Transition single-quantum

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. 

Most molecular vibrations are well described as hannonic oscillators with small anlrannonic perturbations [5]. Por an hannonic oscillator, all single-quantum transitions have the same frequency, and the intensity of single-quantum transitions increases linearly with quantum number v. Por the usual anhannonic oscillator, the single-quantum transition frequency decreases as v increases. Ultrashort pulses have a non-negligible frequency bandwidth. Por a 1... 

A single-quantum transition involves one spin only, whereas the zero- and doublequantum transitions involve two spins at the same time. The zero- and double-quantum transitions give rise to cross-relaxation pathways, which provide an efficient mechanism for dipole-dipole relaxation. 

The transitions between energy levels in an AX spin system are shown in Fig. 1.44. There are four single-quantum transitions (these are the normal transitions A, A, Xi, and X2 in which changes in quantum number of 1 occur), one double-quantum transition 1% between the aa and j8 8 states involving a change in quantum number of 2, and a zero-quantum transition 1% between the a)3 and fia states in which no change in quantum number occurs. The double-quantum and zero-quantum transitions are not allowed as excitation processes under the quantum mechanical selection rules, but their involvement may be considered in relaxation processes. 

If only single-quantum transitions (h, I2, S], and S ) were active as relaxation pathways, saturating S would not affect the intensity of I in other words, there will be no nOe at I due to S. This is fairly easy to understand with reference to Fig. 4.2. After saturation of S, the fMjpula-tion difference between levels 1 and 3 and that between levels 2 and 4 will be the same as at thermal equilibrium. At this point or relaxation processes act as the predominant relaxation pathways to restore somewhat the equilibrium population difference between levels 2 and 3 and between levels 1 and 4 leading to a negative or positive nOe respectively. 

Nitrogen-14, with its natural abundance of 99.6%, is one of the most ubiquitous and, until recently, least studied NMR-active nuclei. Due to the integer spin number (/ = 1), its single-quantum transitions are affected by first-order quadrupolar broadening, which in most materials is on the order of a few megahertz. A new class of 2D HETCOR protocols has been recently developed, which makes it possible to indirectly observe well-resolved 14N sites via their spin-1/2 neighbors and obtain the related parameters of the quadrupolar tensors. 

Multiple quantum transitions (MQT) in ENDOR spectra may be observed for nuclei with I 1 if two or more (rf) photons of the same or of different frequencies combine to produce an ENDOR transition41,62,99"101). In a MQT the magnetic quantum number nij changes by An = n. The MQT should therefore be clearly distinguished from corresponding forbidden single quantum transitions (SQT) with Am] > 0 discussed in Sect. 3.3. 

These experiments did not resolve the question as to whether the deactivating collision of H2+ in t > 1 results in multiquantum transitions, single quantum transitions, or both. Since /c2 0.2-0.3X 10 9 cm3/sec for reactions of H2+ in higher vibrational levels, this means that kD, the rate coefficient for collisional deactivation, is On the same order of magnitude. 

In order to define and pi(j) it is convenient to refer to Fig. 7.1 and to define w[ and w( as the transition probabilities between two states involving a single quantum transition either of spin 7 or J wo is the zero quantum transition and corresponds to the —I— + — transition and vice versa wj corresponds to... 

In equation (143), the subscript (—1) denotes the subspace of the composite Liouville space which is concerned with the eigenvalue — 1 of the superoperator Ff (the subspace of single-quantum transitions). The vector fx is normalized according to equation (135) and the q, (t) function has the same meaning as in equation (50). The coefficient C is given by [equation (139)1 ... 

For a system of S substances the dimension D of the subspace of the single-quantum transitions is equal to the sum of the dimensions of the subspaces for the individual substances D,. The latter is given by ... 

In a typical situation we are interested in the absorption mode of a dynamic spectrum,/abs (co), which equals the real part of the complex function f(co) given by equation (145). In most cases of unsaturated spectra the relaxation matrix which describes single-quantum transitions can be replaced by a constant — E/T2(effective) which is characteristic of the experimental conditions involved and reflects the inhomogeneity of the external magnetic field B0. The absorption mode spectrum is given by ... 

Equation (10) cannot be solved in the Hilbert space, mainly due to the component describing the exchange processes (Equation (8)). To calculate the NMR spectrum, Equation (10) is first transformed into the vector space spanned by the single quantum transitions (the Liouville space... 

The physical meaning of the (j, k) element (Ajkn) of the A(r) matrix is the amplitude of the i> if//1 single quantum transition. Substituting Equation (47) into Equation (46) results in ... 

Note that the observable operator (the operator representing coherence or net magnetization in the x-y plane) is always written first in the product. Also, we see above some examples where both operators are in the x-y plane, or both operators are on the z axis These products represent nonobservable states which are nonetheless very important in NMR experiments. The only observable product operators are those with only one operator in the x-y plane ( single-quantum transitions ). 

Note that the four single-quantum transitions have energy differences corresponding to their exact frequencies in the 1H spectrum ... 

The four single-quantum transitions relax at a rate Wi, or more specifically for the Ha transitions and W for the Hb transitions. The double-quantum transition (aa /3/3) relaxes with rate W2 and the zero-quantum transition (otft fiot) relaxes at a rate W0 (Fig. 10.2). For... 

Consider first the single-quantum transition between the afi and pp states (Ha(2) transition, Fig. 10.1). This is an Ha transition with relaxation rate W. The equilibrium difference in population for this transition is Pap — Ppp = 28. If this equality does not hold, then the overpopulation of the pp state is given by Ppp — Pap + 28, and the rate of spins dropping down from the pp state to the ap state is Wf(Ppp — Pap + 28). If this were the only transition available (i.e., if there were no double-quantum or zero-quantum pathways), we could write down the rate of change of population as... 

In-phase single-quantum coherence (SQC) is represented by nonzero values for the matrix elements that correspond to the single-quantum transitions. For example, I spin (1H) SQC corresponds to a superposition of the cqa s and /3 (xs states (row 1 and column 3), and the a Ps and A/ s states (row 2 and column 4). Real numbers are used for magnetization on the xf axis, and imaginary numbers are used for magnetization on the y axis. Notice that the downward transition A s — cqa s has a matrix element that is the complex conjugate of the upward transition oqas —> A s-... 

In Eq. 11.48 we saw that the basis functions for our density matrix are divided into three groups with fz = 1,0, and —1, respectively. As we saw in Chapter 6, transitions between energy levels El - E2, E3 E4, Ex E3, and E2 -> E4 each result in Afz = 1 and are called single quantum transitions, while transitions Ex E4 and E2 E3 are termed double quantum and zero quantum transitions, respectively. The usual selection rules from time-dependent perturbation theory show that only single quantum transitions are permitted in such simple experiments as excitation by a 90° pulse. Moreover, for weakly coupled nuclei, the single quantum transitions each involve only a single type of nucleus, I or 5, as indicated in Fig. 6.2. 

From Eqs. 6.29 and 6.34 we know that the frequencies of the single quantum transitions include both the chemical shift difference and the coupling constant, and we saw in Eq. 11.54 that the single quantum coherence terms evolve at those frequencies. From Eq. 6.29 we can see that the expression for the double quantum frequency E4 — E, would not depend on J, and the difference 3 — E2 likewise does not depend on J for weakly coupled spins. Thus zero quantum and double quantum coherences evolve as though there were no spin coupling. 

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