

Chemistry literature. The secular Hamiltonian in the heteronuclear case is then given by (see Equation (2.7.15))  

The resonant frequencies coj and cos include the effects of isotropical chemical shifts. The eigenstates and eigenvalues of this Hamiltonian are easily obtained from the matrix representations of each operator 1 and. As these operators commute and act on different vector spaces [2], the eigenstates of the Hamiltonian above are simply given by the direct product of the eigenstates of the operators /j and S. The eigenstates are then written as mj,ms), where mj = 1/2, = 1/2 are the eigenvalues of and S, respectively. 

The eigenvalues of the Hamiltonian, i.e., the energy levels associated with the Hamiltonian (2.8.1), are directly evaluated from the acting of the operators and on the mi, ms) basis, leading to  

In the case of positive values for J, yi, and ys, and if coj cos, then the levels above are arranged from the top to the bottom in order of increasing energy. Transitions between these levels are allowed (i.e., give rise to NMR signal) according to the selection rules mys = 1. Therefore, there are four peaks in the full NMR spectrum, at the frequencies CO] 7tJ and cos 7tJ. The separation within each doublet is, in angular frequency units, equal to InJ, or, in frequency units, equal to /. 

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As a specific example of generating pseudo-pure states in NMR spin systems, lets consider a case of two /-coupled spin 1/2 (see Section 4.1). The equilibrium density matrix of this system can be written as (4.3.3). Remember that the pulse sequence for the CNOT gate, the operations Uq, Ui, and U2 can be written as. 

In the following, we will discuss heteronuclear polarization-transfer techniques in four different contexts. They can be used as a polarization-transfer method to increase the sensitivity of a nucleus and to shorten the recycle delay of an experiment as it is widely used in 1H-13C or 1H-15N cross polarization. Heteronuclear polarization-transfer methods can also be used as the correlation mechanism in a multi-dimensional NMR experiment where, for example, the chemical shifts of two different spins are correlated. The third application is in measuring dipolar coupling constants in order to obtain distance information between selected nuclei as is often done in the REDOR experiment. Finally, heteronuclear polarization transfer also plays a role in measuring dihedral angles by generating heteronuclear double-quantum coherences. 

The similarity of the preceding first-order ESR treatment to the first-order NMR treatment of two coupled protons is evident. For an unpaired electron interacting with n equivalent nuclei of spin the hyperfine coupling term in the spin Hamiltonian is... 

In liquid-state NMR, spin relaxation due to cross-correlation of two anisotropic spin interactions can provide useful information about molecular structure and dynamics. These effects are manifest as differential line widths or line intensities in the NMR spectra. Recently, new experiments were developed for the accurate measurement of numerous cross-correlated relaxation rates in scalar coupled multi-spin systems. The recently introduced concept of transverse relaxation optimized spectroscopy (TROSY) is also based on cross-correlated relaxation. Brutscher outlined the basic concepts and experimental techniques necessary for understanding and exploiting cross-correlated relaxation effects in macromolecules. In addition, he presented some examples showing the potential of cross-correlated relaxation for high-resolution NMR studies of proteins and nucleic acids. 

J-coupling fact two is not as obvious. The inability of two isochronous spins to display coupling with each other greatly simplifies the appearance of many NMR spectra, and for this we should be grateful. 

Ag and Ag both have 7=1/2 but ° Ag is more suitable for NMR measurements . The ° Ag NMR spectra, using the INEPT (insensitive nuclei enhanced by polarization transfer) sequence [proton polarization transfer sequence between two / coupled spins /( Ag) and s( H), 7( ° ° Ag- H-imine) = 9.3 Hz (5 8.87) and 6.3 Hz (S 8.69)], have been applied to gain insight into the electronic environment and the structural and dynamic behaviour of Ag complexes in solutions . The chemical shift of the ° Ag nucleus is highly sensitive towards changes in its environment . The... 

In all the cases discussed above, only the response of the spin system composed of dipolar coupled spin-1/2 pairs was considered. Proton DQ-NMR spectroscopy on elastomers—static and spinning at the magic angle—proved that the consideration of isolated spin-1/2 pairs is a crude approximation (68). The dipolar couplings between the protons belonging to various fimctional groups are shown to be active in two-dimensional DQ-MAS spectra for high excitation/reconversion times (68,73). 

The association of the spin states with logical labeling can be done in real systems. For example, let us consider a solution of enriched chloroform, CHCl3. This system can be well approximated by two coupled spins Ii, h with spin Ii being the and spin I2 the H. The relevant NMR interactions are the Zeeman, chemical shifts and the /-coupling, which are represented by the following secular Hamiltonian (see Chapter 2) [2] ... 

The spin-spin coupling interaction in Equation 12.13 is usually a small perturbation on the energies of the nuclear spin states experiencing the external field of a modern NMR instrument. It is appropriate to treat this interaction with low-order perturbation theory, particularly first-order perturbation theory. The expression for the first-order correction to an energy is the expectation value of the perturbation. For a system of two nonzero spin nuclei, the spin states are distinguished by the two quantum numbers, ni[ and OT/j. The corrections to the energies of these states are given by... 

Spm-spm splitting of NMR signals results from coupling of the nuclear spins that are separated by two bonds (geminal coupling) or three bonds vicinal coupling)... 

Coupling constant J (Section 13 7) A measure of the extent to which two nuclear spins are coupled In the simplest cases It IS equal to the distance between adjacent peaks in a split NMR signal... 

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