Addition of parahydrogen to dimethyl acetylenedicarboxylate to form dimethyl maleate. When the reaction product contains a single 13C nucleus in the vinyl group, the hyperpolarized molecule can be modeled as a three-spin system.
Energy levels for the three-spin system. Each energy level is an angular-momentum manifold with a well-defined value of F. The degenerate states are labeled with M F , the z component of total angular momentum. On the right side of the figure, energies are expressed in Hz. Since the initial density matrix has no population in the manifold with F = 3/2, the evolution is restricted to the space spanned by the two manifolds with F = 1/2. The system therefore oscillates only at frequency ω/2π, which corresponds to the energy difference between these two manifolds.
Simulated evolution of the scalar spin order in the isotopomer of dimethyl maleate that has 13C in the vinyl group. The molecule is modeled as a three-spin system containing protons I 1 and I 2 and a 13C nucleus S. The curves represent the oscillating coefficients in the expansion of the density matrix given by Eq. (6) . In the lower-right corner of the figure, each curve is labeled with the corresponding operator from the expansion. (a) Coherent evolution starting from the singlet state at time t = 0. (b) Averaging of the spin order during a polarization period in which hydrogenation occurs continuously as parahydrogen is bubbled through the sample. The coupling constants used in the simulations were J S1 = 167.2 Hz, J S2 = −2.2 Hz, and J 12 = 13.0 Hz, consistent with the simulation of the experimental zero-field spectrum shown in Fig. 5 .
Depiction of the evolution occurring in a two-spin system during the pulse and the detection period. Expressing the density matrix ρ and the Hamiltonians H J and H dc in terms of the Cartesian components of a pseudospin yields a simple model of the evolution. The axes of the pseudospin are distinguished from the laboratory-frame axes by the labels , , and . (a) Just before the pulse, the pseudospin that represents the polarized system is colinear with H J . (b) The pulse Hamiltonian is perpendicular to H J , and the pulse rotates the pseudospin through an angle of π/2. (c) Free evolution during the detection period corresponds to precession of the pseudospin about H J . This precession is associated with a periodic exchange between I z and S z , which produces an oscillating magnetization in the sample, since the spins have different gyromagnetic ratios.
Zero-field spectrum resulting from the addition of parahydrogen to dimethyl acetylenedicarboxylate to form dimethyl maleate. Below the experimental spectrum are simulations for the two isotopomers that contribute to the signal. The spectra have been phased so that the low-frequency peaks appear above the horizontal axis. For the vinyl isotopomer, the spectrum has the form predicted by Eq. (62) , which was derived by modeling the isotopomer as a three-spin system. In particular, there are peaks of equal integrated area at the frequencies ω jk /2π, including a pair of antiphase peaks at ∼ J S1. 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. The coupling constants used in the simulation of the vinyl isotopomer were 1 J S1 = 167.2 Hz, 2 J S2 = −2.2 Hz, 3 J 12 = 13.0 Hz, 4 J S3 = −0.45 Hz, 5 J 13 = 0.15 Hz, and 6 J 23 = 0 Hz. For the carboxyl isotopomer, the coupling constants were 2 J S1 = 2.7 Hz, 3 J S2 = 13.2 Hz, 3 J 12 = 11.9 Hz, 3 J S3 = 4.1 Hz, 5 J 13 = 0.15 Hz, and 6 J 23 = 0 Hz. In the notation for the coupling constants, the vinyl protons are numbered 1 and 2, the protons in the methyl group are indicated by the subscript 3, and the superscript shows the number of bonds between nuclei. The couplings were adjusted to yield a visual match to the experimental spectrum.
Structure of the scalar order in a three-spin system obtained by adding parahydrogen to a substrate molecule. The initial density matrix ρ0 is diagonalized by a basis of states |F, M F , I⟩, where I is the summed angular momentum of the two protons. These states can be grouped into the manifolds listed on the left side of the table. The manifold with F = 3/2 is a degenerate energy level with spin energy E 1 = (J S1 + J S2 + J 12)/4. At the end of the polarization period, the density matrix ρ1 is diagonalized by a basis of energy eigenstates |F, M F , E⟩. The corresponding manifolds are listed on the right side of the table. The energies E 2 and E 3 and the population a, which can be evaluated using Eqs. (13) and (14) , depend on the details of the scalar-coupling network.
Approximate energy levels of the three-spin system that has |J S1| ≫ |J S2|, |J 12|. The zero-order eigenstates can be written in the form |F, M F , F 1⟩, where the angular momentum F 1 is the sum of S and I 1. These eigenstates are grouped into the degenerate angular-momentum manifolds listed on the left side of the table. The energies are given to first order in the weak couplings.
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