Vector model of the spin motion in a system containing a heteronucleus S and two protons I A and I B , with I A strongly coupled to S, and I B weakly coupled to the other two spins. The strong coupling and the weak couplings are represented by the Hamiltonians H 0 and H 1, respectively. (a) If the weak couplings involving spin I B are negligible, the strongly coupled spins S and I A precess about a motionless vector that represents F A , the sum of their angular momenta. (b) Weak scalar couplings involving spin I B are averaged over this fast precession, so that I B “sees” the projections S ∥ and rather than the instantaneous states of S and I A . The truncated weak interaction therefore couples I B to F A . (c) The truncated interaction causes I B and F A to precess about the total angular momentum F. (d) The slow precession of F A modulates the fast motion of S and I A , which yields a high-frequency doublet in the spectrum. The precession of I B and F A about F is also detectable as a single low-frequency peak.
Spin vectors associated with the fast dipole oscillations. (a) Under H 0, the strongly coupled spins precess about F A . The components S ∥ and are proportional to F A and thus do not evolve. The precession involves motion of and S ⊥, the components of I A and S that are perpendicular to F A . (b) Since the gyromagnetic ratios for I and S are different, the spin vector has a dipole moment that precesses quickly under H 0. (c) This motion is modulated by the slow precession of F A under the truncated weak coupling. The modulated motion of is responsible for the high-frequency peaks in the spectrum.
Spin vectors associated with the slow dipole oscillations. (a) Under the truncated weak coupling, I B and F A precess about the total angular momentum F. The precession involves motion of and , the components of I B and F A that are perpendicular to F. (b) The dipole moment includes contributions γ I ℏ I B and (γ I + γ S )ℏ/2 F A . Because γ I is different than the effective gyromagnetic ratio associated with F A , the spin vector has a dipole moment that precesses under H 1 and contributes a low-frequency peak.
First-order description (dashed lines) and second-order description (solid lines) of the zero-field spectrum of an example three-spin system. The amplitudes, which were evaluated using zero-order eigenstates, correspond to an experimental protocol in which the molecule is prepolarized in an applied field. After the field is dropped suddenly to zero, the oscillations of the sample dipole are detected. The relative amplitudes of the three peaks are 2:3:6, and the second-order approximations to the frequencies ν k are given by Eqs. (28) , where J SA = 167.2 Hz, J SB = −2.2 Hz, and J AB = 13.0 Hz. These scalar couplings correspond to a system consisting of two 1H nuclei and a 13C nucleus in the vinyl group of dimethyl maleate, 25 shown on the right side of Fig. 5 . In this system, the exact transition frequencies differ from the second-order approximations by about 10 mHz.
Hydrogenation of dimethyl acetylenedicarboxylate (DMAD) to form dimethyl maleate (DMM). When the reaction product contains a single 13C nucleus in the vinyl group, the hyperpolarized molecule can be modeled as a three-spin system.
Zero-field spectrum resulting from the addition of parahydrogen to DMAD to form DMM. The signal is primarily generated by molecules that have a single 13C nucleus in either the vinyl group or the carboxyl group of DMM. The relevant positions of the 13C nucleus are indicated by asterisks in the molecular structure. For the vinyl isotopomer, the first-order description (dashed lines) and second-order description (solid lines) of the spectrum are shown. The transition frequencies were calculated using the coupling constants J SA = 167.2 Hz, J SB = −2.2 Hz, and J AB = 13.0 Hz, as in Fig. 4 . The small splittings of the antiphase peaks are due to weak couplings to the methyl protons, which are not included in the three-spin model of the vinyl isotopomer. For the isotopomer with 13C in the carboxyl group, the network of six coupled spins formed by the 13C nucleus, the vinyl protons, and the methyl protons yields a complicated splitting pattern in the low-frequency region of the spectrum.
Zero-field spectrum and allowed transitions of labeled methyl formate (H13COOCH3) prepolarized by thermal equilibration in an applied field. The molecule is an (XA)B3 system, where X and A correspond to the 13C nucleus and the 1H nucleus of the formyl group, respectively, and where the weakly coupled spins represented by B3 are the 1H nuclei of the methyl group. (a) The trace shows the experimental spectrum, while the dashed lines and the solid lines show the first-order description and second-order description of the spectrum, respectively. Amplitudes were calculated using zero-order eigenstates. For the second-order description of the spectrum, the relative amplitudes of the peaks in the high-frequency multiplet are 1:2:2:4:3, while the relative amplitudes of the eight peaks in the full spectrum are 20:25:27:15:30:30:60:45. The scalar couplings used for the calculations were 1 J SA = 226.81 Hz, 3 J SB = 4.0 Hz, and 4 J AB = −0.8 Hz, chosen by finding a visual match between exact simulations and the experimental data. (b) Energy levels and allowed transition for a strongly coupled XA system. (c) Energy levels and transitions within the subspace obtained by adding F A to I B = 3/2. (d) Energy levels and transitions within the subspace obtained by adding F A to the two manifolds with I B = 1/2. The closely spaced energy states are degenerate. In (c) and (d), arrows showing allowed transitions specify pairs of energy levels involved in a transition.
Approximate energy levels of the three-spin system. The zero-order eigenstates can be grouped into degenerate angular-momentum manifolds labeled with quantum numbers S, I A , F A , I B , and F. All of the manifolds have S = I A = I B = 1/2; the values of F A and F are shown in the table. The zero-order energy is denoted by E (0), while the first-order and second-order energy corrections are denoted by Δ(1) and Δ(2), respectively. The energy level with F A = 0 has Δ(1) = 0 because the projections of S and I A onto F A are zero within this level. Because H 1 is a scalar operator, it does not couple states that have distinct values of F. As a result, Δ(2) = 0 for the energy level with F = 3/2, which is not coupled to the two levels with F = 1/2.
Approximate energy levels of labeled methyl formate (H13COOCH3), an (XA)B3 system. The eigenstates can be grouped into degenerate angular-momentum manifolds labeled with quantum numbers S, I A , F A , I B , and F. All of the manifolds have S = I A = 1/2. The values of F A , I B , and F are shown in the table, along with the zero-order energy E (0) and the energy corrections Δ(1), Δ(2). For energy levels with F A = 0, the first-order correction is zero, since the projections of S and I A onto F A are zero. As shown in the Appendix, second-order energy corrections in (XAn)Bm systems are due to couplings between states that have the same values of I A , I B , and F but distinct zero-order energies. For the energy levels in the table that have Δ(2) = 0, the zero-order energy is uniquely specified by the values of I A , I B , and F.
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