Double-spin transitions

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. 

Fig. 10. Energy diagram of a IS spin system (two spins 1/2). a and P are the usual spin functions. One-quantum transitions (Ji, Sy S2, physically observable) full arrows. ZQ and DQ (dashed arrows) refer to zero-quantum and double-quantum transitions.

The longitudinal cross-relaxation rate (see Eq. (13)) originates solely from the terms in the dipolar Hamiltonian involving both spins, namely those terms corresponding to zero-quantum and double-quantum transitions so that... 

Figure 2 The four-level diagram for a system of two interacting spins, in this case an electron (S) and nucleus with a positive gyromagnetic ratio (/). The intrinsic electron and nuclear spin relaxation are given by p and w°, respectively, and the dipolar and/or scalar interactions between the electron and nuclear spin are represented by p, w0, w, and w2. The transition w0 is known as the zero-quantum transition, while w, is the singlequantum transition and w2 is the double-quantum transition. Nuclear and electronic relaxation through mechanisms other than scalar or dipolar coupling are denoted with w° — 1/Tio and p — 1/Tie, where Ti0 and T1e are the longitudinal relaxation times of the nucleus and electron in the absence of any coupling between them. Since much stronger relaxation mechanisms are available to the electron spin, the assumption p>p can be safely made. Adapted with permission from Ref. [24].

Suppose that we are talking about a double-quantum transition in which both the proton and carbon change from the a state to the p state. This transition is thus from the aH c state to the PuPc state ol l lc two-spin, four-state system. This transition corresponds to DQC. Likewise, if the proton flips from ft to a while the carbon simultaneously flips from a to P, we have a zero-quantum transition (P ac to a Pc) because the total number of spins in the excited (ft) state has not changed. This transition corresponds to ZQC. What can we say about these mysterious coherences In Section 7.10, we encountered ZQC and DQC as intermediate states in coherence transfer, created with pulses from antiphase SQC ... 

Jablonski" diagram, showing, for a molecule in the ground (spin)-singlet state S0, the (induced) absorptions, a double-quantum transition, (spontaneous) fluorescence, (spontaneous) phosphorescence, internal conversion, and intersystem crossing between the singlet manifold of states S0, S1 S2, and S3, and the lowest excited triplet state T-. ... 

Figure 11.47. Magnetic hyperfine and spin-rotation splitting of the v = 17, N= 1 level of HD+. The infrared transitions indicated correspond to the lines observed in figure 11.43. The five radio frequency double resonance transitions observed were all between the G = 0 and 1 levels.

Equation 6.33 is completely general. For the two-spin system, it results in the transitions we identified in Fig. 6.2, while the double quantum transition between and 4, and the zero quantum transition between 02 and 03 are forbidden. Note that this statement is true for this treatment, which employs stationary state wave functions and time-dependent perturbations, but as we shall see in Chapter 11, it is easy with suitable pulse sequences to elicit information on zero quantum and quantum double processes. For our present purposes in the remainder of this chapter we accept the validity of Eq. 6.33. 

In Table 11.1 we sketch the form of the density matrix for the two-spin system to show the significance of the elements. Px —P4 refer to the populations of the four states, I and S represent single quantum I and S transitions, and Z refers to zero quantum transitions and D to double quantum transitions. We saw in Eq. 11.9 that an off-diagonal element pm is nonzero only if there is a phase coherence between states m and n, and in Eq. 10.19 we saw that pmn, evolves with a frequency determined by the difference in energies Em — En. Thus, these off-diagonal elements represent not only transitions, but single quantum, double quantum, and zero quantum coherences, which evolve in free precession at approximate frequencies of v, vs, vt + vs, and vt — vs. In Eq. 11.53 we see that p(r) has... 

As is well known, the nucleophilic addition to the C-C double bond can be promoted very effectively by r-coordination of the olefin in a cationic low-spin transition metal complex. Many examples are described in the literature where amines react smoothly with transition metal-ethylene complexes forming )ff-ammonioethyl complexes [1-3]. Very often these complexes are isolable in the pure state, and in the case of the platinum(II) complex [PtCl2(Et2NH) (CH2CH2NHEt2)j the stmcture has also been proved by X-ray crystal structure analysis [4]. 

Double quantum transition—a transition between nuclear spin states in which two spins are simultaneously excited from the ground state or relaxed to the ground state. 

There are two more transitions in our two-spin system which are not allowed by the usual selection rule. The first is between states 1 and 4 (aa —/ / ) in which both spins flip. The AM value is 2, so this is called a double-quantum transition. Using the same terminology, all of the allowed transitions described above, which have AM = 1, are single-quantum transitions. From the table of energy levels it is easy to work out that its frequency is (—r>o, i — r>o,2) i.e. the sum of the Larmor frequencies. Note that the coupling has no effect on the frequency of this line. 

---